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Variational Analysis for the Black and Scholes Equation with Stochastic

Summary: Variational Analysis for
the Black and Scholes Equation with Stochastic
Yves Achdou  ,
Nicoletta Tchou y
January 28, 2002
We propose a variational analysis for a Black and Scholes equation with stochastic volatil-
ity. 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 revert-
ing 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 nite element and nite di erence methods.
1 Introduction
We consider a nancial asset whose price is given by the stochastic di erential equation
dX t = X t dt +  t X t dW t ; (1)
where X t dt is a drift term, (W t ) is a Brownian motion, and ( t ) is the volatility. The simplest
models (see [16]) for a complete overview) take a constant volatility, but these models are
generally to rough to match real prices. A more realistic model consists in assuming that ( t )


Source: Achdou, Yves - Laboratoire Jacques-Louis Lions, Université Pierre-et-Marie-Curie, Paris 6


Collections: Mathematics