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

Achdou, Yves; Franchi, Bruno; Tchou, Nicoletta

doi:10.1090/S0025-5718-04-01714-4

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.

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