Replicating a Capped Call Payoff With a Put Spread
Summary
The document examines a terminal payoff formed by taking the positive part of the stock’s gain over its current price, subtracting a fixed notional, and capping the result at zero. It asks how to price this payoff under risk-neutral valuation, while noting that expectations cannot simply be moved through nested minimum and maximum operations. The replies identify a static payoff decomposition that avoids directly evaluating the nested expression.
The payoff is equivalent to selling a put at the higher strike, equal to the current stock price plus the notional, and buying a put at the lower strike, equal to the current stock price. Below the lower strike, the spread pays minus the notional; between the strikes, its payoff rises linearly; above the higher strike, it is zero. Thus its value can be obtained from the prices of the two puts under the chosen pricing assumptions. The discussion does not derive a full closed-form formula or specify dividends and other contract conventions, and one response suggests an alternative compound-option interpretation without establishing it. The put-spread decomposition is the clearest practical result.
Key ideas
- The nested payoff is zero above the stock price plus the fixed notional.
- Below the current stock price, the payoff is the negative of the fixed notional.
- Between the two strike levels, the payoff changes linearly with the terminal stock price.
- The payoff matches a short put at the higher strike combined with a long put at the lower strike.
- Pricing can therefore use the values of the two puts instead of taking expectations through nested minimum and maximum functions.
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# Closed form / analytical solution for bespoke (but vanilla) Option # 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. Thank you in advance. ## Answer by stackoverflower (score 1) https://quant.stackexchange.com/a/77522 It is the same a option spread: selling put strike at N+S_0 and buying put at strike S_0 ## Answer by KaiSqDist (score 0) https://quant.stackexchange.com/a/77514 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 by justtryingtolearn (score 0) https://quant.stackexchange.com/a/77521 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.
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