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Solvable Local and Stochastic Volatility Models: Supersymmetric Methods in Option Pricing
[journal article]
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 ... view more
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.... view less
Classification
Economic Statistics, Econometrics, Business Informatics
Basic Research, General Concepts and History of Economics
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
DOI
https://doi.org/10.1080/14697680601103045
Status
Postprint; peer reviewed
Licence
PEER Licence Agreement (applicable only to documents from PEER project)