HullFund8eCh11ProblemSolutions

CHAPTER 11

Trading Strategies Involving Options

Practice Questions

Problem 11.8.

Use put–call parity to relate the initial investment for a bull spread created using calls to the initial investment for a bull spread created using puts.

A bull spread using calls provides a profit pattern with the same general shape as a bull

spread using puts (see Figures 11.2 and 11.3 in the text). Define p1 and c1 as the prices of put and call with strike price K1 and p2 and c2 as the prices of a put and call with strike price K2. From put-call parity

p1?S?c1?K1e?rT p2?S?c2?K2e?rT

Hence:

p1?p2?c1?c2?(K2?K1)e?rT

This shows that the initial investment when the spread is created from puts is less than the initial investment when it is created from calls by an amount (K2?K1)e?rT. In fact as mentioned in the text the initial investment when the bull spread is created from puts is negative, while the initial investment when it is created from calls is positive.

The profit when calls are used to create the bull spread is higher than when puts are used by (K2?K1)(1?e?rT). This reflects the fact that the call strategy involves an additional risk-free

investment of (K2?K1)e?rT over the put strategy. This earns interest of

(K2?K1)e?rT(erT?1)?(K2?K1)(1?e?rT).

Problem 11.9.

Explain how an aggressive bear spread can be created using put options.

An aggressive bull spread using call options is discussed in the text. Both of the options used have relatively high strike prices. Similarly, an aggressive bear spread can be created using put options. Both of the options should be out of the money (that is, they should have relatively low strike prices). The spread then costs very little to set up because both of the puts are worth close to zero. In most circumstances the spread will provide zero payoff. However, there is a small chance that the stock price will fall fast so that on expiration both options will be in the money. The spread then provides a payoff equal to the difference between the two strike prices, K2?K1.

Problem 11.10.

Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7,

respectively. How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that shows the profit and payoff for both spreads.

A bull spread is created by buying the $30 put and selling the $35 put. This strategy gives rise to an initial cash inflow of $3. The outcome is as follows:

Stock Price ST?35 Payoff 0 Profit 3 30?ST?35 ST?30 ST?35 ?5 ST?32 ?2

A bear spread is created by selling the $30 put and buying the $35 put. This strategy costs $3 initially. The outcome is as follows

Stock Price ST?35 Payoff 0 Profit ?3 30?ST?35 35?ST 5 32?ST 2 ST?30

Problem 11.11.

Use put–call parity to show that the cost of a butterfly spread created from European puts is identical to the cost of a butterfly spread created from European calls.

Define c1, c2, and c3 as the prices of calls with strike prices K1, K2 and K3. Define p1, p2 and p3 as the prices of puts with strike prices K1, K2 and K3. With the usual notation c1?K1e?rT?p1?S c2?K2e?rT?p2?S c3?K3e?rT?p3?S Hence c1?c3?2c2?(K1?K3?2K2)e?rT?p1?p3?2p2 Because K2?K1?K3?K2, it follows that K1?K3?2K2?0 and

c1?c3?2c2?p1?p3?2p2

The cost of a butterfly spread created using European calls is therefore exactly the same as the cost of a butterfly spread created using European puts.

Problem 11.12.

A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss?

A straddle is created by buying both the call and the put. This strategy costs $10. The

profit/loss is shown in the following table:

Stock Price ST?60 Payoff Profit ST?60 60?ST ST?70 50?ST ST?60

This shows that the straddle will lead to a loss if the final stock price is between $50 and $70.

Problem 11.13.

Construct a table showing the payoff from a bull spread when puts with strike prices K1 and K2 are used (K2?K1).

The bull spread is created by buying a put with strike price K1 and selling a put with strike price K2. The payoff is calculated as follows: Stock Price ST?K2 K1?ST?K2 Payoff from Long Put 0 0 Payoff from Short Put 0 Total Payoff 0 ST?K2 ST?K2 ?(K2?ST) ?(K2?K1) ST?K1 K1?ST

Problem 11.14.

An investor believes that there will be a big jump in a stock price, but is uncertain as to the direction. Identify six different strategies the investor can follow and explain the differences among them.

Possible strategies are:

Strangle Straddle Strip Strap

Reverse calendar spread Reverse butterfly spread

The strategies all provide positive profits when there are large stock price moves. A strangle is less expensive than a straddle, but requires a bigger move in the stock price in order to provide a positive profit. Strips and straps are more expensive than straddles but provide bigger profits in certain circumstances. A strip will provide a bigger profit when there is a large downward stock price move. A strap will provide a bigger profit when there is a large upward stock price move. In the case of strangles, straddles, strips and straps, the profit

increases as the size of the stock price movement increases. By contrast in a reverse calendar spread and a reverse butterfly spread there is a maximum potential profit regardless of the size of the stock price movement.

Problem 11.15.

How can a forward contract on a stock with a particular delivery price and delivery date be created from options?

Suppose that the delivery price isKand the delivery date is T. The forward contract is created by buying a European call and selling a European put when both options have strike price K and exercise date T. This portfolio provides a payoff of ST?K under all

circumstances where ST is the stock price at time T. Suppose that F0 is the forward price. If K?F0, the forward contract that is created has zero value. This shows that the price of a call equals the price of a put when the strike price is F0.

Problem 11.16.

“A box spread comprises four options. Two can be combined to create a long forward

position and two can be combined to create a short forward position.” Explain this statement.

A box spread is a bull spread created using calls and a bear spread created using puts. With the notation in the text it consists of a) a long call with strikeK1, b) a short call with strikeK2, c) a long put with strikeK2, and d) a short put with strikeK1. a) and d) give a long forward contract with delivery priceK1; b) and c) give a short forward contract with delivery priceK2. The two forward contracts taken together give the payoff ofK2?K1.

Problem 11.17.

What is the result if the strike price of the put is higher than the strike price of the call in a strangle?

The result is shown in Figure S11.1. The profit pattern from a long position in a call and a put is much the same when a) the put has a higher strike price than a call and b) when the call has a higher strike price than the put. But both the initial investment and the final payoff are much higher in the first case.

Figure S11.1 Profit Pattern in Problem 11.17

Problem 11.18.

One Australian dollar is currently worth $0.64. A one-year butterfly spread is set up using European call options with strike prices of $0.60, $0.65, and $0.70. The risk-free interest rates in the United States and Australia are 5% and 4% respectively, and the volatility of the

exchange rate is 15%. Use the DerivaGem software to calculate the cost of setting up the butterfly spread position. Show that the cost is the same if European put options are used instead of European call options.

To use DerivaGem select the first worksheet and choose Currency as the Underlying Type. Select Black-Scholes European as the Option Type. Input exchange rate as 0.64, volatility as 15%, risk-free rate as 5%, foreign risk-free interest rate as 4%, time to exercise as 1 year, and exercise price as 0.60. Select the button corresponding to call. Do not select the implied volatility button. Hit the Enter key and click on calculate. DerivaGem will show the price of the option as 0.0618. Change the exercise price to 0.65, hit Enter, and click on calculate again. DerivaGem will show the value of the option as 0.0352. Change the exercise price to 0.70, hit Enter, and click on Calculate. DerivaGem will show the value of the option as 0.0181. Now select the button corresponding to put and repeat the procedure. DerivaGem shows the values of puts with strike prices 0.60, 0.65, and 0.70 to be 0.0176, 0.0386, and 0.0690, respectively.

The cost of setting up the butterfly spread when calls are used is therefore

0?0618?0?0181?2?0?0352?0?0095

The cost of setting up the butterfly spread when puts are used is

0?0176?0?0690?2?0?0386?0?0094

Allowing for rounding errors, these two are the same.

Problem 11.19

An index provides a dividend yield of 1% and has a volatility of 20%. The risk-free

interest rate is 4%. How long does a principal-protected note, created as in Example 11.1, have to last for it to be profitable for the bank issuing it? Use DerivaGem.

Assume that the investment in the index is initially $100. (This is a scaling factor that makes no difference to the result.) DerivaGem can be used to value an option on the index with the index level equal to 100, the volatility equal to 20%, the risk-free rate equal to 4%, the dividend yield equal to 1%, and the exercise price equal to 100. For different times to

maturity, T, we value a call option (using Black-Scholes European) and calculate the funds available to buy the call option (=100-100e-0.04×T). Results are as follows: Time to Funds Available Value of Option maturity, T 1 3.92 9.32 2 7.69 13.79 5 18.13 23.14 10 32.97 33.34 11 35.60 34.91 This table shows that the answer is between 10 and 11 years. Continuing the calculations we find that if the life of the principal-protected note is 10.35 year or more, it is profitable for the bank. (Excel’s Solver can be used in conjunction with the DerivaGem functions to facilitate calculations.)

Further Questions

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