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7. Black-76 Formula. In addition to futures, options on futures con- tracts are actively traded on exchanges. The expiration date te 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(0, FA(te, T) - K) (a) Derive the commonly used Black's Formula for calls on futures by completing the missing steps below C(0) = e-rte E[max(0, FA(te, T) K)] Ite[FA (0,T)N(dl) KN(D2)] ste e"(T-te) E[max(0, A(te) Ke=r(Tte))] where 0)/K d1,2 orte Note that when T = te, Black's Formula reduces to BSM Formula. (b) On Friday 2020-Nov-13, the SPDR S&P 500 ETF (SPY) settled at 358.10, while S&P 500 December futures contract, ESZO, with final settlement date of 2020-Dec-18 (3rd Friday of quarter-end) settled at 3580, and the 3600-strike End-Of-Month (expiration date 2020-Nov-30) call option on ESZO settled at 43.40. Using r = 0.05% (5 bp's), and Act/365 for fractions of time, find the implied volatility of the call option. as 7. Black-76 Formula. In addition to futures, options on futures con- tracts are actively traded on exchanges. The expiration date te 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(0, FA(te, T) - K) (a) Derive the commonly used Black's Formula for calls on futures by completing the missing steps below C(0) = e-rte E[max(0, FA(te, T) K)] Ite[FA (0,T)N(dl) KN(D2)] ste e"(T-te) E[max(0, A(te) Ke=r(Tte))] where 0)/K d1,2 orte Note that when T = te, Black's Formula reduces to BSM Formula. (b) On Friday 2020-Nov-13, the SPDR S&P 500 ETF (SPY) settled at 358.10, while S&P 500 December futures contract, ESZO, with final settlement date of 2020-Dec-18 (3rd Friday of quarter-end) settled at 3580, and the 3600-strike End-Of-Month (expiration date 2020-Nov-30) call option on ESZO settled at 43.40. Using r = 0.05% (5 bp's), and Act/365 for fractions of time, find the implied volatility of the call option. as

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