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Comparing Expected Taylor Remainders with Remainders at the Mean

Article Quant Q&A · Author: user34971
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Summary

This note asks when the expectation of a Taylor expansion’s remainder can be represented by a remainder evaluated at the expected deviation from the expansion point. It focuses on a payoff function of a random variable and the distinction between averaging a nonlinear error term and applying an order estimate after averaging the deviation.

The author suspects convexity may matter and asks how to quantify the gap between these expressions if they are not equivalent. The document does not provide a derivation, conditions, examples, or a resolution, so it is best read as a mathematical question about approximation and expectation rather than a worked method. For quantitative finance, the issue can arise when approximating expected payoffs or other nonlinear functions, but the note itself does not establish a usable bound or trading implication.

Key ideas

  • The note distinguishes the expectation of a Taylor remainder from an order estimate based on the expected deviation.
  • It asks what assumptions on the function or random variable would justify exchanging these operations.
  • It raises convexity as a possible factor without establishing a result.
  • It also asks how to measure the discrepancy when the expressions differ.

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# Order of expectation versus expectation of order (error terms in Taylor expansion)


# Order of expectation versus expectation of order (error terms in Taylor expansion)












Given a payoff function $F(X)$ of a random variable $X$, and a Taylor expansion of $F(X)$ around $X=a$, then the expecation of $F(X)$ can be written as

$$ E[F(X)] = F(a) + E[ O((X-a))] $$

Under what conditions can I write $$ E[F(X)] = F(a) + O(E[(X-a)]) \,\,? $$

And if there is no condition under which the two are equivalent, any ideas how I could quantify

$$ E[ O((X-a))] - O(E[(X-a)]) \,\, ? $$

I suspect convexity will play a role, but not clear yet to me how.

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