The log‐moment formula for implied volatility

We revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. show that in absence of arbitrage, if underlying stock price at time T admits finite log-moments E [ | log S q ] $\mathbb {E}[|\log S_T|^q]$ for some positive q, arbitrage-free growth left wing implied volatility smile is less constrained than Lee's bound. The result rationalized a market trading discretely monitored variance swaps wherein payoff function squared log-returns, and requires no assumption to admit any negative moment. In this respect, can be derived from model-independent setup. As byproduct, we relax moment assumptions on provide new proof notorious Gatheral–Fukasawa formula expressing terms volatility.