Valuing Max and Min Payoffs on Forward-Start Options

Summary

The document asks how to price several option payoffs that depend on the underlying asset at two future dates, with a fixed constant also appearing in maximum and minimum terms. The valuation time precedes both dates. The author wants to know whether the payoffs can be simplified, whether any have closed-form prices under Black–Scholes, and whether they can be approximated with vanilla options.

The motivation is numerical pricing under a stochastic-volatility model and investigating how risk sensitivities relate to parameters that control forward skew. The document supplies no proposed decomposition, formulas, pricing results, or references, so it establishes a valuation problem rather than a solution. Any simplification would depend on the precise payoff structure and model assumptions; the question itself leaves those analytical possibilities open.

Key ideas

• The stated payoffs depend on the underlying price at two future dates and a constant threshold.
• The author asks whether these payoffs admit closed forms under Black–Scholes.
• The practical motivation is numerical valuation under stochastic volatility.
• The intended analysis relates risk sensitivities to parameters controlling forward skew.
• No decomposition or pricing solution is provided in the document.

Tags

# Multiple max/min forward start option


# 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?

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