Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T22:48:56.231Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite volume methods for the valuationof American options

Published online by Cambridge University Press:  21 June 2006

Julien Berton  and
Robert Eymard
Julien Berton
Affiliation:
Université de Marne-la-Vallée, Champs-sur-Marne, 77454 Marne-la-Vallée, France. eymard@math.univ-mlv.fr
Robert Eymard
Affiliation:
Université de Marne-la-Vallée, Champs-sur-Marne, 77454 Marne-la-Vallée, France. eymard@math.univ-mlv.fr
Get access

Abstract

We consider the use of finite volume methods for the approximation of aparabolic variational inequality arising in financial mathematics.We show, under some regularityconditions, the convergence of the upwind implicit finite volume schemeto a weak solution of the variational inequality in a bounded domain.Some results, obtained in comparison with other methodson two dimensional cases, show that finite volume schemes can be accurate and efficient.

Keywords

American optionvariational inequalityfinite volume methodconvergence of numerical scheme.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 2 , March 2006 , pp. 311 - 330
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amin, K. and Khanna, A., Convergence of American option values from discrete- to continuous-time financial models. Math. Finance 4 (1994) 289–304. CrossRef
Bally, V. and Pages, G., A quantization algorithm for solving multi-dimensional discrete-time optimal stopping problems. Bernoulli 9 (2003) 1003–1049. CrossRef
Barles, G., Daher, Ch. and Romano, M., Convergence of numerical Schemes for problems arising in Finance theory. Math. Mod. Meth. Appl. Sci. 5 (1995) 125–143. CrossRef
Bénard, J., Eymard, R., Nicolas, X. and Chavant, C., Boiling in porous media: model and simulations. Transport Porous Med. 60 (2005) 1–31. CrossRef
A. Bensoussan and J.L. Lions, Applications des inéquations variationnelles en contrôle stochastique, Dunod, Paris (1978). Application of variational inequalities in stochastic control, North Holland (1982).
J. Berton and R. Eymard, 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/
J. Berton, Méthodes de volumes finis pour des problèmes de mathématiques financières. Thèse de l'Université de Marne-la-Vallée, France (in preparation).
Boyle, P., Evnine, J. and Gibbs, S., Numerical evaluation of multivariate contingent claims. Rev. Financ. Stud. 2 (1989) 241–250. CrossRef
Brennan, M.J. and Schwartz, E., The valuation of the American put option. J. Financ. 32 (1977) 449–462. CrossRef
H. Brézis, Analyse fonctionnelle (Théorie et applications). Dunod, Paris (1999).
Broadie, M. and Detemple, J., American option valuation: new bounds, approximations, and a comparison of existing methods securities using simulation. Rev. Financ. Stud. 9 (1996) 1221–1250. CrossRef
Carr, P., Jarrow, R. and Myneni, R., Alternative characterizations of American put options. Math. Financ. 2 (1992) 87–106. CrossRef
Cox, J.C., Ross, S.A. and Rubinstein, M., Options pricing: A simplified approach. J. Financ. Econ. 7 (1979) 229–263. CrossRef
J.N. Dewynne, S.D. Howison, I. Rupf and P. Wilmott, Some mathematical results in the pricing of American options, Eur. J. Appl. Math. 4 (1993) 381–398. CrossRef
R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, in Handb. Numer. Anal., Ph. Ciarlet and J.L. Lions (Eds.) 7 (2000) 715–1022.
R. Eymard, T. Gallouët and R. Herbin, Convergence of finite volume schemes for semilinear convection diffusion equations, Numer. Math. 82 (1999) 90–116. CrossRef
R. Eymard, T. Gallouët, R. Herbin and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations, Numer. Math. 92 (2001) 41–82.
Hemker, P.W., Sparse-grid finite-volume multigrid for 3D-problems. Adv. Comput. Math 4 (1995) 83–110. CrossRef
P. Jaillet, D. Lamberton and B. Lapeyre, Variational inequalities and the pricing of American options. Acta Appl. Math. 21 3 (1990) 263–289.
Kamrad, B. and Ritchken, P., Multinomial approximating models for options with k-state variables. Manage. Sci. 37 (1991) 1640–1652. CrossRef
O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural'tseva, Linear and quasi-linear equations of parabolic type. Translated from the Russian by S. Smith. Transl. Math. Monogr. (AMS) 23 (1968) xi+648.
D. Lamberton and B. Lapeyre, Introduction au calcul stochastique appliqué à la finance. Ellipses, Paris, New York, London (1997) 176.
Y. Saad, Iterative methods for sparse linear systems. First edition, SIAM (1996).
I. Sapariuc, M.D. Marcozzi and J.E. Flaherty, A numerical analysis of variational valuation techniques for derivative securities, Appl. Math. Comput. 159 (2004) 171–198.
S. Villeneuve and A. Zanette, Parabolic A.D.I. methods for pricing American options on two stocks, Math. Oper. Res. 27 (2002) 121–149. CrossRef
R. Zvan, P.A. Forsyth and K.R. Vetzal, A finite volume approach for contingent claims valuation, IMA J. Numer. Anal. 21 (2001) 703–731. CrossRef