# Closed form / analytical solution for bespoke (but vanilla) Option

Question:

I want to derive closed form expression (similar to the Black Scholes formula for a call price) for the payoff below. I would like to do it from first principles starting with Expectations and ending up with an option pricing formulae similar to the BS option pricing formulae.

The payoff is:

$$\min[ [\max(S_T - S_0), 0] - N, 0]$$

Where:

• $$S_T$$ is the stock price at maturity
• $$S_0$$ is the stock price today
• $$N$$ some fixed notional

So the only stochastic part is $$S_T$$ and assume constant/deterministic interest rates.

The inner part (the “MAX” part) on its own is just a vanilla Call, but I don’t have the technical skill to evaluate the outer “MIN” under the risk-neutral expectation. I know that Jenson’s inequality tells me that you can’t simply “take the Expectation into the min/max operands”, but that is as far as I got.

I am not sure what your question is actually, but it seems to me that the payoff is just a compound option - short European call (MIN function on the value of a European call with strike N) on a long European call (MAX function on the value of $$S_T$$ with strike $$S_0$$).