The log‐moment formula for implied volatility

Vimal Raval and Antoine Jacquier

Mathematical Finance, 2023, vol. 33, issue 4, 1146-1165

Abstract: We revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. We show that in the absence of arbitrage, if the underlying stock price at time T admits finite log‐moments E[|logST|q]$\mathbb {E}[|\log S_T|^q]$ for some positive q, the arbitrage‐free growth in the left wing of the implied volatility smile for T is less constrained than Lee's bound. The result is rationalized 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 price to admit any negative moment. In this respect, the result can be 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.

Date: 2023
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