HullFund8eCh11ProblemSolutions(2)

Problem 11.20

A trader creates a bear spread by selling a six-month put option with a $25 strike price for $2.15 and buying a six-month put option with a $29 strike price for $4.75. What is

the initial investment? What is the total payoff when the stock price in six months is (a) $23, (b) $28, and (c) $33.

The initial investment is $2.60. (a) $4, (b) $1, and (c) 0.

Problem 11.21

A trader sells a strangle by selling a call option with a strike price of $50 for $3 and selling a put option with a strike price of $40 for $4. For what range of prices of the underlying asset does the trader make a profit?

The trader makes a profit if the total payoff is less than $7. This happens when the price of the asset is between $33 and $57.

Problem 11.22.

Three put options on a stock have the same expiration date and strike prices of $55, $60, and $65. The market prices are $3, $5, and $8, respectively. Explain how a butterfly spread can be created. Construct a table showing the profit from the strategy. For what range of stock prices would the butterfly spread lead to a loss?

A butterfly spread is created by buying the $55 put, buying the $65 put and selling two of the $60 puts. This costs 3?8?2?5?$1 initially. The following table shows the profit/loss from the strategy. Stock Price Payoff Profit 0 ?1 ST?65 60?ST?65 55?ST?60 65?ST 64?ST ST?55

The butterfly spread leads to a loss when the final stock price is greater than $64 or less than $56.

Problem 11.23.

A diagonal spread is created by buying a call with strike price K2 and exercise date T2 and selling a call with strike price K1 and exercise date T1(T2?T1). Draw a diagram showing the the value of the spread at time T1 when (a) K2?K1 and (b) K2?K1.

There are two alternative profit patterns for part (a). These are shown in Figures S11.2 and S11.3. In Figure S11.2 the long maturity (high strike price) option is worth more than the short maturity (low strike price) option. In Figure S11.3 the reverse is true. There is no ambiguity about the profit pattern for part (b). This is shown in Figure S11.4.

ST?55 0 ST?56 ?1

ProfitSTK1K2

Figure S11.2: Investor’s Profit/Loss in Problem 11.20a when long maturity call is worth more than short maturity call

ProfitSTK1K2

Figure S11.3 Investor’s Profit/Loss in Problem 11.20b when short maturity call is worth more than long maturity call

ProfitSTK2K1

Figure S11.4 Investor’s Profit/Loss in Problem 11.20b

Problem 11.24.

Draw a diagram showing the variation of an investor’s profit and loss with the terminal stock price for a portfolio consisting of

a. One share and a short position in one call option b. Two shares and a short position in one call option c. One share and a short position in two call options d. One share and a short position in four call options

In each case, assume that the call option has an exercise price equal to the current stock price.

The variation of an investor’s profit/loss with the terminal stock price for each of the four strategies is shown in Figure S11.5. In each case the dotted line shows the profits from the components of the investor’s position and the solid line shows the total net profit.

ProfitProfitK(a)STK(b)STProfitProfitK(c)STK(d)ST

Figure S11.5 Answer to Problem 11.21

Problem 11.25.

Suppose that the price of a non-dividend-paying stock is $32, its volatility is 30%, and the risk-free rate for all maturities is 5% per annum. Use DerivaGem to calculate the cost of setting up the following positions. In each case provide a table showing the relationship between profit and final stock price. Ignore the impact of discounting.

a. A bull spread using European call options with strike prices of $25 and $30 and a maturity of six months.

b. A bear spread using European put options with strike prices of $25 and $30 and a maturity of six months

c. A butterfly spread using European call options with strike prices of $25, $30, and $35 and a maturity of one year.

d. A butterfly spread using European put options with strike prices of $25, $30, and $35 and a maturity of one year.

e. A straddle using options with a strike price of $30 and a six-month maturity. f. A strangle using options with strike prices of $25 and $35 and a six-month maturity.

(a) A call option with a strike price of 25 costs 7.90 and a call option with a strike price of 30 costs 4.18. The cost of the bull spread is therefore 7?90?4?18?3?72. The profits ignoring the impact of discounting are

Stock Price Range ST?25 Profit ?3?72 25?ST?30 ST?30 ST?28?72 1.28 (b) A put option with a strike price of 25 costs 0.28 and a put option with a strike price of 30 costs 1.44. The cost of the bear spread is therefore 1?44?0?28?1?16. The profits ignoring the impact of discounting are Stock Price Range Profit ?3?84 ST?25 25?ST?30 ST?30 28?84?ST ?1?16 (c) Call options with maturities of one year and strike prices of 25, 30, and 35 cost 8.92, 5.60, and 3.28, respectively. The cost of the butterfly spread is therefore

8?92?3?28?2?5?60?1?00. The profits ignoring the impact of discounting are Stock Price Range Profit ?1?00 ST?25 25?ST?30 30?ST?35 ST?26?00 34?00?ST (d) Put options with maturities of one year and strike prices of 25, 30, and 35 cost 0.70, 2.14, 4.57, respectively. The cost of the butterfly spread is therefore 0?70?4?57?2?2?14?0?99. Allowing for rounding errors, this is the same as in (c). The profits are the same as in (c).

(e) A call option with a strike price of 30 costs 4.18. A put option with a strike price of 30 costs 1.44. The cost of the straddle is therefore 4?18?1?44?5?62. The profits ignoring the impact of discounting are Stock Price Range Profit 24.38?ST ST?30 ST?30 ST?35?62

(f) A six-month call option with a strike price of 35 costs 1.85. A six-month put option with a strike price of 25 costs 0.28. The cost of the strangle is therefore 1?85?0?28?2?13. The profits ignoring the impact of discounting are

Stock Price Range Profit ST?25 22?87?ST ?2.13 25?ST?35 ST?35 ST?37?13

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