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What It Means to Replicate an American Option

Article Quant Q&A · Author: EOeo
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Summary

The document asks how replication should be understood for an American option in a pricing model under a risk-neutral measure. A self-financing portfolio with an adapted strategy has a martingale value process, while an American option’s payoff process generally does not. This raises a distinction between matching the option’s payoff at every time and matching its payoff when exercised.

The question suggests that replication may refer to reproducing the stopped payoff process under an optimal exercise strategy. It provides no answer, derivation, or example resolving that interpretation. The note is therefore useful as a framing of the issue, but it does not establish conditions for American option replication or explain how to construct a replicating portfolio.

Key ideas

  • A self-financing portfolio with an adapted strategy has a martingale value process under the stated pricing setup.
  • An American option’s payoff process is generally not itself a martingale.
  • The document asks whether replication means matching the payoff at the optimal exercise time.
  • It leaves the interpretation and construction of replication unresolved.

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OptionsDerivatives pricing
Full text
# What is meant by "replicating an American option"?


# What is meant by "replicating an American option"?












What does it mean to say that we "replicate an American option" in the usual structure of an asset-pricing model with a measure Q?

I mean, any portfolio with an adapted and self-financing strategy will be a martingale. But the payoff of an American option will in general not be a martingale.

So when people talk about replicating an American option, what do they really mean? Do they mean replicating an American option's payoff assuming optimal behavior? So really we try to replicate the stopped payoff process?

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