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

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Henry-Labordere, Pierre

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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 {\it integrable superpotentials} introduced recently in supersymmetric quantum mechanics, we obtain new analytical solutions. In particular, by applying {\it 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. These solutions are given by a sum of hypergeometric functions, generalizing the results obtained in the paper "Black-Scholes Goes Hypergeometric" \cite{alb}. 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.
Classification Basic Research, General Concepts and History of Economics; Economic Statistics, Econometrics, Business Informatics
Method theory application
Free Keywords Applied Mathematical Finance; Econophysics; Black-Scholes Model; Stochastic Volatility; Calibration of Stochastic Volatility; Volatility Modelling
Document language English
Publication Year 2007
Page/Pages p. 525-535
Journal Quantitative Finance, 7 (2007) 5
Status Postprint; peer reviewed
Licence PEER Licence Agreement (applicable only to documents from PEER project)