# Variational analysis for the Black and Scholes equation with stochastic volatility

- Volume: 36, Issue: 3, page 373-395
- ISSN: 0764-583X

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topAchdou, Yves, and Tchou, Nicoletta. "Variational analysis for the Black and Scholes equation with stochastic volatility." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.3 (2002): 373-395. <http://eudml.org/doc/245442>.

@article{Achdou2002,

abstract = {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.},

author = {Achdou, Yves, Tchou, Nicoletta},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {degenerate parabolic equations; european options; weighted Sobolev spaces; finite element and finite difference method; mean reverting Orstein-Uhlenbeck process; finite element method; finite difference method},

language = {eng},

number = {3},

pages = {373-395},

publisher = {EDP-Sciences},

title = {Variational analysis for the Black and Scholes equation with stochastic volatility},

url = {http://eudml.org/doc/245442},

volume = {36},

year = {2002},

}

TY - JOUR

AU - Achdou, Yves

AU - Tchou, Nicoletta

TI - Variational analysis for the Black and Scholes equation with stochastic volatility

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2002

PB - EDP-Sciences

VL - 36

IS - 3

SP - 373

EP - 395

AB - 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.

LA - eng

KW - degenerate parabolic equations; european options; weighted Sobolev spaces; finite element and finite difference method; mean reverting Orstein-Uhlenbeck process; finite element method; finite difference method

UR - http://eudml.org/doc/245442

ER -

## References

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