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From implied to spot volatilities

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Abstract

This paper is concerned with the relation between spot and implied volatilities. The main result is the derivation of a new equation which gives the dynamics of the spot volatility in terms of the shape and the dynamics of the implied volatility surface. This equation is a consequence of no-arbitrage constraints on the implied volatility surface right before expiry. We first observe that the spot volatility can be recovered from the limit, as the expiry tends to zero, of at-the-money implied volatilities. Then, we derive the semimartingale decomposition of implied volatilities at any expiry and strike from the no-arbitrage condition. Finally the spot volatility dynamics is found by performing an asymptotic analysis of these dynamics as the expiry tends to zero. As a consequence of this equation, we give general formulas to compute the shape of the implied volatility surface around the at-the-money strike and for short expiries in general spot volatility models.

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Authors and Affiliations

  1. Centre de Mathématiques Appliquées, École Polytechnique–CNRS, Route de Saclay, 91128, Palaiseau, France

    Valdo Durrleman

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  1. Valdo Durrleman
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Correspondence to Valdo Durrleman.

Additional information

I am extremely grateful to my Ph.D. advisor René Carmona and to Nicole El Karoui for their help and support. I should also like to thank the referees and the editor for their constructive remarks which have greatly enhanced the clarity of the paper.

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Durrleman, V. From implied to spot volatilities. Finance Stoch 14, 157–177 (2010). https://doi.org/10.1007/s00780-009-0112-1

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  • DOI: https://doi.org/10.1007/s00780-009-0112-1

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