# 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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## Abstract

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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.

## How to cite

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Achdou, 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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11. [11] O. Pironneau and F. Hecht, FREEFEM. www.ann.jussieu.fr
12. [12] O. Pironneau and F. Hecht, Mesh adaption for the Black and Scholes equations. East-West J. Numer. Math. 8 (2000) 25–35. Zbl0995.91026
13. [13] M.H. Protter and H.F. Weinberger, Maximum principles in differential equations. Springer-Verlag, New York (1984). Corrected reprint of the 1967 original. Zbl0549.35002MR762825
14. [14] E. Stein and J. Stein, Stock price distributions with stochastic volatility: an analytic approach. The review of financial studies 4 (1991) 727–752.
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16. [16] P. Willmott, J. Dewynne and J. Howison, Option pricing: mathematical models and computations. Oxford financial press (1993). Zbl0844.90011

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