Swaps Futures Options Trading
Author: D. SelzerMcKenzie
Introduction
Security C 
Sell 10 units of C at $40. 
+$400 

Synthetic security 
Buy 4 units of A at $35. 
$140 


Buy 6 units of B at $30. 
$180 

Profit 

+$80 
The profit on the transactions is $80 on 10 units of the securities, or $8 profit per unit. The profit could be increased to infinity by scaling up the size of all of these transactions.
E. Explain why we do not have to worry about future obligations in a properly constructed riskless arbitrage
transaction.
Consider the following:
Obligation Associated with Assets Held as a Result of
the Arbitrage Transactions the Arbitrage Transactions Cash Flows
Payoff when the market Short selling 10 units of security The investor holds 10 units of $190 + $190 = $0
moves down C obligates the trader to the synthetic security worth
deliver 10 units of security C $19 per unit.
worth $19 per unit.
Payoff when the market Short selling 10 units of security The investor holds 10 units of $560 + $560 = $0
moves up C obligates the trader to the synthetic security worth
deliver 10 units of security C $56 per unit.
worth $56 per unit.
The arbitrage transactions were structured such that the value of the investor's assets are equal to the value of the investor's obligations. In addition, the synthetic security was constructed such that the payoffs on the synthetic security were equal to the payoffs from investing in security C in both states of nature.
F. Explain why we would not expect such a structure of prices to exist in the marketplace.
If these prices were observed in the market, the investor could earn a certain profit with no commitment of her own resources. Persistence of such mispricing would permit all investors to become infinitely wealthy through riskless arbitrage.
G. Explain the purpose of short selling in riskless arbitrage. Discuss the impact on the investor's ability to engage in riskless arbitrage of regulations that limit an investor's access to the proceeds from a short sale transaction.
The purpose of short selling in arbitrage transactions is to finance the purchase of the "cheap" asset through the sale of the "expensive" asset. In arbitrage we sell high to finance the transaction. Any constraints that limit the investor's access to funds from short selling limit the investor's ability to engage in riskless arbitrage. For example, if margin requirements are such that the investor does not have access to any of the proceeds from short selling, then the investor will have to use her personal wealth to pay for the securities she has purchased "cheap." In this environment, the investor is no longer engaging in riskless arbitrage, because investment is now required and the arbitrage transactions are no long selffinancing.
H. Construct a portfolio consisting of securities B and C that replicates the payoffs on security A in both the up and down states subject to the constraint that the sum of the commitments to the two securities (B and C) is one. In other words, construct a synthetic share of security A. Assume that there are no restrictions associated with short selling any of the securities.
To create a synthetic share of security A, we must construct a portfolio of securities B and C that has the same payoffs as security A in both the up and down states of the market. That is, we must construct a portfolio with the following characteristics
Synthetic Security
(Portfolio of Securities B and C) Security A
Payoff when the market moves down $15 x WB + $19 x WC $25
Payoff when the market moves up $60 xWB + $56 x WC $50
subject to the constraint that WB + WC = 1.
Multiplying the first equation in the table by —4 and adding the resulting equation to the second equation, we can solve the two equations in the table simultaneously for W_{C}.
4 x ($15 xW_{B} + $19 x W_{C}) + $60 x W_{B} + $56 x W_{C} =  4 x $25 + $50
WC = 2.5, and W_{B} = —1.5, where W_{B} + W_{C} = 1. That is, to construct synthetic security A, we must short sell security B and invest the proceeds from the short sale of security B in security C.
I. What are the commitments to securities B and C?
Construction of the synthetic security required the short sale of security B to finance the purchase of additional units of security C. Thus, the investor commits more than 100 percent of his wealth to security C. Specifically 250 percent of the investor's wealth is committed to security C financed by short sales of security B in an amount equal to 150 percent of the investor's wealth, W_{B} = —1.5 and W_{C} = 2.5.
J. How much does it cost to construct a synthetic share of security A? Compare this cost with the market price of security A. Which security is cheaper?
The cost of constructing the synthetic security is $55, 1.5 X $30 + 2.5 X $40 = $55. Security A is trading at a price of $35. Security A is cheaper than the synthetic security.
K. Explain the transactions necessary to engage in a riskless arbitrage. How much profit can an investor make in this riskless arbitrage? Assume that one trades 10 shares of security A in constructing the arbitrage transactions.
These prices present an opportunity for the investor to engage in riskless arbitrage. The synthetic security constructed to produce the same payoffs as security A in all states of nature is more expensive than security A itself. Therefore, we must sell the overpriced synthetic security and buy security A itself.
Riskless arbitrage transactions:
Transactions Cash Flows
Sell synthetic security 
Buy 15 units of B at $30. 
$450 


Sell 25 units of C at $40. 
+$1,000 

Buy security A 
Buy 10 units of A at $35. 
$350 

Profit 

+$200 
Short selling the synthetic security requires the investor to purchase security B, and sell security C. The profit on the transactions is $200 on 10 units of security A, or $20 profit per unit.
L. Discuss the differences between the transactions necessary to capture the arbitrage profit when creating a synthetic share of security A and the arbitrage transactions undertaken to capture the arbitrage profit when creating a synthetic share of security
In the first problem, the construction of the synthetic security required positive commitments by the investor to each security, that is, the portfolio weights were positive. In addition, to capture the profit available from the mispricing of the securities, we sold the actual security short and bought the synthetic security. In the second problem, construction of the synthetic security required the investor to sell security B short to finance purchases of security C. We then short sold the synthetic security and bought the actual security to reap the arbitrage profits.
Futures Markets
1. Explain the different roles of a floor broker and an account executive.A floor broker is located on the floor of the exchange and executes orders for traders offthefloor of the exchange. Typically, the floor broker will either be an independent trader who executes orders on a contractual basis for a futures commission merchant (FCM), or the floor broker may be an employee of an FCM. An account executive is almost always employed by an FCM and is located offthefloor of the exchange. The account executive is the person one typically thinks of as a broker. The account executive could be located in the local office of any major brokerage firm and has customers for whom he or she executes orders by communicating them to the exchange via the communication facilities of the FCM.
2. At a party, a man tells you that he is an introducing broker. He goes on to explain that his job is introducing prospective traders such as yourself to futures brokers. He also relates that he holds margin funds as a service to investors. What do you make of this explanation?
The guy is a fraud. First, a defining characteristic of an introducing broker (IB) is that the IB does not hold customers' funds. Instead, the IB is associated with an FCM who holds the customers' funds. Second, the last person the IB wants his customer to meet is another broker. The IB's income depends on executing orders for his customers, so the IB wants to keep his flock of customers away from the wolves (other brokers) who are hungry for customers.
3. Assume that you are a floor broker and a friend of yours is a market maker who trades soybeans on the floor of the Chicago Board of Trade. Beans are trading at $6.53 per bushel. You receive an order to buy beans and you buy one contract from your friend at $6.54, one cent above the market. Who wins, who loses, and why? Explain the rationale for making such practice illegal.
As described, this transaction costs your customer $.01 per bushel and transfers those funds to the friend from whom you purchase the contract at $6.54. On a 5,000bushel contract, this amounts to $50. Thus, as described, the customer loses and the friend wins. It is important to see the motivation for the floor broker's engaging in this transaction. As described, the floor broker cheats her customer and helps the friend. Presumably, the motivation for such an action is the expectation that the friend will return the favor on another transaction. The rationale for making this transaction illegal is clear; it amounts to a direct theft from the customer.
4. You are back at the party, several hours later. Your buddy from question 2 buttonholes you again and starts to explain his great success as a dual trader, trading both beans and corn. What do you think? This guy is not bright. A dual trader is a person who trades for his or her own account and who executes orders for others at the same time. You are having trouble escaping from your friend in question 4. He goes on to explain that liquidationonly trading involves trading soybean against soyoil to profit from the liquidation that occurs when beans are crushed. Explain how your understanding of liquidationonly trading differs from your friend's.
We hope that your understanding of liquidationonly trading runs as follows: Under liquidationonly trading, each trade must result in a reduction of a trader's open interest. Every trade must be an offsetting trade. Liquidationonly trading essentially amounts to the closing of a market, and this is done during serious market disturbances, such as a manipulation. Liquidationonly trading has nothing in particular to do with beans or any other commodities.
6. In purchasing a house, contracting to buy the house occurs at one time. Typically, closing occurs weeks later. At the closing, the buyer pays the seller for the house and the buyer takes possession. Explain how this transaction is like a futures or forward transaction.
The purchase of a house has many features of a forward contract. Contracting occurs at one date, with performance on the contract occurring later at the closing. In buying a house, there is usually a good faith deposit or earnest money put up at the time of contracting. The contract is tailored to the individual circumstances, with the performance terms and the closing date being agreed on between the buyer and seller. The contract is not like a futures, because there is no organized exchange, the contract terms are not fixed, and settlement can occur at any time instead of at a fixed date.
7. In the futures market, a widget contract has a standard contract size of 5,000 widgets. What advantage does this have over the wellknown forward market practice of negotiating the size of the transaction on a casebycase basis? What disadvantages does the standardized contract size have?
With standardized contract size and other terms, a futures contract avoids uncertainty about what is being traded. If these terms were not specified, traders would have to specify all of the features of the underlying good anew each time there is a contract. The futures style of trading has the disadvantage of losing flexibility due to the standardization. For example, the amount of the contract is fixed, as is the quality of the underlying good and the time the contract will be settled.
8. What factors need to be considered in purchasing a commodity futures exchange seat? What are all the possible advantages that could come from owning a seat? A seat on a commodity exchange is essentially a capital asset. The purchaser would want to consider the risk, including systematic risk, associated with such a purchase. The value of the seat depends mainly on the expected trading volume on the exchange, so we expect seat prices to be sensitive to the business cycle and to competition from foreign exchanges. Owning a seat allows one to trade on the exchange. Also, the seat holder can lease the seat for someone else to trade. Therefore, the seat offers the potential for cash inflows as well as capital appreciation.
9. Explain the difference between initial and maintenance margin. Initial margin is the amount a trader must deposit before trading is permitted. Maintenance margin is the minimum amount that must be held in the trader's account while a futures position is open. If the account value falls below the amount specified as the maintenance margin, the trader must deposit additional funds to bring the account value back to the level of the initial margin.
10. Explain the difference between maintenance and variation margin. Maintenance margin is the amount a trader must keep in the account to avoid a margin call. Variation margin is the payment a trader must make in a margin call. The margin call occurs when the account value drops below the level set for the maintenance margin. Upon receiving a margin call, the trader must make a cash payment of the variation margin. The maintenance margin is a stock variable, while the variation margin is a flow variable.
11. On February 1, a trader is long the JUN wheat contract. On February 10, she sells a SEP wheat futures, and sells a JUN wheat contract on February 20. On February 15, what is her position in wheat futures? On February 25, what is her position? How would you characterize her transaction on February 20? On February 15, the trader holds an intracommodity spread, being long the JUN and short the SEP wheat. On February 25, the trader is short one SEP wheat contract. The transaction on February 20 was a transaction offsetting the original long position in SEP wheat.
12. Explain the difference between volume and open interest. Open interest is the number of contracts currently obligated for delivery. The volume is the number of contracts traded during some period. For every purchase there is a sale, and the purchase and sale of one contract generates one contract of trading volume.
13. Define tick and daily price limit. A tick is the minimum amount a futures contract can change. For example, in the Tbond contract, the tick size is 1/32 of one point of par. This gives a dollar tick value of $31.25 per Tbond futures contract. The daily price limit is the amount the contract can change in price in one day. It is usually expressed as some number of ticks, and it is measured from the previous day's settlement price. No trade can be executed at a price that differs from the previous day's settlement price by more than the daily price limit.
14. A trader is long one SEP crude oil contract. On May 15, he contracts with a business associate to receive 1,000 barrels of oil in the spot market. The business associate is short one SEP crude oil contract. How can the two traders close their futures positions without actually transacting in the futures market? The traders close their futures positions through an exchangeforphysicals transaction (EFP). In an EFP, two traders with futures positions exchange the physical good for cash and report this transaction to the exchange, asking the exchange to offset their futures contracts against each other. This transaction is also called an expit or against actuals transaction.
15. Explain how a trader closes a futures market position via cash settlement. For a contract satisfied by cash settlement, there is no delivery. Instead, when the futures contract expires, the final settlement price on the futures is set equal to the cash price for that date. This practice ensures convergence of the futures price and the cash price. Traders then make or receive payments based on the difference between the previous day's settlement price and the final settlement price on the contract.
16. Explain price discovery. In futures markets, price discovery refers to the revealing of information about future prices that the market facilitates. It is one of the two major social functions of the futures market. (The other is risk transference.) As an example, the futures price for wheat for delivery in nine months reveals information to the public about the expected future spot price of wheat at the time of delivery. While controversial, there is some reason to believe that the futures price (almost?) equals the spot price that is expected to hold at the futures expiration. This price discovery function helps economic agents plan their investment and consumption by providing information about future commodity prices.
17. Contrast anticipatory hedging with hedging in general. In anticipatory hedging, a trader enters the futures market and transacts before (or in anticipation of) some cash market transaction. This differs from a hedge of an existing position. For example, a farmer might sell wheat futures in anticipation of the harvest. Alternatively, a merchant holding an inventory of wheat might hedge the inventory by selling wheat futures. The farmer is engaged in anticipatory hedging, because he or she is expecting to have the cash market position and hedges this anticipation. The wheat merchant already has the cash market position, in virtue of holding the wheat inventory, and therefore is not engaged in anticipatory hedging.
18. What is front running? Front running is a market practice in which a broker holds a customer's order for execution and executes a similar order of his or her own before executing the customer's order. This practice can be particularly pernicious if the customer's order is large, because the order may itself move prices. By front running, the broker seeks to capitalize on the privileged information that the order is coming to market. This practice is unethical and against the rules of the futures exchange.
19. Explain the difference in the roles of the National Futures Association and the Commodity Futures Trading Commission. The National Futures Association (NFA) is an industry selfregulatory body, while the Commodity Futures Trading Commission (CFTC) is an agency of the federal government. The same law that instituted the CFTC also provided for the futures industry to establish selfregulatory bodies. The NFA enforces ethical standards on most futures industry members and provides testing for licensing of brokers and other futures industry professionals. The NFA operates under the supervision of the CFTC.
Futures Prices
1. Explain the function of the settlement committee. Why is the settlement price important in futures markets in a way that the day's final price in the stock market is not so important?
In futures markets, the settlement committee determines the settlement price for each contract each day. The settlement price estimates the true value of the contract at the end of the day's trading. In active markets, the settlement price will typically equal the last trade price. In inactive markets, the settlement price is the committee's estimate of the price at which the contract would have traded at the close, if it had traded. The settlement price is important, because it is used to calculate margin requirements and the cash flows associated with daily settlement. In the stock market, there is no practice comparable to daily settlement, so the closing price in the stock market lacks the special significance of the futures settlement price.
2. Open interest tends to be low when a new contract expiration is first listed for trading, and it tends to be small after the contract has traded for a long time. Explain.
When the contract is first listed for trading, open interest is necessarily zero. As traders take positions, the open interest builds. At expiration, open interest must again be zero. Every contract will have been fulfilled by offset, delivery, or an EFP. Therefore, as the contract approaches the expiration month, many traders will offset their positions to avoid delivery. This reduces open interest. In the expiration month, deliveries that occur further reduce open interest. Also, EFPs typically reduce open interest. This creates a pattern of very low open interest in the contract's early days of trading, followed by increases, followed by diminution, followed by the contract's extinction.
3. Explain the distinction between a normal and an inverted market.
In a normal market, prices for more distant expirations are higher than prices for earlier expirations. In an inverted market, prices for more distant expirations are lower than prices for earlier expirations.
4. Explain why the futures price converges to the spot price and discuss what would happen if this convergence failed.
The explanation for convergence at expiration depends on whether the market features delivery or cash settlement, but in each case, convergence depends on similar arbitrage arguments. We consider each type of contract in turn. For a contract with actual delivery, failure of convergence gives rise to an arbitrage opportunity at delivery. The cash price can be either above or below the futures price, if the two are not equal. If the cash price exceeds the futures price, the trader buys the future, accepts delivery, and sells the good in the cash market for the higher price. If the futures price exceeds the cash price, the trader buys the good on the cash market, sells a futures, and delivers the cash good in fulfillment of the futures. To exclude both types of arbitrage simultaneously, the futures price must equal the cash price at expiration. Minor discrepancies can exist, however. These are due to transaction costs and the fact that the short trader owns the options associated with initiating the delivery sequence. For a contract with cash settlement, failure of convergence also implies arbitrage. Just before delivery, if the futures price exceeds the cash price, a trader can sell the futures, wait for expiration, and the futures price will be set equal to the cash price. This gives a profit equal to the difference between the cash and futures. Alternatively, if the cash price is above the futures price, and expiration is imminent, the trader can buy the futures and wait for its price to be marked up to equal the cash price. Thus, no matter whether the futures price is above or below the cash price, a profit opportunity will be available immediately. In short, the futures and cash price converge at expiration to exclude arbitrage, and failure of convergence implies the existence of arbitrage opportunities.
5. Is delivery, or the prospect of delivery, necessary to guarantee that the futures price will converge to the spot price? Explain. No, delivery is not necessary. As explained in the answer to question 4, cash settlement will also lead to convergence of the cash and the futures at expiration.
6. As we have defined the term, what are the two key elements of academic arbitrage? The two elements are riskless profit and zero investment. Each condition is necessary for academic arbitrage, and the two conditions are jointly sufficient.
7. Assume that markets are perfect in the sense of being free from transaction costs and restrictions on short selling. The spot price of gold is $370. Current interest rates are 10 percent per year, compounded monthly. According to the costofcarry model, what should the price of a gold futures contract be if expiration is six months away? In perfect markets, the costofcarry model gives the futures price as: The cost of carrying gold for six months is (1 + . 10/12)^{6} — 1= .051053. Therefore, the futures price should be: F_{0},_{t} = $370(1.051053) = $388.89
8. Consider the information in question 7. Roundtrip futures trading costs are $25 per 100ounce gold contract, and buying or selling an ounce of gold incurs transaction costs of $1.25. Gold can be stored for $.15 per month per ounce. (Ignore interest on the storage fee and the transaction costs.) What futures prices are consistent with the costofcarry model? Answering this question requires finding the bounds imposed by the cashandcarry and reverse cashandcarry strategies. For convenience, we assume a transaction size of one 100ounce contract. For the cashandcarry, the trader buys gold and sells the futures. This strategy requires the following cash outflows:
Buy gold

$370(100)


Pay transaction costs on the spot

$1.25(100)


Pay the storage cost

$.15(100)(6)


Sell futures

0


Borrow to finance these outlays

+ $37,215


Pay the transaction cost on one futures
$25
Repay the borrowing $39,114.95
Deliver on futures
Net outlays at the outset were zero, and they were $39,139.95 at the horizon. Therefore, the futures price must exceed $391.40 an ounce for the cashandcarry strategy to yield a profit.
The reverse cashandcarry incurs the following cash flows. At the outset, the trader must:
Sell gold +$370(100)
Pay transaction costs on the spot $1.25(100)
Investfunds $36,875
Buy futures 0
These transactions provide a net zero initial cash flow. In six months, the trader has the following cash flows:
Collect on investment +$36,875(1 + .10/12)^{6} = $38,757.59
Pay futures transaction costs $25
Receive delivery on futures ?
The breakeven futures price is therefore $387.33 per ounce. Any lower price will generate a profit. From the cashandcarry strategy, the futures price must be less than $391.40 to prevent arbitrage. From the reverse cashandcarry strategy, the price must be at least $387.33. (Note that we assume there are no expenses associated with making or taking delivery.)
9. Consider the information in questions 7 and 8. Restrictions on short selling effectively mean that the reverse cashandcarry trader in the gold market receives the use of only 90 percent of the value of the gold that is sold short. Based on this new information, what is the permissible range of futures prices?
This new assumption does not affect the cashandcarry strategy, but it does limit the profitability of the reverse cashandcarry trade. Specifically, the trader sells 100 ounces short but realizes only .9($370)(100) = $33,300 of usable funds. After paying the $125 spot transaction cost, the trader has $33,175 to invest. Therefore, the investment proceeds at the horizon are: $33,175(1 + .10/12)^{6} = $34,868.69. Thus, all of the cash flows are:
Sell gold

+$370(100)


Pay transaction costs on the spot

$1.25(100)


Broker retains 10 percent

$3,700


Invest funds

$33,175


Buy futures

0

These transactions provide a net zero initial cash flow. In six months, the trader has the following cash flows:
Collect on investment $34,868.69
Receive return of deposit from broker $3,700
Pay futures transaction costs  $25
Receive delivery on futures ?
The breakeven futures price is therefore $385.44 per ounce. Any lower price will generate a profit. Thus, the noarbitrage condition will be fulfilled if the futures price equals or exceeds $385.44 and equals or is less than $391.40.
10. Consider all of the information about gold in questions 7—9. The interest rate in question 7 is 10 percent per annum, with monthly compounding. This is the borrowing rate. Lending brings only 8 percent, compounded monthly. What is the permissible range of futures prices when we consider this imperfection as well?
The lower lending rate reduces the proceeds from the reverse cashandcarry strategy. Now the trader has the following cash flows:
Sell gold +$370(100)
Pay transaction costs on the spot $1.25(100)
Broker retains 10 percent $3,700
Investfunds $33,175
Buy futures 0
These transactions provide a net zero initial cash flow. Now the investment will yield only $33,175 (1 + .08/12)^{6} = $34,524.31. In six months, the trader has the following cash flows:
Collect on investment $34,524.31
Pay futures transaction costs  $25
Receive delivery on futures ?
Return gold to close short sale 0
Receive return of deposit from broker $ 3,700
Total proceeds on the 100 ounces are $38,199.31. Therefore, the futures price per ounce must be less than $381.99 for the reverse cashandcarry strategy to profit. Because the borrowing rate has not changed, the bound from the cashandcarry strategy remains at $391.40. Therefore, the futures price must remain within the inclusive bounds of $381.99 to $391.40 to exclude arbitrage.
11. Consider all of the information about gold in questions 710. The gold futures expiring in six months trades for $375 per ounce. Explain how you would respond to this price, given all of the market imperfections we have considered. Show your transactions in a table similar to Table 3.8 or 3.9. Answer the same question, assuming that gold trades for $395.
If the futures price is $395, it exceeds the bound imposed by the cashandcarry strategy, and it should be possible to trade as follows:
Borrow $37,215 for 6 months at 10%.

+$37,215.00

Buy 100 ounces of spot gold.

37,000.00

Pay storage costs for 6 months.

90.00

Pay transaction costs on gold purchase.

125.00

Sell futures for $395.

0.00


Total Cash Flow $0

Remove gold from storage.

$0

Deliver gold on futures.

+ 39,500.00

Pay futures transaction cost.

25.00

Repay debt.

39,114.95

If the futures price is $375, the reverse cashandcarry strategy should generate a profit as follows:
t = 0

Sell 100 ounces of gold short. Pay transaction costs. Broker retains 10%. Buy futures. Invest remaining funds for 6 months at 8%.

+$37,000.00 125.00 3,700.00 0 33,175.00



Total Cash Flow $0

t = 6

Collect on investment. Receive delivery on futures. Return gold to close short sale. Receive return of deposit from broker. Pay futures transaction cost.

$34,524.31 37,500.00 0 +3,700.00 25.00



Total Cash Flow +$699.31

12. Explain the difference between pure and quasiarbitrage. In a pure arbitrage transaction, the arbitrageur faces full transaction costs on each transaction comprising the arbitrage. For example, a retail customer with no initial position in the market, who attempts arbitrage, would be attempting pure arbitrage. By contrast, a quasiarbitrage transaction occurs when a trader faces less than full transaction costs. The most common example arises in reverse cashandcarry arbitrage, which requires short selling. For example, in stock index arbitrage, holding a large portfolio allows a trader to simulate a short sale by selling part of the portfolio from inventory. Therefore, this trader faces less than the full transaction costs due to the preexisting position in the market. By contrast, the pure arbitrage trade would require the actual short sale of the stocks, and short selling does not provide the full proceeds to earn interest in the reverse cashandcarry transactions.
13. Assume that you are a gold merchant with an ample supply of deliverable gold. Explain how you can simulate short selling and compute the price of gold that will bring you into the market for reverse cashandcarry arbitrage.
The breakeven price for reverse cashandcarry arbitrage depends principally on the transaction costs the trader faces. With an existing inventory of gold, the trader can simulate short selling by selling a portion of the inventory. Further, because the trader already actually owns the gold, she can have full use of the proceeds of the sale. Therefore, the gold owner's reverse cashandcarry transactions are similar to those in problem 10:
Reverse CashandCarry Arbitrage
t = 0 Sell 100 ounces of gold short.
+$37,000.00
Pay transaction costs. 125.00
Buy futures. 0
Invest funds for 6 months at 8%. 36,875.00
Total Cash Flow $0
t = 6 Collect on investment. $38,374.80
Return gold to close short sale.
0
Pay futures transaction cost. 25.00
Receive delivery on futures. (Note: This is the futures price to give zero cash flow.) +38,349.80
Total Cash Flow +$0
Therefore, if the futures price is $383.498 per ounce, the reverse cashandcarry transactions give a zero cash flow. This is the breakeven price for reverse cashandcarry. If the futures price is less that $383.498 per ounce, reverse cashandcarry arbitrage will be possible for the trader who holds an initial inventory of gold. In problem 10, the price of gold has to be less than $381.99 for reverse cashandcarry arbitrage to work. The trader there faced full transaction costs, due to the lack of a preexisting inventory.
14. Assume that silver trades in a full carry market. If the spot price is $5.90 per ounce and the futures that expires in one year trades for $6.55, what is the implied costofcarry? Under what conditions would it be appropriate to regard this implied costofcarry as an implied repo rate?
If the market is at full carry, then F$ _{t} = S0(1 + C) and C = F0,t/S0 — 1. With our values, C = $6.55/ $5.90 — 1 = .110169. It would be appropriate to regard this implied costofcarry as an implied repo rate if the only carrying cost were the financing cost. This is approximately true for silver.
15. What is normal backwardation? What might give rise to normal backwardation?
Normal backwardation is the view that futures prices normally rise over their life. Thus, prices are expected to rise as expiration approaches. The classic argument for normal backwardation stems from Keynes. According to Keynes, hedgers are short in the aggregate, so speculators must be net long. Speculators provide their riskbearing services for an expected profit. To have an expected profit, the futures price must be less than the expected future spot price at the time the speculators assume their long positions. Therefore, given unbiased expectations regarding future spot prices, we expect futures prices to rise over time to give the speculators their compensation. This leads directly to normal backwardation.
16. Assume that the CAPM beta of a futures contract is zero, but that the price of this commodity tends to rise over time very consistently. Interpret the implications of this evidence for normal backwardation and for the CAPM.
Because futures trading requires no investment, positive returns on long futures positions can be consistent with the CAPM only if futures have positive betas. With a zero beta (by our assumption) and a zero investment to acquire a long futures position (by the structure of the market), the CAPM implies zero expected returns. Therefore, a zero beta and positive returns is inconsistent with the CAPM. Even with zero beta, positive returns are consistent with normal backwardation resulting from speculators assuming long positions and being rewarded for their riskbearing services.
17. Explain why futures and forward prices might differ. Assume that platinum prices are positively correlated with interest rates. What should be the relationship between platinum forward and futures prices? Explain.
Futures are subject to daily settlement cash flows, while forwards are not. If the price of the underlying good is not correlated with interest rates, futures and forward prices will be equal. If the price of the underlying good is positively correlated with interest rates, a long trader in futures will receive daily settlement cash inflows when interest rates are high and the trader can invest that cash flow at the higher rate from the time of receipt to the expiration of the futures. Because forwards have no daily settlement cash flows, they are unable to reap this benefit. Therefore, if a commodity's price is positively correlated with interest rates, there will be an advantage to a futures over a forward. Thus, for platinum in the question, the futures price of platinum should exceed the forward price. The opposite price relationship can occur if there is negative correlation. Generally, this price relationship is not sufficiently strong to be observed in the market.
18. Consider the life of a futures contract from inception to delivery. Explain two fundamental theories on why the futures prices might exhibit different volatility at different times over the life of the contract.
According to the Samuelson hypothesis, price volatility will be greater when more information about the price of the good is being revealed. According to this view, this tends to happen as the futures comes to expiration, particularly for agricultural goods. Therefore, the Samuelson hypothesis suggests that the volatility of futures prices should increase over the life of the contract.
There are several other theories that attempt to relate contract maturity and volatility. First, there seems to be some evidence for believing that volatility is higher for some commodities in certain seasons, particularly at times when information about the harvest of some good is reaching the market. With this view, volatility depends on the time of the year and not so much on the contract's expiration. Second, volatility also differs depending on the day of the week. Third, volatility is autocorrelated. High volatility in one month begets high volatility in the next month.
19. Consider the following information about the CMX gold futures contract:
Contract size: 100 troy ounce
Initial margin: $1,013 per contract
Maintenance margin: $750 per contract
Minimum tick size: 10 cents/troy ounce ($10/contract)
There are four traders, A, B, C, and D, in the market when next year's June contract commences trading. A. Complete the following table showing the open interest for the contract.
July 6

A

B

5

$294.50

July 6

C

B

10

$294.00

July 6

Settlement Price



$294.00

July 7

D

A

10

$293.50

July 7

B

D

5

$293.80

July 7 Settlement Price



$293.80


July 8

B

A

7

$293.70

July 8

Settlement Price



$299.50

When trading commences, each trader has a zero net position. The open interest, which is the sum of the trader's net long positions, is also zero. Looking at each trade sequentially, we can track the traders' net positions and open interest as follows:
Trade

Trader A

Trader B

Trader C

Trader D

Open Interest

1

+ 5

05

0

0

5

2

+ 5

15

+ 10

0

15

3

5

15

+ 10

+ 10

20

4

5

10

+ 10

+ 5

15

5

12

3

+ 10

+ 5

15

B. Calculate the gains and losses for Trader A. Assume that at the time of each change in position, Trader A must bring the margin back to the initial margin amount. Compute the amount in Trader A's margin account at the end of each trading day. Will Trader A get a margin call? If so, when and how much additional margin must be posted?
Date

Buy Seil

Position

Price

(loss)

Variation

Margin

Variation

July 6

5

+ 5

$294.50

_

$0

$5,065

$5,065

July 6








settlement


+ 5

$294.00

($250)

$4,815

$0

$4,815

July 7

10

,5

$293.50

($250)

$4,565

$500

$5,065

July 7








settlement


15

$293.80

($150)

$4,915

$0

$4,915

July 8

7

$12

$293.70

$50

$4,965

$7,191

$12,156

July 8








settlement


$12

$299.50

($6,960)

$5,196

$6,960

$12,156

On July 6, Trader A goes long 5 contracts at a price of $294.50. At this time, she is required to post $1,013 margin per contract for a total of $5,065. The July 6 settlement price of $294.00 results in a $50 per contract loss (100 X —$0.50). Trader A's 5 contracts lose a total of $250 which is taken from the margin account. Trader A's margin account still has $963 per contract which is in excess of the minimum maintenance margin of $750 per contract. Therefore, no margin call occurs. On July 7, Trader A reverses her long position and goes short 5 contracts. A $250 loss occurs on the reversed long position which is assessed against the margin account, bringing it down to $4,565. Since Trader A is now short 5 contracts, she adds $500 to her margin account to bring it back to the initial margin requirement of $5,065. At settlement on July 7, Trader A's position lost $150 which is taken from her margin account bringing it to $4,915. No margin call occurs. On July 8, Trader A sells an additional 7 contracts, bringing her total to short 12 contracts. At the time of the transaction, her original position of 5 short contracts has gained $50 which is credited to her margin account, bringing it to $4,965. The required initial margin for 12 contracts is $12,156, so Trader A adds $7,191 to her margin account. July 8 is not a good day for Trader A. The futures price of gold increases by $5.80 per ounce. Trader A's 12contract short position loses $6,960, bringing her margin account to $5,196 or $433 per contract. Since this is below the maintenance level, Trader A gets a margin call. She must add $6,960 to bring her margin back to the initial margin level. Alternatively, she can close her position.
20. Today is June 30. You have an anticipated liability of $10 million due on December 31. To fund this liability, you plan to sell part of your store of gold. Looking in The Wall Street Journal, you note the following futures prices and bond equivalent yields for the Tbills maturing at or near the expiration of the futures contracts:
Cash price 293.00
AUG 297.40 4.86%
DEC 302.00 5.35%
A. What is the basis for the December futures contract?
The basis is computed as the cash price minus the futures price.
Basis = $293.00  $302.00 = $9
B. Is the market normal or inverted? Explain.
In a normal market, the prices for the more distant contracts are higher than the prices for the nearby contracts. In an inverted market, futures prices decline as we go from the nearby contracts to the more distant contracts.
In this question, prices increase the more distant the delivery date, so the market is normal.
C. If you wanted to eliminate the risk of gold price variation, explain your alternatives. There are several alternatives for eliminating price risk.
Alternative 1: Sell gold via the December futures contract. This effectively locks in a December selling price of $302. Alternative 2: Sell gold today in the spot market and invest the proceeds in the 6month Tbill. Alternative 3: Sell gold today in the spot market and invest the proceeds in the 2month Tbill. At the same time, buy gold by using the August futures contract and sell gold using the December futures contract. The most attractive alternative is the one which allows us to meet the $10 million liability while selling the least amount of gold.
D. Given the above prices, justify which of the alternatives you would prefer. Alternative 1: Sell gold using the December futures contract.
r> + t> c \a $10 million
Ounces to Be Sold = = 33,113 ounces
$302
Alternative 2: Sell gold today in the spot market and invest the proceeds in the 6month Tbill.
Cash Needed Today = ,^{$1}° ^{million} = $9.739 _{m}illi_{o}n
r> D c UT a $9.739million ,,., Ounces to Be Sold T oday =ämi= 33,241 ounces
Alternative 3: Sell gold today in the spot market and invest the proceeds in the 2month Tbill. At the same time, buy gold using the August futures contract and sell gold using the December futures contract. We already know from alternative 1 that 33,113 ounces must be sold in December. Then 33,113 ounces must be bought in August.
Cash Needed in August = 33,113 x $297.40 = $9.848 million
r um a a t a $9.848million acnn ir
Cash Needed Today =;—— = $9.769 million
1 + 0.0486
r> D c UT a $9.769million ,,,.,, Ounces to Be Sold T oday =~i^>^{=} 33,340 ounces
$293
Conclusion: Selling gold using the December futures contract requires the fewest ounces of gold to be sold.
E. At what August and December futures prices would you be indifferent between the alternatives?
Assuming negligible costs other than the time value of money, the following relationship must hold for you to be indifferent between alternatives:
In this relationship the cost of carry, C, is the borrowing and lending rate. For the December contract, the futures price at which you are indifferent between alternatives 1 and 2 is:
F0,dec = 293 I 1 + ^y^ I = $300.84 The August futures price at which you are indifferent between alternatives 1 and 3 is:
F_{0}, _{AUG} = $293 1 + 0.0486( 2) = $295.37
21. It is now the beginning of July. In the cash market, No. 2 heating oil is selling for $0.3655 per gallon. The December futures contract for this commodity is selling at $0.4375. The cost of borrowing for these 6 months is 0.458 percent per month.
A. What is the lowest cost of storage and delivery that would prevent the opportunity for cashandcarry arbitrage? If we borrowed for 6 months at 0.458 percent per month, we could buy heating oil today at $0.3655 per gallon and sell it in December for $0.4375 per gallon. Assuming no cost of storage or delivery, the cashandcarry arbitrage profit would be:
Aribitrage Profit = F_{0},_{DEC}  S_{0}(1 + C)
= 0.43750.3655(1.00458)^{6 }= $0.0618/gallon
Any cost of storage and delivery (paid in December) exceeding 6.18 cents per gallon would prevent cashandcarry arbitrage.
B. The August futures contract for the No. 2 heating oil is $0.3702. You can lockin a 0.442 percent per month borrowing rate for the 4month time period between delivery of the August contract and the delivery of the December contract. What is the lowest cost of storage and delivery that would prevent the opportunity for cashandcarry arbitrage in this case? We could borrow 2 months forward, buy heating oil 2 months forward, and sell heating oil 6 months forward. This is a forward cashandcarry arbitrage. Assuming no cost of storage or delivery, the arbitrage profit would be:
Arbitrage Profit = F_{0},_{DEC}  F0,aug(1 + C
= 0.4375  0.3702(1.00442)^{4 }= $0.0607/gallon
Any cost of storage and delivery (paid in December) exceeding 6.07 cents per gallon would prevent cashandcarry arbitrage.
22. Consider the following June spot and futures prices for the CBOT silver contract:
Spot

$5.32

August

$5.55

October

$5.80

December

$6.13

A. Assuming no storage costs, compute the implied spot repo rates.
The implied repo rate is equal to:
C = F_{0>l}/S_{0}l
For August: C = 5.55/5.32  1 = 4.32%
Annualized C= C (12/t) = 4.32% (12/2) = 25.9%
For October: C = 5.80/5.32  1 = 9.02%
Annualized C= C (12/t) = 9.02% (12/4) = 27.1%
For December: C = 6.13/5.32  1 = 15.23%
Annualized C= C (12/t) = 15.23% (12/6) = 30.5%
B. Assuming no storage costs, compute the implied August forward repo rates. The implied forward repo rates are given by:
Forward C = _F_{oß} / F_{0},_{t1} — 1
For October: Forward C = 5.80/5.55  1 = 4.5%
Annualized Forward C= C (12/t) = 4.5% (12/2) = 27.1%
For December: Forward C = 6.13/5.55  1 = 10.5%
Annualized Forward C= C (12/t) = 10.5% (12/4) = 31.4%
1. Explain how futures markets can benefit individuals in society who never trade futures.
One of the main benefits that the futures market provides is price discovery; futures markets provide information about the likely future price of commodities. This information is available to anyone in the economy, because the prices are publicly available. It is not necessary to trade futures to reap this benefit.
2. A futures price is a market quoted price today of the best estimate of the value of a commodity at the expiration of the futures contract. What do you think of this definition? This claim is intriguing but controversial. If there is no risk premium embedded in the futures price, the statement is likely to be true. The definition implies that random holding of futures positions should earn a zero profit. This seems to be approximately true, but studies such as that by Bodie and Rosansky find positive returns to long futures positions. While the claim may not hold literally, it does seem to be close to correct. Further, those who reject the claim may have a difficult time in identifying futures prices that are above or below the future spot price.
3. Explain the concept of an unbiased predictor. A predictor is unbiased if the average prediction error equals zero. This implies that errors in the prediction are distributed around zero, and that the prediction is equally likely to be high as well as low.
4. How are errors possible if a predictor is unbiased? Saying that a predictor is unbiased merely claims that the predictions do not tend to be too high or too low. They can still be in error. For example, the futures price may provide an unbiased prediction of the future spot price of a commodity. Nonetheless, the errors in such a prediction are often large, because the futures price today can diverge radically from the spot price at the expiration of the futures.
5. Scalpers trade to capture profits from minute fluctuations in futures prices. Explain how this avaricious behavior benefits others. Scalpers trade frequently, attempting to profit by a tick here or there. In pursuing their profit, the scalpers provide the market with liquidity. Thus, a trader who wishes to take or offset a position benefits from the presence of scalpers ready to take the opposite side of the transaction. With many scalpers competing for business, position traders will be able to trade at prices that closely approximate the true value of the commodity. Expressed another way, as scalpers compete for profits, they force the bidasked spread to narrow, therefore contributing to the liquidity of the market.
6. Assume that scalping is made illegal. What would the consequences of such an action be for hedging activity in futures markets? Without scalpers, the liquidity of the futures market would be greatly impaired. This would imply a widening of bidasked spreads. The potential hedger would face having to accept a price that was distant from the true price. Faced with the higher transaction costs represented by wider bidasked spreads, some hedgers might find that hedging is too expensive and they might not hedge. Thus, without scalpers, hedging would be more expensive, and we would observe a lower volume of hedging activity.
7. A trader anticipates rising corn prices and wants to take advantage of this insight by trading an intracommodity spread. Would you advise that she trade long nearby/short distant or the other way around? Explain. The answer depends on the relative responsiveness to nearby and distant futures prices to a generally rising price level for corn. If the nearby contract price rises more than the price of the distant contract, the trader should go long nearby/short distant, for example. For most agricultural commodities, there is no general rule to follow.
8. Assume that daily settlement prices in the futures market exhibit very strong firstorder serial correlation. How would you trade to exploit this strategy? Explain how your answer would differ if the correlation is statistically significant but, nonetheless, small in magnitude. With strong serial correlation, a price rise is likely to be followed by another price rise, and a price drop is likely to be followed by another price drop. Therefore, the trader should buy after a price rise and sell after a price fall. If the correlation is strong, the strategy should generate profits. However, the correlation must be very strong to generate profits sufficient to cover transaction costs. The correlation can be statistically significant, but still too small to be economically significant. To be economically significant, the correlation must be strong enough to generate trading profits that will cover the transaction costs. Studies typically find statistically significant firstorder serial correlation in futures price changes, but they also find that these correlations are not economically significant.
9. Assume that you are a rabid efficient markets believer. A commodity fund uses 20 percent of its funds as margin payments. The remaining 80 percent are invested in riskfree securities. What investment performance would you expect from the fund? For any efficient markets believer, rabid or calm, the expected return on the 80 percent of the funds is the riskfree rate. If there is no risk premium, the expected profit on the futures position is zero. Thus, we define a rabid efficient markets believer as one who denies the existence of a risk premium. Therefore, the rabid theorist expects returns from the funds that would be 80 percent of the riskfree rate.
10. Consider two traders. The first trader is an individual with his own seat who trades strictly for his own account. The other trader works for a brokerage firm actively engaged in retail futures brokerage. Which trader has a lower effective marginal trading cost? Relate this comparison in marginal trading costs to quasiarbitrage. This is a difficult question. The trader who owns a seat incurs the following costs to trade: the capital commitment to the seat, the opportunity cost of foregone alternative employment, and the exchange member's outofpocket transaction costs. These outofpocket costs are quite low. For the broker, the scale is much greater. Behind the broker in the pit stands the entire brokerage firm organization with the overhead it represents. Offsetting this overhead to some extent is the much greater scale associated with the brokerage firm. Also, for the trader associated with the brokerage firm, much of the overhead is associated with retail operations, and the marginal cost of trading an additional contract can be quite low. Thus, we judge that the brokerage firm has the lower marginal cost of trading. This difference in trading costs (whichever is really lower) can be important for quasiarbitrage. Essentially, the fruits of quasiarbitrage can be harvested by the trader with the lowest marginal transaction costs. If our assessment of these costs is correct, the brokerage firm should be able to squeeze out the market maker and capture these quasiarbitrage profits.
11. Consider the classic hedging problems of the farmer who sells wheat in the futures market in anticipation of a harvest. Would the farmer be likely to deliver his harvested wheat against the futures? Explain. If he is unlikely to deliver, explain how he manages his futures position instead. Most farmers who hedge would not deliver against the futures. Often the wheat would not be deliverable, due to differences in grade or type of wheat. Also, the wheat is probably distant from an approved delivery point, and trying to deliver the wheat would involve prohibitively high transportation costs. Instead of actually delivering, the farmer would be much more likely to sell the harvested wheat to the local grain elevator and offset the futures position.
12. A cocoa merchant holds a current inventory of cocoa worth $10 million at present prices of $1,250 per metric ton. The standard deviation of returns for the inventory is .27. She is considering a riskminimization hedge of her inventory using the cocoa contract of the Coffee, Cocoa and Sugar Exchange. The contract size is 10 metric tons. The volatility of the futures is .33. For the particular grade of cocoa in her inventory, the correlation between the futures and spot cocoa is .85. Compute the riskminimization hedge ratio and determine how many contracts she should trade. We know that the hedge ratio is: HR = where S and F indicate the spot and futures, respectively. Therefore, with our data, the hedge ratio is: Currently, the merchant holds $10,000,000/$1,250 = 8,000 metric tons. The hedge ratio indicates trading .6955 of the futures for each unit of the spot. This implies a futures position of 8,000(.6955) = 5,563.64 metric tons. With the futures consisting of 10 tons per contract, the correct futures quantity is 5,564/10 ~ 56 contracts. Because she is long the physical cocoa, she should sell 56 futures contracts.
13. A service station operator read this book. He wants to hedge his risk exposure for gasoline. Every week, he pumps 50,000 gallons of gasoline, and he is confident that this pattern will hold through thick and Hussein. What advice would you offer? The operator should probably not hedge. By construction, the operator faces a fairly small and recurring risk. If the futures price equals the expected future spot price, the expected gains from hedging are zero, ignoring transaction costs. If we consider transaction costs, the hedging program is almost certain to cost money over the long run. Futures hedging is better designed for large risks or special applications. Persistent hedging of repeated small and independent risks will lead to losses equal to the transaction costs the more often the hedge is attempted (assuming the futures price equals the expected future spot price).
14. The Mesa Rosa Tortilla Company is a large producer of tortilla chips whose main ingredient is corn. The demand for Mesa Rosa corn chips is seasonal with the largest demand occurring midNovember through the end of December. Production schedules require acquisition of 25 thousand bushels of corn in late September to meet the holiday season demand. Mesa Rosa management is concerned about the possibility that a rise in the price of corn between now and September could hurt profitability. Corn must be acquired at a price of $2.25 per bushel or less to ensure profitability. The September corn futures contract (5,000bushel quantity) is selling for $2.11 per bushel. A. What can Mesa Rosa do to ensure its profitability? Mesa can acquire corn today and store it until September, or Mesa can acquire corn using the September corn futures contract. Using the futures contract, it would buy 5 September contracts at $2.11 per bushel.
B. What risks does Mesa Rosa face in acquiring corn by its taking delivery of the futures contract? How should Mesa Rosa acquire the corn it needs? When September arrives, Mesa can acquire the corn in one of two ways. First, it can take delivery of the corn via the futures contract. Unfortunately, the short side of the contract chooses the delivery location. This location may or may not be convenient for Mesa. The second alternative for acquiring corn eliminates this risk. In this method, Mesa acquires corn in the spot market and enters a reversing trade in the futures market. If the futures price has moved since the initiation of the hedge, any gains (losses) on the futures contract offset any losses (gains) in the cash market so that the effective price Mesa pays for corn is $2.11.
C. If the September spot price turns out to be $3.15 per bushel, show Mesa Rosa's transactions in the corn cash and futures markets and calculate its net wealth change.
Mesa's Long Hedge
Date Cash Market Futures Market
Today Mesa anticipates need for 25,000 bushels of corn in Buy five 5,000bushel SEP corn futures at $2.11/bushel.
September; wants to pay $2.11/bushel or $52,750 total.
September Spot price of corn is $3.15/bushel. Mesa buys 25,000 At maturity, the futures price equals the spot price. Sell
bushels for $78,750. 5 futures contracts at $3.15/bushel.
Opportunity Loss: $52,750  $78,750 =  $26,000 Futures Profit: 25,000 ($3.15  $2.11) = $26,000
Net Wealth Change = 0
15. It is August 10 and Farmer John is making final estimates of this year's wheat crop. His production is turning out to be much better than expected. This causes concern because if his production is better than expected, other farmers must be experiencing the same situation. The current spot price is $2.25 per bushel, and the September wheat futures (5,000 bushels per contract) price is $2.52 per bushel. At the current spot price, Farmer John would just break even with his anticipated 60 thousand bushels. His wheat will not be ready to harvest until September.
A. What can Farmer John do to ensure his profitability? Is this a long or a short hedge? Why? Farmer John can sell his anticipated wheat production in the futures market. Any opportunity gains (losses) resulting from changing wheat prices will be offset by losses (gains) in the futures market. This is a short hedge because Farmer John is selling his production forward. The counterparty to his contract might be a producer acquiring wheat forward for whom the transaction would be a long hedge. B. At harvest time in September, Farmer John's concerns are realized in that the cash price has dropped to $1.70 per bushel. Compute Farmer John's net wealth change due to the drop in corn prices, assuming he hedged his anticipated production and his final yield was 60,000 bushels.
Date Cash Market Futures Market
August 10 Farmer John anticipates the production of 60,000 Sell twelve 5,000bushel SEP wheat futures contracts at bushels of wheat which he wishes to sell at $2.52/bushel.
$2.52/bushel for a total of $151,200.
September Farmer John sells 60,000 bushels in the spot market at At maturity, the futures price will equal the spot price.
$1.70/bushel for a total of $102,000. Farmer John buys 12 contracts at $1.70/bushel.
Opportunity Loss: ($102,000  $151,200) = $49,200 Futures Profit: 60,000 ($2.52  $1.70) = $49,200
C. Suppose Farmer John's production turned out to be only 50,000 bushels. Compute his net wealth change.
Date Cash Market Futures Market
August 10 Farmer John anticipates the production of 60,000 Sell twelve 5,000bushel SEP wheat futures contracts at
bushels of wheat which he wishes to sell at $2.52/bushel.
$2.52/bushel for a total of $151,200.
September Farmer John sells 50,000 bushels in the spot market at At maturity, the futures price will equal the spot price.
$1.70/bushel for a total of $85,000. Farmer John buys 12 contracts at $1.70/bushel. Opportunity Losses: Futures Profit: $49,200
Price Change = 60,000 ($1.70  $2.52) =  $49,200 Production Variation = 10,000x$1.70 =  $17,000 Total:  $66,200
Net Wealth Change =  $17,000
Farmer John's net wealth change is negative because he had anticipated 60,000 bushels of wheat production, but his final production was 10,000 bushels less. He could have sold those 10,000 bushels at $1.70 per bushel if he had them. This results in $17,000 opportunity loss attributable to production variation.
16. Ace Trucking Lines has a fleet of 10,000 trucks that carry a variety of commodities throughout North America. One of its major costs of operation is diesel fuel. There is no futures contract traded on diesel fuel, but there are futures contracts traded on No. 2 heating oil, closely related to diesel, and unleaded regular gasoline. Both of these contracts are traded in quantities of 42,000 gallons/contract. Identify three factors related to Ace Trucking Line's situation that would make any hedging activity be characterized as crosshedging.
A crosshedge is a hedge in which the commodity being hedged and the hedging instrument have dissimilar characteristics. These characteristics can relate to: the time span of the hedge and delivery data on the hedging instrument; the quantity hedged and the size of the underlying instrument; and the commodity being hedged and the commodity deliverable on the hedging instrument.
In Ace Trucking's case, most likely all three of these characteristics apply. First, Ace Trucking would not maintain much inventory of fuel. Most fuel would be purchased through retailers. This makes it difficult to identify a delivery date, as fuel is purchased on a continual basis. Second, Ace's demand for diesel fuel is likely to vary from exact multiples of 42,000 gallons. Hence, the quantity being hedged and the quantity deliverable on the underlying contracts are likely to vary. Finally, there is no diesel contract. The closest contract would be the No. 2 heating oil. Hence, the commodity being hedged is not the same as that deliverable on the hedging instrument.
Ace Trucking, if it would be interested in hedging its fuel price risk, might investigate riskminimization hedging techniques. In this case, Ace would find the futures contract whose price is most highly correlated with the price of diesel fuel and use that contract for hedging.
17. QT has a network of 150 gasoline outlets throughout the central United States. At any one time, the company has 1.125 million gallons of gasoline inventory. Derek Larkin has suggested that QT hedge the risk of its gasoline inventories. He says that the appropriate hedging technique would be riskminimization.
A. What is riskminimization hedging?
In riskminimization hedging, one trades futures contracts in the amount that will minimize the variation of the value of a portfolio composed of the cash position and the futures position. To determine the riskminimizing hedge ratio, one regresses the price changes of the spot price against the hedging instrument's price changes over the same time period. The slope coefficient from the regression is the riskminimizing hedge ratio. The regression result indicates the units of the hedging commodity to trade for each unit of the spot commodity.
B. Derek estimates the following relationship between spot, S_{t}, and futures, F_{t}, prices using the nearby 42,000gallon unleaded regular gasoline contract:
AS_{t} = a+ ßAF_{t} + e_{t }Derek's estimation gives the following results:
a = 0.5231 ß = 0.9217 R^{2} = 0.88
Based on these results, what should QT do to hedge its inventory price risk?
QT has 1.125 million gallons of gasoline in inventory. Derek's results suggest a hedge ratio of —0.9217 gallons of futures for each gallon of inventory. Computing the number of contracts:
Number of Contracts = 0.9217 (1,125,000/42,000) = 24.7 Derek's recommendation would be to sell 25 contracts to hedge QT's price risk.
C. Derek also estimated the same relationship using the nearby 42,000gallon No. 2 heating oil futures contract with the following results:
a = 0.7261 ß = 0.6378 R^{2} = 0.55
Compare the results from the two regressions and comment on which contract would be most appropriate for hedging purposes. Comparing the results of the two regressions, the most important consideration is the R^{2}. The R^{2} tells the percentage of spot price change variation explained by changes in the futures price. A perfect hedging instrument would explain 100 percent of the price change variation. Failing that, Derek should recommend the hedging instrument with the highest R^{2}. In this case, the unleaded gasoline contract with its R^{2} of 88 percent dominates the No. 2 heating oil contract with its R^{2} of 55 percent.
Interest Rate Futures: Refinements
1. Explain the risks inherent in a reverse cashandcarry strategy in the Tbond futures market.
The reverse cashandcarry strategy requires waiting to receive delivery. However, the delivery options all rest with the short trader. The short trader will initiate delivery at his or her convenience. In the Tbond market, this exposes the reverse cashandcarry trader to receiving delivery at some time other than the date planned. Also, with so many different deliverable bonds, the reverse cashandcarry trader is unlikely to receive the bond he or she desires. (These factors are fairly common for other commodities as well.) In the Tbond futures market, the short trader holds some special options such as the wildcard and endofmonth options. The reverse cashandcarry trader suffers the risk that the short trader will find it advantageous to exploit the wildcard play or exercise the endofmonth option.
2. Explain how the concepts of quasiarbitrage help to overcome the risks inherent in reverse cashandcarry trading in Tbond futures. In pure reverse cashandcarry arbitrage, the trader sells the bond short and buys the future. The trader thereby suffers risk about which bond will be delivered and the time at which it will be delivered. If the trader holds a large portfolio of bonds and sells some bond from inventory to simulate the short sale, these risks are mitigated. Receiving a particular bond on delivery is no longer so crucial to the trader's cash flows; after all, whichever bond is delivered will merely supplement the trader's portfolio. Further, the timing of delivery presents fewer problems to the quasiarbitrage trader. In selling a bond from inventory, as opposed to an actual short sale, the trader did not need to worry about financing the short sale for a particular time. Therefore, the selection of a particular delivery date by the short futures trader is less critical. While quasiarbitrage helps to mitigate the risks associated with the reverse cashandcarry trade, risks still remain, particularly the risks associated with the short trader's options.
3. Assume economic and political conditions are extremely turbulent. How would this affect the value of the seller's options on the Tbond futures contract? If they have any effect on price, would they cause the futures price to be higher or lower than it otherwise would be? Generally, options are more valuable the greater the price risk inherent in the underlying good. This is certainly true for the seller's options on the Tbond futures contract. To see this most clearly, we focus on the wildcard option. Exploitation of the wildcard option depends on a favorable price development on any position day between the close of futures trading and the end of the period to announce delivery at 8 p.m., Chicago time. If markets are turbulent, there is a greater chance that something useful will occur in that time window on some day in the delivery month. The greater value of the seller's options in this circumstance would cause the futures price to be lower than it otherwise would be.
4. Explain the difference between the wildcard option and the endofthemonth option. The wildcard option is the seller's option to initiate the delivery sequence based on information generated between the close of futures trading and 8 p.m., Chicago time, the time by which the seller must initiate the delivery sequence for a given day. The settlement price determined at the close of trading is the price that will be used for computing the invoice amount. Trading of the Tbond futures contract ceases on the eighth to last business day of the expiration month, and the settlement price on that day is used to determine the invoice amount for all deliveries. Any contracts not closed by the end of the trading period must be fulfilled by delivery. Even though the short trader must make delivery in this circumstance, the short trader still possesses an endofthemonth option. The short trader can choose which day to deliver and can choose which bond to deliver. The short trader will deliver late in the month if the rate of accrual on the bond planned for delivery exceeds the shortterm financing rate at which the bond is carried. Also, changing market conditions can change which bond will be cheapesttodeliver, and the right to wait and choose a later delivery date has value to the short trader.
5. Some studies find that interest rate futures markets were not very efficient when they first began but that they became efficient after a few years. How can you explain this transition? The growing efficiency of these markets seems to be due to a market seasoning or maturation process. When these contracts were first initiated, it appears that some of their nuances were not fully appreciated. In particular, the complete understanding of the importance of the seller's options seems to have emerged only slowly.
6. Assume you hold a Tbill that matures in 90 days, when the Tbill futures expires. Explain how you could transact to effectively lengthen the maturity of the bill. Buy the Tbill futures that expires in 90 days. After this transaction, you will be long a spot 90day bill, and you will hold (effectively) a spot position in a 90day bill to begin in 90 days. The combination replicates a 180day bill.
7. Assume that you will borrow on a shortterm loan in six months, but you do not know whether you will be offered a fixed rate or a floating rate loan. Explain how you can use futures to convert a fixed to a floating rate loan and to convert a floating rate to a fixed rate loan. For convenience, we assume that the loan will be a 90day loan. If the loan is to be structured as a floating rate loan, you can convert it to a fixed rate loan by selling a shortterm interest rate futures contract (Eurodollar or Tbill) that expires at the time the loan is to begin. The rate you must pay will depend on rates prevailing at the time of the loan. If rates have risen you must pay more than anticipated. However, if rates have risen, your short position in the futures will have generated a profit that will offset the higher interest you must pay on the loan. Now assume that you contract today for a fixed interest rate on the loan. If rates fall, you will be stuck paying a higher rate than the market rate that will prevail at the time the loan begins. To convert this fixed rate loan to a floating rate loan, buy an interest rate futures that expires at the time the loan is to begin. Then, if rates fall, you will profit on the futures position, and these profits will offset the higher than market rates you are forced to pay on your fixed rate loan.
8. You fear that the yield curve may change shape. Explain how this belief would affect your preference for a strip or a stack hedge. If the yield curve is to change shape, rates on different futures expirations for the same interest rate futures contract may change by different amounts. In this case, it is important to structure the futures hedge so that the futures cash flows match the exposure of the underlying risk more closely. Thus, if the cash market exposure involves the same amount at regular intervals over the future, a strip hedge will be more effective against changing yield curve shapes.
9. A futures guru says that tailing a hedge is extremely important because it can change the desired number of contracts by 30 percent. Explain why the guru is nuts. How much can the tailing factor reasonably change the hedge ratio?
To tail a hedge, one simply reduces the computed hedge ratio by discounting it at the riskfree rate for the time of the hedge. For convenience, we assume that the untailed computed hedge ratio is 1.0. If the hedging period is one year, a 30 percent effect would require an interest rate of 43 percent, because 0.7 = (1/1.43). If the hedging horizon is long, say a full two years, the interest rate would still have to be 19.52 percent to generate the 30 percent effect, because (1.1952) = 1/0.7. Thus, it seems extremely improbable that the tailing effect could be so large.
10. We have seen in Chapter 4 that regressionbased hedging strategies are extremely popular. Explain their weaknesses for interest rate futures hedging. First, regressionbased hedging (the RGR model) involves statistical estimation, so the technique requires a data set for both cash and futures prices. This data may sometimes be difficult to acquire, particularly for an attempt to hedge a new security. Second, the RGR model does not explicitly consider the differences in the sensitivity of different bond prices to changes in interest rates, and this can be a very important factor. The regression approach does include the different price sensitivities indirectly, however, since their differential sensitivities will be reflected in the estimation of the hedge ratio. Third, any cash bond will have a predictable price movement over time, and the RGR model does not consider this change in the cash bond's price explicitly. However, the sample data used to estimate the hedge ratio will reflect this feature to some extent. Fourth, the RGR hedge ratio is chosen to minimize the variability in the combined futurescash position over the life of the hedge. Since the RGR hedge ratio depends crucially on the planned hedge length, one might reasonably prefer a hedging technique focusing on the wealth position of the hedge when the hedge ends. After all, the wealth change from the hedge depends on the gain or loss when the hedge is terminated, not on the variability of the cashfutures position over the life of the hedge.
11. You estimate that the cheapesttodeliver bond on the Tbond futures contract has a duration of 10.2 years. You want to hedge your mediumterm Treasury portfolio that has a duration of 4.0 years. Yields are 9.5 percent on the futures and on your portfolio. Your portfolio is worth $120,000,000, and the decimal futures price is 68.91. Using the PS model, how would you hedge? From the text, the PS hedge ratio is: N= FPFMDF) RYC For this problem, we are entitled to assume that RYC =1.0 since no other value is specified. Applying this equation to our data gives:
$120,000,000x3.652968 N = $68,910x9.315068 = 682.902707 Therefore, the PS hedge would require selling about 683 Tbond futures.
12. Explain the relationship between the bank immunization case and hedging with the PS model. Both bank immunization and the PS model rely essentially on the concept of duration. A PS hedge finds the futures position to make the combined cash/futures position have a duration of zero. Similarly, in bank immunization with equal asset and liability amounts, the asset duration is set equal to the liability duration. For the combined balance sheet, the overall duration is effectively zero as well. Therefore, the two techniques are quite similar in approach, even if they use different instruments to achieve the risk reduction.
13. Compare and contrast the BP model and the RGR model for immunizing a bond portfolio. The BP model essentially is an immunization model that is suitable for the bank immunization case. The BP hedge ratio is found empirically, but it is the hedge ratio that gives a price movement on the futures position that offsets the price movement on the cash position. As such, it is effectively reflecting the duration of the two instruments. (Notice that the BP model does not really help with the planning period case, because it
considers only the effect of a current change in rates, not a change over some hedging horizon.) The RGR Model does not really take duration into account in any direct fashion, so it is not oriented toward immunizing at all.
14. It was a hot day in August, and William had just completed the purchase of $20 million of Tbills maturing next March and $10 million of Tbills maturing in one month. The phone rang, and William was informed that the firm had just made a commitment to provide $30 million in capital to a client in midDecember. If William had known this 20 minutes earlier, he would have invested differently.
A. What risks does William face by using his present investments to meet the December commitment? William faces several interest rate risks using his present investments to fund the anticipated cash outflow. These risks stem from the fact that the maturity of his present investments do not match the timing of the cash outflow. The proceeds of the $10 million of September Tbills must be reinvested from September to December. William faces the risk that interest rates may fall between now and September which would reduce the December proceeds from the reinvestment. The $20 million of March Tbills are subject to price risk. If interest rates rise between now and December, the proceeds from the sale of the March Tbills will be less than anticipated. Had he known about the firm's commitment earlier, he could have invested in $30 million of December maturity Tbills.
B. Using the futures markets, how can William reduce the risks of the December commitment? Show what transactions would be made. William would like to lock in a reinvestment rate for September and a selling price for December. He can accomplish this by buying $10 million September futures contracts and selling $20 million December futures contracts. He is in effect lengthening the maturity of his September bills and shortening the maturity of his March bills. His transactions would be: Date
Cash Market Futures Market Today
Buy $10 million September
Tbill futures; sell $20 million December Tbill futures
September Receive proceeds from $10 million Reverse September Tbill futures
September Tbills; reinvest $10 million position by selling $10 million
into 90day Tbills September Tbill futures
December Receive proceeds from maturing Reverse December Tbill futures
December Tbills; sell March Tbills position by buying $20 million
in cash market December Tbill futures
Gains and losses in the futures market will offset losses and gains in the cash market so that the total proceeds available to meet the December commitment will be as anticipated.
15. Handcraft Ale, Ltd. has decided to build additional production capacity in the US to meet increasing demand in North America. Uma Peele has been given the responsibility of obtaining financing for the project. Handcraft Ale will need $10 million to carry the firm through the construction phase. This phase will last two years, at which time the $10 million debt will be repaid using the proceeds of a longterm debt issue.
Ms. Peele gets rate quotes from several different London banks. The best quote is: Variable Fixed
LIBOR+ 150 bp 8.5%
Each of these loans would require quarterly interest payments on the outstanding loan amount. Ms. Peele looks up the current LIBOR rate and finds that it is 5.60%. The variable rate of 7.1% (5.60 + 1.5) looks very attractive, but Ms. Peele is concerned about interest rate risk over the next two years.
A. What could Ms. Peele do to take advantage of the lower variable rate while at the same time have the comfort of fixed rate financing? Ms. Peele could hedge each of the anticipated interest rate adjustments using the Eurodollar futures contract. This would be a strip hedge. She would sell $10 million 3month Eurodollar futures contracts with expirations in the months when interest rates are adjusted. For longer term loans this can present a problem. Contracts may not be traded with the proper expiration date, or the market may be very thin making it difficult to find a counterparty. Alternatively, Ms. Peele could employ a stack hedge. In this case she would stack the hedges for all interest rate adjustments on a single contract whose expiration is not so far into the future. For example, she could sell $70 million Eurodollars futures contracts with expiration in the month of the first interest rate adjustment.
B. Consider the following 3month Eurodollar quotes:
Delivery Month Rate
AUGx0 94.32
SEP 94.34
OCT 94.31
NOV 94.33
DEC 94.35
JANxl 94.43
MAR 94.40
JUN 94.43
SEP 94.40
DEC 94.25
MARx2 94.30
JUN 94.27
SEP 94.23
DEC 94.15
Handcraft Ale takes out a floating rate note with the first interest rate adjustment coming in December. LIBOR at the time of loan initiation is 5.70%. Design a strip hedge to convert the Handcraft Ale floatingrate note to a fixedrate note. What is the anticipated fixed rate? Peele would hedge each of her interest payments by selling $10 million of 3 month Eurodollars in the following months:
Anticipated Loan
Expiration Price Rate
DEC 94.35 7.15
MARxl 94.40 7.10
JUN 94.43 7.07
SEP 94.40 7.10
DEC 94.25 7.25
MARx2 94.30 7.20
JUN 94.27 7.23
The anticipated loan rate for each adjustment date is computed as: Anticipated Rate = 100  ^y^ +1.5
While the hedge is not the same as pure fixed rate financing, there is very little variation anticipated. The range of anticipated rates are from 7.07 to 7.25 percent. The average rate over the life of the loan is 7.15 percent, including the initial period.
C. Suppose that Handcraft Ale's quarterly interest payments were in November, February, May and August. Would a strip hedge be possible? Design a hedge that Ms. Peele could use in this case. If Handcraft's interest payment cycle were November, February, May and August, Ms. Peele would have a problem. It would only be possible to match the timing of the nearest interest rate adjustment. This is a situation in which Ms. Peele could use a stack hedge. She would stack hedge all seven interest rate adjustments by selling $70 million in Eurodollar futures with November expiration. In November Handcraft would make its interest payment. At that time there will be a market for February Eurodollars, and Ms. Peele would stack the six remaining interest rate adjustments on the February contract. This process would continue until maturity with each interest rate adjustment reducing the size of the subsequent stack hedge by $10 million. Alternatively, Ms. Peele could hedge each rate adjustment using the Eurodollar futures contract that expires in the month just following the date the interest rate adjustment is made.
16. Jim Hunter is preparing to hedge his investment firm's decision to purchase $100 million of 90day Tbills 60 days from now in June. The discount yield on the 60day Tbill is 6.1%, and the June Tbill futures contract is trading at 94.80. Jim views these rates as very attractive relative to recent history, and he would like to lock them in. His first impulse is to buy 100 June Tbill futures contracts, but his recent experience leads him to believe that he should be buying something less.
A. Why is a onetoone hedge ratio inappropriate in Jim's situation? A onetoone hedge ratio is not appropriate because the daily settlement gains and losses can be invested or must be financed. If no interest were earned on settlement cash flows then a onetoone hedge would be appropriate because the settlement cash flows would exactly offset the change in value of the underlying instrument. Because interest can be earned on the settlement cash flows, the delivery date value of the settlement cash flows will not exactly equal the change in value of the underlying instrument. The terminal value of the settlement cash flows will exceed, in magnitude, the change in value of the underlying instrument. The difference will be the interest earned and/or assessed between the settlement date and the delivery date.
B. Compute an appropriate hedge ratio given the market conditions faced by Jim. The hedge ratio is slightly reduced to account for the interest that can be earned between daily settlement and the delivery date. This is called tailing the hedge. The amount by which the hedge is reduced is called the tailing factor. The tailing factor is the present value factor between the settlement date and the delivery date. When Jim initiates the hedge the discount yield for the 60day Tbill (his hedging time horizon) is 6.1%. Computing the tailing factor, which is the price per $1 of face value we have:
( 061^6() j =0.
Tailing Factor =H 36q ) =0.9898
The tailed hedge for the $100 million of 90day Tbills to be purchased in 2 months is:
Tailed Hedge = $100 (0.9898) = $98.98 million Jim would buy 99 June Tbill futures contracts.
C. Under what conditions might Jim need to adjust his hedge ratio between now and June?
The tailing factor is a function of the time to delivery and the level of interest rates. If interest rates change significantly, then Jim might need to adjust the hedge. The longer the time to settlement, the more influential interest rate movements will be. Even if interest rates do not move, the passage of time will call for adjustment as the tailing factor will move toward the value of one on the delivery date.
17. Alex Brown has just returned from a seminar on using futures for hedging purposes. As a result of what he has learned, he reexamines his decision to hedge $500 million of longterm debt that his firm plans to issue in May. His current hedge is a short position of 5,000 Tbond futures contracts ($100,000 each). If the debt could be issued today, it would be priced at 11922 to yield 6.5%. With its 8% coupon and 30 years to maturity, the duration of the debt would be 13.09 years. On the futures side, the futures prices are based on the cheapesttodeliver bonds, which are trading at 12414 to yield 5.6%. These bonds have a duration of 9.64 years. A. List and briefly describe possible strategies Alex Brown could use to hedge his impending debt issue. There are a number of different methods Alex could employ to hedge his impending debt issue: Face Value Naive Model—In this method Alex would trade one dollar of nominal futures contract per one dollar of debt face value. The major benefit of this method is the ease of implementation. Unfortunately, it ignores market values and the differential responses of the bond and futures contract prices to interest rates. Market Value Naive Model—In this method Alex would hedge one dollar of debt market value using one dollar of futures price value. That is, the hedge ratio is determined by the market prices instead of nominal and face values. Unfortunately, it does not consider the price sensitivities of the two instruments. Conversion Factor Model—This model can be used when the hedging instrument is a Tnote or Tbond futures contract. The conversion factor adjusts the prices of deliverable bonds and notes that do not have a 6% coupon to make them "equivalent" to the 6% coupon bond or note that is called for in the contract. The hedge ratio is determined by multiplying the Face Value Naive hedge ratio by the conversion factor. The appropriate conversion factor to use is the conversion factor of the cheapest to deliver Tbond or Tnote. This model still ignores price sensitivity differences between the hedging and hedged instruments. Basis Point Model—This model uses the price changes of the futures and cash positions resulting from a one basis point change in yields to determine the hedge ratio. It is calculated as:
BPCS
BPC
HR =
This model works well if the cash and futures instruments face the same rate volatility. If they face different volatilities and that relationship can be quantified, then the basis point model can be adjusted to account for the differing volatilities. Regression Model—In the regression model the historic relationship between cash market price changes and futures market price changes is estimated. This estimation is accomplished by regressing price changes in the cash market on futures price changes. The slope coefficient from this regression is then used as the hedge ratio. Alex may not find this model useful, as he is trying to hedge a new debt issue. Even if Alex had an historic price stream on 30year corporate debt issues, the historic relationship with the futures price might prove to be an unreliable indicator of the present or future relationship. This stems from the fact that the price response of the futures contract is determined by the cheapesttodeliver bond. The cheapesttodeliver bond can vary in maturity from 15 years to 30 years. This means that the futures contract can have very different price responses to interest rates at different points in time. Price Sensitivity Model—This may be a good model for Alex to use. It is designed for interest rate hedging, and it accounts for the differential price responses of the hedging and the hedged instruments. The model is durationbased so that it accounts for maturity and coupon rate differences of the cash and the futures positions. It is computed as: where: FPF and Pi are the respective futures contract and cash instrument prices; MDi and MDF are the modified durations for the cash and futures instruments, respectively, and RYC is the change in the cash market yield relative to the change in the futures yield.
B. What strategy is Alex Brown currently using? What are the strengths and weaknesses of this strategy? Currently Alex has employed a Face Value Naive hedge. For each dollar of debt principal he plans to issue, he is short $1 of nominal Tbond futures. The benefit of the strategy is its ease of implementation. The drawback is that cash instrument and the Tbond futures may have differential price responses to interest rate changes.
C. Based on the knowledge Alex gained at the hedging seminar, he feels that a price sensitivity hedge would be most appropriate for his situation. Design a hedge using the price sensitivity method. Assume that the relative volatility between the corporate interest rate and the Tbond interest rate, RYC, is equal to one. The price sensitivity hedge ratio is:
FPF = 124.4375% x 0.1 million
MDF = 9.128788 Pi = 119.6875% x 500 million
MDi = 12.29108 $598,437,500  12.29108
6,475 contracts
N= $124,437.509.128788 To hedge the risk, 6,475 contracts should be sold.
D. At the time of refinancing, the Tbond futures price is 12127 and B.I.G.'s new debt issue is priced at 11608.
Compute the net wealth change resulting from the naive hedge. In the cash market, Alex suffers an opportunity loss because he anticipated issuing debt at 11922, but he is only to get 11608 for the debt. The opportunity loss is: Opportunity loss = (1.1625 1.196875) $500,000,000 = $17,187,500 In the futures market Alex realizes a gain because he was short Tbond futures. His gain is:
Futures gain = (1.244375  1.2184375) $500,000,000 = $12,968,750 Net wealth change = $17,187,500 + $12,968,750 = $4,218,750
E. Compute the net wealth change resulting from the price sensitivity hedge. Examining the price sensitivity hedge, the cash market loss will still be $17,187,500. The futures market gain will now be:
Futures gain = (1.244375  1.2184375) $647,500,000 = $16,794,531
Net wealth change = $17,187,500 + $16,794,531 = $392,969 The price sensitivity hedge is much more effective than the face value naive hedge at reducing the risk.
Stock Index: Futures: Introduction
1. Assume that the DJIA stands at 8340.00 and the current divisor is 0.25. One of the stocks in the index is priced at $100.00 and it splits 2:1. Based on this information, answer the following questions:
a. What is the sum of the prices of all the shares in the index before the stock split? The equation for computing the index is: Index = ^rr—.1 Divisor If the index value is 8340.00 and the divisor is 0.25, the sum of the prices must be 8,340.00(0.25) = $2,085.00.
b. What is the value of the index after the split? Explain. After the split, the index value is still 8,340.00. The whole purpose of the divisor technique is to keep the index value unchanged for events such as stock splits.
c. What is the sum of the prices of all the shares in the index after the split? The stock that was $100 is now $50, so the sum of the share prices is now $2,035.00.
d. What is the divisor after the split? With the new sum of share prices at $2,035.00, the divisor must be 0.244005 to maintain the index value at 8340.00.
2. What is the main difference in the calculation of the DJIA and the S&P 500 index? Explain. The S&P 500 index gives a weight to each represented share that is proportional to the market value of the outstanding shares. The DJIA simply adds the prices of all of the individual shares, so the DJIA effectively weights each stock by its price level.
3. For the S&P 500 index, assume that the company with the highest market value has a 1 percent increase in stock prices. Also, assume that the company with the smallest market value has a 1 percent decrease in the price of its shares. Does the index change? If so, in what direction?
The index value increases. The share with the higher market value has a greater weight in the index than the share with the smallest market value. Therefore, the 1 percent increase on the high market value share more than offsets the 1 percent decrease on the low market value share.
4. The S&P 500 futures is scheduled to expire in half a year, and the interest rate for carrying stocks over that period is 11 percent. The expected dividend rate on the underlying stocks for the same period is 2 percent of the value of the stocks. (The 2 percent is the halfyear rate, not an annual rate.) Ignoring the interest that it might be possible to earn on the dividend payments, find the fair value for the futures if the current value of the index is 945.00. Assuming the halfyear rate is 0.11 /2, the fair value is: Fair Value = 945.00(1.055  0.02) = 978.075 Assuming semiannual compounding, the interest factor would be 1.0536 and the fair value would be:
Fair Value = 945.00(1.0536  0.02) = 976.752
5. Consider a very simple index like the DJIA, except assume that it has only two shares, A and B. The price of A is $100.00, and B trades for $75.00. The current index value is 175.00. The futures contract based on this index expires in three months, and the cost of carrying the stocks forward is 0.75 percent per month. This is also the interest rate that you can earn on invested funds. You expect Stock A to pay a $3 dividend in one month and Stock B to pay a $1 dividend in two months. Find the fair value of the futures. Assume monthly compounding.
Fair value = 175.00(1.0075)3  $3(1.0075)2  $1(1.0075) = 174.91
6. Using the same data as in Problem 5, now assume that the futures trades at 176.00. Explain how you would trade with this set of information. Show your transactions. At 176.00, the futures is overpriced. Therefore, the trader should sell the futures, buy the stocks and carry them forward to expiration, investing the dividend payments as they are received. At expiration, the total cost incurred to carry the stocks forward is 174.91, and the trader receives 176.00 as cash settlement, for a profit of 1.09 index units. (This ignores interest on daily settlement flows.)
7. Using the same data as in Problem 5, now assume that the futures trades at 174.00. Explain how you would trade with this set of information. Show your transactions. At 174.00 the futures is underpriced. Therefore, the trader should buy the futures and sell the stocks short, investing the proceeds. The trader will have to borrow to pay the dividends on the two shares. At expiration, the total outlays, counting interest, have been:
$175.00(1.0075)3  $3(1.0075)2  $1(1.0075) = 174.91
With the convergence at expiration, the trader can buy the stocks for 174.00 and return them against the short sale. This gives a profit of $.91.
8. For a stock index and a stock index futures constructed like the DJIA, assume that the dividend rate expected to be earned on the stocks in the index is the same as the cost of carrying the stocks forward. What should be the relationship between the cash and futures market prices? Explain. The cash and futures prices should be the same. In essence, an investment in the index costs the interest rate to carry forward. This cost is offset by the proceeds from the dividends. If these are equal, the effective cost of carrying the stocks in the index forward is zero, and the cash and futures prices should then be the same.
9. Your portfolio is worth $100 million and has a beta of 1.08 measured against the S&P index, which is priced at 350.00. Explain how you would hedge this portfolio, assuming that you wish to be fully hedged.
The hedge ratio is:
— ß/> P V P = number of contracts
With our data we have: The cash value of the futures contract is 500 times the index value of 350.00, or $175,000. Therefore, the complete hedge is to sell 617 contracts.
10. You have inherited $50 million, but the estate will not settle for six months and you will not actually receive the cash until that time. You find current stock values attractive and you plan to invest in the S&P 500 cash portfolio. Explain how you would hedge this anticipated investment using S&P 500 futures. Buy S&P 500 index futures as a temporary substitute for actually investing the cash in the stock market. Probably the best strategy is to buy the contract that expires closest in time to the expected date for receiving the cash. If the S&P 500 index value is 300.00, then the dollar value of one contract will be $150,000 (300.00 x $500). Therefore, you should achieve a good hedge by purchasing about 333 ($50,000,000/ $150,000) contracts.
11. William's new intern, Jessica, is just full of questions. She is particularly inquisitive about stock index futures. She notices that the futures price is consistently higher than the current index level and that the difference gets smaller as the contracts near their expiration dates.
A. Explain the relationship between the futures price, the spot price, interest rates, and dividends. The futures contract effectively allows one to commit to the sale or the purchase of the Dow index stocks at a specific point in the future at a price agreed upon today. There are costs and benefits related to using the futures market as opposed to the cash market to buy the Dow index stocks. Take for example an individual who wishes to own the Dow stocks in three months. The alternatives are to: buy the stocks today and hold them, or buy a futures contract with three months to delivery. The benefit of using the futures contract is that there is no cash outlay today. The purchase price of the stocks can be invested for three months to earn interest. The downside of using the futures contract is that since the stocks are not held over the next three months, no dividends are received. This suggests the following relationship between the spot and the futures market prices:
i= i where: Fo i is the futures price today for delivery at time t, S0 is the spot price for the index today, C is the cost of carry, Di is the ith dividend paid between now and time t, and ri is the rate of return received on the ith dividend between the payment date and time t. In general, the dividend yield is smaller than the cost of carry so the index futures markets are generally normal (futures index above the spot index). As the delivery date approaches the futures index converges to the spot index.
B. Jessica asks William to explain the Dow index to her. What type of index is the Dow? How is it constructed? How could she build a portfolio of stocks to replicate it?
The Dow is a priceweighted index. The stocks are represented in the index in proportion to their price. The index is computed as: 30 Dow Index = Divisor where the Pi are the prices of the stocks comprising the Dow, and the Divisor is a number used to compute the "average." When the Dow first appeared with 30 stocks in 1928 (the Dow was first published in 1884 with 11 stocks), the divisor was 30. As stocks split or the components of the Dow changed, the divisor was adjusted to maintain continuity in the index. To form a portfolio that would replicate the Dow, Jessica should buy an equal number of shares of each stock in the Dow.
C. Jessica wants a numerical example of the relationship between a price weighted index and the futures contract based on that index. She supposes the following example. A futures contract is based on a priceweighted index of three stocks A, B, and C. The futures contract expires in 3 months. Stock A pays a dividend at the end of the first month, and Stock C pays a dividend at the end of month two. The term structure is flat over this time period with the monthly interest rate equal to 0.5%. The stock prices and dividends are summarized below Stock Price Dividend
$30 $0.11 in one month
$50 $0
$40 $0.15 in two months
A B
C
Compute the index assuming a divisor of 3. How many shares of each stock should be bought to replicate the index?The index is computed as: 3 y p
T , f~x i 30 + 50 + 40 .„
Index = =r^ = = 40
Divisor 3
To replicate the index, you would buy 1 /Divisor shares of each stock. In our example onethird share of each stock would replicate the index.
D. Suppose the divisor had been 0.5. Compute the index. How many shares of each stock must be bought to replicate the index?
With the divisor equal to 0.5 the index is:
Index = ^— = '»^„Tn, = 24Q
Divisor 0.5
The number of shares of each stock to buy to replicate the index is ^ = . Two shares of each stock will replicate the index.
E. Assuming the divisor is 0.5, compute the fair value for the 3month futures contract. The fair market value is computed using:
F0,t = 240(1 + 0.005)3  2(0.11)(1.005)2  2(0.15)(1.005) = 243.09
F. Right now the Dow Jones Industrial Average is at 8,635. Its dividend yield is 1.76%. The 90day Tbill rate is 5.6% bond equivalent yield. Compute a fair price today for the index futures contract expiring in 90 days.
For an index composed of a large number of securities it is sometimes helpful to express the relationship between the futures and spot indexes as:
F0,t = S0(1 + CDIVYLD)
where DIVYLD is the dividend yield on the index stocks. The fair price for the futures contract expiring in 90 days is:
F0,t = 8,717
12. Casey Mathers manages the $60 million equity portion of Zeta Corporation's pension assets. This past Friday, August 7, Zeta announced that it was downsizing its workforce and would be offering early retirement to many of its older employees. The impact on the portfolio Casey manages would be an anticipated $10 million withdrawal over the next 4 months. The stock market has been good for the past 5 years, but recently there have been signs of weakness. Casey is concerned about a drop in asset prices before the $10 million is withdrawn from the portfolio. Casey runs a fairly aggressive portfolio with a beta of 1.2, relative to the S&P 500 Index. Casey sees the following S&P 500 index futures prices:
Contract Value
Expiration $250 x Index
SEP 1088.50
DEC 1100.00
MARxl 1110.50
A. How much of the portfolio should Casey hedge? Justify your answer. Casey anticipates $10 million being withdrawn from the portfolio over the next 4 months. Casey is concerned about the risk that falling stock prices will result in more shares having to be sold to raise the $10 million. It is just the anticipated withdrawal that should be hedged.
B. Design a hedge based on your answer to part A above. To hedge the risk, Casey should use the futures contract with expiration as soon after the anticipated withdrawal as possible. This would be the December futures contract. Since Casey's portfolio has a beta of 1.2 relative to the S&P 500, Casey should sell $1.2 of future contract value per $1 of portfolio hedged. Then the number of contracts to trade is computed by:
VP $10,000,000
N=  ß,^=  12(
To hedge the risk of the $10 million withdrawal, Casey should sell 44 contracts. As the assets are withdrawn from the portfolio, the hedge should be gradually unwound. The transactions would be as follows:
Date Cash Market Futures Market
Today Anticipate the withdrawal of $10 million Sell 44 December S&P 500 futures
from the portfolio over the next contracts.
four months.
Between now Sell securities to meet portfolio withdrawal Enter reverse trades to unwind the
and December demands. hedge as assets are withdrawn from
portfolio.
13. Byron Hendrickson manages the $30 million equity portion of Fredrick and Sons' pension plan assets. Byron has been trying to get the management of Fredrick and Sons to move more of their assets from their fixed income portfolio (market value of $60 million) to the equity portfolio in order to achieve the objectives
that management had set forth for growth of the plan. Management has decided to invest the proceeds of several bond issues that will be maturing over the next three months. The total proceeds from the bond issues will be $10 million. It is now December 15. Byron believes that the January price runup will be particularly strong this year. Since his portfolio is not particularly aggressive, beta equal to 0.85, he would really like to have that $10 million working for him in January. Design a hedge that will prevent Byron from missing the January action, based on the following current market prices for the S&P 500 index futures:
Contract Value
Expiration $250 x Index
MAR 1157.00
JUN 1170.80
Byron wishes to commit $10 million to the stock market today, but the cash will not be available until March. Therefore, he is implicitly short in the cash market at present. Byron should make a long hedge using the March S&P 500 futures index. The number of contracts Byron should buy is computed as:
n  $10,000,000 N=°85 (i,i57)$250 = 2
Byron's implicit short position is reflected in the preceding equation by the —$10,000,000 cash market position. Byron should buy 29 March S&P futures contracts. This will generate gains to replace the opportunity loss from not having the $10 million invested between now and March. As the bonds mature and the proceeds are transferred to the equity side, the hedge should be unwound. Byron's transactions are summarized below.
Date Cash Market Futures Market
Today Anticipate the investment of $10 million Buy 29 March S&P 500 futures into the equity portfolio over the index contracts. next three months.
March Buy stocks using $10 million plus any Unwind the futures position as the gains/losses from the futures position. Investments are made into the equity portfolio.
Stock Index Futures: Refinements
2. The NoDividend Index consists only of stocks that pay no dividends. Assume that the two stocks in the index are priced at $100 and $48, and assume that the corresponding cash index value is 74.00. The cost of carrying stocks is 1 percent per month. What is the fair value of a futures contract on the index that expires in one year?
Fair Value = 74.00(1.01)^{12} = 83.3851
3. Using the same facts as in Problem 2, assume that the roundtrip transaction cost on a futures is $30. The contract size, we now assume, is for 1,000 shares of each stock. Trading stocks costs $.05 per share to buy and the same amount to sell. Based on this additional information, compute the noarbitrage bounds for the futures price. From the cashandcarry transactions we would buy the stocks, carry them to expiration, and sell the futures. This strategy would cost:
Purchase and carry stock: $148,000(1.01)^{12} = $166,770
Stock transaction cost: +1,000(2)($.05) = $100 Futures transaction cost: $30
For this strategy to generate a profit, the futures must exceed 83.450 per contract. For the reverse cashandcarry, we would sell the stocks, invest the proceeds, and buy the futures:
Sell stock; invest proceeds: $148,000(1.01)^{12} = $166,770 Stock transaction cost: +1,000(2)($.05) = $100 Futures transaction cost: $30
4. Using the facts in Problems 2 and 3, we now consider differential borrowing and lending costs. Assume that the 1 percent per month is the lending rate and assume that the borrowing rate is 1.5 percent per month. What are the noarbitrage bounds on the futures price now? From the cashandcarry transactions we would buy the stocks, carry them to expiration, and sell the futures. Now the financing cost is 1.5 percent per month. This strategy would cost:
Purchase and carry stock: $148,000(1.015)^{12} = $176,951 Stock transaction cost: +l,000(2)($.05) = $100 Futures transaction cost: $30
8. Index arbitrageurs must consider the dividends that will be paid between the present and the futures expiration. Explain how overestimating the dividends that will be received could affect a cashandcarry Assume a trader estimates a dividend rate that is higher than the actual dividend rate that will be achieved. Further assume that the market as a whole correctly forecasts the dividend rate. For this investor, a strategy of cashandcarry arbitrage will appear to be more attractive than it really is. This trader will be expecting to receive more dividends than will actually be forthcoming, so the trader will underestimate the net cost of carrying stocks forward. This overestimate of the dividend rate could lead the trader to expect a profit from the trade that will evaporate when adjusted for the actual dividends that will be received.
9. Explain the difference between the beta in the Capital Asset Pricing Model and the beta one finds by regressing stock returns against returns on a stock index. The beta of the CAPM is a theoretical entity. The CAPM beta is a measure based on the relationship between a particular security and an unobserved and probably unobservable market portfolio. The beta estimated by regressing stock returns against the returns on an index is an estimate of that ideal CAPM beta. Because the index fails to capture the true market portfolio, the actually estimated beta must fail to capture the true CAPM beta. Nonetheless, the estimated beta may be a useful approximation of the true CAPM beta.
10. Explain the difference between an exante and an expost minimum risk hedge ratio. The exante minimum risk hedge ratio is estimated using historical data. In hedging practice, this estimated hedge ratio is applied to a future time period. Almost certainly the hedge ratio that would have minimized risk in the future period (the expost hedge ratio) will not equal the estimated exante hedge ratio. However, the expost minimum risk hedge ratio can only be known after the fact. Therefore, we must expect some inaccuracy in estimating a hedge ratio exante and comparing it with the ideal expost hedge ratio.
11. Assume you hold a welldiversified portfolio with a beta of 0.85. How would you trade futures to raise the beta of the portfolio? Buy a stock index futures. In effect, this action levers up the initial investment in stocks, effectively raising the beta of the stock investment. In principle, this levering up can continue to give any level of beta a trader desires.
12. An index fund is a mutual fund that attempts to replicate the returns on a stock index, such as the S&P 500. Assume you are the manager of such a fund and that you are fully invested in stocks. Measured against the S&P 500 index, your portfolio has a beta of 1.0. How could you transform this portfolio into one with a zero beta without trading stocks? Sell S&P 500 Index futures in an amount equal to the value of your stock portfolio. After this transaction you are effectively long the index (your stock holdings) and short the index by the same amount (your short position in the futures). As a result, you are effectively out of the stock market, and the beta of such a position must be zero.
13. You hold a portfolio consisting of only Tbills. Explain how to trade futures to create a portfolio that behaves like the S&P 500 stock index. Buy S&P 500 Index futures. You should buy an amount of futures that equals the value of funds invested in Tbills. The resulting portfolio will replicate a portfolio that is fully invested in the S&P 500.
14. In portfolio insurance using stock index futures, we noted that a trader sells additional futures as the value of the stocks falls. Explain why traders follow this practice. The goal of portfolio insurance is to keep the value of a portfolio from falling below a certain level or, alternatively expressed, to ensure that the return achieved on a portfolio over a given horizon achieves a certain minimum level. At the same time, portfolio insurance seeks to retain as much potential for beating that minimum return as is possible. The difference between the portfolio's current value and the value it must have to meet the minimum target we will call the cushion. If the portfolio has no cushion, the only way to ensure that the portfolio will achieve the target return, or the target value, is for the portfolio to be fully hedged. We now consider the trader's response if the portfolio value is above the minimum level, that is, if there is some cushion and stock prices fall. The drop in stock prices reduces the cushion, so the trader must move to a somewhat more conservative position. This requires hedging a greater portion of the portfolio, which the trader does by selling futures. Therefore, an initial drop in prices requires the selling of futures, and each subsequent drop in prices requires the sale of more futures.
15. Casey Mathers, manager of the Zeta Corporation's equity portfolio, hires a new assistant, Alec. Alec is pretty sharp and immediately questions Casey's decision to hedge an anticipated $10 million withdrawal. Casey had hedged the portfolio using the S&P 500 index futures contract. In calculating the hedge, Casey used the portfolio beta of 1.2 which was computed using the S&P 500 Index.
A. Explain to Casey why his hedge may not be a risk minimization hedge. Casey's hedge may not be a risk minimization hedge because the beta used in calculating the hedge ratio was computed using the S&P 500 index. It is the index futures contract, though, that is used for hedging. So theoretically the portfolio's beta computed using the futures contract prices is what should be used in calculating the hedge. Additionally, the S&P 500 index futures contract is not the only possible hedging vehicle available. For example, there are the Dow Jones Industrial Average index futures and the NYSE Composite index futures. The portfolio's returns may be more highly correlated with one of these other contracts than it is with the S&P 500 index futures contract. Alec calculates possible riskminimizing betas for the Zeta portfolio using the S&P 500 index futures, the Dow Jones Industrial Average futures, and the NYSE Composite index futures, with the following results:
Contract size 
$250 x index 
$10 x index 
$500 x index 


Current quote (MAR) contract 
1110.59 
8715.00 
559.30 

Beta_{RM} 
1.30 
1.35 
1.10 

R^{2} 
0.83 
0.75 
0.90 
C. Design a risk minimization hedge using Alec's results. Using Alec's results the risk minimizing hedge would be accomplished by selling NYSE Composite index futures. The number of contracts to sell is computed as:
$10,000,000 ^{N=} ^{U}°559.30$10,000,000($500) = " ^{39}.^{3contracts }
Alec would recommend selling 39 contracts. D Will Alec's hedging strategy turn out to be superior to Casey's hedging strategy? Justify your answer. While Alec's hedging strategy will probably be superior to Casey's, this is not guaranteed, because the hedge calculation is based on historic relationships that are measured with error. This relationship will not hold exactly in the future.
16. Raymond J. Johnson, Jr. manages a $20 million equity portfolio. It has been designed to mimic the S&P 500 index. Ray has a hunch that the market is going south during the coming month. He has decided that he wants to eliminate his exposure for the next month and take off for Montana to go fishing. Ray has the following information at hand:
S&P 500 index futures with 1 month to delivery: 1084.50 Dividend yield on Ray's portfolio: 2.1% S&P 500 index today: 1081.40
A. Design a hedge to eliminate Ray's market risk for the next month. Ray, in effect, would like to sell his portfolio for a month and put the money into Tbills. The transactions costs make this strategy cost prohibitive though. Alternatively, Ray could sell futures contracts. Then over the next month any losses in the cash market will be offset by gains in the futures market. To eliminate his exposure to the market, Ray would calculate the number of S&P 500 futures contracts as:
$20,000,000 ^{N=} ^{10} (1084.50)$20,000,000 ($250) = ^{738} ^{contracts }
Ray should sell 74 S&P 500 futures contracts.
B. Compute the return he can expect to receive over the next month. The futures index pricing relationship is:
F0,t = S_{0}(1 + CDIVYLD) where:
F0, t = index futures value S_{0} ^{=} ^{s}p°t price C = costofcarry DIVYLD = dividend yield
The costofcarry can also be viewed as an implied repo rate. This is the return Ray will receive over the next month:
74(1084.50)(250) ._{02}1 \ \ 20,000,000 12 J
C = 5.9% Ray can expect to earn a 5.9% annual return over the next month.
17. Remember Ray? He is the guy running the S&P 500 index fund who wanted to go fishing. Ray has changed his mind. The fishing reports from Montana were not favorable so he has decided not to go. Since he is not leaving, he had decided to devise a portfolio insurance strategy for his $20 million portfolio. His objective is to not let his portfolio value fall below $18 million.
A. Design a portfolio insurance strategy that applies no hedges for portfolio values at or above $20 million and is fully hedged at or below $18 million. Portfolio insurance is a dynamic hedging strategy that applies hedges as the hedged asset falls in value and removes hedges as the hedged asset increases in value. When the asset value is falling, hedges are applied until the assets are fully hedged. Theoretically a money manager could prevent the value of his positions from falling below some prespecified level because further decline in the assets' value is offset by futures gains. Ray wishes to apply hedges as the value of this portfolio falls from $20 million (no hedges) down to $18 million (fully hedged). Assume that Ray will apply the hedges in a linear fashion. That is, the percentage of assets hedged will be given by: When Vp< 18, the percentage of the portfolio hedged is 100%. When V_{P}> 20, the percentage of the portfolio hedged is 0%.
B. On day one, the stock portfolio value falls from $20 million to $19.4 million, and the S&P 500 futures price falls to 1052. What action should Ray take?
If the portfolio value falls from $20 million to 19.4 million on day 1, the percentage of the assets that should be hedged is:
% hedged = / T20 max (19.4,18) \ \l ^{20} ^{18} J ,
% hedged = 30%
Thirty percent of the assets should be hedged. This can be accomplished by selling contracts calculated as follows:
N= 1.0(0.30)^g_{r}22.13 At the end of day 1 Ray should sell 22 S&P 500 futures contracts.
C. On day two, the value of Ray's portfolio increases by 2%. The S&P 500 futures contract increases to 1073. What would be the change in value of Ray's portfolio (including any hedges that may be in place)? What action should Ray take?
Cash Market Gain Assets increase by 2% to $19,788,000
Future Market Loss 22(1052  1073) ($250) =  $115,500
End of Day 2 Assets $19,788,000  $115,500 =$19,672,500
The change in value of Ray's portfolio on day 2 is $272,500. Part of the portfolio must be liquidated to mark the futures contract to market. At the end of day 2 Ray needs to adjust his hedge. The new percentage to hedge is:
o/ u a a 20 19.67 ,, _{C0}/
% hedged = ^r— =16.5%
ZU — lo
The number of contracts to achieve this hedge is: $19,672,500
D. Is Ray really protected against his portfolio value falling below $18 million in value? Explain. The protection Ray has depends upon several factors. First, the amount of protection depends on how closely Ray monitors his position. If Ray does not closely monitor his position, the portfolio value could fall below $18 million before Ray places a hedge. The more closely Ray monitors, and the more frequently he adjusts the hedge, the better the protection. Second, if market frictions prevent Ray from applying hedges in a timely fashion, the portfolio value could fall below $18 million. For many investors, this occurred during the stock market crash in 1987. The order flow overwhelmed the order handling and reporting systems of the stock market. Also, the price information coming from the stock markets was stale. This kind of event could mislead Ray in his hedging decisionmaking process. In this kind of extreme situation, Ray's assets could fall below $18 million before Ray knew it
Foreign Currency Futures
1. The current spot exchange rate for the dollar against the Japanese yen is 146 yen per dollar. What is the corresponding US dollar value of one yen? The dollar value per yen is simply the inverse of the yen per dollar rate:
1/146 = $.0068 per yen
2. You hold the current editions of The Wall Street Journal and The Financial Times, the British answer to the WSJ. In the WSJ, you see that the dollar/pound 90day forward exchange rate is $2.00 per pound. In The Financial Times, the pound 90day dollar/pound rate is £.45 per US dollar. Explain how you would trade to take advantage of these rates, assuming perfect markets. These rates are inconsistent because a rate of $2.00 per pound implies that the cost of one dollar should be £.50. Therefore, an arbitrage opportunity is available by trading as follows:
f=0

In New York, using the WSJ rates, sell $2.00 for £1.00 90 days forward.

$0



In London, using The Financial Times rates, sell £1.00 for $2.22 90 days forward

$0




Total Cash Flow $0


t = 90

In New York, fulfill the forward contract by delivering

 $2.00



$2.00 and collecting £1.00

+ £1.00



In London, fulfill the forward contract by delivering £1.00

 £1.00



and collecting $2.22

+ $2.22




Total Cash Flow + $.22

3. In problem 2, we assumed that markets are perfect. What are some practical impediments that might frustrate your arbitrage transactions in problem 2? Transaction costs would be the major impediment. Every trade of foreign exchange faces a bidasked spread. In addition, there is likely to be some commission to be paid, either in the form of an outright commission or in the form of an implicit commission for maintaining a trading function. In addition, forward contracts sometimes require margin, and this would be an additional cost that the potential arbitrageur must bear.
4. In the WSJ, you see that the spot value of the German mark is $.63 and the Swiss franc is worth $.72. What rate of exchange do these values imply for the Swiss franc and German mark? Express the value in terms of marks per franc. The rate of $.63 per mark implies a value of the mark equal to DM 1.5873 per $. The rate of $.72 per franc implies a value of the franc equal to SF 1.3889 per $. Therefore, DM 1.5873 and SF 1.3889 are equivalent amounts, both equal to $1. As a consequence, the value of the DM per SF must equal 1.5873/1.3889 = 1.1429.
5. Explain the difference between a pegged exchange rate system and a managed float. In a pegged exchange rate system, the value of a pegged currency is fixed relative to another currency. For example, many Caribbean countries peg the value of their currency to the US dollar. In a managed float, the value of the currency is allowed to fluctuate as market conditions require. This is the floating part of the policy. In a managed float, the central bank intervenes in the market to influence the value of the currency by buying or selling its own currency.
6. Explain why covered interest arbitrage is just like our familiar cashandcarry transactions from Chapter 3. In a cashandcarry transaction, a trader sells the futures and buys the underlying good. The trader carries the underlying good to the expiration of the futures, paying the carrying cost along the way, and delivers the good against the futures. In covered interest arbitrage, the transaction has a similar structure. The trader sells the futures and buys the foreign currency. The trader carries the foreign currency to the expiration of the futures, paying the carrying cost along the way, and delivers the good against the futures. The carrying cost for the foreign currency consists of two components. First, there is the financing cost in the home currency for the funds borrowed to buy the foreign currency. Second, the foreign currency that is carried forward to delivery against the futures earns interest. This interest on the foreign currency offsets the first component of the carrying cost.
7. For covered interest arbitrage, what is the costofcarry? Explain carefully. The costofcarry is the difference between the home currency interest rate and the foreign currency interest rate. For covered interest arbitrage, the trader borrows the home currency and pays the domestic interest rate for these funds. The trader uses these funds to buy the foreign currency in the spot market, and invests the foreign currency to earn the foreign interest rate. Therefore, the costofcarry is the domestic interest rate minus the foreign interest rate.
8. The spot value of the German mark is $.65, and the 90day forward rate is $.64. If the US dollar interest factor to cover this period is 2 percent, what is the German rate? What is the cost of carrying a German mark forward for this period? From the interest rate parity theorem, we know that $1 invested in the United States must earn the same rate as the $1 converted into a foreign currency, investing at the foreign rate and converting the proceeds back into dollars via a forward contract initiated at the outset of the transactions. For our data:
$1(1.02) = ($1/$.65)(1 + r_{DM})$.64
where r_{DM} = the German interest rate for this 90day period. Therefore, r_{DM} = .0359. This is also the cost to carry a German mark forward for the 90 days.
9. The French franc is worth $.21 in the spot market. The French franc futures that expires in one year trades for $.22. The US dollar interest rate for this period is 10 percent. What should the French franc interest rate be?
1.10 = (1/.21)(1 +r_{FF}).22
where r_{FF} = the French franc interest rate for this period. Thus, r_{FF} = .05.
10. Using the data in problem 9, explain which country is expected to experience the higher inflation over the next year. If the expected inflation rate in the United States is 7 percent, what inflation rate for the French franc does this imply? The franc is expected to increase in value against the dollar from being worth $.21 now to $.22 in one year. Assuming PPP, this implies that the purchasing value of the dollar will decline relative to the franc. If the expected inflation rate in the United States is 7 percent, the real rate of interest is given by the equation:
1.10 = (1.07) (1 + r*)
where r * is the real rate of interest in the United States, and r* = .028. Assuming identical real rates in the United States and France, the expected French inflation rate is given by:
1.05 = [1 + E(I)](1.028) where E(I) is the expected inflation rate in France, and it equals .0214.
11. Using the data of problem 9, assume that the French franc interest rate for the year is also 10 percent. Explain how you might transact faced with these values. Faced with the exchange rates of problem 9 and interest rates in both the United States and France of 10 percent, we could sell dollars for francs in the spot market, invest the franc proceeds at 10 percent, and arrange now to convert the franc funds in one year at the forward rate of $.22. Assuming an initial amount of $100, we would:
ollar versus Franc Arbitrage 




f = 0 Borrow $100 for one year at 10% 
+ $100.00 


Sell $100 for FF 476.19 in the spot market. 
+ FF 476.19 



 $100.00 


Invest FF 476.19 at 10% in France 
 FF 476.19 

Sell FF 523.81 one year forward for $115.24

0



Total Cash Flow $0


t = 1 year



Collect FF 523.81 on investment

+ FF 523.81


Deliver FF 523.81 on forward contract; collect $115.24

 FF 523.81



+ $115.24


Repay debt from borrowing $100.00

 $110.00



Total Cash Flow + $5.24

12. Many travelers say that shoes in Italy are a big bargain. How can this be, given the purchasing power parity theorem? Travelers are wrong as a matter of fact, but we still must answer the question. If PPP held with perfection, shoes would have the same cost in any currency, and there would be no bargain shoes anywhere. Bargains can arise, however, due to market imperfections. First, transportation is costly. As a consequence, shoes in Italy could be cheaper than the same shoes in New York. The New York shoes must include the transportation cost. Second, even ignoring transportation costs, there are barriers to the free flow of shoes around the world. Governments impose tariffs and quotas, which can affect the price. Thus, if the United States protects its shoe industry by imposing tariffs or quotas on the Italian shoes, the shoes can cost more in the United States, thereby making shoes in Italy a bargain.
13. For the most part, the price of oil is denominated in dollars. Assume that you are a French firm that expects to import 420,000 barrels of crude oil in six months. What risks do you face in this transaction? Explain how you could transact to hedge the currency portion of those risks. Here we assume that the price of oil is denominated in dollars. Further, contracts traded on the NYMEX in oil are also denominated in dollars. Therefore, hedging on the NYMEX will not deal with the currency risk the French firm faces. However, the French firm can hedge the currency risk it faces by trading forwards for the French franc. To see how the French firm can control both its risk with respect to oil prices and foreign exchange, consider the following data. We assume a futures delivery date in six months for the oil and for foreign exchange forward contracts. The futures price of oil is $30 barrel, and the sixmonth forward price of a French franc is $.20. With these prices, the French firm must expect a total outlay of $12.6 million for the oil, and a total franc outlay of FF 63 million. By trading oil futures and French franc forwards, it can lock in this French franc cost. Because the crude oil contract is for 1,000 barrels, the French firm should buy 420 contracts. This commits it to a total outlay of $12.6 million. The French firm then sells FF 63 million in the forward market for $12.6 dollars. These two transactions lockin a price of FF 63 million for the oil.
14. A financial comptroller for a US firm is reviewing the earnings from a German subsidiary. This sub earns DM 1 million every year with exactitude, and it reinvests those earnings in its own German operations. This plan will continue. The earnings, however, are translated into US dollars to prepare the US parent's financial statements. Explain the nature of the foreign exchange risk from the point of view of the US parent. Explain what steps you think the parent should take to hedge the risk that you have identified. This risk is entirely translation risk, because we assume that the funds stay strictly in Germany. If the firm enters the futures or forward market to hedge the dollar value of the DM 1 million, it undertakes a transaction risk to hedge a translation risk. In other words, the firm increases its economic risk to hedge a purely accounting risk. From an economic point of view, this hedge would not make sense.
15. Joel Myers works for a large international bank. He has been watching the trading screen on this hot August morning and is disappointed in the lack trading activity. He is just about to take a break when a flurry of activity in the French bond and currency markets catches his attention. He quickly pulls up the following quotes:
Spot exchange rate:

$0.1656/FF


1month forward:

$0.1659/FF


3month forward:

$0.1665/FF


6month forward:

$0.1673/FF

A. Compute the 1month (30day), 3month (91day), and 6month (182day) yields Joel should expect to see in the French money market. For interest rate parity to hold, the French interest rates should be such that Joel would be indifferent between investing in the US money market and the French money market. The interest rate parity relationship is:
where r_{DC} and r_{FC} are, respectively, the domestic and foreign interest rates, FC is the spot exchange rate expressed as the cost of one unit of foreign currency in terms of domestic currency, and F_{0}, _{t} is the forward exchange rate today for a transaction at time t expressed as the domestic currency cost of one foreign currency unit.
Joel has already determined that the noarbitrage 6month return in the French money market would be 3.02 percent. If the 6month yield in the French market is 4 percent, then Joel could borrow domestically, exchange the dollars for French francs, and invest in the French money market. At the same time, he would lock in a 6month forward exchange rate to convert the francs back to dollars so the borrowing can be repaid. The profit on a $1 million arbitrage would be computed as:
Date Cash Market Forward Market
Today Borrow $1 million for 6 months at 5.11%
Sell FF 6.1591 million 6 months forward at $0.1673/FF
Invest FF 6.0386 million for 6 months at 4% Anticipated proceeds are FF 6.1591 million
Net Investment = 0
6 months Receive anticipated FF 6.1591 million Deliver FF 6.1591 million and receive $1.0304 million
Repay borrowing; amount due is $1.0255 million
Profit = ($1.0304  $1.0255) million = $4,936
16. As the fall semester starts, David McElroy is making arrangements for Oklahoma State University's Summer in London program for the next summer. This is a program in which OSU faculty teach courses to OSU students at Regents College in London, England. Room and board is £1,500 per participant to be paid May 15th. The enrollment is capped at 42 people, and OSU always operates at the cap. In the past, the Summer in London program has been burned by adverse movements in exchange rates. This happens because OSU has borne the exchange rate risk between the dollar denominated room and board rate quoted to the students and the British pound rate paid to Regents College. David wonders if there is some way that OSU could pass this risk off to someone else.
A. Does OSU face translation or transaction exposure? A trader faces transaction exposure when one currency must be converted into another. This differs from translation exposure in which one currency is restated in but not converted to another currency. OSU faces transaction exposure because it will be converting dollars to pounds in May.
B. What could OSU do to reduce this exchange rate risk? There are several ways OSU could reduce its exchange rate risk. First, OSU could negotiate a room and board contract denominated in dollars. This would transfer the risk to Regents College. This may be a viable alternative for future years, but it is too late for this year, as the contract has already been made. The second alternative is to buy British pounds forward using the futures market. This transfers the risk to a third party.
C. David asks a finance professor for advice. The professor pulls up the following $/£ quotes on the £62,500 futures contract:
SEP (this year)

1.6152


DEC (this year)

1.6074


MAR (next year)

1.6002


JUN (next year)

1.5936

What strategy might the professor recommend to reduce OSU's exchange rate exposure? (Make a recommendation.) The professor might suggest buying British pounds using the June futures contract. The amount of exposure OSU has is equal to the enrollment in the program multiplied by the pounddenominated room and board rate. The exposure will be:
Exposure = 42 x £1500 = £63,000
To hedge this exposure, OSU should buy one June British pound futures contract at $1.5936 per £. D. May 15th arrives, and the following situation is realized:
Compute OSU's gains and losses in the cash market and the futures market. Was the hedging strategy successful? When May arrives, exchange rates have risen. That is, the British pound has become more expensive in dollar terms. Luckily, the June futures price has also increased, resulting in gains from OSU's futures position. The gains and losses are as follows:
Date Cash Market Futures Market
Today Anticipate the need for $100,397 on Buy one June £62,500 futures contract
May 15th to make £63,000 room at $1.5936 per £.
and board payment.
May 15th Buy £63,000 in the spot market at Sell one June futures contract at $1.6451 per £.
$165 per £ for $103,950
Opportunity loss =  $3,553 Profit = $3,219
Net Loss = $334
While the hedge did not totally eliminate OSU's transaction exposure, it did reduce it. Therefore, the hedge was a success.
17. Viva Soda is an upandcomer in the highly competitive sports drink market. Viva owns three regional bottling facilities in the United States and one Canadian subsidiary that meets the demand for Viva in the Canadian provinces. Great North Bottling, the Canadian subsidiary, accounts for 25 percent of Viva's total sales and net earnings at the present exchange rates. Dave Baker, CFO for Viva, is very concerned about Viva's translation exposure. Viva will be in the debt refinancing market in one year. Dave is acutely aware of the relationship between the cost of debt and earnings results. Dave's assistant has made the following forecasts of Great North's earnings before taxes for the next four quarters:
Great North Earnings before Taxes
DEC

0.6603


MAR99

0.6609


JUN

0.6615


SEP

0.6621

Design a hedge that will solve Dave's problem. Assume one futures contract is for $100,000 Canadian. To hedge the translation exposure, Dave should sell each of Great North's anticipated pretax earnings in the futures market. To do this, Dave would sell 100 December contracts, 75 March contracts, 85 June contracts, and 120 September contracts.
C. Assuming the spot prices shown in the following table are realized, compute the translated earnings each quarter and the net impact on Viva's results considering the hedging activities.
DEC

0.6271


MAR99

0.6827


JUN

0.5961


SEP

0.7100

The anticipated contribution of Great North to pretax earnings each quarter is:
DEC

0.6603


MAR99

0.6609


JUN

0.6615


SEP

0.6621

The realized contribution of Great North to earnings before taxes each quarter is:
Quarter

Realized US$/CAN$


DEC

0.6271


MAR99

0.6827


JUN

0.5961


SEP

0.7100

The impact of the exchange rate changes on the translated quarterly earnings is the anticipated earnings before taxes minus the realized earnings before taxes. These are:
Quarter Realized EBT (US$ millions) Anticipated EBT (US$ millions) Gain (Loss) (US$ millions)
The gains and losses in the futures market are calculated as:
100,000(Selling price — Buying price) X # contracts
DEC

0.6603

0.6271

100


MAR99

0.6609

0.6827

75


JUN

0.6615

0.5961

85


SEP

0.6621

0.7100

120

The gains (losses) in the futures markets offset the translation losses (gains). One thing to keep in mind is that the translation gains and losses are accounting in nature. As long as Great North cash flows are not converted back into US dollars, the gains and losses are only on paper. The futures trading gains and losses are cash gains and losses. Unless there is some cash benefit to reducing the translated earnings volatility, hedging the translation exposure might not be a good idea.
The Options Market:
1. State the difference between a call and a put option. Call and put options are the two fundamental kinds of exchange traded options. They differ in the rights and privileges that ownership conveys. The owner of a call option has the right to buy the good that underlies the option at a specified price, with this right lasting until a stated expiration date. The put owner has the right to sell the good that underlies the option at a specified price with this right lasting until a stated expiration date. Thus, owning a call gives the right to buy and owning a put gives the right to sell. Correlatively, the seller of a call receives a payment and must sell the underlying good at the option of the call owner. The seller of a put receives a payment and must buy the underlying good at the option of the put owner.
2. How does a trader initiate a long call position, and what rights and obligations does such a position involve? To initiate a long call position, a trader buys a call option. At the time of purchase, the trader must pay the price of the option, which the seller of the call collects. Upon purchase, the owner of a call has the right to purchase the underlying good at the specified call price with that right lasting until the stated expiration date. The owner of a call has no obligations, once he or she pays the purchase price.
3. Can buying an option, whether a put or a call, result in any obligations for the option owner? Explain. The owner of a call or put has already paid the purchase price. After buying the option, the owner has only rights and no obligations. The option owner may exercise the option, sell it, or allow it to expire worthless, but the option owner is not compelled to do anything.
4. Describe all of the benefits that are associated with taking a short position in an option. Taking a short position in an option involves selling an option. Upon the sale, the seller receives a cash payment. This is the only benefit associated with selling an option. After receiving payment for the option, the seller has only potential obligations, because the seller may be required to perform at the discretion of the option owner.
5. What is the difference between a short call and a long put position? Which has rights associated with it, and which involves obligations? Explain. The short call position is obtained when a trader sells a call option. The seller of a call may be required to surrender the underlying good in exchange for the payment stated in the option contract. The short call position has a maximum benefit equal to the price that the seller received to enter the short call position. The short call position is most favorable when the price of the underlying good remains below the exercise price. Then the seller of the call retains the full price of the option as profit. The higher the stock price above the exercise price, the worse for the call seller. In a long put position, the trader buys a put option. Owning the put gives the trader the right to sell the underlying good at the stated exercise price until the option expires. The put purchaser profits when the price of the underlying good falls below the exercise price. Then the owner of the put can require the put seller to buy the underlying good at the exercise price. When the underlying good has a price above the exercise price, the long put trader cannot exercise and loses the entire purchase price of the option. In contrasting the short call and the long put positions, we note that the short call trader has a maximum profit equal to the original sales price, and the long put trader has a maximum loss equal to the original sales price. For the long put position, there is the chance of a virtually unlimited profit as the stock price falls to zero. For the short call position, there is the chance of a theoretically unlimited loss, as the stock price rises toward infinity.
6. Consider the following information. A trader buys a call option for $5 that gives the right to purchase a share of stock for $100. In this situation, identify: the exercise price, the premium, and the striking price. The premium is the same as the option price and equals $5. The exercise price is the same as the striking price and equals $100.
7. Explain what happens to a short trader when the option he or she has sold expires worthless. What benefits and costs has the trader incurred? At the time of trading, the short trader of a put or call receives a payment. This is the only benefit the short trader receives from trading. If the option expires worthless, then the option was not exercised and the short trader attains the maximum possible profit. In selling an option, the short trader exposes himself or herself to the risk that the purchaser will exercise. For accepting this risk, the seller has received the option premium. If the option expires worthless, the short trader has escaped that risk.
8. Explain why an organized options exchange needs a clearinghouse. The clearinghouse guarantees the financial integrity of the market and oversees the performance of traders in the market. If there were no clearinghouse, each trader would have to be concerned with the financial integrity of his or her trading partner. Assuring that the opposite trading party will perform as promised is difficult and expensive. With a clearinghouse, each trader has an assurance that the opposite side of his or her transaction will be fulfilled. The clearinghouse guarantees it.
9. What is the difference between an American and a European option? A European option can be exercised only at expiration, while an American option can be exercised at any time prior to expiration. This difference implies that the American option must be at least as valuable as the European option.
10. Assume a trader does not want to continue holding an option position. Explain how this trader can fulfill his or her obligations, yet close out the option position. The trader will be either long or short. If the trader is long, he or she can close the position by selling the exact same option. The option that will be sold must be on the same underlying good, have the same expiration, and have the same striking price as the original option that is to be closed. If the trader were short initially, the trader would close the position by buying the identical option. In essence, the trader closes the position by trading to bring his or her net position back to zero. Again, making sure that all characteristics of the option match is an essential condition.
11. A developer has purchased 60 acres of rural property just north of Augusta, Georgia, to develop a golf course. The golf course development will also include a housing development. In order to generate operating capital, the developer is selling rights. The rights give the holder of the contract the right to purchase lots in the housing development for a fixed price. Each lot in the housing development is half an acre. The agreements expire six months after they are signed. The developer is offering the following inducement. A potential homeowner can purchase a lot for $25,000 at the end of six months if the homeowner enters into the contract this week. The purchase price for a lot increases to $40,000 on all contracts signed after this week. A. Describe the type of option being sold by the developer. The developer is selling European call options.
B. Describe the position held by the potential homeowner as an option. The potential homeowner has a long position in the European call option sold by the developer. The potential homeowner is the person who decides whether to exercise the right to purchase the property.
C. Discuss the risks associated with this transaction. This is an overthecounter transaction between the developer and the potential homeowner. The principal risk associated with this transaction is default (credit) risk. Both the developer and homeowner must be concerned with the possibility of the other party defaulting on the contract. However, the default risk is much lower for the developer. If the homeowner's check clears, then the developer has no default risk. However, the homeowner must be concerned with default by the developer at the end of six months.
D. Suppose we purchased the rights on a corner lot on the eighteenth hole during the inducement period, and we have just found out that legendary golfer Tiger Irons is building a house on the same block. Explain what you think will happen to the value of the right that you own. Is this contract inthemoney? This news will in all likelihood dramatically increase the value of the property in the neighborhood and thus the contract you own. A contract that permitted you to purchase the land at a below market value would be inthemoney.
E. Suppose the developer was selling two contracts. One contract permits you to purchase a lot anytime during the sixmonth period, and the other allows you to purchase the lot only at the end of six months. Which of the two contracts is worth more? Explain why. The developer is offering both American and European call options to potential homeowners. The American option must be at least as valuable as the European. It can be worth more if the underlying good has some payoff, analogous to a dividend on a stock, before the option expires. In this situation, it is not so clear that there is a "dividend" before expiration. With the revelation that a celebrity is moving into the neighborhood, the American option might be more valuable to the homeowner. The homeowner could exercise the option, acquire the property at below market value, and hold the property. The homeowner then has the option of selling the property immediately or at some date in the future, or the option to build on the property. The excess value of the American option over the European one depends on some incentive to early exercise. It is not obvious that there is such an incentive.
F. To reduce your cash outflows shortly before it became public knowledge that Tiger was going to build a house in the development, you signed a contract with a colleague. This contract gives you the right to sell the lot anytime in the next six months to your friend for $35,000. Describe your position and that of your friend. You have a long position in an American put option that gives you the right to sell the property anytime in the next six months for $35,000. Your friend has a short position in an American put option.
G. Describe potential obligations associated with the options involving the developer and the two friends. The original contract requires the developer to deliver the deed on the property for $25,000 if the potential homeowner chooses to exercise the contract. Exercising the contract requires the potential homeowner to deliver $25,000 in exchange for the deed to the property. The second contract gives the potential homeowner the opportunity to deliver the deed to the property to their colleague in exchange for $35,000. Your friend must take delivery of the deed and make a payment of $35,000 if you choose to exercise your right
H. Suppose the contract that you signed required you to sell the property to your colleague at the end of six months for $35,000. What type of contract would this be? This is a forward contract, since you must deliver the deed to the property at the end of six months to your colleague. You have a short position in a forward contract that requires you to deliver the deed to the property in exchange for $35,000.
I. Suppose that you signed the contract that required you to sell the property to your friend, but once you found out that Tiger was going to be your neighbor, you did not want to sell the property. What recourse is available to you? One option would be to purchase the contract back from your colleague. However, your friend will have the power to capture all the gains associated with the increase in the value of the property. Thus, you will be negotiating from a position of weakness. Alternatively, you could repurchase the property from your friend at the current market price. If you were to default on the contract, your friend would take you to court and force you to live up to the terms of the contract. In addition, you would probably have to pay all the legal costs associated with your friend's court case, and any additional damages assigned by the court.
12. When you purchase or sell an option on the Chicago Board of Options Exchange you contract with the Options Clearing Corporation. Discuss the advantages of this type of trading arrangement. The OCC acts as the buyer to every seller and the seller to every buyer. This arrangement increases the liquidity and marketability of exchange traded options contracts. If an investor wanted to reverse an options position, he or she could do so by submitting a reversing order to the options market. The investor is not forced to approach the original counterparty to negotiate a reversing transaction. In addition, every investor does not have to evaluate the credit (default) risk of other market participants. This reduces the cost of transacting in this marketplace, as each investor is concerned only with the credit risk of the OCC.
A. If all trades in a day match, describe the net position of the clearing corporation. If all trades match, then the OCC has a net zero option position. B. When a retail customer makes a margin payment, to which organization is the margin paid? Trace the flow of margin funds from the retail customer to the ultimate recipient.
A retail customer makes margin payments to his broker. If his broker is a clearing member firm, the broker will make margin payments to the OCC. If the broker is not a clearing member firm, then the broker will make margin payments to a clearing member firm and the clearing member firm will make the payments to the OCC. It is clearing member firms that make margin payments to the OCC.
13. Describe the services provided by a market maker, floor broker, and order book official. Discuss the differences between a market maker and an order book official. A broker represents customer orders on the floor of the exchange. The broker is paid a commission for the provision of these services. The broker tries to fill orders as rapidly as possible at the best price available at the time of the transaction. The more orders the broker handles, the more commissions the broker earns. Order book officials represent public orders that are in the limit order book. They represent the exchange and have no equity positions in the market. They are responsible for maintaining an orderly and fair market in the handling of customer orders. A market maker holds a portfolio of options. A market maker manages her portfolio with the objective of maximizing her wealth. A market maker is required to trade a minimum number of options contracts at their quoted bid and ask price. Both market makers and order book officials maintain a portfolio of options. However, market makers trade to make a profit, and order book officials represent the orders of public customers (not professional traders).
14. During an extended period of financial difficulty, a firm's CFO offers the firm's treasurer the following contract in lieu of 25 percent of her salary. The contract permits the treasurer to "clip up to ten coupons" that entitle her to convert each coupon into 10,000 shares of stock at no cost anytime during a threeyear period. Suppose the treasurer chooses to accept the contract. Explain what type of contract the treasurer holds. Since the treasurer of the firm has the right to convert the coupons into a maximum of 100,000 shares of common stock anytime, the treasurer has a long position in an American call option written on the firm's shares. However, since the exercise price of the option is $0, the treasurer effectively owns 100,000 shares of the firm's stock.
15. Explain the "rights" attached to a stock option contract traded on the CBOE. Discuss the economic motivation for exercising a stock option at expiration. A call option is a contract that extends to the holder of the contract the right to buy 100 shares of the underlying stock at the specified strike price anytime during the life of the contract. The stock options traded on the CBOE are American options. A put option is a contract that extends to the holder of the contract the right to sell 100 shares of the underlying stock at the specified strike price anytime during the life of the contract. When making the decision to exercise an option at expiration, the owner of a stock option is in effect asking himself the following questions.
Call option: Is it cheaper to buy 100 shares of the stock underlying the option in the market at the market (spot) price or via the contract (call option) at the contracted buying price? The contracted buying price is the strike price of the option.
Put option: Do I make more money by selling 100 shares of the stock underlying the option in the market at the market (spot) price or via the contract (put option) at the contracted selling price? The contracted selling price is the strike price of the option. If the owner of the option finds it more profitable to trade the underlying stock according to the terms of the contract, then he will exercise the option.
16. Explain the obligations associated with establishing a long position in an option contract. The purchaser of an option (the long position) must pay for the option before the market opens for business on the trading day following the purchase of the option. If payment has not been received by the market's opening, the OCC closes the position.
17. Explain the obligations associated with a short position in call and put options. The writer of an option (short position) does not have any obligations if the option is not exercised. If the person has written a call option and the option is exercised against her, the writer must deliver the quantity of the underlying commodity specified by the option's contract. Upon delivery, the writer of the option will receive payment in the amount specified in the contract. If the person has written a put option and the option is exercised against her, the writer must take delivery of the quantity of the underlying commodity specified by the option's contract. Upon delivery, the writer of the option must make payment in the amount specified in the contract to the owner of the option's contract. When an option is exercised, the OCC randomly selects individual(s) from the pool of individuals who have open short positions in the relevant options contract. Thus, it is possible that an individual may have a short position in an option and not have the option exercised against her.
18. A trader purchases an option for $6.50 that gives him the right to sell 100 shares of stock for $50 per share. Identify the type of option, the option's price, and the option's strike price. The investor is long a put option. The strike price is $50, and the price of the option is $6.50 on a per share basis. The total cost of the option is $650 ($6.50 X 100).
19. Consider the following "opportunities." Determine if the opportunity is an option, and if it is an option, explain what type of option the opportunity represents. Describe positions of the parties involved in the opportunity. A. An old college classmate calls and offers you the opportunity to purchase automobile insurance. The insurance is renewable semiannually. Auto insurance is an American put option. If there is an accident involving an insured driver, then the insurance company makes a payment to the insured driver. The insurance company has written the put option. The insured is long the put option and "chooses" to exercise the option because of an accident.
B. You have just received your SAT scores from ETS. It turns out that you had a good day and scored a perfect 1600. One week later, you receive the following offer in a letter from Private School U. If you enroll in PSU in the fall, PSU will guarantee that your total annual cost of attending PSU will be $1. The offer applies only if you enroll at PSU for the upcoming fall semester. You hold a long position in a European call option. PSU has written a call to you. If you enroll in PSU in the fall, then you have the right to purchase your education for the fixed price of $1 per year.
C. The following day your parents, who attended the local statesupported university, receive a letter from their alma mater, Basketball Power U. The letter makes the following offer. If their child enrolls in BPU for the fall semester, BPU will guarantee that they can purchase 10 courtside basketball seats for the face value of the tickets as long as their child is enrolled in BPU. Your parents have nothing, but may come to have a long position in a portfolio of European call options. BPU has written the call options to the parents, contingent on the child's enrollment. If their child enrolls in BPU in the fall and remains enrolled at BPU, then they will have the right to purchase 10 valuable courtside basketball tickets at BPU each year their child is enrolled at BPU. The offer applies only if their child enrolls at BPU for the upcoming fall semester and only for each year their child maintains enrollment at BPU.
D. A high school classmate calls during dinner from his firm, High Pressure Telemarketing. He offers you the opportunity to enter into a longterm, fixed price, noncancellable lease on a condo in Miami Beach. This opportunity is not a time share, and you cannot sublet the condo. This is not an option. There is no right associated with the contract. The contract requires you to make the fixed payments over the life of the contract. Once you enter into the contract, you have an obligation to make the payments, and you cannot choose to avoid a payment.
E. You have just purchased 40 acres of heavily wooded land in the nearby hills. To minimize fire danger and relieve workrelated stress, you plan to remove all the underbrush and dead trees from the property. The local hardware store is advertising a sale on chain saws. At the store, a sales consultant informs you that they are temporarily out of the saw that you are interested in. The salesperson offers you a rain check that gives the "opportunity" to purchase the saw at the sale price for two weeks. You hold a long position in an American call option. You can return to the store anytime in the next two weeks and purchase the saw at the sales price if desired. The store has written the call giving you the right to purchase the saw at the sales price during the next two weeks.
E As usual things are running late. After work you run home to get dinner. Arriving home, you frantically search for the telephone book. In the middle of the telephone book, you find a coupon from the local pizza parlor, My Pies, that gives you the opportunity to buy a large supreme pizza and two liters of pop for $15.99. You are long an American call giving the right to purchase the pizza and pop for $15.99. My Pies has written the call option.
20. Compare the obligations associated with exercising stock index options and stock options. The fundamental difference between the two contracts is that the index option is a cash settlement contract. An index option contract is settled at exercise by an exchange of cash, while an option on an individual stock is settled by the delivery of shares in exchange for cash.
21. An investor holds a long option position that she wishes to close. Explain the different means available to the investor to close this options position. Explain how your answer changes if the option position was entered on an exchange or overthecounter. There are three ways this long option position can be closed. The investor can exercise the option, the investor can allow the option to expire worthless, and, if the option was entered on an exchange, the investor can enter into a reversing (offsetting) trade. To reverse the position, the investor must sell the same quantity of the option she owns. The options that she sells must have the same strike price and the same expiration date. With OTC options, if the investor wishes to reverse her position, she must arrange a reversing trade with the same counterparty with whom she initially contracted. If the investor must sell the option, the counterparty will have market power when negotiating the terms of the reversing transaction.
22. Suppose that there are two call options written on the same share of stock, XYZ. Both options have the same expiration date and strike price. Explain why the American option is always worth at least as much as the European option. The American option contract has an additional right attached to the contract. This contract gives the option's owner the right to exercise the option anytime prior to the option's expiration. If this right to exercise early is worthless, then the value of the American option is the same as the value of a comparable European option.
23. Explain why the bidasked spread quoted by the market maker on the floor of the CBOE represents a real cost to an investor trading options. Explain why the investor is willing to trade at the quoted spread prices in a competitive market. If a trader wishes to trade an option listed on the CBOE, the trader must trade that option through the floor of the CBOE. In addition, the trader must trade the listed option at the price (spread) quoted by the market makers. The trader purchases the option at the market maker's ask price and sells options at the market maker's bid price. If market makers are informed and are correctly valuing the traded option, their quoted ask price should be above the fair value of the option and the quoted bid price should be below the fair value of the option. If there were no market makers in the market and the market was competitive with fully informed investors who correctly valued the traded options, then investors would trade options at the fair value of the option. The bidasked spread represents compensation paid by market participants to the market maker. In a competitive marketplace, the market maker earns a fair return for bearing the risk associated with holding an inventory of option contracts, and for providing liquidity to the market.
24. Mr. Smith holds a certificate of deposit (CD) from his local bank that matures in three months. Mr. Smith is well aware of the current bull market for stocks, and the opportunity cost of having his money tied up in a CD earning 4.5 percent. When the CD matures, Mr. Smith is planning to invest in the stock market. Mr. Smith feels that the market will continue its strong bull run for the next three months, and he does not want to wait three months to invest in the market. Discuss how Mr. Smith may use options to invest in the market today. Explain the type of contracts and the positions Mr. Smith could use to undertake this investment. Since Mr. Smith expects the value of the market to increase during the next three months, he should take a long position in a call option on the stock market. There are several different index options traded in the marketplace, including options written on the S&P 500, the S&P 100, the Russell 2000, and the Dow Jones Industrial Average. With a long position in an index call option, Mr. Smith can participate in the upward movements in the market with limited downside risk and a limited investment.
25. It is late in the afternoon of the third Friday of the expiration month, and an investor has an open long position in a deep inthemoney European call option. The investor must decide whether to sell the option in the market while the market is open or to exercise the option at the close of the market. For an option contract, the oneway transaction cost is $25. The exercise fee for an option is $25, and the commission for selling 100 shares of stock is $20. Discuss the impact of transactions costs on the investor's decision to either exercise the option or sell the option. Exercising the option requires two transactions by the investor, exercising the option followed by selling the stock. The cost of exercise is $25, and the cost of selling the shares acquired by the exercise is $20 for a total of $45. Simply selling the option to capture the same underlying value will cost only $25.
Option








IBM

Strike

Exp.

Vol.

Last

Vol.

Last


1011

115

Sep

1632

3

10

131


1011

115

Oct

861

u

10

141


1011

115

Jan

225

4

3

14

A. What is the cost of 15 IBM Oct 115 call options?
15 x $1.25 x 100 = $1,875
B. Assume the clearing corporation is using the following schedule for the calculation of margin requirements: The maximum of: 100% of the proceeds from the sale of the options plus 20% of the value of the underlying stock position minus the dollar amount the option's contract is outofthemoney, or 100% of the proceeds from the sale of the options plus 10% of the value of the underlying stock position.
What are the initial margin requirements for the buyer and seller of 3 IBM Jan 115 call options? Explain how the answer changes if the seller owns 300 shares of IBM. If the investor owns 300 shares of IBM and chooses to lend them to his broker to collateralize his options position, then his margin requirement is $0. This investor will have written covered calls. If the investor has written naked calls, then his initial margin is the maximum of:
First Method Second Method
100% of the proceeds from 3 x $4 x 100 = $1,200 100% of the proceeds from 3 x $4 x 100 = $1,200
the sale of the option the sale of the option
20% of the value of the 3 x ($101.625 x 100) x 0.2 10% of the value of the 3 x ($101.625 x 100)
underlying stock position =$6,097.50 underlying stock position x 0.1 = $3,048.75
The dollar amount that the ($101.625  $115) x 3 x 100
options contract is = $4,012.5
outofthemoney
Required margin $3,285.00 $4,248.75
The margin requirement is the larger amount of $4,248.75.
C . Explain what happens to your option position if you are unable to meet a margin call. A margin call requires the investor to recollateralize her margin position. If you do not have sufficient cash to recollateralize the position, the OCC reverses your position at the opening of the market on the next trading day. The existing balance in your margin account is used to cover your losses. The OCC also subtracts the relevant transactions costs from your margin account balances.
Option Payoffs and Options Strategies
1. Consider a call option with an exercise price of $80 and a cost of $5. Graph the profits and losses at expiration for various stock prices.
2. Consider a put option with an exercise price of $80 and a cost of $4. Graph the profits and losses at expiration for various stock prices.
3. For the call and put in questions 1 and 2, graph the profits and losses at expiration for a straddle comprising these two options. If the stock price is $80 at expiration, what will be the profit or loss? At what stock price (or prices) will the straddle have a zero profit?
With a stock price at $80 at expiration, neither the call nor the put can be exercised. Both expire worthless, giving a total loss of $9. The straddle breaks even (has a zero profit) if the stock price is either $71 or $89.
4. A call option has an exercise price of $70 and is at expiration. The option costs $4, and the underlying stock trades for $75. Assuming a perfect market, how would you respond if the call is an American option? State exactly how you might transact. How does your answer differ if the option is European? With these prices, an arbitrage opportunity exists because the call price does not equal the maximum of zero or the stock price minus the exercise price. To exploit this mispricing, a trader should buy the call and exercise it for a total outofpocket cost of $74. At the same time, the trader should sell the stock and deliver the stock just acquired through exercise for a $75 cash inflow. This produces a riskless profit without investment of $1. Because the option is at expiration, both the American and European options have the same right to exercise. Therefore, the American or European character of the option has no effect on the trading strategy.
5. A stock trades for $120. A put on this stock has an exercise price of $140 and is about to expire. The put trades for $22. How would you respond to this set of prices? Explain. At expiration, the put price must equal the maximum of zero or the exercise price minus the stock price to avoid arbitrage. Therefore, the put price should be $20 in this situation, but it trades for $22. This difference gives rise to an arbitrage opportunity, because the put is priced too high relative to its theoretical value. To exploit this, the trader should simply sell the put and receive $22. Now the option can be exercised against the trader or not. If it is not exercised, the put expires worthless, the obligation is complete, and the trader retains the $22 as total profit. However, the purchaser of the option may choose to exercise immediately. In this case, the seller of the put must buy the stock for the exercise price of $140. The trader then sells the stock for $120 in the market, giving a $20 loss on the exercise. But the put seller already received $22, so he or she still has a $2 profit. In summary, selling the put leads to a $22 profit if the put buyer foolishly fails to exercise. Alternatively, if the put buyer exercises, the put seller still has a $2 profit. This is an arbitrage opportunity, because selling the overpriced put gives a profit without investment in all circumstances.
6. If the stock trades for $120 and the expiring put with an exercise price of $140 trades for $18, how would you trade? As in the previous problem, these prices violate the noarbitrage condition. Now, however, the put is underpriced relative to the other values. To conduct the arbitrage, the trader should buy the stock and buy and exercise the put. In this sequence of transactions, the trader pays $120 to acquire the stock, pays $18 to acquire the put, and receives $140 upon exercise of the put. These transactions yield a profit of $2 with no risk and no investment.
7. Consider a call and a put on the same underlying stock. The call has an exercise price of $100 and costs $20. The put has an exercise price of $90 and costs $12. Graph a short position in a strangle based on these two options. What is the worst outcome from selling the strangle? At what stock price or prices does the strangle have a zero profit? The worst outcomes occur when the stock price is very low or very high. First, the strangle loses $1 for each dollar the stock price falls below $58. With a zero stock price, the strangle loses $58. If the stock price is too high, the strangle also loses money. Because the stock could theoretically go to infinity, the potential loss on the strangle is unbounded. For stock prices of $58 or $132, the strangle gives exactly a zero profit.
8. Assume that you buy a call with an exercise price of $100 and a cost of $9. At the same time, you sell a call with an exercise price of $110 and a cost of $5. The two calls have the same underlying stock and the same expiration. What is this position called? Graph the profits and losses at expiration from this position. At what stock price or prices will the position show a zero profit? What is the worst loss that the position can incur? For what range of stock prices does this worst outcome occur? What is the best outcome and for what range of stock prices does it occur? This position is a bull spread with calls, because it is designed to profit if the stock price rises. The entire position has a zero profit if the stock price is $104. At this point, the call with the $100 exercise price can be exercised for a $4 exercise profit. This $4 exercise value exactly offsets the price of the spread. The worst loss occurs when the stock price is $100 or below, because the option with the $100 exercise price cannot be exercised, and the entire position is worthless. This gives a $4 loss. The best outcome occurs for any stock price of $110 or above and the total profit is $6.
9. Consider three call options with the same underlying stock and the same expiration. Assume that you take a long position in a call with an exercise price of $40 and a long position in a call with an exercise price of $30. At the same time, you sell two calls with an exercise price of $35. What position have you created? The value of this position at expiration. What is the value of this position at expiration if the stock price is $90? What is the position's value for a stock price of $15? What is the lowest value the position can have at expiration? For what range of stock prices does this worst value occur? This is a long position in a butterfly spread. If the stock price is $90, the value of the spread is zero. For a $15 stock price, the spread is worth zero. The entire spread can be worth zero at expiration. This zero value occurs for any stock price of $30 or below and $40 or above.
10. Assume that you buy a portfolio of stocks with a portfolio price of $100. A put option on this portfolio has a striking price of $95 and costs $3. Graph the combined portfolio of the stock plus a long position in the put. What is the worst outcome that can occur at expiration? For what range of portfolio prices will this worst outcome occur? What is this position called? The worst result is a portfolio value of $95. The purchase of the put for $3 gives a loss of $8. This worst outcome occurs for a terminal stock portfolio value of $95 or less. This combined position is an insured portfolio. The position insures against any terminal portfolio value less than $95 or any loss greater than $8.
11. Consider a stock that sells for $95. A call on this stock has an exercise price of $95 and costs $5. A put on this stock also has an exercise price of $95 and costs $4. The call and the put have the same expiration. Graph the profit and losses at expiration from holding the long call and short put. How do these profits and losses compare with the value of the stock at expiration? If the stock price is $80 at expiration, what is the portfolio of options worth? If the stock price is $105, what is the portfolio of options worth? Explain why the stock and option portfolio differ as they do. No matter what stock price results, the option portfolio will have $1 less profit than the stock itself. For example, the option portfolio costs $1, but both options are worthless at a stock price of $95. Therefore, at a stock price of $95, the stock has a zero profit, and the option portfolio has a $1 loss. Further, the option portfolio will be worth exactly $95 less than the stock at every price. With a stock price of $80, the call is worthless and the put will be exercised against the option holder for an exercise loss of $15. Therefore, the option portfolio is worth —$15 for an $80 stock price. If the stock trades for $105, the option portfolio will be worth $10.
12. Assume a stock trades for $120. A call on this stock has a striking price of $120 and costs $11. A put also has a striking price of $120 and costs $8. A riskfree bond promises to pay $120 at the expiration of the options in one year. What should the price of this bond be? Explain. A portfolio consisting of one long call, one short put, and a riskless investment equal to the common exercise price of the two options gives exactly the same payoffs as a share of the underlying stock on the common expiration date. This putcall parity relationship requires that this portfolio of long call, short put, plus riskless investment should have the same price as the stock. With our data, the riskless bond must therefore cost $120  $11 + $8 = $117. The riskless interest rate must be 2.53 percent
13. In the preceding question, if we combine the two options and the bond, what will the value of this portfolio be relative to the stock price at expiration? Explain. What principle does this illustrate? As the previous answer already indicated, the described portfolio (long call, short put, plus long bond) must have the same value as the stock itself. This illustrates the putcall parity relationship.
14. Consider a stock that is worth $50. A put and call on this stock have an exercise price of $50 and expire in one year. The call costs $5 and the put costs $4. A riskfree bond will pay $50 in one year and costs $45. How will you respond to these prices? State your transactions exactly. What principle do these prices violate? These prices violate putcall parity. The long call, plus short put, plus riskless investment of the present value of the exercise price must together equal the stock price: Instead, we have $50 S= $5  $4 + $45 = $46. Therefore, the stock is overpriced relative to the duplicating righthand side portfolio. Accordingly, we transact as follows, with the cash flows being indicated in parentheses: Sell stock (+$50), buy call ($5), sell put ($4), and buy the riskless bond ($45). This gives a positive cash flow of $4 at the time of trading. To close our position, we collect $50 on the maturing bond. If the stock price is above $50, we exercise our call and use our $50 bond proceeds to acquire the stock, which we can then repay to close our short position. The put cannot be exercised against us, so we conclude the transaction with our original $4 profit. If the stock price is below $50, the put will be exercised against us. If so, we lose $50 — S on the exercise, paying our $50 bond proceeds to acquire the stock. Now with the stock in hand, we close our short position and the call expires worthless. As a result, we still have our $4 original cash inflow as profit. No matter what the stock price may be at expiration, our profit will be $4.
15. A stock sells for $80 and the riskfree rate of interest is 11 percent. A call and a put on this stock expire in one year and both options have an exercise price of $75. How would you trade to create a synthetic call option? If the put sells for $2, how much is the call option worth? (Assume annual compounding.) A synthetic call option consists of the following portfolio: long the stock, long the put, and short a riskfree bond paying the exercise price at the common maturity date of the call and put. Therefore, the following relationship must hold: C=S+P(1 + r)t Therefore, with the information given: C = $80 + $2  $67.67 = $14.43
16. A stock costs $100 and a riskfree bond paying $110 in one year costs $100 as well. What can you say about the cost of a put and a call on this stock that both expire in one year and that both have an exercise price of $110? Explain. Put—call parity implies: In this case, the stock and bond have the same price, so the lefthand side of the equation equals zero. For the righthand side to equal zero, the call and put must have the same price as well. However, from the information given, we cannot determine what that price would be.
17. Assume that you buy a strangle with exercise prices on the constituent options of $75 and $80. You also sell a strangle with exercise prices of $70 and $85. Describe the payoffs on the position you have created. this portfolio of options have a payoff pattern similar to that of any of the combinations explored in this chapter? This position will profit for very low (say below $70) or very high (say above $85) stock prices at expiration. For prices in the range of $75 to $85, the position will lose. The exact breakeven points cannot be determined from the information given, however. In effect, the pair of strangles described is a short condor position.
18. If a stock sells for $75 and a call and put together cost $9 and the two options expire in one year and have an exercise price of $70, what is the current rate of interest? From the information given, and applying put—call parity, we know: S ^_{r} =CP = $9 The stock price is $75 and must exceed the bond price by $9, so the price of the bond is $66. Thus, the present value of the $70 exercise price is $66, implying an interest rate of 6.06 percent.
19. Assume you buy a bull spread with puts that have exercise prices of $40 and $45. You also buy a bear spread with puts that have exercise prices of $45 and $50. What will this total position be worth if the stock price at expiration is $53? Does this position have any special name? Explain. The bull spread with puts and these exercise prices implies buying the put with X = $40 and selling the put with X = $45. To buy the bear spread implies buying a put with X = $50 and selling a put with X = $45. If the stock price at expiration is $53, all of the puts expire worthless. In making this trade, one has bought puts with X = $40 and X = $50, and sold two puts with X = $45. This is equivalent to a long butterfly spread with puts.
20. Explain the difference between a box spread and a synthetic riskfree bond. The box spread gives a riskless payoff, so it is equivalent to a synthetic riskfree bond.
21. Within the context of the putcall parity relationship, consider the value of a call and a put option. What will the value of the put option be if the exercise price is zero? What will the value of the call option be in the same circumstance? What can you say about potential bounds on the value of the call and put option? If the exercise price of a put is zero, there can be no payoff from the put so the price must be zero. In this circumstance, the call must be worth the stock price. With respect to bounds on the option prices, the call price can never exceed the stock price and the put price can never exceed the exercise price.
22. Using the putcall parity relationship, write the value of a call option as a function of the stock price, the riskfree bond, and the put option. Now consider a stock price that is dramatically in excess of the exercise price. What happens to the value of the put as the stock price becomes extremely large relative to the exercise price? What happens to the value of the call option? The put price declines as the ratio S/X becomes large. The call option must increase in value. As we will see, the call must always be worth at least the stock price minus the present value of the exercise price.
23. The CBOE is thinking of opening a new market center in London that will trade only European options and has hired you as a consultant. The board of directors for the CBOE believes that there is demand in the market for options contracts that can be used by investors to insure their portfolios. A unique feature of this market is the fact that only put options will be traded.
A. Explain why it will not be necessary to trade call options in this market. Because the options that will be traded in this market are European options. Arbitrage, in the form of putcall parity, guarantees that the call options will be priced fairly. Any investor can use putcall parity to construct the corresponding synthetic call option c = p + S — Xe^ ~ ^ that will have the same value as an exchange traded call option.
B. Discuss the expected clientele in this market and explain why these traders might not desire to use American options traded in Chicago on the CBOE. The target clientele for this market is hedgers. Hedgers desire to protect the value of an asset that they currently own for a specified time period. That is, the hedger has an investment horizon and desires to protect the value of her investment over that investment horizon. The price of the option contract traded in Chicago on the CBOE includes the value of the early exercise premium. Hedgers do not expect to exercise their options prior to expiration, and they are not interested in paying for the right to exercise the option prior to expiration. The value of this early exercise premium on an American option effectively increases the cost of the insurance available to hedgers.
C. The CBOE has asked you as a consultant to identify the important characteristics of the market and the option contracts that will ensure the success of this new market. Identify those characteristics that you feel will contribute to the success of the market. For this marketplace to be successful, the contracts and services offered must be demanded by the market. The target clientele for these contracts is hedgers. Hedgers have an investment horizon and desire to protect the value of their investment over that investment horizon. To do an effective job, you must understand the demographics of the marketplace. The following questions might be relevant in determining the nature of the marketplace. What types of assets are owned by the investors who are likely candidates to be the clients of this market? Who are the likely users of the market, individuals or professional funds managers? Should the option contracts be written on individual shares of stock or should they be written on stock indices (portfolios)? Should the market include equities traded globally or should they focus on a specific geographic location (e.g., the United States)? Should the contracts be settled in cash or call for delivery of the underlying security? Should the market be open continuously? How frequently should the options expire— weekly, bimonthly, monthly, and so on? Should the options have expiration dates longer than one year? What is the optimal size of the contract (e.g., 100 shares)? What is the optimal differential between the strike prices of the traded options? Should there be any restriction on the size of the options position held by an investor? What is the optimal structure of transactions costs for the market?
24. Suppose an investor purchases a call option on XYZ with a $50 strike price and sells a put option on XYZ with the same strike price. Both options are European options that expire in one month. Describe the investor's position. The investor has created a synthetic long position in a forward contract written on the underlying stock. The payoff on this position mimics the payoff on a long position in a share of the underlying stock (S — X, where X is the purchase price of the stock, and S is the spot price of the stock).
25. DRP is currently selling at $58 per share. An American call option written on DRP with six weeks until expiration has a strike price of $50. A. If DRP does not pay a dividend, explain why it is not economically rational to exercise this American call option prior to expiration. Equivalently, explain why a call option is worth more alive than dead. The price of an option consists of its intrinsic value (S — X), and its time value (the option's price  the intrinsic value of the option). When you exercise the call option, you capture its intrinsic value (S — X). (The option must be inthemoney for you to consider exercising it.) However, you lose the time value when you exercise. You can capture the time value by selling the option in the marketplace. For example, consider the following information: S = $50, X = $40, and C = $11.5. If the call were exercised, the payoff to the investor would be ($50  $40) X 100 = $1,000. If the call were sold, the payoff to the investor would be $11.50 x 100 = $1,150.
B. If DRP pays a quarterly dividend, but the dividend will not be paid for eight weeks, explain why it is not economically rational to exercise this American call option prior to expiration. In this problem, the dividend is irrelevant to the early exercise decision. The dividend is to be paid after the option's expiration and will not affect your decision regarding the early exercise of the option.
26. Call prices are directly related to the volatility of the underlying stock. That is, the more volatile the underlying stock, the more valuable the call option. However, higher volatility means that the stock price may decrease by a large amount. That is, the probability of a large decrease in the stock price has increased. Explain this apparent paradox. Examination of the components of the price of an option reveals the answer to this apparent paradox. The price of an option consists of its intrinsic value, and its time value. A call option pays off when the market price of the stock exceeds the exercise price (contracted purchase price). That is, the call option pays off in only one direction, when the stock price rises (the up state). An option that is outofthemoney at expiration is worthless. In other words, it does not matter how far outofthemoney the option is at expiration. A call option that is deepoutofthemoney is just as worthless as a call option that is barely outofthemoney. Increases in the volatility of the stock underlying the option contract increase the expected payoff on the option in the up state, which increases the value of the option. Thus, although increases in the volatility of the underlying stock imply that the stock is more likely to experience a large decrease in price, it is the corresponding increase in the probability of a large increase in the stock price that increases the value of a call option.
27. An investor has just obtained the following quotes for European options on a stock worth $30 when the threemonth riskfree interest rate is 10 percent per annum. Both options have a strike price of $30 and expire in three months. European Call: $3 European put: $13 A. Given the information above, determine whether the prices conform to the put—call parity rule. Put—call parity statesp = c — S + Xe^^{T} ~ ^{t}); however, $1.75 # $3  $30 + $30e~^{lu} ^{x} °^{.25 }$1.75 # $2.2593
B. If there is a violation, suggest a trading strategy that will generate riskless arbitrage profits. The synthetic put costs $2.2593, and the exchange traded put is trading at $1.75. To capture the potential profits from this arbitrage opportunity, we must simultaneously sell the synthetic put and purchase the traded put. Selling the synthetic put requires one to sell the call for $3, purchase the stock for $30, and sell the present value of $30 of the riskfree bond, $29.2593, resulting in a cash inflow of $2.2593. Purchasing the put will cost $1.75.
C. Indicate how much profit you will make from the arbitrage transactions, if such an opportunity exists. The profit from the arbitrage transactions will be the difference between the cash inflow from selling the synthetic put and the cash outflow from purchasing the traded put, $2.2593 — $1.75 = $.5093.
28. Suppose you are an information services professional with contracts throughout the industry. In conversations with colleagues, you get the feeling that Computer Associates International is likely to attempt to acquire Computer Sciences Corporation. You also remember from your finance courses in college that in the course of an acquisition, the share price of the target firm normally increases and the share price of the acquiring firm decreases. (You are confident that trading on the information gleaned in these conversations violates no law and is ethical.)
A. Discuss the advantages of using options to speculate on the expected stock price changes of the firms involved in an acquisition. There are several factors favoring the use of options to speculate on expected changes in the prices of stocks involved in an acquisition. It is possible to replicate the payoffs to a particular stock position with the appropriate positions in option contracts. That is, one can replicate the payoffs to a long stock position by holding a call option and selling a put with the same strike price and expiration date. Additionally, you can replicate this payoff with options at a lower cost than purchasing the stock itself. However, options pay off in only one direction, calls in the up state, puts in the down state. Thus, if one expected the price of a particular stock to increase, one could hold a long call position and participate in the upside gains in the stock with a limited investment. A long call position offers the same action as a long stock position at a lower cost. Chapter 13 shows that a long call option is equivalent to a levered stock position. Since call options are cheaper than the stock underlying the option, an investor with a fixed amount to invest can increase his leverage by purchasing several call options. Consider an investor with $10,000 available to speculate in a $100 stock that has a call trading at $10. The investor could purchase 100 shares of stock, 100 X $100 = $10,000, or he could purchase 10 call options [(10 contracts X 100 shares per contract) X $10 = $10,000], significantly increasing his exposure to the market. The fact that an investor faces lower transactions costs in the options market is an important factor motivating investors to trade options. Additionally, there may be tax implications of trading options that induce particular investors to trade options rather than trading the underlying stock. The tax factor will be unique to the investor and will depend on the individual's tax obligations. Because it is possible to create synthetic securities with options, it may be possible to avoid stock market restrictions by trading options. In particular, there are specific regulations regarding the short selling of stock that may be avoided by creating a synthetic short position using the Treasury bill and traded options.
B. Based on your understanding of option payouts, discuss and explain the option positions that you would establish to speculate on these expected price changes. (Assume that traded options exist for both firms.) Our expectations are the stock price of CSC will increase, and the stock price of CAI will decrease because of the expected acquisition. There are many strategies the investor could undertake to take advantage of the expected movements in the stock prices of CSC and CAI. One basic strategy would be to purchase call options written on CSC, and to purchase put options written on CAI. One could reduce the cost of this investment by writing puts on CSC, and writing calls on CAI. In addition, the investor must choose expiration dates and strike prices for the options. The decision regarding the maturity of the option contract will be influenced by expectations regarding the timing of the acquisition.
29. After watching a late night infomercial, a colleague comes to work professing the gospel of income enhancement via covered calls. The pitch man in the infomercial, Mr. Oracle, says that writing covered calls enhances the return on a stock investment with no cost. Discuss the sources of the apparent costless gains and the risks associated with writing covered calls. When writing an outofthemoney covered call, you contract to sell the stock you currently own at a price that is higher than the current stock price. As compensation for writing this contract, you receive premium income. The investor owns the underlying stock and is entitled to receive any distributions associated with owning the stock. These may include dividends, or additional shares of stock associated with a stock split or dividend. If the option is exercised against the investor, the investor sells the stock underlying the contract at the strike price. If the strike price is higher than the purchase price of the stock, then the investor will have earned a capital gain on the transaction. Thus, there are three potential sources of gains to the investor: premium income received from writing the call options, dividends, and capital gains from the sale of the stock. These potential returns do not, however, come without corresponding risk. By writing covered calls, the investor is exposed to an opportunity cost. If the option goes intothemoney, the potential gains associated with owning the stock will be offset by the obligations associated with the call option. Writing the covered call has reduced the upside potential of the long stock position. By writing the call option, the investor has given the owner of the call the right to purchase the underlying stock according to the terms of the option contract. It is the owner of the call who determines if and when an option is exercised. If the investor is holding the stock as a longterm investment, having a call exercised against him forces the investor to sell an asset that he wished to hold in his portfolio. If the investor desires to continue to hold the asset in his portfolio, he must return to the market to purchase the stock at the market price bearing all the transactions costs associated with reestablishing an existing position in the stock. Besides limiting the upside potential associated with owning a stock, a covered call position provides no protection against stock price decreases.
30. Late one Friday afternoon in March, an investor receives a call from her broker. Her broker tells her that the March options on Microsoft will expire in ten minutes, that Microsoft is currently trading at $94.50, and that the March 85 call is selling at $8. The investor tells her broker that he is mistaken and must have read his screen incorrectly. Explain which individual, the broker or the investor, is correct. Supposing that the broker is in fact correct, state the transactions that the investor would undertake to take advantage of this situation. Suppose the broker tells the investor that these options are European options. Discuss the impact of the fact that the options are European options on the actions of the investor. Microsoft options are traded on the CBOE, and the stock options traded on the CBOE are American options. This means that the March 85 MSFT call option must trade at a price that is at least the intrinsic value of the option prior to the option's expiration. The intrinsic value of the call is $9.50 and the option is selling for $8.00, according to the broker. This mispricing represents a classic arbitrage opportunity. To capture the arbitrage profit of $1.50, the investor will simultaneously sell the appropriate number of Microsoft shares and purchase the appropriate amount of the March 85 MSFT call option. The investor will receive $94.50 from the sale of the stock and will pay $8.00 for the call (all on a per share basis). The investor will then exercise the call option paying $85 per share for MSFT. Exercising the option provides the investor with the shares necessary to cover her obligation from selling the stock (short position). The total cash outflows are $93 ($85 + $8), and the cash inflows are $94.5, resulting in $1.50 in riskless profit for the investor. While an investor cannot exercise a European call until the call option expires, the call options in question will expire in 10 minutes. The call should be trading at a price that is above the intrinsic value of the option, $9.50, at this point in the life of the European call option. That is, in ten minutes the call will be inthemoney and be worth S — X. The investor will undertake the same arbitrage transactions to profit from this mispricing.
31. As a finance major in college, you were taught the efficient market hypothesis. Because you believe that it is not possible for a mutual fund manager to consistently outperform the market, you hold a portfolio of the 30 stocks that make up the Dow Jones Industrial Average (DJIA). Your broker has just called you with the following offer. He can provide you with insurance that will guarantee that the value of your portfolio will not fall below an indexed level of 8,900. This insurance is evaluated at the end of each quarter and costs $500 per quarter. A quick search of the CBOE web site shows that the Dow is currently trading at an indexed level of 9,005, and that a threemonth put option on the DJIA with a strike price of 89 is trading at 2. The DJIA options traded on the CBOE are quoted at strike prices that are j^ of the level of the DJIA. Each premium point is multiplied by $100 to determine the total cost of the option. A. Should you accept the insurance contract offered by your broker? Explain. It is possible to use the put option on the DJIA traded on the CBOE to create an insured portfolio. The cost of the traded put is less than the cost of the insurance offered by the broker. Therefore, a rational investor will use the option market to insure the value of his portfolio.
B. Explain how you can provide your own insurance for your portfolio. What is the cost of this insurance? What is the maximum loss on your insured portfolio? The insured portfolio is constructed by combining your long position in the Dow with a long position in the DJIA put option with a strike price of 89 and an expiration in three months (the strike price of 89 translates to a Dow level of 8,900). The cost of this traded option is $206.25 (2.0625 x $100). If the Dow drops below 8,900 at the end of three months, then the threemonth DJIA put option with a strike price of 89 will be inthemoney. The investor will exercise the option at expiration, and be paid cash equal to the difference between the strike price of 89 (DJIA level of 8,900) and the level of the market at the expiration of the option contract. This payoff will create a price floor for the portfolio at an index level of 8,900. The maximum loss on the portfolio would be 105 index points, 9,005  8,900.
32. You own shares of AGH that are currently trading at $100. A European put option written on AGH with a $100 strike price that expires in three months is priced at $4. The equivalent call option is priced at $5. A. What is the price of a threemonth Tbill that pays par ($100) implied by the prices given above? S = $100, X = $100, c = $5,p = U,Tt = 0.25 years Given these prices, the put—call parity relationship implies that the price of the synthetic Tbill is $99. cp = S Xe«? '\ $5  $4 = 100  X_{e}^{r{T}'\ Xe'^'^ = $99.
B. What is the continuously compounded threemonth interest rate implied by these prices? The continuously compounded threemonth interest rate implied by these prices is .0402. $99 = SlOCte"*^{25}' ln(99/100) = ln(^{f(}^{25)}) 0.01005 = r(0.25) r = 0.0402
C. A quick check of the price of threemonth Tbills on The Wall Street Journal web site reveals that this bill is trading at $98.50. Explain your actions upon finding this information. The synthetic bill is overpriced according to the prices given above. Thus, there is an arbitrage opportunity for the investor. To capture the $.50 profit implied by these prices, the investor should simultaneously sell the synthetic Treasury bill and purchase the traded Treasury bill. The synthetic bill position is created by constructing a portfolio that consists of a long position in the stock, a short position in the call, and a long position in the put, S — c + p = Xe ^{r<}^{T} ~ '\ To capture the profit opportunities, the investor must sell the stock short, buy the call, and sell the put resulting in a cash inflow of $99 (+$100  $5 + $4 = $99). As a result of these transactions, the investor has effectively borrowed at a low interest rate, the short synthetic bond position, and lent at a higher rate, the long traded Tbill position.
33. A protective put position is created by combining a long stock position with a long position in a put option written on the stock (S + P). Construct the equivalent position using call options, assuming all the options are European. An equivalent position can be created by combining a long position in a call option with an investment (long position) in the appropriate amount of a Treasury bill. The call and put option must be written on the same stock and have the same strike price and expiration date. The appropriate investment in the Tbill is determined by calculating the present value of the strike price of the put option used to create the protective put position, Xe~ ^{r(T} ~ ^{t)}.
34. TMS is currently trading at $40, which you think is above its true value. Given your knowledge of the firm, its products, and its markets, you believe that $35 per share is a more appropriate price for TMS. One means of purchasing TMS at $35 per share would be to write a limit order. A limit order is an order directed to a broker on the floor of the exchange that specifies, among other things, the purchase price of the stock. You also notice that TMS has traded options and that a $40 put option is selling for $5. An alternative investment strategy involves purchasing TMS for $40 and writing a TMS put option for $5.
A. If at the expiration of the option, TMS's stock is trading at a price above $40 per share, discuss the benefits of the buy stock/sell put investment strategy. When TMS is trading above $40 per share at the put's expiration, the put option that you have written is worthless. In this environment, you will have effectively purchased TMS for $35 per share, $40 to purchase TMS in the spot market less $5 received from writing the $40 put option.
B. If at the expiration of the option, TMS's stock is trading at a price below $40 per share, discuss the benefits of this alternative investment strategy. When TMS is trading below $40 per share at expiration, the short put option position is inthemoney and will be exercised against the investor. In this environment, the impact of the lower stock price on the cost of your position will vary linearly with the stock price. For prices between $35 and $40, the $5 premium income received from writing the put option will offset losses on the option position dollar for dollar. That is, at $39, the $1 loss on the option position will be offset by the $5 premium income resulting in an effective purchase price of $36 for a share of TMS. At stock prices lower than $35 per share, the losses from the option position will offset the gains from the premium income received resulting in an effective purchase price greater than $40 per share.
35. When you purchase property insurance, you must choose the dollar amount of the deductible on the policy. Similarly, when you construct a protective put position, you must choose the strike price on the put option. Discuss the similarities between the choice of the deductible on an insurance policy and the selection of a strike price for the protective put position. When selecting the deductible on the insurance policy, the investor is deciding the dollar amount of losses that she is willing to selfinsure. That is, the choice of the deductible determines how much of a loss must be covered by the resources of the insured prior to any payment by the insurer. In general, the greater the risk that is selfinsured, the lower the cost of the insurance. A protective put position is constructed to limit an investor's exposure to losses arising from decreases in the price of the stock held by the investor. The investor must choose the strike price on the put option when constructing the position. If the investor chooses a strike price that is well below the current stock price, the investor is bearing more risk from holding the stock, and the price of the insurance will be less. That is, the investor has selfinsured more of the risk of a stock price decline. Therefore, the price of a deepoutofthemoney put should be less than the price of a nearthemoney put.
36. A client has recently sold a stock short for $100. The short sale agreement requires him to cover the short position in one month. The client wants to protect his position against stock price increases. You notice that the stock has puts and calls traded on the CBOE. Explain how your client can use either the put or call option to construct a portfolio that will limit his risk exposure to stock price increases. Compare the two alternative strategies. In this situation, increases in the stock price represent a risk to the investor. If at the end of the month the stock is selling above $100, the investor will have to cover the position and will incur a loss. In this hedging problem, the investor needs to construct a portfolio such that increases in the stock price will result in an increase in the value of the hedging instrument that can offset the losses incurred on the short stock position. The first strategy would be to combine the short stock position with a long call position. Increases in the price of the stock will increase the value of the call option. The increases in the value of the call option position will offset losses in the short stock position. One decision the investor must make is the selection of the strike price of the option—that is, the selection of the amount of loss (risk) the investor is willing to bear. The price of a call option with a $110 strike price will be less than the price of a call with a $100 exercise price. The investor would choose to hold a call that expired in one month (the same time until the short position must be covered). For example, assume the investor held a $100 call that expired in one month. If the stock price at the end of the month was $105, then the investor would exercise the call and purchase the stock at $100 to cover the short position. The cost of the protection would be the price of the call option. The other strategy would combine a short put position with the short stock position. In this case, the investor would use the premium income received from writing the put to cover any losses associated with stock price increases. As happens with a covered call position, this premium income would provide limited protection against adverse stock price movements, and would limit potential gains associated with stock price decreases. For example, assume the investor wrote a $100 put that expired in one month. If the stock price at the end of the month was $95, then the owner of the put would exercise the put against your client, forcing your client to purchase the stock at $100. This loss of $5 on the option contract will offset the $5 gain in the value of the short stock position. At prices above $100, the put will expire worthless and the premium income will offset some of the losses in the short stock position
Bounds on Options Prices
1. What is the maximum theoretical value for a call? Under what conditions does a call reach this maximum value? Explain. The highest price theoretically possible for a call option is to equal the value of the underlying stock. This happens only for a call option that has a zero exercise price and an infinite time until expiration. With such a call, the option can be instantaneously and costlessly exchanged for the stock at any time. Therefore, the call must have at least the value of the stock itself. Yet it cannot be worth more than the stock, because the option merely gives access to the stock itself. As a consequence, the call must have the same price as the stock.
2. What is the maximum theoretical value for an American put? When does it reach this maximum? Explain. The maximum value of a put equals the potential inflow of the exercise price minus the associated outflow of the stock price. The maximum value for this quantity occurs when the stock price is zero. At that time, the value of the put will equal the exercise price. In this situation, the put gives immediate potential access to the exercise price because it is an American option.
3. Answer question 2 for a European put. As with the American put, the European put attains its maximum value when the stock price is zero. However, before expiration, the put cannot be exercised. Therefore, the maximum price for a European put is the present value of the exercise price, when the exercise price is discounted at the riskfree rate from expiration to the present. This discounting reflects the fact that the owner of a European put cannot exercise now and collect the exercise price. Instead, he or she must wait until the option expires.
4. Explain the difference in the theoretical maximum values for an American and a European put. The exercise value of a put option equals the exercise price (an inflow) minus the value of the stock at the time of exercise (an outflow). In our notation, this exercise value is X — S. For any put, the maximum value occurs when the stock is worthless, S = 0. The American and European puts have different maximum theoretical values because of the different rules governing early exercise. Because an American put can be exercised at any time, its maximum theoretical value equals the exercise price, X. If the stock price is zero at any time, an American put gives its owner immediate access to amount X through exercise. This is not true of a European put, which can be exercised only at expiration. If the option has time remaining until expiration and the stock is worthless, the European put holder must wait until expiration to exercise. With the stock worthless, the exercise will yield X to the European put holder. Because the exercise must wait until expiration, however, the put can be worth only the present value of the exercise price. Thus, the theoretical maximum value of a European put is Xe ^{r<}^{T} ^{)}. In the special case of an option at expiration, t = 0, the maximum value for a European and an American put is X.
5. How does the exercise price affect the price of a call? Explain. The call price varies inversely with the exercise price. The exercise price is a potential liability that the call owner faces, because the call owner must pay the exercise price in order to exercise. The smaller this potential liability, other factors held constant, the greater will be the value of a call option.
6. Consider two calls with the same time to expiration that are written on the same underlying stock. Call 1 trades for $7 and has an exercise price of $100. Call 2 has an exercise price of $95. What is the maximum price that Call 2 can have? Explain. A noarbitrage condition places an upper bound on the value of Call 2. The price of Call 2 cannot exceed the price of the option with the higher exercise price plus the $5 difference in the two exercise prices. Thus, the upper bound for the value of Call 2 is $12. If Call 2 is priced above $12, say, at $13, the following arbitrage becomes available. Sell Call 2 for cash flow +$13 and buy Call 1 for cash flow $7. This is a net cash inflow of +$6. If Call 2 is exercised against you, you can immediately exercise Call 1. This provides the stock to meet the exercise of Call 1 against you. On the double exercise, you receive $95 and pay $100, for a net cash flow of —$5. However, you received $6 at the time of trading for a net profit of $1. This is the worst case outcome. If Call 1 cannot be exercised, the profit is the full $6 original cash flow from the two trades. Also, if the stock price lies between $95 and $100 when Call 1 is exercised against you, it may be optimal to purchase the stock in the market rather than exercise Call 2 to secure the stock. For example, assume the stock trades for $98 when Call 1 is exercised against you. In this case, you buy the stock for $98 instead of exercising Call 2 and paying $100. Then your total cash flows are +$6 from the two trades, +$95 when Call 1 is exercised against you, and —$98 from purchasing the stock to meet the exercise. Now your net arbitrage profit is $3. In summary, stock prices of $95 or below give a net profit of $6, because Call 1 cannot be exercised. Stock prices of $100 or above give a net profit of $1, because you will need to exercise Call 2 to meet the exercise of Call 1. Prices between $95 and $100 give a profit equal to +$6 +$95  stock price at the time of exercise.
7. Six months remain until a call option expires. The stock price is $70 and the exercise price is $65. The option price is $5. What does this imply about the interest rate? We know from the noarbitrage arguments that: C> S — Xe ^{r<}^{T} ^{l}\ In this case, we have C = S — Xexactly. Therefore, the interest rate must be zero.
8. Assume the interest rate is 12 percent and four months remain until an option expires. The exercise price of the option is $70 and the stock that underlies the option is worth $80. What is the minimum value the option can have based on the noarbitrage conditions studied in this chapter? Explain. We know from the noarbitrage arguments that: C>5 —Xe^{r<}^{Tt}\ Substituting the specified values gives C> $80  $70^^{12}(°^{33}> = $80  $67.28 = $12.72. Therefore, the call price must equal or exceed $12.72 to avoid arbitrage.
9. Two call options are written on the same stock that trades for $70, and both calls have an exercise price of $85. Call 1 expires in six months, and Call 2 expires in three months. Assume that Call 1 trades for $6 and that Call 2 trades for $7. Do these prices allow arbitrage? Explain. If they do permit arbitrage, explain the arbitrage transactions. Here we have two calls that are identical except for their time to expiration. In this situation, the call with the longer time until expiration must have a price equal to or exceeding the price of the shorterlived option. These values violate this condition, so arbitrage is possible as follows: Sell Call 2 and buy Call 1 for a net cash inflow of $1. If Call 2 is exercised at any time, the trader can exercise Call 1 and meet the exercise obligation for a net zero cash flow. This retains the $1 profit no matter what happens. It may also occur that the profit exceeds $1. For example, assume that Call 2 cannot be exercised in the first three months and expires worthless. This leaves the trader with the $1 initial cash inflow plus a call option with a threemonth life, so the trader has an arbitrage profit of at least $1, and perhaps much more.
10. Explain the circumstances that make early exercise of a call rational. Under what circumstances is early exercise of a call irrational? Exercising a call before expiration discards the time value of the option. If the underlying stock pays a dividend, it can be rational to discard the time value to capture the dividend. If there is no dividend, it will always be irrational to exercise a call, because the trader can always sell the call in the market instead. Exercising a call on a nodividend stock discards the time value, while selling the option in the market retains it. Thus, only the presence of a dividend can justify early exercise. Even in this case, the dividend must be large enough to warrant the sacrifice of the time value.
11. Consider a European and an American call with the same expiration and the same exercise price that are written on the same stock. What relationship must hold between their prices? Explain. Because the American option gives every benefit that the European option does, the price of the American option must be at least as great as that of the European option. The right of early exercise inherent in the American option can give extra value if a dividend payment is possible before the common expiration date. Thus, if there is no dividend to consider, the two prices will be the same. If a dividend is possible before expiration, the price of the American call may exceed that of the European call.
12. Before exercise, what is the minimum value of an American put? The minimum value of an American put must equal its value for immediate exercise, which is X  S. A lower price results in arbitrage. For example, assume X = $100, S = $90, and P = $8. To exploit the arbitrage inherent in these prices, buy the put and exercise for a net cash outflow of$98. Sell the stock for +$100 for an arbitrage profit of $2.
13. Before exercise, what is the minimum value of a European put? For a put, the exercise value is X — S. However, a European put can be exercised only at expiration. Therefore, the present value of the exercise value is Xe^^{T} ^ — S, and this is the minimum price of a European put. For example, consider X = $100, S = $90, T — t = 0.5 years, and r = 0.10. The noarbitrage condition implies the put should be worth at least $5.13. Assume that the put actually trades for $5. With these prices, an arbitrageur could trade as follows. Borrow $95 at 10 percent for six months and buy the stock and the put. This gives an initial net zero cash flow. At expiration, the profit depends upon the price of the stock. First, there will be a debt to pay of $99.87 in all cases. If the stock price is $100 or above, the put is worthless and the profit equals S — $99.87. Thus, the profit will be at least $.13, and possibly much more. For stock prices below $100, exercise of the put yields $100, which is enough to pay the debt of $99.87 and keep $.13 profit.
14. Explain the differences in the minimum values of American and European puts before expiration. The difference in minimum values for American and European puts stems from the restrictions on exercising a put. An American put offers the immediate access to the exercise value X if the put owner chooses to exercise. Because the European put cannot be exercised until expiration, the cash inflow associated with exercise must be discounted to Xe ^{r<}^{T} ^{)}. The difference in minimum values equals the time value of the exercise price.
15. How does the price of an American put vary with time until expiration? Explain. The value of an American put increases with the time until expiration. A longerlived put offers every advantage that the shorterlived put does. Therefore, a longerlived put must be worth at least as much as the shorterlived put. This implies that value increases with time until expiration. Violation of this condition leads to arbitrage.
16. What relationship holds between time until expiration and the price of a European put? For a European put, the value may or may not increase with time until expiration. Upon exercise, the put holder receives X  S. If the European put holder cannot exercise immediately, the inflow represented by the exercise price is deferred. For this reason, the value of a European put can be lower the longer the time until expiration. However, having a longer term until expiration also adds value to a put, because it allows more time for something beneficial to happen to the stock price. Thus, the net effect of time until expiration depends on these two opposing forces. Under some circumstances, the value of a European put will increase as time until expiration increases, but it will not always do so.
17. Consider two puts with the same term to expiration (six months). One put has an exercise price of $110, and the other has an exercise price of $100. Assume the interest rate is 12 percent. What is the maximum price difference between the two puts if they are European? If they are American? Explain the difference, if any. For two European puts, the price differential cannot exceed the difference in the present value of the exercise prices. With our data, the difference cannot exceed ($110  $100)^{12(0} ^{5)} = $9.42. If the price differential on the European puts exceeds $9.42, we have an arbitrage opportunity. To capture the arbitrage profit, we sell the relatively overpriced put with the exercise price of $110 and buy the put with the $100 exercise price. If the put we sold is exercised against us, we accept the stock and dispose of it by exercising the put we bought. This will always guarantee a profit. For example, assume that the put with X = $100 trades for $5 and the put with X = $110 sells for $15, giving a $10 differential. We sell the put with X = $110 and buy the put with X = $100, for a net inflow of $10. We invest this until expiration, at which time it will be worth $10^ '^{12}('^{5}) = $10.62. If the put we sold is exercised against us, we pay $110 and receive the stock. We can then exercise our put to dispose of the stock and receive $10. This gives a $10 loss on the double exercise. However, our maturing bond is worth $10.62, so we still have a profit of $.62. For two American puts, the price differential cannot exceed the difference in the exercise prices. If it does, we conduct the same arbitrage. However, we do not have to worry about the discounted value of the differential, because the American puts carry the opportunity to exercise immediately and to gain access to the value of the mispricing at any time, not just at expiration.
18. How does the price of a call vary with interest rates? Explain. For a call, the price increases with interest rates. The easiest way to see this is to consider the noarbitrage condition: C > S  Xe~^^{T} '\ The higher the interest rate, the smaller will be the present value of the exercise price, a potential liability. With extremely high interest rates, the exercise price will have an insignificant present value and the call price will approach the stock price.
19. Explain how a put price varies with interest rates. Does the relationship vary for European and American puts? Explain. Put prices vary inversely with interest rates. This holds true for both American and European puts. For the put owner, the exercise price is a potential inflow. The present value of this inflow, and the market value of the put, increases as the interest rate falls. Therefore, put prices rise as interest rates fall.
20. What is the relationship between the risk of the underlying stock and the call price? Explain in intuitive terms. Call prices rise as the riskiness of the underlying stock increases. A call option embodies insurance against extremely bad outcomes. Insurance is more valuable the greater the risk it insures against. Therefore, if the underlying stock is very risky, the insurance embedded in the call is more valuable. As a consequence, call prices vary directly with the risk of the underlying good.
21. A stock is priced at $50, and the riskfree rate of interest is 10 percent. A European call and a European put on this stock both have exercise prices of $40 and expire in six months. What is the difference between the call and put prices? (Assume continuous compounding.) From the information supplied in this question, can you say what the call and put prices must be? If not, explain what information is lacking From putcall parity, S  Xe ^{r<}^{T} ) = C  P. Therefore, the C P must equal: $50 _ $40e°.^{5}(°.^{1}) = $11.95 We cannot determine the two option prices. Information about how the stock price might move is lacking.
22. A stock is priced at $50, and the riskfree rate of interest is 10 percent. A European call and a European put on this stock both have exercise prices of $40 and expire in six months. Assume that the call price exceeds the put price by $7. Does this represent an arbitrage opportunity? If so, explain why and state the transactions you would make to take advantage of the pricing discrepancy. From question 21, we saw that the call price must exceed the put price by $11.95 according to putcall parity. Therefore, if the difference is only $7, there is an arbitrage opportunity, and the call price is cheap relative to the put. The long call/short put position is supposed to be worth the same as the long stock/short bond position. But the long call/short put portfolio costs only $7, not the theoretically required $11.95. To perform the arbitrage, we would buy the relatively underpriced portfolio and sell the relatively overpriced portfolio. Specifically, we would: buy the call, sell the put, sell the stock, and buy the riskfree bond that pays the exercise price in six months. From these transactions, we would have the following cash flows: from buying the call and selling the put —$7; from selling the stock +$50, and from buying the riskfree bond —$38.05, for a net cash flow of $4.95. This net cash flow exactly equals the pricing discrepancy. At expiration, we can fulfill all of our obligations with no further cash flows. If the stock price is below the exercise price, the put we sold will be exercised against us and we must pay $40 and receive the stock. We will have the $40 from the maturing bond, and we use the stock that we receive to repay our short sale on the stock. If the stock price exceeds $40, we exercise our call and use the $40 proceeds to pay the exercise price. We then fulfill our short sale by returning the share.
23. Cursory examination of the table below shows a violation of a basic option pricing rule. Which pricing rule has been violated? Discuss the limitations of using a newspaper as the source of prices used to make inferences about pricing
Option IBM

Strike

Exp.

Call



Put


Vol.

Last

Vol.

Last


1011

115

Sep

1632

3 16

10

131


1011

115

Oc

861


10

141


1011

115

Jan

225

4

3

14

The pricing rule violated is the rule that the price of an American put option increases as the maturity of the option increases. That is, the longer the time until expiration, the more valuable a put option. The only characteristic that varies across the put options in the table is the time until expiration. This is a classic problem of stale prices. The information reported in the paper is historical information. Examining the reported trading volume reveals the likely source of this apparent problem. Very few put options were traded that day. In fact, only 3 Jan IBM 115 put options were traded on the day reported in the table. It is highly likely that the prices reported in the table represent prices recorded at different times during the day. To take advantage of mispricings, one needs to know the prices of all the relevant options at the same point in time.
24. Explain why an American call is always worth at least as much as its intrinsic value. Explain why this is not true for European calls. The owner of an American call can exercise the option anytime over the life of the option and capture the option's intrinsic value (S — X). Therefore, the price of the American call can never be less that the intrinsic value of the option. The owner of a European call cannot exercise the option until the option's expiration. If the option is inthemoney, the investor cannot capture this gain until the option's expiration; therefore, it is possible for the value of a European call to be less than the intrinsic value of the option. This is likely to happen if the option is deepinthemoney and has a long time until expiration.
25. Give an intuitive explanation of why the early exercise of an inthemoney American put option becomes more attractive as the volatility of the underlying stock decreases, and the riskfree interest rate increases. If the stock's volatility decreases, then the probability of a very large change in the stock price decreases. It is a large stock price move that will take the put deeperintothemoney, making the put more valuable. If large price decreases are not very likely, then the probability of a large payoff from holding the put is very small and the value of the put will not increase. If at the same time that volatility is increasing the riskfree interest rate is increasing, then the opportunity cost of holding the put will increase. If the put were exercised, the exercise proceeds could be invested at the new higher riskfree rate.
Portfolio 
Current Value of Portfolio 

X1 


X_{2} < S < X_{3} 
S> X_{3} 

A B Relative value of the portfolio at expiration 
2c_{2} ^{c}1 + ^{c}3 
0 0 


0 S X_{1} 14 > i/, 
2(S X2) 
2( 
Since the value of portfolio B is never less than the value of portfolio A at the expiration of the options, then the current value of portfolio B must be at least as large as the current value of portfolio A. If this were not the case, arbitrage would be possible. Thus, c1 + c_{3} — 2 c_{2} > 0 and c_{2} < 0.5 X (c1 + c3).
27. Suppose that p_{h} p_{2}, and p_{3} are the prices of three European put options written on the same share of stock that are identical in all respects except their strike prices. The strike prices for the three put options are X_{1 }X_{2}, and X_{3}, respectively, where X_{3}> X_{2}> X1 and X_{3}  X_{2} = X_{2}  X1. All options have the same maturity. Assume that all portfolios are held to expiration. Show that: Consider two portfolios, A and B. Portfolio A consists of a long position in two put options with a strike price of X_{2}. Portfolio B consists of a long position in one put option with a strike price X1 plus a long position in one put option with a strike price of X_{3}. Consider the payoffs that are possible at the option's expiration. Remember that there is a fixed dollar difference between the strike prices of the options, for example,X_{1} = 50, X_{2} = 55, and X_{3} = 60 Portfolio Current Value of Portfolio S < X_{1} X_{1} < S < X_{2} X_{2} < S < X_{3} S > X_{3 }A 2 p_{2} 2(X_{2} S) 2(X_{2} S) 0 0 B p1 + p3 (X1  S) + X_{3} S X_{3} S 0 Relative value of the portfolio at expiration V_{A} = V_{B} V_{B}> V_{A} V_{B} >V_{A} V_{A} = V Because the value of portfolio B is never less than the value of portfolio A at the expiration of the options, the current value of portfolio B must be at least as large as the current value of portfolio A. If this were not the case, arbitrage would be possible. Thus, p_{1} + p_{3}  2p_{2} ^ 0 andp_{2} =£ 0.5 X (p1 + p_{3}).
European Option Pricing


Therefore, C = 0.0556($120)  $5.03 = $1.64. The probability of a stock price increase is: (R  D)/{UD) = (1.06  0.8)/(1.1  0.8) = 0.8667
4. Based on the data in question 3, what would you hold to form a riskfree portfolio? Because C = N*S  B*, the portfolio of C  N*S + B* should be a riskless portfolio.
5. Based on the data in question 3, what will the price of the call option be if the option expires in two years and the stock price can move up 10 percent or down 20 percent in each year? Terminal stock prices in two periods are given as follows: UUS = $145.20, DDS = $76.80, and UDS = DUS = $105.60. The probabilities of these different terminal stock prices are: ir_{m} = (0.8667) (0.8667) = 0.7512; TT_{ai}= (0.8667)(0.1333) = 0.1155; ir_{ia} = (0.1333)(0.8667) = 0.1155; and n_{M} = (0.1333)(0.1333) = 0.0178. The call price at expiration equals the terminal stock price minus the exercise price of $100, or zero, whichever is larger. Therefore, we have C_{m} = $15.20, C^ = 0, C_{du} = C_{ud} = 0. We have already found that the probability of an increase is 0.8667, so the probability of a down movement is 0.1333. Because the option pays off only with two increases, we need consider only that path. Thus, the value of the call is: C = t^uuCJR^{1} = (0.7512) ($15.20)/(1.06)^{2} = $10.16
6. Based on the data in question 3, what would the price of a call with one year to expiration be if the call has an exercise price of $135? Can you answer this question without making the full calculations? Explain. From question 3, we see that US = $132. This is not enough to bring the call intothemoney. Therefore, we know that the call must expire worthless, so its current price is zero.
7. A stock is worth $60 today. In a year, the stock price can rise or fall by 15 percent. If the interest rate is 6 percent, what is the price of a call option that expires in three years and has an exercise price of $70? What is the price of a put option that expires in three years and has an exercise price of $65? (Use OPTION! to solve this problem.) The call is worth $6.12 and the put is worth $3.04. The two trees from OPTION! are shown here:
8. Consider our model of stock price movements given in Equation 13.8. A stock has an initial price of $55 and an expected growth rate of 0.15 per year. The annualized standard deviation of the stock's return is 0.4. What is the expected stock price after 175 days? Substituting values for our problem, and realizing that the expected value of a drawing for a N(0,1) distribution is zero, gives: S_{t+1}S_{t} = $55 (0.15) (175/365) = $3.96 Adding this amount to the initial stock price of $55 gives $58.96 as the expected stock price in 175 days.
9. A stock sells for $110. A call option on the stock has an exercise price of $105 and expires in 43 days. If the interest rate is 0.11 and the standard deviation of the stock's returns is 0.25, what is the price of the call according to the BlackScholes model? What would be the price of a put with an exercise price of $140 and the same time until expiration?
105J

[.11 + .5(.25)(.25)]

L365J

= .7361

The Black—Scholes put pricing model is: p_{t} = Xe^{MT}^ N(d_{2})  S_{t} N(d1) N(2.7035) = 0.996569 and N(2.6177) = 0.995574. Therefore, p_{t} = $140e°^{.}11^{(43/365)}(0.996569)  $110(0.995574) = $28.21
10. Consider a stock that trades for $75. A put and a call on this stock both have an exercise price of $70 and they expire in 150 days. If the riskfree rate is 9 percent and the standard deviation for the stock is 0.35, compute the price of the options according to the BlackScholes model.
d_{2} = d1 5845  .35 A = .3601 N(0.5845) = 0.720558, N(0.3601) = 0.640614, N(0.5845) = 0.279442, and N(0.3601) = 0.359386. c_{t} = $75(0.720558)  $70_{°09(150/365) (0.640614) = $54.04  $43.21 = $10.83 pt = $70°09(150/365) (0.359386)  $75 (0.279442) = $24.24  $20.96 = $3.29 }dM = df aVt = .5113  .35 / = .2869 N(0.5113) = 0.695430, N(0.2869) = 0.304570, N(0.5113) = 0.612906, and N(0.2869) = 0.387094.
2. Consider a Treasury bill with 173 days until maturity. The bid and asked yields on the bill are 9.43 and 9.37. What is the price of the Tbill? What is the continuously compounded rate on the bill? Therefore, the price of the bill is 95.48 percent of par. To fi e^{r{T}'^{)}=\/PTB r = 0.0962 Thus, the continuously compounded rate on the bill is 9.62 percent.
13. Consider the following sequence of daily stock prices: $47, $49, $46, $45, $51. Compute the mean daily logarithmic return for this share. What is the daily standard deviation of returns? What is the annualized standard deviation? Let P_{t} = the price on day t, PR_{t} = P_{t}/P_{t}  1, PR^ = the mean daily logarithmic return, and er = the standard deviation of the daily logarithmic return. Then, PR_{t} ln(PR_{t}) [InPff,
1.0426 0.0417 .000454
0.9388  0.0632 .006989
0.9783  0.0219 .001789
1.1333 0.1251 .010962
PR^ = (0.0417  0.0632  0.0219 + 0.1251)/4 = 0.0204 VAR(PR) = (1/3) (0.000454 + .006989 + .001789 + .010962) = 0.006731 er = the square root of 0.006731 = 0.082045. The annualized a = a times the square root of 250 = 1.2972.
14. A stock sells for $85. A call option with an exercise price of $80 expires in 53 days and sells for $8. The riskfree interest rate is 11 percent. What is the implied standard deviation for the stock? (Use OPTION! to solve this problem.) er = 0.332383. It is also possible to find this value by repeated application of the Black—Scholes formula. For example, with this option data, different trial values of er give the following sequence of prices:
Call Price 
Trial Value of 
er is: 



$6.29 
.1 
too low 


9.84 
.5 
too high 


7.67 
.3 
too low 


8.72 
.4 
too high 


8.18 
.35 
too high 


7.98 
.33 
too low 


8.03 
.335 
too high 


8.01 
.333 
too high 


8.00 
.332 
very close 
Note that 0.5 (1.09) + 0.5 (0.91) = 1.0, so the expected return on the stock is zero. The expected return on the stock must equal the riskfree rate in the riskneutral setting of the binomial model. Therefore, these up and down factors imply a zero interest rate.
16. For a stock that trades at $120 and has a standard deviation of returns of 0.4, use the Black—Scholes model to price a call and a put that expire in 180 days and that have an exercise price of $100. The riskfree rate is 8 percent. Now assume that the stock will pay a dividend of $3 on day 75. Apply the known dividend adjustment to the Black—Scholes model and compute new call and put prices. With no dividends, the call price is $27.54, and the put price is $3.67. With the known dividend adjustment, the call price is $25.14, and the put price is $4.23.
17. A call and a put expire in 150 days and have an exercise price of $100. The underlying stock is worth $95 and has a standard deviation of 0.25. The riskfree rate is 11 percent. Use a threeperiod binomial model and stock price movements consistent with the Black—Scholes model to compute the value of these options. Specify U, D, and tt„, as well as the values for the call and put. The call price is $5.80, and the put price is $6.38. U = 1.0969, and D = 0.9116.
18. For the situation in problem 17, assume that the stock will pay 2 percent of its value as a dividend on day 80. Compute the value of the call and the put under this circumstance. Recalling that these are European options, the call is worth $4.66, and the put is worth $7.14.
21. Consider the second tree in Figures 13.10 and 13.11. If the stock price increases in the first period, the price is $88.35. For the second tree in Figure 13.11, the call price is $12.94 in this case. Given that the exercise price on the put is $75, does this present a contradiction? Explain. One of the arbitrage conditions we have considered says that the call price must equal or exceed S — X. In this situation, S  X = $88.35  $75.00 = $13.35, which is greater than the call price of $12.94. Thus, it appears that an arbitrage opportunity exists. The apparent contradiction dissolves when we realize that the call price reflects the dividend that will occur before the option can be exercised.
22. As a costcutting measure, your CFO, an accountant, decides to cancel your division's subscription to Bloomberg. You rely on Bloomberg for realtime quotes on Treasury bill prices to value options using the Black—Scholes option pricing model. You discover that you can obtain realtime quotes on commercial paper rates from Reuters, to which you still have a subscription. Discuss the implications of using the yield on AAA rated commercial paper instead of Treasury bill yields to value options using the Black—Scholes model. It is assumed that the debt of the United States government is free of default risk. The yield on commercial paper, which is an unsecured debt obligation of a corporation, includes a default risk premium. Thus, the return on AAA rated commercial paper, r_{cp}, will be greater than the return on an equivalent Treasury bill, ^{r}Tbill. The difference in the two returns, r_{c}p — rrbiib '^{s} ^{equa}l ^{to} the default risk premium. Using the commercial paper rate instead of the Treasury bill rate will lead to the systematic overvaluation of call options, and the undervaluation of put options. The magnitude of the pricing error will be a function of the magnitude of the de fault risk premium. However, Chapter 14 shows that option prices are not very sensitive to changes in interest rates. Consequently using the commercial paper rate rather than the Treasury bill rate should not produce significant differences in the prices of options valued with the BlackScholes model.
23. Assume that stock returns follow a random walk with a drift equal to the expected return on the stock. You are modeling stock returns using a binomial process for the purpose of valuing a European call option. Explain why creating an initial position in the stock and the call option that will remain riskless for the entire life of the option is not possible. Valuing options using the binomial model requires the construction of a synthetic option. The synthetic option is nothing more than a levered stock position. That is, a portfolio with an appropriate investment in the stock underlying the option, N*, and Treasury bills, B*. The synthetic option is combined with the traded option to create a riskless hedge portfolio. When the price of the underlying stock changes, the value of the traded option changes. Maintaining the riskless hedge requires periodic rebalancing of the synthetic option position as the value of the underlying stock changes. That is, one must alter one's position in the underlying stock and Treasury bill as the value of the traded option changes in concert with changes in the price of the stock.
24. WMM is currently priced at $117.50 per share. The 50day options on WMM are currently being traded at three different strike prices, $110, $115, and $120. The 50day Treasury bill is priced to yield an adjusted annual return of 6 percent compounded continuously. The prices and implied volatilities for the three different
If the WMM options are priced by the BlackScholes model, then the implied volatility of each of the 50day option contracts would be the same.
A. Which of the three implied volatilities would you use as an estimate for the true volatility? The recommended practice in the literature is to calculate a weighted average of several estimates of implied volatility. Many factors will influence the decision regarding the appropriate weighting scheme used in calculating a weighted implied volatility, including the trading volume for a particular option. Most weighting systems assign greater weights to options nearthemoney.
27. We generally assume that the price of a share of stock decreases by the dollar amount of a dividend on the day when the stock goes exdividend, at least approximately. The exdividend date is the date on which the purchaser of a share of stock is not entitled to the next dividend paid by the firm. That is, the stock does not carry the right to the next dividend. Develop an arbitrage based argument why in a competitive market without frictions the price of a stock must fall by the dollar amount of the dividend on the day the stock goes exdividend. The price of a stock the instant before it goes exdividend, S_{h} must be equal to the price of the stock the instant after it goes exdividend, S_{a}, plus the dollar value of the dividend, D. If the price of the stock before it goes exdividend, S_{h} is greater than the sum of the price of the stock the instant after it goes exdividend, S_{a}, and the dollar value of the dividend, D, then it is possible to profit by selling the stock before it goes exdividend and purchasing the stock after it goes exdividend. Selling the stock obligates the seller to deliver a share of stock for which they are paid, $Sb. The seller must deliver the stock and the dollar value of the dividend, D. However, the cost of the stock when purchased after the exdividend date, S_{a}, plus the dollar amount of the dividend, D, is less than the selling price, S_{h} leaving the investor with a profit of Sb ~ (Sa + D)>0. If the price of the stock before it goes exdividend, Sb, is less than the sum of the price of the stock the instant after it goes exdividend, S_{a}, and the dollar value of the dividend, D, then it is possible to profit by buying the stock before it goes exdividend and selling the stock after it goes exdividend. Purchasing the share of stock entitles the investor to the next dividend to be paid to shareholders. Selling the stock obligates the seller to deliver a share of stock for which they are paid S_{a}. The purchased stock will be used to deliver against the obligation created by selling the stock. The cost of the stock, Sb, is less than the sum of the cash received from selling the stock, S_{a}, and the dividend paid to the owners of the stock, (S_{a} + D) — S/,>0. Thus, only stock prices that are consistent with S_{b} — D = S_{a} will prevent arbitrage.
28. A quick check of the wire reveals that TMS is trading at $50 per share. Earlier in the day you were having lunch with a colleague and her husband Will, the CFO of TMS. During the lunch discussion, you talked about TMS's recently introduced new products. Will made it clear that if the products are well received by the market, TMS will be trading at $60 in six months, and if the market does not respond to the products, TMS will be trading at $42 in six months. The current sixmonth riskfree interest rate is 6 percent. Calculate the price of a sixmonth European call option written on TMS with a $50 strike price. Show that the price calculated using the oneperiod riskneutral pricing model, c = (cjj^u + Cd^d)^, is the same as the price calculated using the singleperiod noarbitrage binomial pricing model, c = N*S — B*.The first step in determining the value of this call option is to examine what is expected to happen to the price of TMS's stock over the next six months. The value of the $50 call option will depend on the movements in the price of TMS's stock. At the end of six months, two outcomes are possible. TMS's stock price will either rise to $60 with the introduction of successful products or decrease to $42 if the new products are not successful. Based on these stock prices, we can then calculate the value of the call option at expiration. The value of the call option at expiration is given by the intrinsic value function, MAX [0, S — X]. With successful products, the stock price will be $60, generating an intrinsic value of $10, c_{u} = MAX [0, $60  $50]. With unsuccessful products, the stock price will be $42, generating an intrinsic value of $0, c_{d} = MAX [0, $42  $50].
Singleperiod binomial model t/= $60/$50 = 1.2 D = $42/$50 = 0.84 # = 1.06 c_{u}Dc_{D}U_{=} $10(0.84) $0(1.2) _{=} $8.4 ^{=} (UD)R ^{=} (1.20.84)1.06 0.3816 * (UD)S (1.2 0.84)$50 18 c = N*S B* = 0.5556($50)  $22.01 = $5.77 Singleperiod riskneutral pricing model, c =(c_{U}'nu + c_{d}tt_{d})/R _ RD _{D} 1.06 0.84 _{U} _UR_ 1.2 1.06 __{n}™ , ^"IT^D 1.20.84 ~°^{61} ^ ~ "fT^D ~ 1.20.84 ~°^{39}' Tu+VdI The difference in the calculated values of the options is due to rounding error.
29. You are paired with the president of NYB to play golf in a tournament to raise money for the local children's hospital. After playing golf in the tournament, you learn that the president of NYB expects the price of his firm to increase by 6 percent per quarter if their new stores are successful in Seattle. If the stores are unsuccessful, he expects the stock price of NYB to decrease 5 percent per quarter. Checking Quote.com you find NYB trading at $30 per share. The current threemonth riskfree interest rate is 3 percent, and you expect this rate to remain unchanged for the next six months.
A. Calculate the price of a sixmonth European call and put option written on NYB with $31 strike prices using the twoperiod riskneutral pricing model.
C/= 1.06 0 = 0.95 # = 1.03 _ 0.0 x 0.7273^{2} + 0.79 x (0.7273 x 0.2727) + 0.79 x (0.2727 x 0.7273) + 3.925 x 0.2727^{2} _
B. Confirm that the putcall parity relationship generates the same price for the put option. p = cS + X/R^{2} = $1.35  $30 + 31/1.03^{2} = $0.57 ,
C. Construct the twoperiod stock price tree. Working backward through the stock price tree, use the oneperiod riskneutral pricing model, c ^{=}(cjj^u + Cd^d)^, to calculate the value of a oneperiod call option, given that the stock price has increased in period one, c_{up}. Calculate the value of a oneperiod call option, given that the stock price has decreased in period one, c_{dow}n. Calculate the current value of a oneperiod call option that has a value of c_{up} if the stock price increases or a value of c_{dow}n if the stock price decreases. (This process of valuing an option is known as the recursive valuation process.) Compare the price of the call option calculated using the recursive process with the price of the call option calculated using the twoperiod riskneutral pricing model. Do the same for a put. c_{up} = (2.708 x 0.7273 + 0.0 x 0.2727)/1.03 = $1.91 c_{down} = (0.0 x 0.7273 + 0.0 x 0.2727) /1.03 = $0 c = (c_{up} x 0.7273 + c_{dom} x 0.2727)/1.03 = (1.91 x 0.7273 + 0 x 0.2727)/1.03 = $1.35 p_{up} = (0.0 x 0.7273 + .79 x 0.2727)/1.03 = $.21 pdown = (0.79 x 0.7273 + 3.925 x 0.2727)/1.03 = $1.60 p = (p_{up} x 0.7273 + p_{iam} x 0.2727)/1.03 = (0.21 x 0.7273 + 1.59 x 0.2727)/1.03 = $.57 The calculated prices for the options are the same.
D. Repeat the process of part C for a call using the singleperiod noarbitrage binomial pricing model, c = N*S — B* to calculate the price of the singleperiod call option. Discuss what happens to N* at each branch in the tree. (1.060.95)30 c = 0.5788x30 16.02 = $1.34 When the price of the underlying stock changes, the value of the traded option changes, and the appropriate investment in the stock underlying the option, N*, changes. In the binomial model, we create a mimicking portfolio consisting of a stock position and a Treasury bill position. The outcome we are modeling changes as the underlying stock price changes. Thus, we must change our stock position in the mimicking portfolio. For example, when the stock price is $30.00, then N* = 0.5788, and when the stock price is $31.80, then N* = 0.7742. Thus, as the stock price increases, we increase our holdings of the underlying stock, and as the stock price decreases, we reduce our holdings of the underlying stock.
30. Both put and call options on HWP are traded. Put and call options with an exercise price of $100 expire in 90 days. HWP is trading at $95 and has an annualized standard deviation of 0.3. The threemonth riskfree interest rate is 5.25 percent per annum. A. Use a threeperiod binomial model to compute the value of the put and call options using the recursive procedure (singleperiod binomial option pricing model). Be sure to specify the values of U, D, and iTy. Call price, c = $4.21
B. What is the riskneutral probability of a stock price increase in period one, period two, and period three? A characteristic of the multiplicative process used to model stock price movements in the binomial model is that the probability of a stock price increase in any period is independent of previous stock price changes. This process produces prices that follow a random walk. Thus, iTy does not change from period to period, and tt_{v} = 0.5036.
31. Consider a stock, ABM, trading at a price of $70. Analysis of ABM's recent returns reveals that ABM has an annualized standard deviation of return of 0.4. The current riskfree rate of interest is 10 percent per annum.
A. What is the price of a European call written on ABM with a $75 strike price that expires in 180 days? Price the option according to the BlackScholes model. Putcall parity The price of the call calculated using the putcall parity equation, p = c — S + Xe • , is the same as the put price calculated using the Black—Scholes put option pricing model. p = cS + Xe^V = $7.22  $70 + $75^^{aixa4932} = $8.61
32. Consider a stock with a price of $72 and a standard deviation of 0.4. The stock will pay a dividend of $2 in 40 days and a second dividend of $2.50 in 130 days. The current riskfree rate of interest is 10 percent per annum.
A. What is the price of a European call written on this stock with a $70 strike price that expires in 145 days? Price the option with the BlackScholes model. To calculate the value of this option using the Black—Scholes model, we must adjust the current stock price of $72 downward by the present value of the dividends to be received prior to the option's expiration. In this problem, both dividends are paid before the option's expiration. The first dividend will be received in 40 days, and the second will be received in 130 days. .40 V(0.3973) d_{2} = 0.1458  0.40 V(0.3973) = 0.1063 N(d_{x}) = 0.442042 N(d_{2}) = 0.542336 p = Xe^^Nirdz)  SN(d_{x}) = vo^^{01}^^{3973} x 0.542336  67.61 x 0.442042 = $6.60
33. Consider a stock that trades for $75. A put and a call on this stock both have an exercise price of $70, and they expire in 145 days. The riskfree rate is 9 percent per annum, and the standard deviation of the stock is 0.35. Assume that the stock pays a continuous dividend of 4 percent. .35V.3973 dM = 0.5131  0.35V.3973 = 0.2925 N(d1M) = .696056 N(df) = 0.615045 ^{1} .35V.3973 dM = 0.5131  0.35V.3973 = 0.2925 N(df) = .303944 N(df) = 0.384955 _{p}M _{=} _{Xe}r(T_{1)} _{N}(df)  e^V SN{df) _{p}M _{=} _{7O(?}o.o9(o.3973) _{x} 0.384955  _{e}°^{M{}^^{m)} 75 x 0.303944 = $3.56
34. Your broker has just told you about TXF. He describes the firm as the real innovator in the entertainment industry. You search the web and discover that TXF has both put and call options trading on the exchange. Put and call options with an exercise price of $70 expire in 145 days. TXF is currently trading at $75, and has an annualized standard deviation of 0.35. The threemonth riskfree interest rate is 9 percent per annum. A quick, backoftheenvelope calculation reveals that TXF is paying dividends at a continuous rate of 4 percent. The call price is $10.06, and the put price is $3.79.
B. Compare the values of the put and call options calculated in this problem with the values of the put and call options calculated using Merton's model. Explain the source of the differences in the calculated values of the options. In both cases, the prices for the options calculated using the binomial model are greater than the prices calculated using Merton's model. The prices calculated using the binomial model for the call and put options are $10.06 and $3.79, respectively, while the prices for the same options calculated with Merton's model are $9.84 and $3.56. This difference in valuation arises because we modeled a 145day option using a fourperiod model. If we were to model the price movements in TXF using more periods in the binomial model, then the prices calculated using the binomial model would be very close to the prices calculated using the Merton model.
35. CSM is trading at $78 and has an annualized standard deviation of return of 30 percent. CSM is expected to pay a dividend equal to 3 percent of the value of its stock price in 70 days. The current riskfree rate of interest is 7 percent per annum. Options written on this stock have an exercise price of $80 and expire in 120 days.
36. One year later than the time of the previous question, the management of CSM announces that they are changing their dividend policy. CSM will now pay a fixed dollar dividend each quarter. CSM's management declares that the payout for the current year will be $6.00 to be paid equally each quarter. CSM now trades at $85. However, the firm's risk has increased. CSM's annualized standard deviation of return is now 40 percent. CSM will pay the first quarterly dividend in 10 days with the second quarterly dividend coming 90 days after the first dividend. The current riskfree rate of interest is 5.5 percent per annum. Options written on CSM have an exercise price of $80 and expire in 120 days.
A. Using a fourperiod binomial model, calculate the value of a European put option written on CSM using the recursive procedure (singleperiod binomial option pricing model). To construct the stock price tree necessary to calculate the value of this option, we must adjust the current stock price of $85 downward by the present value of the dividends to be received prior to the option's expiration. In this problem, both dividends of $1.50 will be paid prior to the option's expiration. The first dividend will be received in 10 days, and the second will be received in 100 days. ^{)} = $1.50 x ^»55 x (10/36_{5}) _{=} _{$1} _{4977 }'^{)} = $1.50 x ^oossx(100/365) _{=} _{$1} _{4776 }S' = $85  $1.4977  $1.4776 = $82.0247
A. Determine the number of days until the January HCJ options expire. Determining the number of days until an option expires requires one to count both the weekdays and weekends between the current date and the option's expiration date in January. The options expire in 42 days.
B. Determine the continuously compounded interest rate on the Treasury bill. When calculating the continuously compounded interest rate on the Treasury bill, it is important to remember that the Treasury bill matures one day prior to the expiration of the option. That is, the Treasury bill matures on the third Thursday of the month. In this calculation, we must adjust quoted yields on the discount instrument. The price of the Treasury bill is P = 1 0.01((5.09 + 5.05)/2) x (41/360) = 0.994226 The
C. Calculate the annualized standard deviation of return on HCJ's stock using the time series of HCJ stock returns. Assume that there are 252 trading days in a typical year. To calculate the historical volatility of HCJ, we must convert the price series into a return series. To do this, we construct a price relative series, PR_{t} = P_{t}/P_{t} _ 1, for using the data for days 1 through 31. The returns series is created by taking the natural logarithm of the 30 price relatives. The daily standard deviation of returns, cr_{d}, is 0.01762. The annualized standard deviation of returns, ,_{i}s0.27967. is 0.27967.
D. Using the Black—Scholes option pricing model, calculate the current price of the January 140 call and put options written on HCJ. The stock price is 143.125.
E. Use the following information, as of December 4, to calculate the implied standard deviations for the January 135 and 145 HCJ call options. Stock price 142.5625 142.5625 Exercise price 135 145 Days until expiration 43 43 Riskfree rate 5.15% 5.15% Call price $12.00 $4.75
F. Use the equally weighted average of the implied standard deviation on the January 135 and 145 call options in the BlackScholes option pricing model to calculate the current price on the January 140 call and put options. Compare the option prices calculated using the historical volatility and the implied volatility.
Option Sensitivities and Option Hedging
3. Consider a straddle comprised of Call A and Put A. Compute the price, DELTA, GAMMA, THETA, VEGA, and RHO for this straddle. price = c+p = $12.61 DELTA = 0.2735  0.7265 = 0.4530 GAMMA = 0.0279 + 0.0279 = 0.0558 THETA = 8.9173  4.47790 = 13.6963 VEGA = 9.9144 + 9.9144 = 19.8288 RHO = 3.5985  13.4083 = 9.8098
4. Consider Call A. Assuming the current stock price is $60, create a DELTAneutral portfolio consisting of a short position of one call and the necessary number of shares. What is the value of this portfolio for a sudden change in the stock price to $55 or $65? As we saw for this call, DELTA = 0.2735. The DELTAneutral portfolio, given a short call component, is 0.2735 shares —1 call, costs: .2735 ($60)  $1.82 = $14.59 If the stock price goes to $55, the call price is $.77, and the portfolio will be worth: .2735 ($55)  $.77 = $14.27 With a stock price of $65, the call is worth $3.55, and the portfolio value is: .2735 ($65)  $3.55 = $14.23 Notice that the portfolio values are lower for both stock prices of $55 and $65, reflecting the negative GAMMA of the portfolio.
5. Consider Call A and Put A from above. Assume that you create a portfolio that is short one call and long one put. What is the DELTA of this portfolio? Can you find the DELTA without computing? Explain. Assume that a share of stock is added to the short call/long put portfolio. What is the DELTA of the entire position? The DELTA of the portfolio is —1.0 = —0.2735 — 0.7265. This is necessarily true, because the DELTA of the call is N(d_{1}), the DELTA of the put is N(d_{2}), and N(d_{1}) + N(d_{2}) = 1.0. If a long share of stock is added to the portfolio, the DELTA will be zero, because the DELTA of a share is always 1.0.
6. What is the GAMMA of a share of stock if the stock price is $55 and a call on the stock with X = $50 has a price c = $7 while a put with X = $50 has a price p = $4? Explain. The GAMMA of a share of stock is always zero. All other information in the question is irrelevant. The GAMMA of a share is always zero because the DELTA of a share is always 1.0. As GAMMA measures how DELTA changes, there is nothing to measure for a stock, since the DELTA is always 1.0.
7. Consider Call B written on the same stock as Call A with: X = $50; r = 0.06; T  t = 90 days; a = 0.4; and S = $60. Form a bull spread with calls from these two instruments. What is the price of the spread? What is its DELTA? What will the price of the spread be at expiration if the terminal stock price is $60? From this information, can you tell whether THETA is positive or negative for the spread? Explain. As observed in problem 1, for Call A, c = $1.82, DELTA = 0.2735, and THETA = 8.9173. For Call B, c = $11.64, DELTA = 0.8625, and THETA = 7.7191. The long bull spread with calls consists of buying the call with the lower exercise price (Call B) and selling the call with the higher exercise price (Call A). The spread costs $11.64  $1.82 = $9.82. The DELTA of the spread equals DELTAB  DELTAA = 0.8625 0.2735 = 0.5890. If the stock price is $60 at expiration, Call B will be worth $10, and Call A will expire worthless. If the stock price remains at $60, the value of the spread will have to move from $9.82 now to $10.00 at expiration, so the THETA for the spread must be positive. This can be confirmed by computing the two THETAs and noting: THETA_{A} = 8.9173 and THETA_{B} = 7.7191. For the spread, we buy Call B and sell Call A, giving a THETA for the spread of 7.7191  (8.9173) = 1.1982.
8. Consider again the sample options, C2 and P2, of the chapter discussion as given in Table 14.7. Assume now that the stock pays a continuous dividend of 3 percent per annum. See if you can tell how the sensitivities will differ for the call and a put without computing. Now compute the DELTA, GAMMA, VEGA, THETA, and RHO of the two options if the stock has a dividend. The presence of a continuous dividend makes d1 smaller than it otherwise would be, because the continuous dividend rate, er, is subtracted in the numerator of d1. With a smaller d1, N(d1) is also smaller. But, N(d_{1}) = DELTA for a call, so the DELTA of a call will be smaller with a dividend present. By the same reasoning, the DELTA of the put must increase.
Sensitivity 
C2 
P2 



DELTA 
0.5794 
0.4060 


GAMMA 
0.0182 
0.0182 


THETA 
10.3343 
5.5997 


VEGA 
26.93 
26.93 


RHO 
23.9250 
23.4823 
9. Consider three calls, Call C, Call D, and Call E, all written on the same underlying stock; S = $80; r = 0.07; ct = 0.2. For Call C, X = $70, and T  t = 90 days. For Call D, X = $75, and T  t = 90 days. For Call E, X = $80, and T  t = 120 days. Compute the price, DELTA, and GAMMA for each of these calls. Using Calls C and D, create a DELTAneutral portfolio assuming that the position is long one Call C. Now use calls C, D, and E to form a portfolio that is DELTAneutral and GAMMAneutral, again assuming that the portfolio is long one Call C. Measure Call C Call D Call E
Price $11.40 $7.16 $4.60
DELTA .9416 .8088 .6018
GAMMA .0147 .0343 .0421
For a DELTAneutral portfolio comprised of Calls C and D that is long one Call C, we must choose a position of Z shares of Call D to satisfy the following equation: 0.9416+ 0.8088Z = 0 Therefore, Z = —1.1642, and the portfolio consists of purchasing one Call C and selling 1.1642 units of Call D. To form a portfolio of Calls C, D, and E that is long one Call C and that is also DELTAneutral and GAMMAneutral, the portfolio must meet both of the following conditions, where Yand Zare the number of Call Cs and Call Ds, respectively. DELTAneutrality: 0.9416 + 0.8088 Y + 0.6018 Z = 0 GAMMAneutrality: 0.0147 + 0.0343 Y + 0.0421 Z = 0 Multiplying the second equation by (0.8088/.0343) gives: 0.3466 + 0.8088 Y + 0.9927 Z = 0 Subtracting this equation from the DELTAneutrality equation gives: 0.5950  0.3909 Z = 0 Therefore, Z = 1.5221. Substituting this value of Z into the DELTAneutrality equation gives: 0.8088 Y + 0.9416 + 0.6018 (1.5221) = 0 Y = —2.2968. Therefore, the DELTAneutral and GAMMAneutral portfolio consists of buying one unit of Call C, selling 2.2968 units of Call D, and buying 1.5221 units of Call E.
10. Your largest and most important client's portfolio includes option positions. After several conversations it becomes clear that your client is willing to accept the risk associated with exposure to changes in volatility and stock price. However, your client is not willing to accept a change in the value of her portfolio resulting from the passage of time. Explain how the investor can protect her portfolio against changes in value due to the passage of time. Your client wants to avoid changes in the value of her portfolio due to the passage of time. THETA measures the impact of changes in the time until expiration on the value of an option. Your client should create a THETAneutral portfolio to protect the value of her option positions against changes in the time until expiration. To protect her portfolio against the wasting away effect associated with option contracts, she must first determine the THETA for her current portfolio. Given the THETA value of her portfolio, she should construct a position in option contracts that has a THETA value that is of an equal magnitude and opposite sign of the THETA of her portfolio. Thus, the THETA for the hedge portfolio, the original portfolio plus the additional options contracts used to create the hedge, is zero. The value of this portfolio should not change with the passage of time. However, the portfolio will have exposure to changes in other market variables, that is, interest rates, volatility, and stock price changes.
11. Your newest client believes that the Asian currency crisis is going to increase the volatility of earnings for firms involved in exporting, and that this earnings volatility will be translated into large stock price changes for the affected firms. Your client wants to create speculative positions using options to increase his exposure to the expected changes in the riskiness of exporting firms. That is, your client wants to prosper from changes in the volatility of the firm's stock returns. Discuss which "Greek" your client should focus on when developing his options positions. Your client wants to create exposure to changes in the volatility of stock returns. VEGA measures the change in the value of an option contract resulting from changes in the volatility of the underlying stock. Once you have identified stocks with traded options that have significant Asian exposures, you want to construct positions based on the VEGA of the option. Because your client wants exposures to volatility risk, you would construct a portfolio with a large VEGA.
12. A longtime client, an insurance salesperson, has noticed the increased acquisition activity involving commercial banks. Your client wishes to capitalize on the potential gains associated with this increased acquisition activity in the banking industry by creating speculative positions using options. Your client realizes that bank cash flows are sensitive to changes in interest rates, and she believes that the Federal Reserve is about to increase shortterm interest rates. Realizing that an increase in the shortterm interest rates will lead to a decrease in the stock prices of commercial banks, your client wants the value of her portfolio of options to be unaffected by changes in shortterm interest rates. Explain how the investor can use option contracts to protect her portfolio against changes in value due to changes in the riskfree rate, and to capitalize on the expected price changes in bank stocks. Your client wants to create exposure to changes in bank stock prices. DELTA measures the change in the value of an option contract resulting from changes in the underlying stock price. Additionally, your client has a preference to construct a portfolio such that the value of the portfolio will not change as interest rates change. RHO measures the change in the value of an option contract resulting from changes in the riskfree rate of interest. Because your client wants exposure to stock price changes, you would construct a portfolio with a large DELTA. However, the RHO for the portfolio should be constrained to equal zero. Thus, the resulting portfolio would be RHOneutral hedge with a large DELTA.
13. Your brotherinlaw has invested heavily in stocks with a strong Asian exposure, and he tells you that his portfolio has a positive DELTA. Give an intuitive explanation of what this means. Suppose the value of the stocks that your brotherinlaw holds increases significantly. Explain what will happen to the value of your brotherinlaw's portfolio. DELTA measures the change in the value of an option due to a change in the price of the underlying asset, which is usually a stock. If an investor holds a portfolio consisting of a single stock, the DELTA of the portfolio is one, because a one dollar increase in the stock price will produce a one dollar per share increase in the value of the portfolio. If the asset in question is an option, then the DELTA of the option measures the change in the value of the option contract because of a change in the underlying stock price. If your brotherinlaw's portfolio has a positive DELTA, the value of his portfolio will move in the same direction as the value of the underlying asset. If the value of the stocks he holds increases, then the value of his portfolio will increase at a rate of DELTA times the dollar change in the asset price.
14. Your motherinlaw has invested heavily in the stocks of financial firms, and she tells you that her portfolio has a negative RHO. Give an intuitive explanation of what this means. Suppose the Federal Reserve increases shortterm interest rates. Explain what will happen to the value of your motherinlaw's portfolio. RHO measures the change in the value of an asset due to changes in interest rates. If the investor holds an option, then the RHO of the option measures the change in the value of the option contract because of a change in the riskfree interest rate. If your motherinlaw's portfolio has a negative RHO, that implies that the value of the portfolio moves in the opposite direction as changes in the interest rate. If the shortterm interest rate is increased by the Federal Reserve, then the value of her portfolio will decrease.
15. Your brother, Daryl, has retired. With the free time necessary to follow the market closely, Daryl has established large option positions as a stock investor. He tells you that his portfolio has a positive THETA. Give an intuitive explanation of what this means. Daryl is also a big soccer fan, and is heading to France to watch the World Cup for a month. He believes that there is not sufficient liquidity in the market to close out his open option positions, and he is going to leave the positions open while he is in France. Explain what will happen to the value of your brother's portfolio while he is in France. THETA measures the change in the value of an option because of changes in the time until expiration for the option contract. That is, with the passage of time, the value of an option contract will change. In most cases, the option will experience a decrease in value with the passage of time. This is known as time decay. Formally, THETA is the negative of the first derivative of the option pricing model with respect to changes in the time until expiration. Since your brother has constructed a portfolio with a positive THETA, the passage of time should increase the value of his portfolio. Thus, he should, all things being equal, return from his vacation to find that the value of his portfolio has increased.
16. Consider the following information for a call option written on Microsoft's stock. S = $.96 DELTA = 0.2063 X = $100 GAMMA = 0.0635 T  t = 5 days THETA = 48.7155 er = 0.4 VEGA = 3.2045 r = 0.1 RHO = 0.2643 Price = $.5 If in two days Microsoft's stock price has increased by $1 to $97, explain what you would expect to happen to the price of the call option. Two variables are changing in this problem, the underlying stock price, S, and the time until expiration, T — t. Thus, one needs to assess the impact of both DELTA and THETA on the value of the Microsoft option. DELTA is 0.2063, and THETA is —48.7155. A one dollar increase in the price of Microsoft would be expected to increase the price of the call option by $.2063 = $1 X 0.2063. However, as an option contract approaches expiration, the passage of time has a significant adverse effect on the value of the option. Here two days represent 40 percent of the life of the option. The THETA effect is equal to $.2669 = (2/365) X —48.7155, which is a larger negative effect than the positive impact of a stock price increase on the value of the option. The combined DELTA and THETA effects are $.0606 = $.2063  $.2669. Thus, the expected price of the call option is $.4394. The price of the call option according to the BlackScholes model is $.4162.
17. Consider a stock, CVN, with a price of $50 and a standard deviation of 0.3. The current riskfree rate of interest is 10 percent. A European call and put on this stock have an exercise price of $55 and expire in three months (0.25 years). A. If c = $1.61057 and N(d_{1}) = 0.3469, then calculate the put option price. B. Suppose that you own 3,000 shares of CBC, a subsidiary of CVN Corporation, and that you plan to go Christmas shopping in New York City the day after Thanksgiving. To finance your shopping trip, you wish to sell your 3,000 shares of CBC in one week. However, you do not want the value of your investment in CBC to fall below its current level. Construct a DELTAneutral hedge using the put option written on CVN. Be sure to describe the composition of your hedged portfolio. Construction of a DELTAneutral hedge using the put option requires the investor to hold — 1 /DELTA_{put }put options per 100 shares of stock held by the investor. The put option extends the right to sell 100 shares of stock. The DELTA for the put option is 0.6531 = 0.3469  1. Thus, the hedge ratio is 1.5312 = 1/ —0.6531, put options per 100 shares of stock. Because you hold 3,000 shares of CBC, you must purchase 45.9348 = (3,000/100) X 1.5312, put options written on CVN that expire in three months with a strike price of $55. Since purchasing a fraction of an option's contract is not possible, you would round this up to 46 options contracts purchased.
18. An investor holds a portfolio consisting of three options, two call options and a put option, written on the stock of QDS Corporation with the following characteristics.
DELTA 0.8922 0.2678 0.6187
GAMMA 0.0169 0.0299 0.0245
THETA 5.55 3.89 3.72
The investor is long 100 contracts of option C1, short 200 contracts of option C^ and long 100 contracts of option P1. The investor's options portfolio has the following characteristics. DELTA = 0.8922 (100)  0.2678 (200)  0.6187 (100) = 26.21 GAMMA = 0.0169 (100)  0.0299 (200) + 0.0245 (100) =1.84 THETA = 5.55 (100) + 3.89 (200)  3.72 (100) = 149 The investor wishes to hedge this portfolio of options with two call options written on the stock of QDS Corporation with the following characteristics.
DELTA 0.5761 0.6070
GAMMA 0.0356 0.0247
THETA 9.72 7.04
A. How many contracts of the two options, C_{a} and Cb, must the investor hold to create a portfolio that is DELTAneutral and has a THETA of 100? The investor must determine how many of the call options a and b to hold to create a portfolio that has a DELTA of 26.21 and a THETA of 249. That is, the DELTA of the combined portfolio of options should be zero, and the THETA of the combined portfolio should be 100. We must solve this for N_{a} and N_{b} subject to the constraint that the DELTA of the portfolio is 26.21 and the
THETA is 249. 0.5671 N_{a} + 0.607 N_{b} = 26.21 9.72 N_{a} 7.04 N_{b} = 249 N_{a}= 175.955 N_{b} = 207.568 To create a portfolio with a DELTA of 26.21 and a THETA of 249, the investor must sell 176 contracts of option C_{a} and purchase 208 contracts of option C_{b}.
B. If QDS's stock price remains relatively constant over the next month, explain what will happen to the value of the portfolio created in part A of this question. If QDS's stock price remains relatively constant, then the passage of time should increase the value of the portfolio. The DELTA of the portfolio is zero and the THETA is positive. Thus, small changes in the stock price will have little impact on the value of the portfolio of options, and the passage of time should increase the value of the option portfolio.
C. How many of the two options, C_{a} and C_{h} must the investor hold to create a portfolio that is both DELTAand GAMMAneutral? The investor must determine how many of the call options C_{a} and Cb to hold to create a portfolio that has a DELTA of 26.21 and a GAMMA of 1.84. That is, the DELTA and GAMMA of the combined portfolio of options are zero. We must solve the equation for N_{a} and Nb subject to the constraint that the DELTA of the portfolio is 26.21 and the GAMMA is 1.84. 0.5671 N_{a} + 0.607 N_{b} = 26.21 0.0356 N_{a} + 0.0247 N_{b}= 1.84 N_{a} = 61.7573 N_{b}= 14.51667 To create a portfolio with a DELTA of 26.21 and a GAMMA of 1.84, the investor must purchase 62 contracts of option C_{a} and sell 15 contracts of option C_{b}.
D Suppose the investor wants to create a portfolio that is DELTA, GAMMA, and THETAneutral. Could the investor accomplish this objective using the two options, C_{a} and C_{b}, which have been used in the previous problems? Explain. No. Creating neutrality in three dimensions requires at least three different option contracts. We need at least as many option contracts as parameters we are trying to hedge. To create a DELTA, GAMMA, and THETAneutral portfolio would require an additional new option contract.
19. Both put and call options trade on HWP. Put and call options with an exercise price of $100 expire in 90 days. HWP is trading at $95, and has an annualized standard deviation of return of 0.3. The threemonth riskfree interest rate is 5.25 percent per annum.
A. Use a fourperiod binomial model to compute the value of the put and call options using the recursive procedure (singleperiod binomial option pricing model). The call price is $4.33, and the put price is $8.05.
B. Increase the stock price by $.25 to $95.25 and recalculate the value of the put and call options using the recursive procedure (singleperiod binomial option pricing model). If the stock price is increased by $.25, then the call price is $4.43 and the put price is $7.89.
C. For both the call and put options, calculate DELTA as the change in the value of the option divided by the change in the value of the stock.
D. Decrease the stock price by $.25 to $94.75 and recalculate the value of the put and call options using the recursive procedure (singleperiod binomial option pricing model). When the stock price is decreased by $.25, then the call price is $4.24 and the put price is $8.20.
E. For both the call and put options, calculate DELTA as the change in the value of the option divided by the change in the value of the stock.
F:For both the call and put options, calculate GAMMA as the difference between the DELTA associated with a stock price increase and the DELTA associated with a stock price decrease, divided by the change in the value of the stock.
G. Increase the riskfree interest rate by 20 basis points to 5.45 percent per annum and recalculate the value of the put and call options using the recursive procedure (singleperiod binomial option pricing model). If the interest rate is increased by 20 basis points to 5.45 percent, then the call price is $4.35 and the put price is $8.02.
H. For both the call and put options, calculate RHO as the change in the value of the option divided by the change in the riskfree interest rate.
I. Increase the volatility of the underlying stock to 33 percent and recalculate the value of the put and call options using the recursive procedure (singleperiod binomial option pricing model). If the volatility is increased to 33%, then the call price is $4.89 and the put price is $8.60.
J. For both the call and put options, calculate VEGA as the change in the value of the option divided by the change in the volatility of the stock.
K. Notice that each branch in the binomial tree represents the passage of time. That is, as one moves forward through the branches in a binomial tree, the life of the option wastes away. Also notice that the initial stock price of $95 reappears in the middle of the second branch of the tree. Using the parameter values for U, D, TT[/, IT/), At, and e~^{rAt} calculated using the initial information given, recalculate the value of the call and put options using a twoperiod model. That is, calculate the option prices after 45 days, assuming the stock price is unchanged.
L. For both the call and put options, calculate THETA as the difference in the value of the option with two periods to expiration less the value of the option with four periods to expiration, divided by twice the passage of time associated with each branch in the tree, that is, 2 X At.
20. Consider a stock, ABM, trading at a price of $70. Analysis of ABM's recent returns reveals that ABM has an annualized standard deviation of return of 0.4. The current riskfree rate of interest is 10 percent per annum.
A. What is the price of a European call and put written on ABM with a $75 strike price that expires in 180 days? Price the options using the BlackScholes model. 0.40 V(0.4932) d_{2} = 0.0704  0.40 V(0.4932) =  0.2105 N(d_{1}) = 0.528061 N(d_{2}) = 0.416638 c = 70 x 0.528061  7501 ^{xa4932} x 0.416638 = $7.2201 p = cS + Xe«?^ = $7.22  $70 + $75^°'^{lxa4932} = $8.6111 The call price is $7.2201, and the put price is $8.6111.
B. Increase the stock price by $.25 to $70.25 and recalculate the value of the put and call options. If the stock price is increased by $.25, then the call price is $7.3527 and the put price is $8.4938.
C. For both the call and put options, calculate DELTA as the change in the value of the option divided by the change in the value of the stock.
D. Decrease the stock price by $.25 to $69.75 and recalculate the value of the put and call options. If the stock price is decreased by $.25, then the call price is $7.0887 and the put price is $8.7298.
E. For both the call and put options, calculate DELTA as the change in the value of the option divided by the change in the value of the stock. (7.2201  7.0887)/0.25 = 0.5256 (8.6111  8.7298)/0.25 = 0.4748
F. For both the call and put options, calculate GAMMA as the difference between the DELTA associated with a stock price increase and the DELTA associated with a stock price decrease divided by the change in the value of the stock. (0.5304  0.5256)/0.5 = 0.0096 (0.4692  (0.4748))/0.5 = .0112
G. Increase the riskfree interest rate by 25 basis points to 10.25 percent per annum and recalculate the value of the put and call options. If the interest rate is increased by 25 basis points to 10.25 percent, then the call price is $7.2568 and the put price is $8.5599.
H. For both the call and put options, calculate RHO as the change in the value of the option divided by the change in the riskfree interest rate. (7.2568  7.2201 )/0.0025 = 14.68 (8.5599  8.6111)/0.0025 = 20.48
I. Increase the volatility of the underlying stock to 44 percent and recalculate the value of the put and call options. If the volatility is increased to 44 percent, then the call price is $8.0019 and the put price is $9.3930.
J. For both the call and put options, calculate VEGA as the change in the value of the option divided by the change in the volatility of the stock (8.0019  7.2201 )/0.04 = 19.5450 (9.3930  8.6111)/0.04 = 19.5475
K. Decrease the life of the option by 10 percent to 162 days and recalculate the value of the put and call options. If the life of the option is decreased by 10 percent to 162 days, then the call price is $6.6721 and the put price is $8.4161.
L. For both the call and put options, calculate THETA as the difference in the value of the option with 162 days to expiration less the value of the option with 180 days to expiration divided by the passage of time. (6.6721  7.2201)/(18/365) = 11.1122 (8.4161  8.6111)/(18/365) = 3.9542
21. Consider a stock that trades for $75. A put and a call on this stock both have an exercise price of $70, and they expire in 145 days. The riskfree rate is 9 percent per annum and the standard deviation of return for the stock is 0.35. Assume that the stock pays a continuous dividend of 4 percent. N(df) = 0.303944 N(df) = 0.384955 _{pM} _{=} _{Xe}r(T_{t)} _{N}(_{d}M) _ _{e}w_{1)} s_{t}N(df) _{p}M _{=} _{70(?}0.09(0.3973) _{x} q 0.384955 _ ,0.04(0.3973) 75 _{x} 0.303944 = _{$3}. The call price is $9.8402, and the put price is $3.5641.
B. Increase the stock price by $.25 to $75.25 and recalculate the value of the put and call options. When the stock price is increased by $.25, then the call price is $10.0121 and the put price is $3.4899.
C. For both the call and put options, calculate DELTA as the change in the value of the option divided by the change in the value of the stock. (10.0121  9.8402)/0.25 = 0.6876 (3.4899  3.5641 )/0.25 = 0.2968
D. Decrease the stock price by $.25 to $74.75 and recalculate the value of the put and call options. If the stock price is decreased by $.25, then the call price is $9.6696 and the put price is $3.6395.
E. Calculate DELTA for both the call and put options as the change in the value of the option divided by the change in the value of the stock. (9.8402  9.6696)/0.25 = 0.6824 (3.5641  3.6395)/0.25 = 0.3016
F. Calculate GAMMA for both the call and put options as the difference between the DELTA associated with a stock price increase and the DELTA associated with a stock price decrease divided by the change in the value of the stock. (0.6876  0.6824)/0.5 = 0.0104 (0.2968  (0.3016))/0.5 = 0.0096
G. Increase the riskfree interest rate by 25 basis points to 9.25 percent per annum and recalculate the value of the put and call options. When the interest rate is increased by 25 basis points to 9.25 percent, then the call price is $9.8815 and the put price is $3.5383.
H. Calculate RHO for both the call and put options as the change in the value of the option divided by the change in the riskfree interest rate.
I. Increase the volatility of the underlying stock to 38.5 percent and recalculate the value of the put and call options. When the volatility is increased to 38.5 percent, the call price is $10.4136 and the put price is $4.1374.
J. Calculate VEGA for both the call and put options as the change in the value of the option divided by the change in the volatility of the stock.
K. Decrease the life of the option by 20 percent to 116 days and recalculate the value of the put and call options. When the life of the option is decreased by 20 percent to 116 days, then the call price is $9.1028 and the put price is $3.0764.
L. Calculate THETA for both the call and put options as the difference in the value of the option with 116 days to expiration less the value of the option with 145 days to expiration divided by the passage of time.
22. CSM is trading at $78 and has an annualized standard deviation of return of 30 percent. CSM is expected to pay a dividend equal to 3 percent of the value of its stock price in 70 days. The current riskfree rate of interest is 7 percent per annum. Options written on this stock have an exercise price of $80 and expire in 120 days. The stock prices in the tree must be adjusted for the dividend to be paid in 70 days before we can calculate the value of the options. Therefore, the stock prices in the tree in periods three and four must be adjusted downward by one minus the dividend yield paid by the firm (1 — 3%). Dividendadjusted stock price tree
B. Increase the stock price by $.25 to $78.25 and recalculate the value of the put and call options using the recursive procedure (singleperiod binomial option pricing model). When the stock price is increased by $.25, then the call price is $4.42 and the put price is $6.69.
C. Calculate DELTA for both the call and put options as the change in the value of the option divided by the change in the value of the stock. (4.42  4.32)/0.25 = 0.40 (6.69  6.84)/0.25 = 0.60
D. Decrease the stock price by $.25 to $77.75 and recalculate the value of the put and call options using the recursive procedure (singleperiod binomial option pricing model). When the stock price is decreased by $.25, then the call price is $4.22 and the put price is $6.99.
E. Calculate DELTA for both the call and put options as the change in the value of the option divided by the change in the value of the stock. (4.32  4.22)/0.25 = 0.40 (6.69  6.84)/0.25 = 0.60
F. Calculate GAMMA for both the call and put options as the difference between the DELTA associated with a stock price increase and the DELTA associated with a stock price decrease divided by the change in the value of the stock. (0.3869  0.3869)/0.5 = 0 (0.5831  (0.5831 ))/0.5 = 0 Calculating the price of each option to two decimal places using a fourperiod binomial process to model the price movements in a stock over a 90day period produces symmetric price changes in the value of the options. There is not sufficient information in our modeling process to capture the nonlinearity of the pricing function for small symmetric changes in the stock price. This generates a value of zero for GAMMA.
G. Increase the riskfree interest rate by 20 basis points to 7.20 percent per annum and recalculate the value of the put and call options using the recursive procedure (singleperiod binomial option pricing model). When the interest rate is increased by 20 basis points to 7.20 percent, then the call price is $4.34 and the put price is $6.81.
H. Calculate RHO for both the call and put options as the change in the value of the option divided by the change in the riskfree interest rate. (4.34  4.32)/0.002 = 10 (6.81  6.84)/0.002 = 15
I. Increase the volatility of the underlying stock to 33 percent and recalculate the value of the put and call options using the recursive procedure (singleperiod binomial option pricing model). When the volatility is increased to 33 percent, then the call price is $4.83 and the put price is $7.35.
K. Notice that each branch in the binomial tree represents the passage of time. That is, as one moves forward through the branches in a binomial tree, the life of the option wastes away. Also notice that the initial stock price of $78 reappears in the middle of the second branch of the tree. Using the parameter values for U, D, TT[/, IT/), At, and e^^{1} calculated using the initial information given, recalculate the value of the call and put options using a twoperiod model. Assume that the dividend will be paid before period one and make the appropriate adjustments to the stock price tree. That is, calculate the option prices after 60 days, assuming that the stock price is unchanged.
L. Calculate THETA for both the call and put options as the difference in the value of the option with two periods to expiration less the value of the option with four periods to expiration divided by twice the passage of time associated with each branch in the tree, that is, 2 X A.t.
Call THETA Put THETA
(2.56  4.32)/[2 x (30/365)] = 10.7067 (5.98  6.84)/[2 x (30/365)] = 5.2317
23. DCC exports highspeed digital switching networks, and their largest and most important clients are in Asia. A recent financial crisis in Asia has diminished the prospects of new sales to the Asian market in the near term. However, you believe that DCC is a good investment for the long term. A quick check of Quote.com reveals that DCC is trading at $78.625 per share. Your previous calculation of the historical volatility for DCC indicated an annual standard deviation of return of 27 percent, but examining the implied volatility of several DCC options reveals an increase in annual volatility to 32 percent. There are two traded options series that expire in 245 days. The options have $75 and $80 strike prices respectively. The current 245day riskfree interest rate is 4.75 percent per annum, and you hold 2,000 shares of DCC.
A. Construct a portfolio that is DELTA and GAMMAneutral using the call options written on DCC. The investor owns 2,000 shares of DCC. An option contract extends the right to buy or sell 100 shares of the underlying stock at the strike price. Thus, by scaling the stock position by 100, we can convert the stock position to a scale that matches a single option contract. On a scaled basis, the shareholder has a long position in 20 = 2000/100, contract units of stock. The investor must determine how many $75 call options and $80 call options to hold to create a portfolio that is DELTA and GAMMAneutral. The DELTA of the stock, DELTA_{s}, is one, and the GAMMA of the stock, GAMMA_{s}, is zero. N_{s} DELTA_{s} + N_{75} DELTA_{75} + N_{80} DELTA_{80} = 0 N_{s} GAMMA_{s} + N_{75} GAMMA_{75} + N_{80} GAMMA_{80} = 0 20 x 1 + N_{75}(0.6674) + N_{80} (0.5740) = 0 20 x 0 + N_{75}(0.0176) + N_{80} (0.0190) = 0 We must solve the equation for N_{7}5 and N80 subject to the constraint that the DELTA of the portfolio is 20 and the GAMMA is 0. N_{75} (0.6674) + N_{80} (0.5740) = 20 N_{75} (0.0176) + N_{80} (0.0190) = 0 jV_{75}= 147.39 N_{80} = 136.53 To create a portfolio with a DELTA of 20 and a GAMMA of 0, the investor must sell 147.39 of the $75 call option contracts and purchase 136.53 of the $80 call option contracts.
B. Construct a portfolio that is DELTA and GAMMAneutral using the put options written on DCC. The investor owns 2,000 shares of DCC. An option contract extends the right to buy or sell 100 shares of the underlying stock at the strike price. Thus, by scaling the stock position by 100, we can convert the stock position to a scale that matches a single option contract. On a scaled basis, the shareholder has a long position in 20 = 2000/100, contract units of stock. The investor must determine how many $75 put options and $80 put options to hold to create a portfolio that is DELTA and GAMMAneutral. The DELTA of the stock, DELTA_{s}, is one, and the GAMMA of the stock, GAMMA_{s}, is zero. N_{s} DELTA_{s} + N_{75} DELTA_{75} + N_{80} DELTA_{80} = 0 N_{s} GAMMA_{s} + N_{75} GAMMA_{75} + N_{80} GAMMA_{80} = 0 20 x 1 + N_{75} (0.3326) + N_{80} (0.4260) = 0 20 x 0 + N_{75} (0.0176) + N_{80} (0.0190) = 0 We must solve the equation for N_{75} and N_{80} subject to the constraint that the DELTA of the portfolio is 20 and the GAMMA is 0. N_{75} (0.3326) + N_{80} (0.426) = 20 N_{75} (0.0176) + N_{80} (0.0190) = 0 N_{75} = 322.53 N_{80} = 298.76 To create a portfolio with a DELTA of 20 and a GAMMA of 0, the investor must sell 322.53 of the $75 put option contracts and purchase 298.76 of the $80 put option contracts.
C. Construct a portfolio that is DELTA and THETAneutral using the call options written on DCC. The investor owns 2,000 shares of DCC. An option contract extends the right to buy or sell 100 shares of the underlying stock at the strike price. Thus, by scaling the stock position by 100, we can convert the stock position to a scale that matches a single option contract. On a scaled basis, the shareholder has a long position in 20 = 2000/100, contract units of stock. The investor must determine how many $75 call options and $80 call options to hold to create a portfolio that is DELTA and THETAneutral. The DELTA of the stock, DELTA_{s}, is one, and the THETA of the stock, THETA_{s}, is zero. N_{s} DELTA_{s} + N_{75} DELTA_{75} + N_{80} DELTA_{80} = 0 N_{s} THETA_{s} + N_{75}THETA_{75} + N_{80}THETA_{80} = 0 20 x 1 + N_{75} (0.6674) + N_{80} (0.5740) = 0 20 x 0 + N_{75} (7.5372) + N_{80} (7.7495) = 0 We must solve the equation for N_{75} and N_{80} subject to the constraint that the DELTA of the portfolio is 20 and the THETA is 0. N_{75} (0.6674) + N_{80} (0.5740) = 20 N_{75} (7.5372) + N_{80} (7.7495) = 0 jV_{75}= 183.28 N_{80} = 178.26 To create a portfolio with a DELTA of 20 and a THETA of 0, the investor must sell 183.28 of the $75 call option contracts and purchase 178.26 of the $80 call option contracts. D. Construct a portfolio that is DELTA and THETAneutral using the put options written on DCC. The investor owns 2,000 shares of DCC. An option contract extends the right to buy or sell 100 shares of the underlying stock at the strike price. Thus, by scaling the stock position by 100, we can convert the stock position to a scale that matches a single option contract. On a scaled basis, the shareholder has a long position in 20 = 2000/100, contract units of stock. The investor must determine how many $75 put options and $80 put options to hold to create a portfolio that is DELTA and THETAneutral. The DELTA of the stock, DELTA_{s}, is one, and the THETA of the stock, THETA_{s}, is zero. N_{s} DELTA_{s} + N_{75} DELTA_{75} + N_{80} DELTA_{80} = 0 N_{s} THETA_{s} + N_{75}THETA_{75} +jV_{80}THETA_{80} = 0 20 x 1 + N_{75} (0.3326) + N_{80} (0.426) = 0 20 x 0 + N_{75} (4.0865) + N_{80} (4.0687) = 0 We must solve the equation for N_{75} and N_{80} subject to the constraint that the DELTA of the portfolio is 20 and the THETA is 0. N_{75} (0.3326) +jV_{80} (0.426) = 20 N_{75} (4.0865) +jV_{80} (4.0687) = 0 N_{75} = 209.94 N_{80} = 210.86 To create a portfolio with a DELTA of 20 and a THETA of 0, the investor must sell 209.94 of the $75 put option contracts and purchase 210.86 of the $80 put option contracts.
E. How effective do you expect the DELTA and GAMMAneutral hedges to be? Explain. The effectiveness of the DELTA, GAMMAneutral hedges will be conditional on movements in DCC's stock price over the hedge period. If DCC's stock remains relatively constant during the hedge period, then the hedge should be effective. However, if DCC's stock price is volatile during the hedge period, which is very likely to happen, then it will be necessary to rebalance the portfolio periodically to maintain the effectiveness of the hedge.
24. Consider a stock, PRN, that trades for $24. A put and a call on this stock both have an exercise price of $22.50, and they expire in 45 days. The riskfree rate is 5.5 percent per annum, and the standard deviation of return for the stock is .28.
A. Calculate the price of the put and call option using the Black—Scholes model.
ln(24/22.5) + ([0.055 + 0.5(0.28^{2})](0.1233))
d, =,= 0.7746
0.28V(0.1233)
d_{2} = 0.7746  0.28 V(0.1233) =  0.6763 N(d_{1}) = 0.780705 N(d_{2}) = 0.750563
c = 24 x 0.780705  22.5^^{a055x0Am} x 0.750563 = $1.96 p = cS + Xe^^{7}^ = $1.96  $24 + $22.5^°'^{055x0}'^{1233} = $.31 The call price is $1.96, and the put price is $.31.
Note: Use the following information for the remaining parts of this problem. Suppose that you own 1,500 shares of PRN and you wish to hedge your investment in PRN using the traded PRN options. You are going on vacation in 45 days and want to use your shares to finance your vacation, so you do not want the value of your PRN shares to fall below $22.50.
B. Construct a hedge using a covered call strategy. In a covered call strategy, the investor sells call options to hedge against the risk of a stock price decline. The investor owns 1,500 shares of PRN. An option contract extends the right to buy or sell 100 shares of the underlying stock at the strike price. Thus, by scaling the stock position by 100, we can convert the stock position to a scale that matches a single option contract. On a scaled basis, the shareholder has a long position in 15 = 1,500/100, contract units of stock. The investor wants to create a hedge position that expires in 45 days and the PRN options expire in 45 days. In this hedge, the investor will hold the option contracts to their expiration date and use the moneyness of the options to hedge the investment in PRN. The investor's objective is to create a price floor of $22.50 for the 1,500 shares of PRN. The minimum value desired for the PRN shares in 45 days is $33,750 = 1,500 x $22.50. In this covered call hedge, the investor will sell one call option of every share of stock that he owns. On a scaled basis, the shareholder has a long position in 15 PRN contract units of stock. Thus, the investor should sell 15 PRN call options at $1.96. The cash inflow from the sale of the 15 call options is $2,940 = $1.96 X 15 x 100.
C. Construct a hedge using a protective put strategy. In a protective put strategy, the investor purchases put options to hedge against the risk of a stock price decline. In this protective put hedge, the investor will buy one put option of every share of stock that he owns. On a scaled basis, the shareholder has a long position in 15 PRN shares. Thus, the investor should buy 15 PRN put options at $.31. The cash outflow from the purchase of the 15 put options is $465 = $.31 X 15 X 100.
D. If PRN is trading at $19 in 45 days, analyze and compare the effectiveness of the two alternative hedging strategies. Covered call: When PRN is trading at $19, the call options that the investor sold expire worthless. The investor keeps the proceeds of $2,940 from the sale of the call options. The investor's stock position is worth $28,500 = $19 X 1,500. The total value of the portfolio consisting of the stock position and the proceeds from the sale of the options is $31,440 = $28,500 + $2,940. Thus, the value of the portfolio is $2,310 less than the $33,750 minimum value for the portfolio established by the investor at the start of the hedge. Protective put: When PRN is trading at $19, the put options owned by the investor are inthemoney. The investor will exercise the 15 put options, selling 1,500, 15 X 100, shares of stock at $22.50 per share. The value of the investor's portfolio is $33,750, the minimum value for the portfolio established by the investor at the start of the hedge. The covered call hedge strategy resulted in a loss of value to the investor's portfolio that was greater than the loss the investor was willing to accept at the start of the hedge period. The protective put strategy produced a portfolio that had a value that was equal to the minimum set by the investor at the start of the hedge.
E. If PRN is trading at $26 in 45 days, analyze and compare the effectiveness of the two alternative hedging strategies. Covered call: When PRN is trading at $26, the call options sold by the investor are inthemoney. The owner of the options will exercise the 15 call options against the investor. The investor will have to sell his 1,500 shares of PRN stock at $22.50 per share to the owner of the option. The investor will receive $33,750 from the sale of the stock. The proceeds of $2,940 from the sale of the call options plus the $33,750 from the sale of the stock result in a position that is worth $36,690. The value of 1,500 shares of PRN at $26 per share is $39,000. Thus, the investor incurs an opportunity cost of $2,310 from the sale of 15 PRN call options. Protective put: When PRN is trading at $26, the put options that the investor purchased expire worthless. The investor's stock position is worth $39,000, $26 X 1,500. The total value of the portfolio consisting of the stock position and the cost of the put options is $38,535, $39,000  $465 When the stock price exceeds the exercise price of an option at expiration, the investor incurs an opportunity cost when implementing a covered call hedge strategy. The protective put hedging strategy is not subject to this opportunity cost. 25. A friend, Audrey, holds a portfolio of 10,000 shares of Microsoft stock. In 60 days she needs at least $855,000 to pay for her new home. You suggest to Audrey that she can construct an insured portfolio using Microsoft stock options. You explain that an insured portfolio can be constructed several different ways, but the basic notion is to create a portfolio that consists of a long position in Microsoft's stock and a long position in put options written on Microsoft. If at the end of the hedge period Microsoft's stock is trading at a price below the strike price on the put option, Audrey has the right to sell her Microsoft stock to the owner of the put option for the strike price. Thus, at the end of the hedge period, Audrey has sufficient assets to cover her needs or obligations. This hedging strategy can be implemented using traded options. However, Audrey may not be able to find an option with the desired strike price or expiration date. Dynamic hedging permits Audrey to overcome these limitations associated with traded option contracts. In dynamic hedging, Audrey constructs a portfolio that consists of a long position in stock and a long position in Treasury bills. As the underlying stock price changes, Audrey dynamically alters the allocation of assets in the portfolio between stock and Treasury bills. Therefore, we can view dynamic hedging as an asset allocation problem where Audrey determines how much of her resources are allocated to the stock, and how much of her resources are allocated to Treasury bills. The amount of resources available for investment is simply the current cash value of Audrey's stock position. The proportion of resources committed to the stock, w, are calculated as where S is the current stock price, P is the price of the relevant put option, and N(d1) comes from the Black—Scholes model. The proportion of resources committed to the Treasury bill is 1 — w.
A. Microsoft is currently trading at $90. The annualized riskfree interest rate on a 60day Treasury bill is 5 percent. The current volatility of Microsoft's stock is 0.32. Audrey wishes to create an insured portfolio. Since she needs $855,000 in 60 days, she decides to establish a position in a 60day Microsoft put option with an $85.50 strike price. Audrey calls her broker, who informs her there is no 60day Microsoft put option with a strike price of $85.50. Thus, Audrey must construct a dynamic hedge to protect the value of her investment in Microsoft stock. Using the BlackScholes model, calculate the value of a put option with an $85.50 strike price with 60 days to expiration. Determine the allocation of assets in Audrey's insured portfolio. That is, find the proportion of resources committed to Microsoft stock, w, and the proportion of resources committed to Treasury bills, 1 — w. Determine the dollar amount of her resources committed to Microsoft stock, and the dollar amount of her resources committed to Treasury bills.
ln(90/85.5) + ([0.05 + .5(0.32^{2})](0.1644))
d 1. = 0.5236
0.32 V(0.1644) d_{2} = 0.5236  0.32 V(0.1644) = 0.3938 N(d_{1}) = 0.699711 N(d_{2}) = 0.653146 c = 90 x 0.699711  85.5^°^{05x01644} x 0.653146 = $7.59 p = cS + Xe'^{iT}'^{)} = $7.59  $90 + $85.5a05XCU644 = $2.39 Audrey holds 10,000 shares of Microsoft with a market value of $90 per share, giving her a portfolio with a cash value of $900,000. Audrey will commit 68.16 percent of her resources to Microsoft stock and 31.84 percent of her resources to Treasury bills. Audrey holds $613,468 worth of Microsoft stock, $900,000 x 0.6816, and $286,532 worth of Treasury bills, $900,000 x 0.3184.
B. Twenty days later Microsoft is trading at $92. The annualized riskfree interest rate on a 40day Treasury bill is 5 percent, and the volatility of Microsoft's stock is 0.32. Using the Black—Scholes model, calculate the value of a put option with an $85.50 strike price with 40 days to expiration. Determine the allocation of assets in Audrey's insured portfolio. That is, find the proportion of resources committed to Microsoft stock, w, and the proportion of resources committed to Treasury bills, 1 — w.
ln(92/85.5) + ([0.05 + 0.5(0.32^{2})](0.1096))
d =. := 0.7964
0.32 V(0.1096) d_{2} = 0.7964  0.32 V(0.1096) = 0.6904 N(d1) = 0.787092 N(d_{2}) = 0.755041 c = 92 x 0.787092  85.5^°^{05x01096} x 0.755041 = $8.21 p = cS + Xe~'^{(T}^{t)} = $8.21  $92 + $85.5^{0}^{05} ^{x} °^{.1096} = $1.24 S + P $92+ $1.24 The proportion of resources committed to Treasury bills is 1  w = 1  0.7766 = 0.2234
C. Twenty days later Microsoft is trading at $86. The annualized riskfree interest rate on a 20day Treasury bill is 5 percent, and the volatility of Microsoft's stock is 0.32. Using the Black—Scholes model, calculate the value of a put option with an $85.50 strike price with 20 days to expiration. Determine the allocation of assets in Audrey's insured portfolio. That is, find the proportion of resources committed to Microsoft stock, w, and the proportion of resources committed to Treasury bills, 1 — w. In dynamic hedging, the investor's commitment to Microsoft stock increases as Microsoft's stock price increases and decreases as Microsoft's stock price decreases. With 60 days to the end of the hedge period, Audrey holds 68.16 percent of her resources in Microsoft stock. With 40 days to the end of the hedge period and Microsoft trading at $92, Audrey holds 77.66 percent of her resources in Microsoft stock. At 20 days to the end of the hedge period and Microsoft trading at $86, Audrey holds 54.63 percent of her resources in Microsoft stock. The trading strategy associated with dynamic hedging requires the investor to purchase additional Microsoft shares as the market price of the stock goes up and to sell additional Microsoft shares as the market price of the stock goes down. This buying of stock at successively higher prices and selling at successively lower prices is a direct cost of the insurance created.
American Option Pricing
An American option is just like a European option, except the American option carries the right of early exercise. Exercising a call before expiration discards the time value inherent in the option. The only offsetting benefit from early exercise arises from an attempt to capture a dividend. If there is no dividend, there is no incentive to early exercise, so the early exercise feature of an American call on a nondividend stock has no value.
2. Explain why American and European puts on a nondividend stock can have different values. The exercise value of a put is X— S. On a European put, this value cannot be captured until the expiration date. Therefore, before expiration, the value of the European put will be a function of the present value of these exercise proceeds: e~^{r{}^{T} ~ '\X— S). The American put gives immediate access at any time to the full proceeds, X— S, through exercise. In certain circumstances, notably on puts that are deepinthemoney with time remaining until expiration, this differential in exercise conditions can give the American put extra value over the corresponding European put, even in the absence of dividends.
3. Explain the circumstances that might make the early exercise of an American put on a nondividend stock desirable. Early exercise of an American put provides the holder with an immediate cash inflow of X  S. These proceeds can earn a return from the date of exercise to the expiration date that is not available on a European put. However, early exercise discards the time value of the put. Therefore, the early exercise decision requires trading off the sacrificed time value against the interest that can be earned by investing the exercise value from the date of exercise to the expiration date of the put. For deepinthemoney puts with time remaining until expiration, the potential interest gained can exceed the time value of the put that is sacrificed.
4. What factors might make an owner exercise an American call? The key factor is an approaching dividend, and exercise of an American call should occur only at the moment before an exdividend date. The dividend must be "large" relative to the share price, and the call will typically also be deepinthemoney.
5. Do dividends on the underlying stock make the early exercise of an American put more or less likely? Explain. Dividends make early exercise of an American put less likely. Dividends decrease the stock price and increase the exercise value of the put. Thus, the holder of the American put has an incentive to delay exercising and wait for the dividend payments.
6. Do dividends on the underlying stock make the early exercise of an American call more or less likely? Explain. Dividends increase the likelihood of early exercise on an American call. In fact, if there are no dividends on the underlying stock, early exercise of an American call is irrational.
7. Explain the strategy behind the pseudoAmerican call pricing strategy. In pseudoAmerican call pricing, the analysis treats the stock price as the current stock price reduced by the present value of all dividends to occur before the option expires. It then considers potential exercise just prior to each exdividend date, by reducing the exercise price by the present value of all dividends to be paid, including the imminent dividend. (The dividends are a reduction from the exercise price because they represent a cash inflow if the option is exercised.) For each dividend date, the analysis values a European option using the Black—Scholes model. The pseudoAmerican price is the maximum of these European option prices. Implicitly, the pricing strategy assumes exercise on the date that gives the highest European option price.
8. Consider a stock with a price of $140 and a standard deviation of 0.4. The stock will pay a dividend of $2 in 40 days and a second dividend of $2 in 130 days. The current riskfree rate of interest is 10 percent. An American call on this stock has an exercise price of $150 and expires in 100 days. What is the price of the call according to the pseudoAmerican approach? First, notice that the second dividend is scheduled to be paid in 130 days, after the option expires. Therefore, the second dividend cannot affect the option price and it may be disregarded. To apply the pseudoAmerican model, we begin by subtracting the present value of the dividend from the stock price to form the adjusted stock price: Adjusted Stock Price = $140  $2_{(?}°^{10}(^{40/365}) = $138.02 For the single dividend date, we reduce the exercise price by the $2 of dividend so the adjusted exercise price is $148. Applying the BlackScholes model with S = $138.02, E = $148, and T  t = 40 days gives a price of $4.05. Applying the BlackScholes model with S = $138.02, E = $150, and T  t = 100 gives a price of $8.29. The higher price, $8.29, is the pseudoAmerican option price.
9. Could the exact American call pricing model be used to price the option in question 8? Explain. Yes. Once we notice that the second dividend falls beyond the expiration date of the option, the exact American model fits exactly and gives a price of $8.28, almost the same as the pseudoAmerican price of$8.29.
10. Explain why the exact American call pricing model treats the call as an "option on an option." The exact American model applies to call options on stocks with a single dividend occurring before the option expires. Early exercise of an American call is optimal only at the exdividend date. At the exdividend date, the holder of an American call has a choice: exercise and own the stock or do not exercise and hold what is then equivalent to a European option that expires at the original expiration date of the American call. (The option that results from not exercising is equivalent to a European call because there are no more dividends occurring before expiration.) Thus, the exact American call model recognizes that the call embodies an option to own a European option at the dividend date. It also embodies the right to acquire the stock at the stated exercise price at the exdividend date.
11. Explain the idea of a bivariate cumulative standardized normal distribution. What would be the cumulative probability of observing two variables both with a value of zero, assuming that the correlation between them was zero? Explain. The bivariate cumulative distribution considers the probability of two standardized normal variates having values equal to or below a certain threshold at the same time given a certain correlation between the two. Consider first a univariate standardized normal variate. The probability of its value being zero or less equals the chance that it is below its mean of zero, which is 50 percent. Considering two such variates, with a zero correlation between them, the probability that both have a value of zero or less equals 0.5 X 0.5 = 0.25. If the two variables had a correlation other than zero, this probability would be different.
12. In the exact American call pricing model, explain why the model can compute the call price with only one dividend. The exact American model uses the cumulative bivariate standardized normal distribution, which considers the correlation between a pair of variates. The formula, for example, evaluates the probability of not exercising and the option finishing inthemoney, and of not exercising and the option finishing outofthemoney. If there were more dividends, the bivariate distribution would be inadequate to handle all of the possible combinations, and higher multivariate normal distributions would have to be considered. For these, no solution has yet been found.
13. What is the critical stock price in the exact American call pricing model? The critical stock price, S*, is the stock price that makes the call owner indifferent regarding exercise at the exdividend date. If the option is not exercised at the exdividend date, the American call effectively becomes a European call and the value is simply given by the BlackScholes model. If the owner exercises, she receives the stock price, plus the dividend, less the exercise price. Therefore, where D_{1} is the dividend, the critical stock price makes the following equation hold: S* + D_{1}  X = European Call
14. Explain how the analytical approximation for American option values is analogous to the Merton model. Both models pertain to underlying goods with a continuous dividend rate.
15. Explain the role of the critical stock price in the analytic approximation for an American call. The critical stock price is the stock price that makes the owner of an American call indifferent regarding exercise. If the stock price exceeds the critical stock price, the owner should exercise. Otherwise, the option should not be exercised.
16. Why should an American call owner exercise if the stock price exceeds the critical price? If the stock price exceeds the critical stock price, the owner should exercise to capture the exercise proceeds. These can be invested to earn a return from the date of exercise to the expiration of the option. The critical stock price is the price at which the benefits of earning that interest just equal the costs of discarding the time value of the option. If the stock price exceeds the critical stock price, the potential interest proceeds are worth more than the time value of the option, and the option should be exercised.
17. Consider the binomial model for an American call and put on a stock that pays no dividends. The current stock price is $120, and the exercise price for both the put and the call is $110. The standard deviation of the stock returns is 0.4, and the riskfree rate is 10 percent. The options expire in 120 days. Model the price of these options using a fourperiod tree. Draw the stock tree and the corresponding trees for the call and the put. Explain when, if ever, each option should be exercised. What is the value of a European call in this situation? Can you find the value of the European call without making a separate computation? Explain. U = 1.1215; D = 0.8917; iTy = 0.5073. The American call is worth $18.93, while the American put is worth $5.48. With no dividend, the American call should not be exercised at any time. The put should be exercised if the stock price drops three times from $120.00 to $85.07. Then the exercisable proceeds would be $24.93, but the corresponding European put would be worth only $24.03. The asterisk in the option tree indicates a node at which exercise should occur.
18. Consider the binomial model for an American call and put on a stock whose price is $120. The exercise price for both the put and the call is $110. The standard deviation of the stock returns is 0.4, and the riskfree rate is 10 percent. The options expire in 120 days. The stock will pay a dividend equal to 3 percent of its value in 50 days. Model and compute the price of these options using a fourperiod tree. Draw the stock tree and the corresponding trees for the call and the put. Explain when, if ever, each option should be exercised. U= 1.1215; D = 0.8917; tt_{v} = 0.5073. The call is worth $16.14, and the put is worth $6.28. The call should never be exercised. The put should be exercised if the stock price drops three straight times to $82.52. This gives exercisable proceeds of $27.48, compared to a computed value of $26.58. The asterisk in the option tree indicates a node at which exercise should occur.
19. Consider the binomial model for an American call and put on a stock whose price is $120. The exercise price for both the put and the call is $110. The standard deviation of the stock returns is 0.4, and the riskfree rate is 10 percent. The options expire in 120 days. The stock will pay a $3 dividend in 50 days. Model and compute the price of these options using a fourperiod tree. Draw the stock tree and the corresponding trees for the call and the put. Explain when, if ever, each option should be exercised. U= 1.1215; D = 0.8917; tt_{v} = 0.5073. The call is worth $16.63, while the put is worth $6.14. The call should never be exercised. The put should be exercised if the stock price drops three straight times to $82.97. This gives exercisable proceeds of $27.03, which exceeds the computed value of $26.13. The asterisk in the option tree indicates a node at which exercise should occur.
20. Consider the analytic approximation for American options. A stock sells for $130, has a standard deviation of 0.3, and pays a continuous dividend of 3 percent. An American call and put on this stock both have an exercise price of $130, and they both expire in 180 days. The riskfree rate is 12 percent. Find the value of the call and put according to this model. Demonstrate that you have found the correct critical stock price for both options. For the call, the critical price is S* = $604.08. For the put, the critical price is S** = $103.88. To verify that these critical prices are correct, we need to show that they satisfy the following two equations After constructing the stock price tree, the prices in the tree must be adjusted upward by the present value of the dividend yet to be received. That is, we must add the present value of $5.00 to be received in 40 days to the stock prices at node one, and we must add the present value of $5.00 to be received in 10 days to the stock prices at node two. Thus, we add $4.9050 to the node zero stock price, $4.9455 to the node one stock prices, and $4.9863 to the node two stock prices. = $5.00 x e~^{0A} ^{x} ^{(70/365)} = $4.9050 = $5.00 x e~^{0A} ^{x} ^{(40/365)} = $4.9455 = $5.00 x e~°^{A} ^{x} ^{(10/365)} = $4.9863
B .Calculate the values of European and American call and put options written on this stock. Value the options using the recursive procedure. Construct the price trees for each option.
C. Compare the prices of the European and American options. How much value does the right to exercise the option before expiration add to the value of the American options? In both cases the American options are more valuable than the equivalent European options. The American call is worth $.71 more than the European call, and the American put is worth $.29 more than the European put.
D. Explain when, if ever, each option should be exercised. Theory tells us that it will only be rational for the investor to exercise an American call option immediately before a dividend is paid, and that the rational exercise of an American put will occur immediately after a dividend is paid. The dividend will be paid in 70 days, which is between the second and third branches in the stock price tree used to value the options. Examination of both the stock price and option pricing trees reveals the following: The call option should be exercised early if the stock price rises to $94.18 after two periods. At this stock price, the intrinsic value of the American option, $14.18, is greater than the value of the European option, $11.58. The put option should be exercised if the stock price falls to $68.91 or lower. It should also be exercised at a stock price of $58.02 or less after three periods. At this stock price, $68.91, the intrinsic value of the American put option, $11.09, is greater than the value of the European option, $10.44.
23.
Consider a stock with a price of $70 and a standard deviation of 0.4.
The stock will pay a dividend of $2 in 40 days and a second dividend of
$2 in 130 days. The current riskfree rate of interest is 10 percent.
An American call on this stock has an exercise price of $75 and expires
in 180 days. What is the price of the call according to the
pseudoAmerican approach? Theory
suggests that the early exercise of a call will occur immediately
before a dividend. The pseudoAmerican pricing "model" views each
dividend date as a potential date for early exercise and estimates the
value of the American call by evaluating a portfolio of European call
options. Because there are two dividends paid during the life of the
option, we must determine the value of three European call options to
price this call option using the pseudoAmerican pricing methodology.
The valuation technique requires an adjustment to the current stock
price equal to the present value of all dividends to be received over
the life of the option. In addition, at each potential early exercise
date, that is, the dividend date, we decrease the strike price of the
option by the present value of the dividend yet to be received. In
other words, for the option that expires in forty days, we reduce the
strike price of the option by $2 for the dividend to be paid that day,
and the present value of the $2 dividend that will be paid 90 days in
the future. The estimated value of the American call option is equal to
the value of the European call option with the largest value. To
calculate the value of the European options using the BlackScholes
model, we must adjust the current stock price of $70 downward by the
present value of the dividends to be received before the option's
expiration. In this problem, both dividends are paid before the
option's expiration. The first dividend will be received in 40 days,
and the second will be received in 130 days. DiXefO = $2 x „ei x(40/365) _{=} _{$L98 }DiXe^Q = $2 x ^ix(130/365) _{=} _{$L93 }S' = $70  $1.98  $1.93 =$66.09 Option #1 that expires in 180 days ln(66.09/75) + ([0.10 + 0.5(0.4^{2})](0.4932)) d, =, = —0.1341 0.40 V(0.4932) d_{2} = 0.1341  0.40 V(0.4932) =  0.4150 N(d_{1}) = 0.446650 N(d_{2}) = 0.339061 c = 66.09 x 0.446650  75^{lx} ^{0.}4^{932} x 0.339061 = $5.31 Option #2 that expires in 40 days To
calculate the value of the European option that expires in 40 days
using the Black—Scholes model, we must decrease the strike price of the
option, $75, by the present value of the dividends to be received after 40
days, but before the option's expiration. In this case, both dividends
are paid before the option's expiration. The first dividend will be
received immediately and is equal to $2, and the second $2 dividend
will be received in 90 days. n V„rtTf) Mv „0.1 x (90/365) — »1 OC X = $75 $2 $1.95 = $71.05 _{d} _ ln(66.09/71.05) + ([0.10 + 0.5(0.4^{2})](0.1096)) _ _{Q} 0 0.40 V(0.1096) d_{2} =  0.3972  0.40 V(0.1096) =  0.5296 N(d_{1}) = 0.345612 N(d_{2}) = 0.298190 c = 66.09 x 0.345612  71.05^°'^{x01096} x 0.298190 = $1.89 Option #3 that expires in 130 days To
calculate the value of the European option that expires in 130 days
using the Black—Scholes model, we must decrease the strike price of the
option, $75, by the amount of the dividends to be received prior to the
option's expiration. In this case, the second dividend of $2 is paid on
day 130. Thus, the strike price of $75 will be reduced by $2 to $73. X = $75  $2 = $73 ln(66.09/73) + ([0.10 + 0.5(0.4^{2})](0.3562)) ^{1} 0.40 V(0.3562) d_{2} = 0.1479  0.40 V(0.3562) =  0.3866 N(d_{1}) = 0.441212 N(d_{2}) = 0.349521 c = 66.09 x 0.441212  73^{x} °^{0}.^{3562} x 0.349521 = $4.54 The value of the American call using the pseudoAmerican pricing methodology is the largest of the three option values, $5.31. C = MAX ($5.31, $1.89, $4.54) = $5.31
24. Consider a stock with a price of $140 and a standard deviation of 0.4. The stock will pay a dividend of $5 in 40 days and a second dividend of $5 in 130 days. The current riskfree rate of interest is 10 percent. An American call on this stock has an exercise price of $150 and expires in 100 days.
A. What is the price of the call according to the pseudoAmerican approach? Theory
suggests that the early exercise of a call will occur immediately
before a dividend. The pseudoAmerican pricing "model" views each
dividend date as a potential date for early exercise and estimates the
value of the American call by evaluating a portfolio of European call
options. Since there is only one dividend paid during the life of the
option, we must determine the value of two European call options to
price this call option using the pseudoAmerican pricing methodology.
The valuation technique requires an adjustment to the current stock
price equal to the present value of all dividends to be received over
the life of the option. In addition, at each potential early exercise
date, that is, the first dividend date, we decrease the strike price of
the option by the present value of the dividend yet to be received
before the option expires. The estimated value of the American call
option is equal to the value of the European call option with the
largest value. To
calculate the value of the European options using the Black—Scholes
model, we must adjust the current stock price of $140 down ward by the
present value of the dividends to be received before the option's
expiration. In this problem, the second dividend is paid after the
option expires and is irrelevant in the pricing of this option. The
dividend of $5 will be received in 40 days. D_{l}Xe^{T{T}'^{)} = $5 x ^o.ix(40/365) _{=} _{$4} _{95 }S' = $140  $4.95 = $135.05 Option #1 that expires in 100 days ln(135.05/150) + ([0.10 + 0.5(0.4^{2})](0.2740)) d =. := — 0.2658 0.40V(0.2740) d_{2} = 0.2658  0.40 V(0.2740) = 0.4751 N(d1) = 0.395212 N(d_{2}) = 0.317348 c = 135.05 x 0.395212  150^°'^{x02740} x 0.317348 = $7.06 Option #2 that expires in 40 days To
calculate the value of the European option that expires in 40 days
using the BlackScholes model, we must decrease the strike price of the
option, $150, by the amount of the dividends to be received after day
40 but before the option expires. In this case, the second dividend of
$5 is paid on day 40. Thus, the strike price of $150 will be reduced by
$5 to $145. X' = $150$5 = ln(135.05/145) + ([0.10 + 0.5(0.4^{2})](0.1096)) d = :0.3876 0.40 V(0.1096) d_{2} = 0.3876  0.40 V(0.1096) = 0.5201 N(d1) = 0.349143 N(d_{2}) = 0.301514 c = 135.05 x 0.349143  \ASe~^{0A} ^{xCU096} x 0.301514 = $3.91 The value of the American call using the pseudoAmerican pricing methodology is the largest of the two option values, $7.06. C = MAX ($7.06, $3.91) = $7.06 B. What is the price of the call according to the compound option pricing model? The first step necessary to value this American call option is to determine the critical stock price, S*. The
critical stock price is the stock price that makes the investor
indifferent between holding an option until expiration, and exercising
the option—thereby receiving the stock and the dividend. The critical
stock price, S^{*}, is determined by solving the following equation, S* + D — X = c, where the European call option, c, has
a life of 60 days beginning 40 days in the future. That is, if the
American call option is not exercised on the dividend date, then the
investor holds an American call option written on a stock that does not
pay a dividend. We can then value the American call option as a
European call option. This option has 60 days until expiration, and we
assume that the interest rate and the volatility of the stock remain
constant. The critical stock price, S*, is $170.90. N(b_{1}) = 0.051681 N(b_{2}) = 0.039104 jV_{2}[0.2658; 1.6288; 0.6325] = 0.348740 jV_{2}[0.4751; 1.7612; 0.6325] = 0.283474 The price of the American call using the compound option pricing model is $7.10.
C. What is the critical stock price, S*? Discuss the implications of this finding on the likelihood of exercising the call option early. As computed earlier, the critical stock price is $170.90, which is $30.90 higher than the current stock price of $140. Thus, it is highly unlikely that the call option will be exercised early. That is, it is very unlikely that the stock price will rise $30.90 in forty days. Thus, the value of the early exercise premium is very small.
D. Compare the prices calculated using the three option pricing methods. Since the critical stock price of $170.90 is $30.90 higher than the current stock price of $140, it is highly unlikely that the call option will be exercised early. That is, it is very unlikely that the stock price will rise $30.90 in forty days. In this problem, the approximations provided by the pseudoAmerican call option pricing model and the dividendadjusted Black—Scholes European call pricing model are very close to the value produced by the compound option pricing model. Consequently, the prices for the American call options calculated using the three different option pricing models differ by four cents. Thus, the value of the right to exercise this option early is very small.
25. Consider the binomial model for an American call and put on a stock that pays no dividends. The current stock price is $120, and the exercise price for both the put and the call is $110. The standard deviation of the stock returns is 0.4, and the riskfree rate is 10 percent. The options expire in 120 days. Model the price of these options using a fourperiod tree.
B. What is the value of each of the two options? Value the options using the recursive procedure. The value of the American call option is $18.93, and the American put option is $5.48.
C. Explain when, if ever, each option should be exercised. Since the stock underlying the options does not pay dividends, it would never be rational to exercise the American call early. However, this is not true for the American put option. If the stock price falls to $85.07, then the investor should exercise the option.
D. What is the value of a European call written on this stock? Can you find the value of the European call without making a separate computation? Explain. Since exercising the American call early is not rational, the right to exercise the option before expiration is worthless. Thus, the price of the European call is the same as the price of the American call, $18.93.
26. Consider the binomial model for an American call and put on a stock whose price is $50. The exercise price for both the put and the call is $55. The standard deviation of the stock returns is 0.35, and the riskfree rate is 10 percent. The options expire in 160 days. The stock will pay a dividend equal to 4 percent of its value in 65 days. Model and compute the price of these options using a fourperiod tree. The stock prices in the tree must be adjusted for the dividend to be paid in 65 days before calculating the value of the options. Therefore, the stock prices in the tree in periods two, three, and four must be adjusted downward by one minus the dividend yield paid by the firm (1 — 4%). Dividendadjusted stock price tree The price of the American call option is $2.89, and the price of the American put option is $7.94.
C. Explain when, if ever, each option should be exercised. Theory tells us that it will only be rational for the investor to exercise an American call option immediately before a dividend is paid, and that the rational exercise of an American put will occur immediately after a dividend is paid. The dividend will be paid in 65 days, which is between the first and second branches in the stock price tree used to value the options. Examination of both the stock price and option pricing trees reveals the following. The call option should not be exercised early. The put option should be exercised if the stock price falls to $42.75 or lower after two periods. At the stock prices of $42.75, $38.07, and $33.91, the intrinsic values of the put options are greater than the value of the corresponding options, and the option should be exercised early if the stock price falls to these levels.
Option on Stock Indexes, Foreign Currency and Futures
6. For a call and a put option on a foreign currency, compute the Merton model price, the binomial model price for a European option with three periods, the BaroneAdesi and Whaley model price, and the binomial model price with three periods for American options. Data are as follows: The foreign currency value is 2.5; the exercise price on all options is 2.0; the time until expiration is 90 days; the riskfree rate of interest is 7 percent; the foreign interest rate is 4 percent; the standard deviation of the foreign currency is .2. The prices in the following table show that the American call and put have no exercise potential. The difference between the Merton and Whaley model put prices, on the one hand, and the binomial model put prices, on the other, is due to the very few periods being employed.
Merton Model European Binomial Whaley Model American Binomial
Call .5104 .5097 .5104 .5097
Put .0008 0 .0008 0
7. Consider a call and a put on a stock index. The index price is 500.00, and the two options expire in 120 days. The standard deviation of the index is .2, and the riskfree rate of interest is 7 percent. The two options have a common exercise price of 500.00. The stock index will pay a dividend of 20.00 index units in 40 days. Find the European and American option prices according to the binomial model, assuming two periods. Be sure to draw the lattices for the stock index and for all of the options that are being priced.
£7=^=1.0845; D = ^^ = .9221 _ 1.0116.9221 _ ,,_{1fl }^"1.0845.9221 ~^{SS1}°
As the following price lattices show, the European and American calls are worth $19.20. The European put is worth $27.67, and the American put is worth $30.21.
8. Consider two European calls and two European put options on a foreign currency. The exercise prices are $.90 and $1.00, giving a total of four options. All options expire in one year. The current riskfree rate is 8 percent, the foreign interest rate is 5 percent, and the standard deviation of the foreign currency is .3. The foreign currency is priced at $.80. Find all four option prices according to the Merton model. Compare the ratio of the option prices to the ratio of the exercise prices. What does this show? With X = $.90, the call is worth $.0640, and the put is worth $.1338. With X = $1.00, the call is worth $.0392, and the put is worth $.2014. Comparing the ratio of the option prices to the ratio of exercise prices shows that the call is more sensitive to a change in the exercise price than is the put. A change of about 10 percent in the exercise price causes about a 63 percent change in the call price, but only about a 51 percent change in the put price.
The Options Approach to Corporate Securities
1. Explain why common stock is itself like a call option. In the option analysis of common stock, what plays the role of the exercise price and what plays the role of the underlying stock? Common stock is like a call option on the entire firm. To see how this can be the case, consider a firm with a single bond issue outstanding and assume that the bond is a pure discount bond. When the bond matures, the common stockholders have a choice: They can pay the bondholders the promised payment, or they can surrender the firm to the bondholders. If the firm is worth more than the amount due to the bondholders, the stock owners will pay the bondholders and keep the excess. If the firm is worth less than the amount due to the bondholders, the stock owners will abandon the firm to the bond owners. In this situation, the amount due to the bond owners plays the role of the exercise price. The maturity date of the bond is the expiration date of the call option represented by the common stock. The common stock is like a call option. At expiration, the stock owners can exercise their call option by paying the claim of the bondholders (the exercise price). Upon exercising, the stockholders receive the underlying asset (the entire firm).
2. Consider a firm that issues a pure discount bond that matures in one year and has a face value of $1,000,000. Analyze the payoffs that the bondholders will receive in option pricing terms, assuming the only other security in the firm is common stock. When the bond matures, the stock owners decide whether to pay the bonds or surrender the firm to the bondholders in lieu of payment. If the value of the firm exceeds the amount owed to the bond owners, $1,000,000, the bondholders receive full payment and the stock owners retain the excess. If the firm's value is less than the promised payment, the stock owners abandon the firm and the bondholders receive a payment equal to the value of the entire firm. However, by hypothesis, this is less than the promised payment of $1,000,000. This pattern of payment is like the payments on a short put position with an exercise price that equals the face value of the bond. However, a short position in a put can give a payoff at expiration that is negative. This is not true of a bond. The worst payoff for the bond is zero. Therefore, the payoff has the same pattern as a short position in a put with an exercise price that equals the face value of the bond plus a long position in a riskless bond.
3. Consider a firm with common stock and a pure discount bond as its financing. The total value of the firm is $1,000,000. There are 10,000 shares of common stock priced at $70 per share. The bond matures in ten years and has a total face value of $500,000. What is the interest rate on the bond, assuming annual compounding? Would the interest rate become higher or lower if the volatility of the firm's cash flows increases? The $1,000,000 value of the firm equals the sum of the stock and bond values. As the outstanding stock is worth $700,000, the bonds must be worth $300,000. Therefore, the interest rate is 5.24 percent. If the volatility of the firm's cash flows increases, the total value of the firm will not change. However, because the common stock can be analyzed as a call option on the firm, the value of the common stock must increase. This means that the value of the bonds must decrease. If the bond value decreases, its yield must increase. This makes sense, because the bonds should be worth less if the firm's cash flows become more risky.
4. A firm has a capital structure consisting of common stock and a single bond. The managers of the firm are considering a major capital investment that will be financed from internally generated funds. The project can be initiated in two ways, one with a high fixed cost component and the other with a low fixed cost component. Although both technologies have the same expected value, the high fixed cost approach has the potential for greater payoffs. (If the product is successful, the high fixed cost approach gives much lower total costs for large production levels.) What does option theory suggest about the choice the managers should make? Explain. Assuming that the managers perform in the interest of their shareholders, they must make the decision that increases the value of the stock. As the stock represents a call option on the total firm value, the managers should prefer the higher operating leverage/higher operating risk strategy.
5. In a firm with common stock, senior debt, and subordinated debt, assume that both debt instruments mature at the same time. What is the necessary condition on the value of the firm at maturity for each security holder to receive at least some payment? With two classes of debt, does option theory counsel managers to increase the riskiness of the firm's operations? Would there be any difference on this point between a firm with a single debt issue and two debt issues? Which bondholders would tend to be more riskaverse as far as choosing a risk level for the firm's operations? Explain. For the senior debtholders to receive some payment, the value of the firm must exceed zero. For the subordinated debtholders to receive some payment, the value of the firm must exceed the total owed to the senior debtholders. For the common stockholders to receive any payment, the value of the firm must exceed the amount owed on both classes of debt. If the managers perform in the interest of the stockholders, the mere presence of two classes of debt does not suggest a change in operating policy. The stockholders get paid only after all the bondholders are paid, so it does not matter to the stockholders how the debt is split up, but only how much the total amount of debt payments is. Given that the junior debtholders have already purchased the junior debt, they are (by revealed preference) more risktolerant than the holders of the senior debt. However, increasing operating risk transfers wealth away from bondholders to stockholders. Thus, the junior debtholders would probably prefer a lowrisk operating strategy if funds would be certain to sufficiently cover their holdings. However, consider an operating policy that would only generate enough cash to pay the senior debtholders. In this situation, it is clear that the junior debtholders would prefer a more risky operating policy that might give sufficient payoffs to repay their obligations.
6. Consider a firm financed solely by common stock and a single callable bond issue. Assume that the bond is a pure discount bond. Is there any circumstance in which the firm should call the bond before the maturity date? Would such an exercise of the firm's call option discard the time premium? Explain. The stockholders should wait until the maturity date. The stockholders' situation here is analogous to a call on a nondividend stock. Early payment of the bond discards the time premium inherent in the option they hold.
7. Consider a firm financed only by common stock and a convertible bond issue. When should the bondholders exercise? Explain. If the common shares pay a dividend, could it make sense for the bondholders to exercise before the bond matures? Explain by relating your answer to our discussion of the exercise of American calls on dividendpaying stocks If the common stock pays no dividend, the bondholders should not exercise until the last possible date. However, if the stock pays a sufficiently large dividend, it might pay the bondholders to convert earlier. The bondholder holds a call option on the firm's shares. If those shares pay dividends, then they are leaking value. The bondholders must decide whether it is worthwhile to discard the time premium in favor of securing the dividend