Approximation of the Snell envelope and american options prices in dimension one
ESAIM: Probability and Statistics, Volume 6  (2002), p. 1-19

We establish some error estimates for the approximation of an optimal stopping problem along the paths of the Black-Scholes model. This approximation is based on a tree method. Moreover, we give a global approximation result for the related obstacle problem.

DOI : https://doi.org/10.1051/ps:2002001
Classification:  49L20,  60G40,  65M15,  91B28
Keywords: dynamic programming, snell envelope, optimal stopping
@article{PS_2002__6__1_0,
     author = {Bally, Vlad and Saussereau, Bruno},
     title = {Approximation of the Snell envelope and american options prices in dimension one},
     journal = {ESAIM: Probability and Statistics},
     publisher = {EDP-Sciences},
     volume = {6},
     year = {2002},
     pages = {1-19},
     doi = {10.1051/ps:2002001},
     zbl = {0998.60037},
     mrnumber = {1888135},
     language = {en},
     url = {http://www.numdam.org/item/PS_2002__6__1_0}
}
Bally, Vlad; Saussereau, Bruno. Approximation of the Snell envelope and american options prices in dimension one. ESAIM: Probability and Statistics, Volume 6 (2002) , pp. 1-19. doi : 10.1051/ps:2002001. http://www.numdam.org/item/PS_2002__6__1_0/

[1] C. Baiocchi and G.A. Pozzi, Error estimates and free-boundary convergence for a finite-difference discretization of a parabolic variational inequality. RAIRO Anal. Numér./Numer. Anal. 11 (1977) 315-340. | Numdam | MR 464607 | Zbl 0371.65020

[2] V. Bally, M.E. Caballero and B. Fernandez, Reflected BSDE's, PDE's and Variational Inequalities. J. Theoret. Probab. (submitted).

[3] A. Bensoussans and J.-L. Lions, Applications of the Variational Inequalities in Stochastic Control. North Holland (1982). | MR 653144 | Zbl 0478.49002

[4] A.N. Borodin and P. Salminen, Handbook of Brownian Motion Facts and Formulae. Birkhauser (1996). | MR 1477407 | Zbl 0859.60001

[5] M. Broadie and J. Detemple, American option valuation: New bounds, approximations, and a comparison of existing methods. Rev. Financial Stud. 9 (1995) 1211-1250.

[6] N. El Karoui, C. Kapoudjan, E. Pardoux, S. Peng and M.C. Quenez, Reflected Solutions of Backward Stochastic Differential Equations and related Obstacle Problems for PDE's. Ann. Probab. 25 (1997) 702-737. | Zbl 0899.60047

[7] W. Feller, An Introduction to Probability Theory and its Applications, Vol. II. John Wiley and Sons (1966). | MR 210154 | Zbl 0138.10207

[8] D. Lamberton, Error Estimates for the Binomial Approximation of American Put Options. Ann. Appl. Probab. 8 (1998) 206-233. | MR 1620362 | Zbl 0939.60022

[9] D. Lamberton, Brownian optimal stopping and random walks, Preprint 03/98. Université de Marne-la-Vallée (1998). | MR 1885822 | Zbl 1040.60032

[10] D. Lamberton and G. Pagès, Sur l'approximation des réduites. Ann. Inst. H. Poincaré Probab. Statist. 26 (1990) 331-335. | Numdam | Zbl 0704.60042

[11] D. Lamberton and C. Rogers, Optimal Stopping and Embedding, Preprint 17/99. Université de Marne-la-Vallée (1999). | Zbl 0981.60049

[12] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Springer Verlag, Berlin Heidelberg (1991). | MR 1083357 | Zbl 0731.60002

[13] A.W. Roberts and D.E. Varberg, Convex Functions. Academic Press, New York (1973). | MR 442824 | Zbl 0271.26009

[14] B. Saussereau, Sur une classe d'équations aux dérivées partielles. Ph.D. Thesis of the University of Le Mans, France (2000).