How do I calculate option payoff before its expiration date? For example, if I long a 6 month call with K = 11100, T = 0.5, p = 150, what would be the payoff of the option if I exercise it in 3 months time and if the spot price is 11300.?
If you exercise the option (assuming that is an american option) you would receive the intrinsic value, which is for a Call option $\max(S-K, 0) $, and for a Put option $\max(K-S, 0)$. Hence, 11300.00 - 11100.00 = 200.
If you are talking about selling the option instead of exercising it, I recommend to have a look at the Black & Scholes model, John C Hull "Options, Futures and other Derivatives" is a good reference here.