The formula for the price c of a European call futures option in terms of the futures
Chapter 19, Problem 19.28(choose chapter or problem)
The formula for the price c of a European call futures option in terms of the futures price F0 is given in Chapter 18 as c e rT F0Nd1 KNd2 where d1 lnF0=K 2 T=2 ffiffiffi T p and d2 d1 ffiffiffi T p and K, r, T, and are the strike price, interest rate, time to maturity, and volatility, respectively. (a) Prove that F0N0 d1 KN 0 d2. (b) Prove that the delta of the call price with respect to the futures price is e rTNd1. (c) Prove that the vega of the call price is F0 ffiffiffi T p N0 d1e rT . (d) Prove the formula for the rho of a call futures option given in Section 19.12. The delta, gamma, theta, and vega of a call futures option are the same as those for a call option on a stock paying dividends at rate q, with q replaced by r and S0 replaced by F0. Explain why the same is not true of the rho of a call futures option. 19.29.
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