Chapter 5: Q25I (page 516)
You buy a share of stock, write a one-year call option with X = \(10, and buy a one-year put option with X = \)10. Your net outlay to establish the entire portfolio is $9.50. What must be the risk-free interest rate? The stock pays no dividends.
Answer
a. 5.26%
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Show that Black-Scholes call option hedge ratios increase as the stock price increases. Consider a one-year option with exercise price \(50 on a stock with annual standard deviation 20%. The T-bill rate is 3% per year. Find N (d1) for stock prices \)45, \(50, and \)55.
Return to Problem 37. What will be the payoff to the put, Pu, if the stock goes up?
What will be the payoff, Pd, if the stock price falls? Value the put option using the riskneutralshortcut described in the box on page 533. Confirm that your answer matchesthe value you get using the two-state approach.
Consider a stock that will pay a dividend of D dollars in one year, which is when a futures contract matures.
Consider the following strategy: Buy the stock, short a futures contract on the stock, and borrow dollars, where is the current price of the stock.
a. What are the cash flows now and in one year? (Hint: Remember the dividend the stock will pay.)
b. Show that the equilibrium futures price must beto avoid arbitrage.
c. Call the dividend yield d = D / , and conclude that .
In what ways is owning a corporate bond similar to writing a put option? A calloption?
A collar is established by buying a share of stock for \(50, buying a six-month put option with exercise price \)45, and writing a six-month call option with exercise price \(55. Based on the volatility of the stock, you calculate that for an exercise price of \)45 and maturity of six months, N (d1) = .60, whereas for the exercise price of \(55, N (d1) = .35.
a. What will be the gain or loss on the collar if the stock price increases by \)1?
b. What happens to the delta of the portfolio if the stock price becomes very large? Very small?
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