Options, Futures, and Other Derivatives (10th Edition) 作业二

Problem 10.11.

Describe the terminal value of the following portfolio: a newly entered-into long forward contract on an asset and a long position in a European put option on the asset with the same maturity as the forward contract and a strike price that is equal to the forward price of the asset at the time the portfolio is set up. Show that the European put option has the same value as a European call option with the same strike price and maturity. 

Problem 10.12.

A trader buys a call option with a strike price of $45 and a put option with a strike price of $40. Both options have the same maturity. The call costs $3 and the put costs $4. Draw a diagram showing the variation of the trader’s profit with the asset price. 

Problem 11.11.

A four-month European call option on a dividend-paying stock is currently selling for $5. The stock price is $64, the strike price is $60, and a dividend of $0.80 is expected in one month. The risk-free interest rate is 12% per annum for all maturities. What opportunities are there for an arbitrageur? 

Problem 11.13.

Give an intuitive explanation of why the early exercise of an American put becomes more attractive as the risk-free rate increases and volatility decreases. 

The early exercise of an American put is attractive when the interest earned on the strike price is greater than the insurance element lost. When interest rates increase, the value of the interest earned on the strike price increases making early exercise more attractive. When volatility decreases, the insurance element is less valuable. Again this makes early exercise more attractive.

 

Problem 11.14.

The price of a European call that expires in six months and has a strike price of $30 is $2. The underlying stock price is $29, and a dividend of $0.50 is expected in two months and again in five months. Risk-free interest rates (all maturities) are 10%. What is the price of a European put option that expires in six months and has a strike price of $30? 

Problem 11.16.

The price of an American call on a non-dividend-paying stock is $4. The stock price is $31, the strike price is $30, and the expiration date is in three months. The risk-free interest rate is 8%. Derive upper and lower bounds for the price of an American put on the same stock with the same strike price and expiration date. 

Problem 11.18.

Prove the result in equation (11.7). (Hint: For the first part of the relationship consider (a) a portfolio consisting of a European call plus an amount of cash equal to  and (b) a portfolio consisting of an American put option plus one share.) 

Problem 12.8.

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. 

 

Problem 12.23.

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? 

 

Problem 12.25.

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

1. One share and a short position in one call option
2. Two shares and a short position in one call option
3. One share and a short position in two call options
4. 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. 

Problem 12.26.

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.

1. A bull spread using European call options with strike prices of $25 and $30 and a maturity of six months.
2. A bear spread using European put options with strike prices of $25 and $30 and a maturity of six months
3. A butterfly spread using European call options with strike prices of $25, $30, and $35 and a maturity of one year.
4. A butterfly spread using European put options with strike prices of $25, $30, and $35 and a maturity of one year.
5. A straddle using options with a strike price of $30 and a six-month maturity.
6. A strangle using options with strike prices of $25 and $35 and a six-month maturity.

In each case provide a table showing the relationship between profit and final stock price. Ignore the impact of discounting.

 

Problem 12.28. (Excel file)

Describe the trading position created in which a call option is bought with strike price K₁ and a put option is sold with strike price K₂ when both have the same time to maturity and K₂ > K₁. What does the position become when K₁ = K₂?