Upward and Downward Bias in American Option Monte Carlo Estimates

Summary

The document asks how two estimates from Monte Carlo methods for American options can have different biases. It describes a high estimate that may be pushed upward by look-ahead, because exercise decisions use future information, and a low estimate that may be pushed downward by following a suboptimal exercise policy. The central question is why the high estimate would not also inherit the downward bias associated with a suboptimal policy when its calculation uses backward induction.

The author says the issue may apply to random trees, stochastic mesh, and regression-based approaches, then focuses on the random tree method described in Glasserman and in a paper by Broadie and Glasserman. No answer, derivation, numerical comparison, or supporting evidence is included in the document. It is therefore useful as a statement of a methodological question, rather than as guidance that resolves the bias properties of these estimators. Any conclusion about the high estimator requires analysis beyond the material presented here.

Key ideas

• The document distinguishes an upward look-ahead bias from a downward bias caused by a suboptimal exercise policy.
• It asks whether backward induction makes the high estimator vulnerable to both biases.
• The question is framed around simulation methods for pricing American options.
• Random trees are the specific method of interest, with stochastic mesh and regression approaches also mentioned.

Tags

# Suboptimality bias in least squares Monte Carlo for American options


# Suboptimality bias in least squares Monte Carlo for American options












In Monte Carlo pricing of American options we form two estimators:

- A high estimator that is biased upward because of "look-ahead" bias (i.e., at any given time we uses future information to decide whether to exercise).

- A low estimator that is biased downward because of using a suboptimal exercise policy.

I understand the upward bias of the high estimator, but why does it not also suffer from downward suboptimality bias? The implementation of the high estimator is based on backward induction which uses a suboptimal exercise policy.

Edit: I believe all the standard simulation methods to price American options (random trees, stochastic mesh, regression-based approaches) suffer from the above biases.

However, if it helps, I am specifically thinking about the random tree approach. It is the first approach described in Chapter 8 of the Monte Carlo book by Glasserman. The approach was initially presented in a 1997 paper by Broadie and Glasserman: called "Pricing American-style securities using simulation".

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