  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.
