Solvable Local and Stochastic Volatility Models: Supersymmetric Methods in Option Pricing

21 Pages Posted: 9 Aug 2005

Date Written: July 31, 2005

Abstract

In this paper, we provide an extensive classification of one- and two-dimensional diffusion processes which admit an exact solution to the Kolmogorov (and hence Black-Scholes) equation (in terms of hypergeometric functions). By identifying the one-dimensional solvable processes with the class of integrable superpotentials introduced recently in supersymmetric quantum mechanics, we obtain new analytical solutions. In particular, by applying supersymmetric transformations on a known solvable diffusion process (such as the Natanzon process for which the solution is given by a hypergeometric function), we obtain a hierarchy of new solutions.

For two-dimensional processes, more precisely stochastic volatility models, the classification is achieved for a specific class called gauge-free models including the Heston model, the 3/2-model and the geometric Brownian model. We then present a new exact stochastic volatility model belonging to this class.

Keywords: Solvable diffusion process, supersymmetry, differential geometry

JEL Classification: C60

Suggested Citation

Henry-Labordere, Pierre, Solvable Local and Stochastic Volatility Models: Supersymmetric Methods in Option Pricing (July 31, 2005). Available at SSRN: https://ssrn.com/abstract=773568 or http://dx.doi.org/10.2139/ssrn.773568

Pierre Henry-Labordere (Contact Author)

Natixis - Paris, France ( email )

Paris, Paris 75
France

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