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

DOI:
https://doi.org/10.1090/S0025-5718-04-01714-4

Published electronically:
October 5, 2004

MathSciNet review:
2137004

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Abstract | References | Similar Articles | Additional Information

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

Email:
achdou@math.jussieu.fr

**Bruno Franchi**

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

Email:
bfranchi@dm.unibo.it

**Nicoletta Tchou**

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

Email:
nicoletta.tchou@univ-rennes1.fr

DOI:
https://doi.org/10.1090/S0025-5718-04-01714-4

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