The Log Moment formula for implied volatility

12 Pages Posted: 18 Feb 2021

See all articles by Antoine (Jack) Jacquier

Antoine (Jack) Jacquier

Imperial College London; The Alan Turing Institute

Vimal Raval

affiliation not provided to SSRN

Date Written: January 20, 2021


We revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. We show that when the underlying stock price martingale admits finite log-moments E[|log (S)|^q] for some positive q, the arbitrage-free growth in the left wing of the implied volatility smile is less constrained than Lee's bound. The result is rationalised by a market trading discretely monitored variance swaps wherein the payoff is a function of squared log-returns, and requires no assumption for the underlying martingale to admit any negative moment. In this respect, the result can derived from a model-independent setup. As a byproduct, we relax the moment assumptions on the stock price to provide a new proof of the notorious Gatheral-Fukasawa formula expressing variance swaps in terms of the implied volatility.

Suggested Citation

Jacquier, Antoine and Raval, Vimal, The Log Moment formula for implied volatility (January 20, 2021). Available at SSRN: or

Antoine Jacquier (Contact Author)

Imperial College London ( email )

South Kensington Campus
London SW7 2AZ, SW7 2AZ
United Kingdom


The Alan Turing Institute ( email )

British Library, 96 Euston Road
London, NW12DB
United Kingdom

Vimal Raval

affiliation not provided to SSRN

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