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Short-Maturity Links Between Implied and Local Volatility

Article Quant Q&A · Author: stackoverflower
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Summary

The document asks how to interpret the short-maturity result associated with Berestycki, Busca, and Florent: implied volatility is related to local volatility through a harmonic mean. It connects that result to a derivation of the half-slope rule and asks whether the derivation relies on treating the harmonic and arithmetic means as approximately equal.

It also compares this volatility statement with a result in Gatheral’s book, which is expressed in terms of implied and local variance. The document provides no derivation or resolution, so it does not establish the approximation’s validity or explain the half-slope rule. Its value is as a focused research question about the assumptions needed when translating between local and implied volatility asymptotics.

Key ideas

  • Short-maturity asymptotics relate implied volatility to local volatility through a harmonic-mean expression.
  • The document asks how that relation leads to the half-slope rule.
  • It questions whether approximating a harmonic mean by an arithmetic mean explains the derivation.
  • A related result is cited in variance terms, highlighting a distinction between volatility and variance formulations.

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Full text
# How to derive the half slope rule from asymptotic relation between implied and local volatility?


# How to derive the half slope rule from asymptotic relation between implied and local volatility?












In their paper(https://www.sciencedirect.com/science/article/pii/S0764444200017493) BERESTYCKI, BUSCA, and FLORENT proved an asymptotical relation(for short maturity) between implied and local volatity: implied vol is the harmonic mean of the local volatility.

This result is used in https://www.math.nyu.edu/faculty/avellane/Avellaneda.pdf. to derive equation (16) from (11). Are they just saying that the harmonic mean is close to the arithmetic mean?

A similar result is demonstrated in the Gatheral book(lemma page 88) but it is in terms of implied and local variance not implied and local volatilty. https://www.amazon.com/Volatility-Surface-Practitioners-Guide/dp/0471792519/ref=sr_1_1?keywords=gatheral&qid=1567079638&s=gateway&sr=8-1

Thanks

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