The Heat-Kernel Most-Likely-Path Approximation

19 Pages Posted: 23 Aug 2010 Last revised: 22 Jun 2014

See all articles by Jim Gatheral

Jim Gatheral

CUNY Baruch College

Tai-Ho Wang

Baruch College, CUNY

Date Written: September 22, 2011


In this article, we derive a new most-likely-path (MLP) approximation for implied volatility in terms of local volatility, based on time-integration of the lowest order term in the heat-kernel expansion. This new approximation formula turns out to be a natural extension of the well-known formula of Berestycki, Busca and Florent. Various other MLP approximations have been suggested in the literature involving different choices of most-likely-path; our work fixes a natural definition of the most-likely-path. We confirm the improved performance of our new approximation relative to existing approximations in an explicit computation using a realistic S&P500 local volatility function.

Keywords: implied volatility, local volatility, approximation, heat-kernel expansion

Suggested Citation

Gatheral, Jim and Wang, Tai-Ho, The Heat-Kernel Most-Likely-Path Approximation (September 22, 2011). International Journal of Theoretical and Applied Finance, Vol. 15, No. 1, 1250001, 2012. Available at SSRN:

Jim Gatheral (Contact Author)

CUNY Baruch College ( email )

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

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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