Monte Carlo algorithms for optimal stopping and statistical learning

Annals of Applied Probability

Volumn 15, Issue 2, 2005, Pages 1396-1432

  Egloff, Daniel ^a,b  

^a Zurich Cantonal Bank ^*  (Switzerland)

^b Cantonal Bank of Zurich   (Switzerland)

Author keywords

American options; Concentration inequalities; Empirical processes; Monte Carlo methods; Optimal stopping; Statistical learning; Uniform law of large numbers; Vapnik Chervonenkis classes

Indexed keywords

EID: 21244500615     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/105051605000000043     Document Type: Article

References (57)

1. Anthony, M. and Bartlett, P. L.: 1999, Neural Network Learning, University Press.
2. Barone-Adesi, G. and Whaley, R.: 1987, Efficient analytic approximation of American option values, J. of Finance 42, 301-320.
3. Barron, A., Erigé, L. and Massart, P.: 1999, Risk bounds for model selection via penalization, Probab. Theory Relat.Fields 113, 301-413.
4. Bartlett, P., Bousquet, O. and Mendelson, S.: 2002, Local Rademacher complexities. Preprint.
5. Bensoussan, A.: 1984, On the theory of option pricing, Acta Appl. Math. 2, 139-158.
6. Bensoussan, A. and Lion, J.: 1982, Application of Variational Inequalities in Stochastic Control, Vol. 12 of Studies in Math, and its Appl., North-Holland.
7. Benveniste, A., Metiver, M. and Priouret, P.: 1990, Adaptive Algorithms and Stochastic Approximations, Sringer-Verlag, Berlin.
8. α, Math. USSR Sb. 73, 331-355.
9. Boessarts, P.: 1989, Simulation estimators of optimal early exercise. Working paper, Carnegie-Mellon University.
10. Brigé, L. and Massart, P.: 1998, Minimum constrast estimators on sieves: exponential bounds and rates of convergence, Bernoulli 4, 329-375.
11. Broadie, M. and Glasserman, P.: 1997, Pricing American-style securities using simulation, J. of Economic Dynamics and Control 21, 1323-1352.
12. Broadie, M. and Glasserman, P.: 1998, Monte Carlo methods for pricing high-dimensional American options: An overview, Monte Carlo Methodologies and Applications for Pricing and Risk Management, Risk Books, pp. 149-161.
13. Carl, B. and Stephani, I.: 1990, Entropy, compactness and the approximation of operators, Cambridge Univ. Press.
14. Chow, Y. S., Robbins, H. and Siegmund, D.: 1971, Great Expectations: The Theory of Optimal Stopping, Houghton Mifflin, Boston.
15. Clément, E., Lamberton, D. and Protter, P.: 2002, An analysis of the Longstaff-Schwartz algorithm for American option pricing, Finance and Stochastics 6(4), 449-471.
16. Cox, J., Ross, S. and Rubinstein, M.: 1979, Option pricing: A simplified approach, J. Financial Economics 7, 229-263.
17. Cucker, F. and Smale, S.: 2001, On the mathematical foundations of learning, Bull. Amer. Math. Soc. 39(1), 1-49.
18. DeVore, R. A. and Lorentz, G. G.: 1993, Constructive Approximation, Springer Verlag.
19. Dudley, R. M.: 1999, Uniform Central Limit Theorems, Vol. 63 of Cambridge Studies in advanced mathematics, Cambridge Univ. Press.
20. Glasserman, P.: 2004, Monte Carlo methods in financial engineering, Vol. 53 of Applications of Mathematics, Springer.
21. Glasserman, P. and Yu, B.: 2003, Number of paths versus number of basis functions in American option pricing. Preprint.
22. Györfi, L., Kohler, M., Krzyżak, A. and Walk, H.: 2002, A distribution-free theory of nonparametric regression, Springer Series in Statistics, Springer.
23. Haussler, D.: 1995, Sphere packing numbers for subsets of the boolean n-cube with bounded Vapnik-Chervonenkis dimension, Journal of Combinatorial Theory 69(2), 217-232.
24. Huang, J. and Pang, J.: 1998, Option pricing and linear complementarity.
25. Jaillet, P., Lamberton, D. and Lapeyre, B.: 1990, Variational inequalities and the pricing of American options, Acta Appl. Math. 21, 263-289.
26. Karatzas, I.: 1988, On the pricing of American options, Applied Mathematics and Optimization 17, 37-60.
27. Karatzas, I. and Shreve, S. E.: 1998, Methods of Mathematical Finance, Vol. 39 of Applications of Math., Springer-Verlag.
28. Kohler, M.: 2000, Inequalities for uniform deviations of averages from expectations with applications to nonparametric regression, Journal of Statistical Planning and Inference 89, 1-23.
29. Kolmogorov, A. N. and Tikhomirov, V. M.: 1959, ε-entroyp and ε-capacity of function spaces, Usbecki 14, 3-86.
30. Kushner, H. J.: 1997, Numerical methods for stochastic control in finance, Mathematics of Derivative Securities, Cambridge Univ. Press, pp. 504-527.
31. Kushner, H. J. and Clark, D. S.: 1978, Stochastic Approximation Methods for Constrained and Unconstrained Systems, Springer-Verlag, New York.
32. Kushner, H. J. and Yin, G. G.: 1997, Stochastic Approximation Algorithms and Applications, Vol. 35 of Stoch. Modelling and Appl. in Prob., Springer.
33. Laprise, S. B., Su, Y., Wu, R., Fu, M. C. and Madan, D. B.: 2001, Pricing American options: A comparision of Monte Carlo simulation approaches, Journal Comput. Finance 4(3), 39-88.
34. Ledoux, M.: 1996, On Talagrand's deviation inequality for product measures, ESIAM Prob. & Stat. 1, 63-87.
35. Lee, W., Bartlett, P. and Williamson, R. C.: 1996, Efficient agnonstic learning of neural networks with bounded fan-in, IEEE Transactions on Information Theory 42(6), 2118-2132.
36. Longstaff, F. A. and Schwartz, E. S.: 2001, Valuing American options by simulation: A simple least-square approach, Review of Financial Studies 14(1), 113-147.
37. Massart, P.: 2000, On the constants in Talagrand's concentration inequalities for empirical processes, Annals of Prob. 28(2), 863-885.
38. Mendelson, S.: 2003a, A few notes on statistical learning theory, in S. Mendelson and A. Smola (eds), Advanced Lectures in Machine Learning, Vol. 2600 of LNCS, Springer, pp. 1-40.
39. Mendelson, S.: 2003b, Geometric parameters in learning theory. GAFA lecture notes.
40. Pollard, D.: 1984, Convergence of Stochastic Processes, Springer Series in Statistics, Springer-Verlag.
41. Pollard, D.: 1990, Empirical Processes: Theory and Applications, NSF-CBMS Regional Conference Series in Statistics, Inst. of Maths, and Am. Stat. Assoc.
42. Rio, E.: 2001, Une ineqalité de Bennet pour les maxima de processus empirques, in D. D. Castelle & I. Ibragimov (ed.), Colloque en l'honneur de J. Bretagnolle.
43. Rogers, L. C. G.: 2002, Monte Carlo valuing of American options, Mathematical Finance 12, 271-286.
44. Shen, X.: 1997, On methods of sieves and penalization, Annals of Stat. 25(6), 2555-2591.
45. Shen, X. and Wong, W. H.: 1994, On methods of sieves and penalization, Annals of Stat. 22(2), 580-615.
46. Shiryayev, A. N.: 1978, Optimal Stopping Rules, Vol. 8 of Applications of Mathematics, Springer Verlag.
47. Stone, C. J.: 1982, Optimal rates of convergence for nonparametric regression, Annals of Statist. 10(4), 1040-1053.
48. Talagrand, M.: 1994, Sharper bounds for Gaussian and empirical processes, Annals of Prob. 22(1), 28-76.
49. Tilley, J. A.: 1993, Valuing American options in a path simulation model, Transactions of the Society of Actuaries 45, 83-104.
50. Tsitsiklis, J. N. and Van Roy, B.: 1999, Optimal stopping of Markov processes: Hilbert space theory, approximation algorithms, and an application to pricing high-dimensional financial derivatives, IEEE Trans Autom. Control 44(10), 1840-1851.
51. Van der Vaart, A. W. and Wellner, J. A.: 1996, Weak Convergence and Empirical Processes with Applications to Statistics, Springer Series in Statistics, Springer.
52. Van Moerbeke, P. L. J.: 1976, On optimal stopping and free boundary problems, Archive Rat. Mech. Anal. 60, 101-148.
53. Vapnik, V. N.: 1982, Estimation of Dependences Based on Empirical Data, Springer Series in Statistics, Springer-Verlag.
54. Vapnik, V. N.: 2000, The Nature of Statistical Learning Theory, Statistics for Engineering and Information Science, 2nd edn, Springer.
55. Vapnik, V. N. and Chervonenkis, A. Y.: 1971, On the uniform convergence of relative frequencies of events to their probabilities, Theory of Probability and its Applications 16(2), 264-280.
56. Vidyasagar, A.: 2003, The theory of learning and generalization with applications to Neural Networks, 2nd edition edn. Springer.
57. Wald, A. and Wolfowitz, J.: 1948, Optimum character of the sequential probability ratio tests, Ann. Math. Statist. 19, 326-339.