Variational analysis for the Black and Scholes equation with stochastic volatility
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 3, p. 373-395
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.
DOI : https://doi.org/10.1051/m2an:2002018
Classification:  91B28,  91B24,  35K65,  65M06,  65M60
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {3},
     year = {2002},
     pages = {373-395},
     doi = {10.1051/m2an:2002018},
     zbl = {1137.91421},
     mrnumber = {1918937},
     language = {en},
     url = {http://www.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, Volume 36 (2002) no. 3, pp. 373-395. doi : 10.1051/m2an:2002018. http://www.numdam.org/item/M2AN_2002__36_3_373_0/

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