The formula for the price c of a European call futures option in terms of the futures price Fo is given in Chapter 14 as where c=e[FoN(d)- KN(d)] d = = In(Fo/K)+0T/2 and dd-oT and K, r, T, and are the strike price, interest rate, time to maturity, and volatility, respectively.

(a) Prove that FoN'

(d) = KN'(d).

(b) Prove that the delta of the call price with respect to the futures price is er N(d).

(c) Prove that the vega of the call price is FoTN'(d)e

(d) Prove the formula for the rho of a call futures option given at the end of Section 15.9. 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 So replaced by Fo. Explain why the same is not true of the rho of a call futures option.

AppendixLO1