 
Summary: Variational Analysis for
the Black and Scholes Equation with Stochastic
Volatility
Yves Achdou ,
Nicoletta Tchou y
January 28, 2002
Abstract
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 OrsteinUhlenbeck 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 dierence methods.
1 Introduction
We consider a nancial asset whose price is given by the stochastic dierential 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 )
