Implied Volatility from Local Volatility: A Path Integral Approach

Large Deviations and Asymptotic Methods in Finance, Peter K. Friz, Jim Gatheral, Archil Gulisashvili, Antoine Jacquier, Josef Teichmann eds., pp. 247-271, Springer, 2015

25 Pages Posted: 23 Jun 2014 Last revised: 10 Jul 2015

See all articles by Tai-Ho Wang

Tai-Ho Wang

Baruch College, CUNY

Jim Gatheral

CUNY Baruch College

Date Written: June 22, 2014

Abstract

Assuming local volatility, we derive an exact Brownian bridge representation for the transition density; an exact expression for the transition density in terms of a path integral then follows. By Taylor-expanding around a certain path, we obtain a generalization of the heat kernel expansion of the density which coincides with the classical one in the time-homogeneous case, but is more accurate and natural in the time inhomogeneous case. As a further application of our path integral representation, we obtain an improved most-likely-path approximation for implied volatility in terms of local volatility.

Keywords: Small time asymptotic expansion, heat kernels expansion, implied volatility, local volatility model, most likely path, path integral

Suggested Citation

Wang, Tai-Ho and Gatheral, Jim, Implied Volatility from Local Volatility: A Path Integral Approach (June 22, 2014). Large Deviations and Asymptotic Methods in Finance, Peter K. Friz, Jim Gatheral, Archil Gulisashvili, Antoine Jacquier, Josef Teichmann eds., pp. 247-271, Springer, 2015 , Available at SSRN: https://ssrn.com/abstract=2457618 or http://dx.doi.org/10.2139/ssrn.2457618

Tai-Ho Wang

Baruch College, CUNY ( email )

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

Jim Gatheral (Contact Author)

CUNY Baruch College ( email )

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

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