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Variational Analysis for the Black and Scholes Equation with Stochastic Volatility

Published online by Cambridge University Press:  15 August 2002

Yves Achdou  and
Nicoletta Tchou
Yves Achdou
Affiliation:
UFR Mathématiques, Université Paris 7, 2 Place Jussieu, 75252 Paris cedex 5, France. Laboratoire d'Analyse Numérique, Université Paris 6. achdou@math.jussieu.fr.
Nicoletta Tchou
Affiliation:
IRMAR, Université de Rennes 1, Rennes, France.
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Abstract

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.

Keywords

Degenerate parabolic equationseuropean optionsweighted Sobolev spacesfinite element and finite difference method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 3 , May 2002 , pp. 373 - 395
Copyright
© EDP Sciences, SMAI, 2002

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