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@article{ Henry-Labordere2007, title = {Solvable Local and Stochastic Volatility Models: Supersymmetric Methods in Option Pricing}, author = {Henry-Labordere, Pierre}, journal = {Quantitative Finance}, number = {5}, pages = {525-535}, volume = {7}, year = {2007}, doi = {https://doi.org/10.1080/14697680601103045}, urn = {https://nbn-resolving.org/urn:nbn:de:0168-ssoar-220959}, 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.}, }