Variational Analysis for the Black and Scholes Equation with Stochastic Volatility
UFR Mathématiques, Université Paris 7, 2 Place Jussieu, 75252 Paris cedex 5, France. Laboratoire d'Analyse Numérique, Université Paris 6.
2 IRMAR, Université de Rennes 1, Rennes, France.
Revised: 4 February 2002
We propose a variational analysis for a Black and Scholes equation with stochastic volatility. This equation 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 variational analysis involves weighted Sobolev spaces. It enables to prove qualitative properties of the solution, namely a maximum principle and additional regularity properties. Finally, we make numerical simulations of the solution, by finite element and finite difference methods.
Mathematics Subject Classification: 91B28 / 91B24 / 35K65 / 65M06 / 65M60
Key words: Degenerate parabolic equations / european options / weighted Sobolev spaces / finite element and finite difference method.
© EDP Sciences, SMAI, 2002