Asymptotics of Implied Volatility in Local Volatility Models

Mathematical Finance, Forthcoming

39 Pages Posted: 27 Jan 2010 Last revised: 30 Jul 2010

Jim Gatheral

CUNY Baruch College

Elton P. Hsu

Northwestern University - Department of Mathmatics

Peter M. Laurence

University of Rome I - Department of Mathematics; Courant Institute, NYU

Cheng Ouyang

Purdue University

Tai-Ho Wang

Baruch College, CUNY

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Date Written: December 23, 2009

Abstract

Using an expansion of the transition density function of a 1-dimensional time inhomogeneous diffusion, we obtain the first and second order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the first order and second order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate. The analysis is extended to degenerate diffusion's using probabilistic methods, i.e. the so called principle of not feeling the boundary.

Keywords: Implied volatility, local volatility, asymptotic expansion, heat kernels

Suggested Citation

Gatheral, Jim and Hsu, Elton P. and Laurence, Peter M. and Ouyang, Cheng and Wang, Tai-Ho, Asymptotics of Implied Volatility in Local Volatility Models (December 23, 2009). Mathematical Finance, Forthcoming. Available at SSRN: https://ssrn.com/abstract=1542077

Jim Gatheral (Contact Author)

CUNY Baruch College ( email )

Department of Mathematics
One Bernard Baruch Way
New York, NY 10010
United States

Elton P. Hsu

Northwestern University - Department of Mathmatics ( email )

2033 Sheridan Rd
Evanston, IL 60208
United States

Peter M. Laurence

University of Rome I - Department of Mathematics ( email )

Roma, I-00185
Italy

Courant Institute, NYU ( email )

Division of Quantitative Finance 251 Mercer Street
New York, NY 10012
United States
212 9983000 (Phone)
212 9954121 (Fax)

Cheng Ouyang

Purdue University ( email )

150 N. University Street
Department of Mathematics
West Lafayette, IN 47907
United States
765-496-3578 (Phone)

Tai-Ho Wang

Baruch College, CUNY ( email )

1 Bernard Baruch Way
New York, NY 10010
United States
+1-646-312-4130 (Phone)

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