Short-Maturity Implied Volatility as a Proxy for Instantaneous Volatility

Summary

The document addresses the theoretical basis for using the implied volatility of a short-maturity, at-the-money option as a proxy for the instantaneous volatility of a stock’s log price. It states the key limiting idea: as option maturity tends to zero, this implied volatility converges to instantaneous volatility under the result being cited.

The answers point readers to a lemma proof by Ledoit and coauthors and to a dissertation by Durrleman as sources for a formal justification. No proof, derivation, empirical evidence, or assumptions are reproduced in the exchange itself, so the document is primarily a pointer to further reading rather than a self-contained explanation. The proxy concerns the short-maturity, at-the-money limit; the exchange does not establish how accurate it is at finite maturities or under alternative market and model conditions.

Key ideas

• The proposed proxy is implied volatility from a short-maturity, at-the-money option.
• The cited theoretical claim is convergence to instantaneous log-price volatility as maturity approaches zero.
• The answers refer to a lemma proof and a dissertation for formal justification.
• The exchange gives no derivation or empirical validation of the limiting result.
• The limiting claim alone does not quantify proxy accuracy at finite maturities.

Tags

# Implied Volatility as proxy for instantaneous volatility


# Implied Volatility as proxy for instantaneous volatility












In many papers and book I have found a reasoning that it is well summarized in this paper as

"The first proxy we use is an unadjusted Black-Scholes proxy in which the implied volatility of a short-maturity at-the-money option is used in place of the true instantaneous volatility state variable. The use of this proxy is justified in theory by the fact that the implied volatility of such an option converges to the instantaneous volatility of the logarithmic stock price as the maturity of the option goes to zero."

Where can I find a formal justification of this assertion?

## Answer by M. Jeunesse (score 1)

https://quant.stackexchange.com/a/25708

This is a result of Ledoit et al

Lemma proof in Appendix of http://www.ledoit.net/9-98.pdf

## Answer by Quantuple (score 0)

https://quant.stackexchange.com/a/25716

See also this dissertation by Durrleman.

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