Stochastic expansions in models
combining local volatility and stochastic
volatility and application to the
European options pricing
Romain Bompis and Emmanuel Gobet
Abstract
In an operational point of view, there is an increasing need for real-time computations
and calibration procedures, while controlling numerical errors with respect to the model
parameters. If it remains important to choose a relevant model according to the situation,
one has to be able to perform quick calculus.
There are no closed formulas for the price of European option with underlying which
combines local volatility and stochastic volatility. Our approach to derive analytical approximations
of the price smartly combines stochastic expansions and Malliavin calculus. That
allows us to obtain explicit formulas and tight error estimates expressed as a function of all
the specific model parameters.
We will show the regularity of the perturbed process for a small perturbation in a Lpsense.
This step allows us in the same time to estimate the Lp distance between the initial
process and its expansions. Then we will perform the derivation of the approximation of
the option price. Finally we will analyse the error, focusing mainly on the regular payo
function case. If the payo function is only once time almost everywhere dierentiable (for
instance the classical puts and calls), the proof becomes more long and tedious. For the
sake of brevity we will just present the ideas which are mainly the use of a regularization
argument and the Malliavin integration by part formula