Problem Set 3
Options, Futures and Derivative Securities
Instructions: This problem set is due on 2/17 at 11:59 pm CST and is an individual assignment. All problems must be handwritten. Scan your work and submit a PDF file.
Problem 1 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 year. You have the following information on European call options written on the stock for different strikes and maturities:
| Strike | 25 | 30 | 35 |
|---|---|---|---|
| Expiration 6 months | 7.90 | 4.18 | 1.85 |
| Expiration 1 year | 8.92 | 5.60 | 3.28 |
Calculate the cost of setting up the following positions. In each case provide a diagram showing the relationship between payoff and profit with respect to final stock price. Use put-call parity to compute the price of the corresponding European put options. In your diagrams, indicate the cutoff prices that lead to a gain/loss.
- A bull spread using European call options with strike prices of $25 and $30 and a maturity of six months.
- A bear spread using European put options with strike prices of $25 and $30 and a maturity of six months.
- A butterfly spread using European call options with strike prices of $25, $30, and $35 and a maturity of one year.
- A butterfly spread using European put options with strike prices of $25, $30, and $35 and a maturity of one year.
- A straddle using options with a strike price of $30 and a six-month maturity.
- A strangle using options with strike prices of $30 and $35 and a six-month maturity.
Problem 2 A 4-month European put option on a non-dividend-paying stock is currently selling for $1.80. The stock price is $96, the strike price is $100, and the risk-free interest rate is 6% per year. What opportunities could an arbitrageur exploit? Think of synthesizing a negative price call.
Problem 3 A 6-month European call option on a non-dividend-paying stock is currently selling for $6.70. The stock price is $104, the strike price is $100, and the risk-free interest rate is 6% per year. What opportunities could an arbitrageur exploit? Think of synthesizing a negative price put.
Problem 4 A non-dividend paying stock trades for $200. The risk-free rate is 5% per year with continuous compounding. European call and put options with strike $200 and maturity 9 months trade for $15 and $8 per share, respectively. Is there an arbitrage opportunity? If so, how an arbitrageur would make a profit?
Problem 5 A non-dividend paying stock trades for $100. The risk-free rate is 6% per year with continuous compounding. European call and put options with strike $100 and maturity 6 months trade for $15 and $13 per share, respectively. Is there an arbitrage opportunity? If so, how an arbitrageur would make a profit?
Problem 6 The price of a non-dividend paying stock is $250. The risk-free rate is 5% per year with continuous compounding. Consider a European put option with strike price $280 and maturity 1 year. What should be the price of the put if the volatility of the stock returns is zero?
Problem 7 What should be the price of a six-month European call option written on a non-dividend-paying stock when the stock price is $160, the strike price is $150, the risk-free rate is 10% per year, and the volatility of the stock returns is zero?
Problem 8 Consider a non-dividend paying asset. There are American and European call and put options written on this asset and available for trade. The risk-free rate is positive for all maturities. Please say whether the following statements are true or false, and briefly explain why.
- It might be optimal to exercise early an American call option written on this asset.
- It might be optimal to exercise early an American put option written on this asset.
- The time-value of a European call written on this asset might be negative for stock prices that are high enough.
- The time-value of a European put written on this asset might be negative for stock prices that are low enough.