We consider the use of finite volume methods for the approximation of a parabolic variational inequality arising in financial mathematics. We show, under some regularity conditions, the convergence of the upwind implicit finite volume scheme to a weak solution of the variational inequality in a bounded domain. Some results, obtained in comparison with other methods on two dimensional cases, show that finite volume schemes can be accurate and efficient.
@article{M2AN_2006__40_2_311_0,
author = {Berton, Julien and Eymard, Robert},
title = {Finite volume methods for the valuation of american options},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {40},
year = {2006},
pages = {311-330},
doi = {10.1051/m2an:2006011},
mrnumber = {2241825},
zbl = {1137.91427},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2006__40_2_311_0}
}
Berton, Julien; Eymard, Robert. Finite volume methods for the valuation of american options. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) pp. 311-330. doi : 10.1051/m2an:2006011. http://gdmltest.u-ga.fr/item/M2AN_2006__40_2_311_0/
[1] and, Convergence of American option values from discrete- to continuous-time financial models. Math. Finance 4 (1994) 289-304. | Zbl 0884.90012
[2] and, A quantization algorithm for solving multi-dimensional discrete-time optimal stopping problems. Bernoulli 9 (2003) 1003-1049. | Zbl 1042.60021
[3] , and, Convergence of numerical Schemes for problems arising in Finance theory. Math. Mod. Meth. Appl. Sci. 5 (1995) 125-143. | Zbl 0822.65056
[4] ,, and, Boiling in porous media: model and simulations. Transport Porous Med. 60 (2005) 1-31.
[5] and, Applications des inéquations variationnelles en contrôle stochastique, Dunod, Paris (1978). Application of variational inequalities in stochastic control, North Holland (1982). | MR 513618 | Zbl 0411.49002
[6] and, Une méthode de volumes finis pour le calcul des options américaines, Congrès d'Analyse Numérique. La Grande Motte, France (2003). http://www.math.univ-montp2.fr/canum03/
[7] , Méthodes de volumes finis pour des problèmes de mathématiques financières. Thèse de l'Université de Marne-la-Vallée, France (in preparation).
[8] , and, Numerical evaluation of multivariate contingent claims. Rev. Financ. Stud. 2 (1989) 241-250.
[9] and, The valuation of the American put option. J. Financ. 32 (1977) 449-462.
[10] , Analyse fonctionnelle (Théorie et applications). Dunod, Paris (1999). | MR 697382 | Zbl 0511.46001
[11] and, American option valuation: new bounds, approximations, and a comparison of existing methods securities using simulation. Rev. Financ. Stud. 9 (1996) 1221-1250.
[12] , and, Alternative characterizations of American put options. Math. Financ. 2 (1992) 87-106. | Zbl 0900.90004
[13] , and, Options pricing: A simplified approach. J. Financ. Econ. 7 (1979) 229-263. | Zbl 1131.91333
[14] ,, and, Some mathematical results in the pricing of American options, Eur. J. Appl. Math. 4 (1993) 381-398. | Zbl 0797.60051
[15] , and, Finite Volume Methods, in Handb. Numer. Anal., Ph. Ciarlet and J.L. Lions (Eds.) 7 (2000) 715-1022. | Zbl 0981.65095
[16] , and, Convergence of finite volume schemes for semilinear convection diffusion equations, Numer. Math. 82 (1999) 90-116. | Zbl 0930.65118
[17] ,, and, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations, Numer. Math. 92 (2001) 41-82. | Zbl 1005.65099
[18] , Sparse-grid finite-volume multigrid for 3D-problems. Adv. Comput. Math 4 (1995) 83-110. | Zbl 0826.65100
[19] , and, Variational inequalities and the pricing of American options. Acta Appl. Math. 21 3 (1990) 263-289. | Zbl 0714.90004
[20] and, Multinomial approximating models for options with k-state variables. Manage. Sci. 37 (1991) 1640-1652. | Zbl 0825.90061
[21] , and, Linear and quasi-linear equations of parabolic type. Translated from the Russian by S. Smith. Transl. Math. Monogr. (AMS) 23 (1968) xi+648. | Zbl 0174.15403
[22] and, Introduction au calcul stochastique appliqué à la finance. Ellipses, Paris, New York, London (1997) 176. | MR 1607509 | Zbl 0949.60005
[23] , Iterative methods for sparse linear systems. First edition, SIAM (1996). | Zbl 1031.65047
[24] , and, A numerical analysis of variational valuation techniques for derivative securities, Appl. Math. Comput. 159 (2004) 171-198. | Zbl 1080.91040
[25] and, Parabolic A.D.I. methods for pricing American options on two stocks, Math. Oper. Res. 27 (2002) 121-149. | Zbl 1082.60515
[26] , and, A finite volume approach for contingent claims valuation, IMA J. Numer. Anal. 21 (2001) 703-731. | Zbl 1004.91032