The log-moment formula for implied volatility |
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Authors: | Vimal Raval Antoine Jacquier |
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Affiliation: | 1. Quantitative Strategies & Data Group, Bank of America, London, UK;2. Department of Mathematics, Imperial College London and the Alan Turing Institute, London, UK |
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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 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. |
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Keywords: | implied volatility moment formula variance swaps |
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