American Options, Early Exercise, and Optimal Stopping

Summary

The document asks whether American and European puts should have the same value because they share the same terminal payoff, and what it means for an American option holder to exercise optimally. The replies clarify that matching terminal payoffs do not guarantee matching values when the American option can be exercised early. They describe the pricing decision as an optimal stopping problem: the holder chooses an exercise time to maximize the option’s value. This makes early exercise rights part of the contract’s value, beyond its final payoff.

The discussion points readers toward optimal stopping and the Snell envelope, and recommends standard martingale pricing texts for a fuller treatment. It does not give a calculation method, derive an exercise boundary, or explain how to find the optimal time in a particular model. Its account is therefore conceptual and brief; practical valuation requires additional assumptions and mathematical detail.

Key ideas

• An American option permits early exercise, which can affect its value relative to a European option with the same terminal payoff.
• Optimal exercise can be framed as choosing a stopping time that maximizes the holder’s value.
• The replies identify optimal stopping and the Snell envelope as relevant pricing concepts.
• The document does not show how to calculate an exercise boundary or value a specific contract.

Tags

# General questions about american options


# General questions about american options












I'm currently reading about how to value an american option and I have a few questions about it. Would be very grateful if anyone can spare the time and answer them.

$1)$ Since an american put and a european put both have the same payoff and they have identical cash flow, don't they have the same value at any point by the Law of One Price?

$2)$ The source I'm using says that you need the condition of "holder chooses the early exercise strategy in order to maximize the option’s value" in order to price it. What does this condition mean? From what I've read is it that once you reach some "optimal" value, you the exercise the option if the value decreases? If yes, then how is this "optimal" value determined?

## Answer by Vanity (score 2, accepted)

https://quant.stackexchange.com/a/39326

1) No, they do NOT have the same Payoff:





## Answer by Vanity (score 2)

https://quant.stackexchange.com/a/39327

2) This boils down to quite a lot of theory about "optimal stopping times / Snell Envelop and so on" and i would advise you to read about it in one of the standard textbooks on martingale pricing. Economically this would mean that the payoff basically is defined as the payoff you get by exercising the option early optimally, by choosing time t such that the payoff is maximised.

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