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I have the following question on futures options:

There is a Black’s model, which is a variant of the Black-Scholes formula that is used to price stock options. The Black’s model prices future options.

https://en.m.wikipedia.org/wiki/Black_model

The approach in the pricing model uses Magrabe’s formula. Both the agreed strike futures price $K_1$ of the future contract and the market future price $K_2$ of the futures contract at the maturity date $T$ of the futures option are used. If $K_1 < K_2$, the futures option is exercised with immediate profit $K_2-K_1$.

However, if the holder exercises the futures option at time $T$ of the futures option, he/she enters into a long position of the futures contract of the underlying asset, with maturity $T’ \geq T$. The problem is that the payoff at time $T’$ is $S_{T’}-K_2$, where $S_{T’}$ is the price of the underlying asset at time $T’$.

The main problem is that $S_{T’}$ does not appear in the pricing formula. More precisely, the value of the futures option does not take the payoff at the maturity of the futures into account. Is that because the option is long gone at time $T’$ and this payoff is not “part” of the option?

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    $\begingroup$ The assumption of the model is S(t)=e^{{-r(T-t)}}F(t) - so if the future and option expire at the same time they have the same value S(T). But they don't have to, you can buy an option expiry T1 on a future expiry T2 (T2 > T1) and at T1 the option exercise into a future with maturity T2-T1. $\endgroup$ – David Waterworth Oct 6 at 2:30

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