Question 1
A stock price is currently $100. It is known that in one year it will be either $146 or $80. The risk-free rate of interest with continuous compounding is 8% per annum.
Compute American put option prices for strikes equal to $100, $110, $120 and $130.
(a) At which strike(s) does early exercise occur?
(b) Use put-call parity to explain why early exercise does not occur at other strikes.
(c) Use put-call parity to explain why early exercise is sure to occur for all strikes greater than that in your answer to (a).
Question 2
Consider a European call option on a non-dividend paying stock where the stock price is $40, the strike price is $40, the risk-free rate is 8% per annum, the volatility is 30% per annum and the time to maturity is 6-months.
(a) Construct a two-step binomial tree, calculating u and d using the Cox-Ross-Rubinstein approach to matching volatility.
(b) Show that the call price is $3.772, using a two-step tree.
(c) Compute the prices of American and European puts, both with the same strike price and time to maturity as the European call, again using a two-step tree.
Question 3
(a) If S(t) follows the process dS(t) = dt + dZ(t) what is the process followed by y = S 1 ?
(b) If S(t) follows the process dS(t) = [ S(t)]dt + dZ(t) what is the process followed by y = pS?
Question 4
A stock price is currently $100 and has a volatility of 30% per annum. The risk-free rate is 8% and the stock does not pay any dividends.
(a) Compute the Black-Scholes price for a call option with a strike price of $120, ?rst for a maturity of one year, and then for a variety of very long times to maturity. What happens to the option price as the time to maturity tends to in?nity?
(b) Suppose now that the stock pays a small dividend yield equal to 0.1%. Repeat part (a). Now what happens to the option price? What accounts for the difference?
(c) Now suppose that the dividend yield is again zero, but now that the risk-free rate is also zero. What is the Black-Scholes price for a call option with a strike price of $120e :08 and a time to maturity of 1 year. How does this price compare to the 1-year maturity call price calculated in (a)?