Why the American Put Payoff Alone Does Not Solve Its Pricing PDE

Summary

The document asks why the immediate-exercise payoff for an American put, the greater of intrinsic value and zero, is not itself a valid solution to the American-option pricing problem. Although that payoff satisfies the terminal payoff condition and never falls below intrinsic value, it omits any value from the possibility of continuing to hold the option. The question concerns how to reconcile this apparent candidate with the inequality form of the Black–Scholes PDE for an American option.

The answer points to risk-neutral valuation on a binomial tree: the expected discounted value of future outcomes can exceed the payoff from exercising immediately. That continuation value must be compared with intrinsic value when determining the option’s value and exercise decision. The exchange gives a conceptual rebuttal rather than a derivation of the variational inequality, boundary conditions, or a worked example. It does not quantify when early exercise is optimal, so further analysis is needed to connect the intuition to a complete PDE solution.

Key ideas

• An American put’s intrinsic payoff alone omits the value of retaining the right to exercise later.
• Risk-neutral expected future value can exceed the value from immediate exercise.
• A binomial tree illustrates the comparison between continuation value and intrinsic value.
• The exchange offers intuition but does not derive the full PDE solution or exercise boundary.

Tags

# American option PDE


# American option PDE












I'm reading the pdf here regarding the PDE associated with the American option. Essentially, one would turn the Black Scholes PDE into an inequality.

Suppose you're pricing an American put where $S$ is the stock price and $E$ is the exercise price. Why can't $V = max(E-S,0)$ be the solution to the American option PDE? It satisfies all the constraints: the terminal condition, never dropping below the stock payoff, and the inequality arising from the PDE. Clearly, $V = max(E-S,0)$ is not the right answer since it convers no extrinsic value. However, how do I reconcile with this wrong solution with the PDE formulated in the pdf?

## Answer by KaiSqDist (score 1, accepted)

https://quant.stackexchange.com/a/79953

I don't think what you said is true. Sometimes, the future expected value of the option is greater than $V=max(E-S,0)$. This can be visualized in the context of a binomial tree, where the risk-neutral probabilities can be used to compute a future expected value of the American put such that it can be used to compare to the current intrinsic value of that American put.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.