Approximation of the Snell Envelope and American Options Prices in dimension one

Vlad Bally; Bruno Saussereau

ESAIM: Probability and Statistics (2010)

  • Volume: 6, page 1-19
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Bally, Vlad, and Saussereau, Bruno. "Approximation of the Snell Envelope and American Options Prices in dimension one." ESAIM: Probability and Statistics 6 (2010): 1-19. <http://eudml.org/doc/104288>.

@article{Bally2010,
abstract = { 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. },
author = {Bally, Vlad, Saussereau, Bruno},
journal = {ESAIM: Probability and Statistics},
keywords = {Dynamic programming; snell envelope; optimal stopping.; dynamic programming; optimal stopping problem; Black-Scholes model; tree method},
language = {eng},
month = {3},
pages = {1-19},
publisher = {EDP Sciences},
title = {Approximation of the Snell Envelope and American Options Prices in dimension one},
url = {http://eudml.org/doc/104288},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Bally, Vlad
AU - Saussereau, Bruno
TI - Approximation of the Snell Envelope and American Options Prices in dimension one
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 6
SP - 1
EP - 19
AB - 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.
LA - eng
KW - Dynamic programming; snell envelope; optimal stopping.; dynamic programming; optimal stopping problem; Black-Scholes model; tree method
UR - http://eudml.org/doc/104288
ER -

References

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  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.  Zbl0371.65020
  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).  
  4. A.N. Borodin and P. Salminen, Handbook of Brownian Motion Facts and Formulae. Birkhauser (1996).  Zbl0859.60001
  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.  Zbl0899.60047
  7. W. Feller, An Introduction to Probability Theory and its Applications, Vol. II. John Wiley and Sons (1966).  Zbl0138.10207
  8. D. Lamberton, Error Estimates for the Binomial Approximation of American Put Options. Ann. Appl. Probab.8 (1998) 206-233.  Zbl0939.60022
  9. D. Lamberton, Brownian optimal stopping and random walks, Preprint 03/98. Université de Marne-la-Vallée (1998).  Zbl1040.60032
  10. D. Lamberton and G. Pagès, Sur l'approximation des réduites. Ann. Inst. H. Poincaré Probab. Statist.26 (1990) 331-335.  Zbl0704.60042
  11. D. Lamberton and C. Rogers, Optimal Stopping and Embedding, Preprint 17/99. Université de Marne-la-Vallée (1999).  Zbl0981.60049
  12. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Springer Verlag, Berlin Heidelberg (1991).  Zbl0731.60002
  13. A.W. Roberts and D.E. Varberg, Convex Functions. Academic Press, New York (1973).  
  14. B. Saussereau, Sur une classe d'équations aux dérivées partielles. Ph.D. Thesis of the University of Le Mans, France (2000).  

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