Lie Symmetry Methods for Local Volatility Models

33 Pages Posted: 10 Sep 2016

See all articles by Mark Craddock

Mark Craddock

University of Technology Sydney (UTS)

Martino Grasselli

University of Padova - Department of Mathematics; Léonard de Vinci Pôle Universitaire, Research Center

Date Written: September 6, 2016

Abstract

We investigate PDEs of the form u_t = 1/2 σ^2 (t, x)u_{xx} − g(x)u which are associated with the calculation of expectations for a large class of local volatility models. We find nontrivial symmetry groups that can be used to obtain standard integral transforms of fundamental solutions of the PDE. We detail explicit computations in the separable volatility case when σ(t, x) = h(t)(α βx γx2), g = 0, corresponding to the so called Quadratic Normal Volatility Model. We also consider choices of g for which we can obtain exact fundamental solutions that are also positive and continuous probability densities.

Keywords: Lie symmetries, fundamental Solution, PDEs, Local Volatility Models, Normal Quadratic Volatility Model

JEL Classification: C6, C63, G1, G12, G13

Suggested Citation

Craddock, Mark and Grasselli, Martino, Lie Symmetry Methods for Local Volatility Models (September 6, 2016). Available at SSRN: https://ssrn.com/abstract=2836817 or http://dx.doi.org/10.2139/ssrn.2836817

Mark Craddock

University of Technology Sydney (UTS) ( email )

15 Broadway, Ultimo
PO Box 123
Sydney, NSW 2007
Australia

Martino Grasselli (Contact Author)

University of Padova - Department of Mathematics ( email )

Via Trieste 63
Padova, Padova
Italy

Léonard de Vinci Pôle Universitaire, Research Center ( email )

Paris La Défense
France

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