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.
Classification : 91B28, 91B24, 35K65, 65M06, 65M60
Mots clés : degenerate parabolic equations, european options, weighted Sobolev spaces, finite element and finite difference method
@article{M2AN_2002__36_3_373_0,
author = {Achdou, Yves and Tchou, Nicoletta},
title = {Variational analysis for the Black and Scholes equation with stochastic volatility},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {373--395},
publisher = {EDP-Sciences},
volume = {36},
number = {3},
year = {2002},
doi = {10.1051/m2an:2002018},
zbl = {1137.91421},
mrnumber = {1918937},
language = {en},
url = {archive.numdam.org/item/M2AN_2002__36_3_373_0/}
}
Achdou, Yves; Tchou, Nicoletta. Variational analysis for the Black and Scholes equation with stochastic volatility. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) no. 3, pp. 373-395. doi : 10.1051/m2an:2002018. http://archive.numdam.org/item/M2AN_2002__36_3_373_0/
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