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

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- by Yves Achdou, Bruno Franchi and Nicoletta Tchou PDF
- Math. Comp.
**74**(2005), 1291-1322 Request permission

## 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.## References

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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
- 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”.
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2137004