# Multiple max/min forward start option

I want to calculate the price at $t$ for such payoff at $T$ $$\max(S_T,S_{T_0},C),$$ $$\max\left(S_T,\min(S_{T_0}, C)\right),$$ $$S_T -\min(S_{T_0}, C),$$ $$t<T_0<T.$$ Is there any way or reference to simplify above payoffs. Since it seems much complicate than a forward start option.

How to obtain the closed forms under BS model (If some of the payoffs have the closed forms)?

Actually I use the stochastic vol model and have to numerically solve the price. I want to investigate the relation between some risk sensitivity and the parameters controlling the forward skew. So is there any way to simplify the payoff or approximate to some vanilla payoffs?

• Could you clarify your question a bit? What model do you want to price these payoffs under? Also, why insisting on simplifying the payoffs, do you want to find a closed form formulae for the pricing? If you just the price and if you're under Black-Scholes, you could use Monte Carlo, in this case, since you have a closed form formula for the call price, there is no need to simplify anything. – byouness Jun 5 '18 at 12:46
• @byouness pls see the updated. – user6703592 Jun 5 '18 at 14:15