# The formula for the price of a European call futures option in terms of the futures price, F 0 ,

## Question:

The formula for the price of a European call futures option in terms of the futures price, F_{0}, is given in Chapter 18 as

where

and K, r, T, and Ïƒ are the strike price, interest rate, time to maturity, and volatility, respectively.

(a) Prove that F_{0}N'(d_{1}) = KN'(d_{2})

(b) Prove that the delta of the call price with respect to the futures price is e^{-rT}N(d_{1}).

(c) Prove that the vega of the call price is F_{0} √T N'(d_{1})e^{-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 S_{0} replaced by F_{0}. Explain why the same is not true of the rho of a call futures option.

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