In addition to futures, options on futures contracts are actively traded on exchanges. The expiration date \(T_{e}\) of the option need not coincide with the forward date \(T\) of the futures contract, and the payoff of a call option on a futures contract is

\[\max \left(0, F_{A}\left(T_{e}, Tight)-Kight)\]

(a) Derive the commonly used Black's formula for calls on futures by completing the missing steps below

\[\begin{aligned}C(0) & =e^{-r T_{e}} E\left[\max \left(0, F_{A}\left(T_{e}, Tight)-Kight)ight] \\& \ldots \\& =e^{-r T_{e}} e^{r\left(T-T_{e}ight)} E\left[\max \left(0, A\left(T_{e}ight)-K e^{-r\left(T-T_{e}ight)}ight)ight] \\& \ldots \\& =e^{-r T_{e}}\left[F_{A}(0, T) N(d 1)-K N(d 2)ight]\end{aligned}\]

where \[d_{1,2}=\frac{\ln \left(F_{A}(0, T) / Kight)}{\sigma \sqrt{T_{e}}} \pm \frac{1}{2} \sigma \sqrt{T_{e}}\]
Note that when \(T=T_{e}\), Black's formula reduces to the BSM formula.

(b) On Friday, November 13, 2020, the S\&P 500 December futures contract, ESZ0, with a final settlement date of December 18, 2020 (third Friday of quarter-end) settled at 3580, and the 3600-strike end-of-month (expiration date November 20, 2020) call option on ESZ0 settled at 43.40. Using \(r=0.75 \%\) ( \(75 \mathrm{bps}\) ), and Act/365 for fractions of time, find the implied volatility of the call option.