• Bernoulli
  • Volume 19, Number 3 (2013), 931-953.

On data-based optimal stopping under stationarity and ergodicity

Michael Kohler and Harro Walk

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The problem of optimal stopping with finite horizon in discrete time is considered in view of maximizing the expected gain. The algorithm proposed in this paper is completely nonparametric in the sense that it uses observed data from the past of the process up to time $-n+1$, $n\in\mathbb{N}$, not relying on any specific model assumption. Kernel regression estimation of conditional expectations and prediction theory of individual sequences are used as tools. It is shown that the algorithm is universally consistent: the achieved expected gain converges to the optimal value for $n\to\infty$ whenever the underlying process is stationary and ergodic. An application to exercising American options is given, and the algorithm is illustrated by simulated data.

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Bernoulli, Volume 19, Number 3 (2013), 931-953.

First available in Project Euclid: 26 June 2013

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American options ergodicity nonparametric regression optimal stopping prediction stationarity universal consistency


Kohler, Michael; Walk, Harro. On data-based optimal stopping under stationarity and ergodicity. Bernoulli 19 (2013), no. 3, 931--953. doi:10.3150/12-BEJ439.

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  • [1] Belomestny, D. (2011). Pricing Bermudan options by nonparametric regression: Optimal rates of convergence for lower estimates. Finance Stoch. 15 655–683.
  • [2] Cesa-Bianchi, N. and Lugosi, G. (2006). Prediction, Learning, and Games. Cambridge: Cambridge Univ. Press.
  • [3] Chow, Y.S., Robbins, H. and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping. Boston, MA: Houghton Mifflin Co.
  • [4] Duan, J.C. (1995). The GARCH option pricing model. Math. Finance 5 13–32.
  • [5] Egloff, D. (2005). Monte Carlo algorithms for optimal stopping and statistical learning. Ann. Appl. Probab. 15 1396–1432.
  • [6] Gänssler, P. and Stute, W. (1977). Wahrscheinlichkeitstheorie. Berlin: Springer.
  • [7] Györfi, L., Kohler, M., Krzyżak, A. and Walk, H. (2002). A Distribution-Free Theory of Nonparametric Regression. Springer Series in Statistics. New York: Springer.
  • [8] Györfi, L., Lugosi, G. and Udina, F. (2006). Nonparametric kernel-based sequential investment strategies. Math. Finance 16 337–357.
  • [9] Györfi, L., Udina, F. and Walk, H. (2008). Nonparametric nearest neighbor based empirical portfolio selection strategies. Statist. Decisions 26 145–157.
  • [10] Kohler, M. (2008). A regression-based smoothing spline Monte Carlo algorithm for pricing American options in discrete time. AStA Adv. Stat. Anal. 92 153–178.
  • [11] Kohler, M. (2010). A review on regression-based Monte Carlo methods for pricing American options. In Recent Developments in Applied Probability and Statistics (L. Devroye, B. Karasözen, M. Kohler and R. Korn, eds.) 37–58. Heidelberg: Physica.
  • [12] Krengel, U. (1985). Ergodic Theorems. de Gruyter Studies in Mathematics 6. Berlin: de Gruyter. With a supplement by Antoine Brunel.
  • [13] Loève, M. (1977). Probability Theory. II, 4th ed. New York: Springer.
  • [14] Longstaff, F.A. and Schwartz, E.S. (2001). Valuing American options by simulation: A simple least-squares approach. Review of Financial Studies 14 113–147.
  • [15] Morvai, G., Yakowitz, S. and Györfi, L. (1996). Nonparametric inference for ergodic, stationary time series. Ann. Statist. 24 370–379.
  • [16] Shiryayev, A.N. (1978). Optimal Stopping Rules. Applications of Mathematics 8. New York: Springer. Translated from the Russian by A.B. Aries.
  • [17] 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. Automat. Control 44 1840–1851.