The formula for the price $c$ of a European call futures option in terms of the futures price $F_0$ is given in Chapter 18 as
$$
c=e^{-r T}\left[F_0 N\left(d_1\right)-K N\left(d_2\right)\right]
$$
Where
$$
d_1=\frac{\ln \left(F_0 / K\right)+\sigma^2 T / 2}{\sigma \sqrt{T}} \quad \text { and } \quad d_2=d_1-\sigma \sqrt{T}
$$
and $K, r, T$, and $\sigma$ are the strike price, interest rate, time to maturity, and volatility, respectively.
(a) Prove that $F_0 N^{\prime}\left(d_1\right)=K N^{\prime}\left(d_2\right)$.
(b) Prove that the delta of the call price with respect to the futures price is $e^{-r T} N\left(d_1\right)$.
(c) Prove that the vega of the call price is $F_0 \sqrt{T} N^{\prime}\left(d_1\right) e^{-r T}$.
(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.