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
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/
[1] and (in preparation).
[2] , Analyse Fonctionnelle, Théorie et Applications. Masson (1983). | MR 697382 | Zbl 0511.46001
[3] and, An introduction to semilinear evolution equations. The Clarendon Press Oxford University Press, New York (1998). Translated from the 1990 French original by Y. Martel and revised by the authors. | MR 1691574 | Zbl 0926.35049
[4] and, Numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element methods or finite difference method. SIAM J. Numer. Anal. 19 (1982) 871-885. | Zbl 0492.65051
[5] , and. Derivatives in financial markets with stochastic volatility. Cambridge University Press, Cambridge (2000). | MR 1768877 | Zbl 0954.91025
[6] , and. Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields. Houston J. Math. 22 (1996) 859-890. | Zbl 0876.49014
[7] and, A finite element approximation for a class of degenerate elliptic equations. Math. Comp. 69 (2000) 41-63. | Zbl 0941.65117
[8] , The identity of weak and strong extensions of differential operators. Trans. Amer. Math. Soc. 55 (1944) 132-151. | Zbl 0061.26201
[9] and, Problèmes aux limites non homogènes et applications. Vol. I and II. Dunod, Paris (1968). | Zbl 0165.10801
[10] , Semi-groups of linear operators and applications to partial differential equations. Appl. Math. Sci.. 44, Springer Verlag (1983). | MR 512912 | Zbl 0516.47023
[11] and, FREEFEM. www.ann.jussieu.fr
[12] and, Mesh adaption for the Black and Scholes equations. East-West J. Numer. Math. 8 (2000) 25-35. | Zbl 0995.91026
[13] and, Maximum principles in differential equations. Springer-Verlag, New York (1984). Corrected reprint of the 1967 original. | MR 762825 | Zbl 0549.35002
[14] and, Stock price distributions with stochastic volatility: an analytic approach. The review of financial studies 4 (1991) 727-752.
[15] , Bi-cgstab: a fast and smoothly converging variant of bi-cg for the solution of nonlinear systems. SIAM J. Sci. Statist. Comput. 13 (1992) 631-644. | Zbl 0761.65023
[16] , and, Option pricing: mathematical models and computations. Oxford financial press (1993). | Zbl 0844.90011
