The Log Moment formula for implied volatility

TL;DR

This paper revisits Lee's Moment Formula for implied volatility, showing that under certain finite log-moment conditions, the constraints on the implied volatility smile's left wing are less strict, with implications for variance swap pricing.

Contribution

It demonstrates that the growth constraints on implied volatility can be relaxed under finite log-moments, extending the model-independent understanding of variance swaps and the Gatheral-Fukasawa formula.

Findings

Less constrained implied volatility growth with finite log-moments

Market trading of variance swaps explained without negative moment assumptions

New proof of the Gatheral-Fukasawa formula

Abstract

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.