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

Question

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.

Thank you in advance.

Answer 1

It is the same a option spread: selling put strike at N+S_0 and buying put at strike S_0

Answer 2

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 $S_0$).

Answer 3

I may have misunderstood the question, but it seems like this payoff is identical to being short the S0/(S0+N) European put spread? If ST > N+S0, the payoff is 0. If ST < S0, the payoff is -N. If S0 < ST < S0+N, the payoff is ST - (S0+N). If the payoff is identical, then the price should be equal to that of the put spread.