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The Log Moment formula for implied volatility

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  • Vimal Raval
  • Antoine Jacquier

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.

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  • Vimal Raval & Antoine Jacquier, 2021. "The Log Moment formula for implied volatility," Papers 2101.08145, arXiv.org.
    • Handle: RePEc:arx:papers:2101.08145
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    References listed on IDEAS

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    1. Peter Carr & Sander Willems, 2019. "A lognormal type stochastic volatility model with quadratic drift," Papers 1908.07417, arXiv.org.
    2. repec:bla:jfinan:v:58:y:2003:i:2:p:753-778 is not listed on IDEAS
    3. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2013. "Shapes of implied volatility with positive mass at zero," Papers 1310.1020, arXiv.org, revised May 2017.
    4. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-777, April.
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    7. Archil Gulisashvili, 2015. "Left-Wing Asymptotics Of The Implied Volatility In The Presence Of Atoms," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(02), pages 1-25.
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