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A partial differential equation connected to option pricing with stochastic volatility: Regularity results and discretization

Authors: Yves Achdou, Bruno Franchi and Nicoletta Tchou
Journal: Math. Comp. 74 (2005), 1291-1322
MSC (2000): Primary 35K65, 65M15, 65M60, 65N30
Published electronically: October 5, 2004
MathSciNet review: 2137004
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Abstract: This paper completes a previous work on a Black and Scholes equation with stochastic volatility. This is a degenerate parabolic equation, which gives the price of a European option as a function of the time, of the price of the underlying asset, and of the volatility, when the volatility is a function of a mean reverting Orstein–Uhlenbeck process, possibly correlated with the underlying asset. The analysis involves weighted Sobolev spaces. We give a characterization of the domain of the operator, which permits us to use results from the theory of semigroups. We then study a related model elliptic problem and propose a finite element method with a regular mesh with respect to the intrinsic metric associated with the degenerate operator. For the error estimate, we need to prove an approximation result.

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Additional Information

Yves Achdou
Affiliation: UFR Mathématiques, Université Paris 7, 2 place Jussieu, 75251 Paris cedex 05, France; and Laboratoire J.L. Lions, Université Paris 6, 4 place Jussieu, 75252 Paris cedex 05, France

Bruno Franchi
Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy

Nicoletta Tchou
Affiliation: IRMAR, Université de Rennes 1, Rennes, France

Keywords: Finance, degenerate parabolic operator, finite elements
Received by editor(s): April 16, 2003
Received by editor(s) in revised form: March 3, 2004
Published electronically: October 5, 2004
Additional Notes: The second author was partially supported by University of Bologna, funds for selected research topics and by GNAMPA of INdAM, Italy, project “Analysis in metric spaces”.
Article copyright: © Copyright 2004 American Mathematical Society