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

[Zeitschriftenartikel]

## 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. Klassifikation Allgemeines, spezielle Theorien und "Schulen", Methoden, Entwicklung und Geschichte der Wirtschaftswissenschaften; Wirtschaftsstatistik, Ökonometrie, Wirtschaftsinformatik Methode Theorieanwendung Freie Schlagwörter Applied Mathematical Finance; Econophysics; Black-Scholes Model; Stochastic Volatility; Calibration of Stochastic Volatility; Volatility Modelling Sprache Dokument Englisch Publikationsjahr 2007 Seitenangabe S. 525-535 Zeitschriftentitel Quantitative Finance, 7 (2007) 5 DOI http://dx.doi.org/10.1080/14697680601103045 Status Postprint; begutachtet (peer reviewed) Lizenz PEER Licence Agreement (applicable only to documents from PEER project)