Consider a six month American put option on index futures where the current futures price is 450, the exercise price is 450, the risk-free rate of interest is 7 percent per annum, the continuous dividend yield of the index is 3 percent, and the volatility of the index is 30 percent per annum. The futures contract underlying the option matures in seven months. Using a three-step binomial tree, calculate

a. the price of the American put option now,

b. the delta of the option with respect to the futures price,

c. the delta of the option with respect to the index level, and

d. the price of the corresponding European put option on index futures.

e. Apply the control variate technique to improve your estimate of the American option price and of the delta of the option with respect to the futures price.

Note that the Black-Scholes price of the European put option is $36.704 and the delta with respect to the futures price given by Black-Scholes is 0.442.