Local Volatility Averages as an Implied Volatility Approximation

Summary

The document discusses an approximation that relates implied volatility for a spot and strike pair to the average of local volatility across the price interval between them. The motivating explanation is that paths ending in the money may spend much of their time between the initial spot and strike, so averaging local volatility over that range might capture relevant behavior.

The question challenges this intuition: distinct paths can have different realized volatilities, and a simple spatial average does not explicitly describe a probability-weighted average over paths. The document raises the issue but supplies no formal derivation, path measure, or answer resolving it. It therefore serves as a conceptual prompt about interpreting the approximation, rather than evidence that the formula is generally exact; its applicability and assumptions remain unspecified.

Key ideas

• The discussed approximation averages local volatility between spot and strike.
• Its informal rationale focuses on in-the-money paths spending time within that price interval.
• Different paths can realize different volatility, raising questions about how path variation enters the approximation.
• The document asks for a formal probabilistic explanation but does not provide one.

Tags

# Implied Volatility Approximated with Average of Local Volatilies: where are the paths?


# Implied Volatility Approximated with Average of Local Volatilies: where are the paths?












In the book The Volatility Smile by Derman & Miller, at pag. 262, is given this approximation of the implied volatility in terms of local ones: $$ \Sigma(S,K) \approx \frac{1}{K-S}\int_S^K\sigma(S')dS' $$

As an explanation is written that most of the paths that ends in the money will take values between $S$ and $K$ and stay there for the most of the time. However, these paths could be very different.

Different paths should have different realized volatilities. But to approximate $\Sigma$, it is sufficient to average the values, in $[S,K]$, that the local volatility can assume.

Question: How can I state more formally this fact that we are averaging among all the possible paths?

I'm thinking about introducing somewhere in the formula the fact that we are averaging over all the possible paths. Or, maybe, the authors are assuming something in terms of probability of the paths being in $[S,K]$? Basically I would like to reverse engineer the formula to see the paths but I am not able to do it.

Let me know if more details are needed. Thanks for the help.

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