From Implied to Spot Volatilities

26 Pages Posted: 21 Jul 2008 Last revised: 23 Dec 2008

See all articles by Valdo Durrleman

Valdo Durrleman

Ecole Polytechnique - Centre de Mathematiques Appliquees - CNRS

Date Written: July 17, 2008

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.

Keywords: implied volatility, model calibration

JEL Classification: G13

Suggested Citation

Durrleman, Valdo, From Implied to Spot Volatilities (July 17, 2008). Available at SSRN: https://ssrn.com/abstract=1162425 or http://dx.doi.org/10.2139/ssrn.1162425

Valdo Durrleman (Contact Author)

Ecole Polytechnique - Centre de Mathematiques Appliquees - CNRS ( email )

Palaiseau, 91128
France

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