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An iterative method for multiple stopping: convergence and stability

Part of: Stochastic processes Sequential methods

Published online by Cambridge University Press:  01 July 2016

Christian Bender  and
John Schoenmakers
Christian Bender*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics
John Schoenmakers*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics
*
∗ Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany.
∗ Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany.
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Abstract

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We present a new iterative procedure for solving the multiple stopping problem in discrete time and discuss the stability of the algorithm. The algorithm produces monotonically increasing approximations of the Snell envelope which coincide with the Snell envelope after finitely many steps. Unlike backward dynamic programming, the algorithm allows us to calculate approximative solutions with only a few nestings of conditional expectations and is, therefore, tailor-made for a plain Monte Carlo implementation.

Keywords

Optimal stoppingpolicy improvementmultiple callable financial derivatives

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 62L15: Optimal stopping
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 3 , September 2006 , pp. 729 - 749
Copyright
Copyright © Applied Probability Trust 2006 

Footnotes

Supported by the DFG Research Center Matheon, Berlin.

References

Andersen, L. (1999). A simple approach to the pricing of Bermudan swaptions in the multifactor LIBOR market model. J. Comput. Finance 3, 5–32.CrossRefGoogle Scholar
Bender, C. and Schoenmakers, J. (2006). Addendum to ‘An iterative method for multiple stopping: convergence and stability’. Available at http://www.wias-berlin.de/people/schoenma/.Google Scholar
Bertsekas, D. P. (2001). Dynamic Programming and Optimal Control, Vol. II, 2nd edn. Athena Scientific, Belmont, MA.Google Scholar
Broadie, M. and Glasserman, P. (2004). A stochastic mesh method for pricing high-dimensional American options. J. Comput. Finance 7, 35–72.CrossRefGoogle Scholar
Broadie, M., Glasserman, P. and Ha, Z. (2000). Pricing American options by simulations using a stochastic mesh with optimized weights. In Probabilistic Constrained Optimization: Methodology and Applications, ed. Uryasev, S., Kluwer, Dordrecht, pp. 26–44.CrossRefGoogle Scholar
Carmona, R. and Touzi, N. (2006). Optimal multiple-stopping and valuation of swing options. To appear in Math. Finance.Google Scholar
Haugh, M. B. and Kogan, L. (2004). Pricing American options: a duality approach. Operat. Res. 52, 258–270.CrossRefGoogle Scholar
Ibáñez, A. (2004). Valuation by simulation of contingent claims with multiple exercise opportunities. Math. Finance 14, 223–248.Google Scholar
Ibáñez, A. and Zapatero, F. (2004). Monte Carlo valuation of American options through computation of the optimal exercise frontier. J. Financial Quant. Anal. 39, 253–276.CrossRefGoogle Scholar
Jaillet, P., Ronn, E. I. and Tompaidis, S. (2004). Valuation of commodity-based swing options. Manag. Sci. 50, 909–921.CrossRefGoogle Scholar
Jamshidian, F. (2006). Duality of optimal stopping times and domineering claims: a Doob–Meyer decomposition approach to American options. Working paper.Google Scholar
Kolodko, A. and Schoenmakers, J. (2006). Iterative construction of the optimal Bermudan stopping time. Finance Stoch. 10, 27–49.CrossRefGoogle Scholar
Kushner, H. J. and Dupuis, P. (2001). Numerical Methods for Stochastic Control Problems in Continuous Time, 2nd edn, Springer, New York.CrossRefGoogle Scholar
Longstaff, F. A. and Schwartz, E. S. (2001). Valuing American options by simulation: a simple least-square approach. Rev. Financial Stud. 14, 113–147.Google Scholar
Meinshausen, N. and Hambly, B. M. (2004). Monte Carlo methods for the valuation of multiple-exercise options. Math. Finance 14, 557–583.CrossRefGoogle Scholar
Milstein, G. N., Reiss, O. and Schoenmakers, J. (2004). A new Monte Carlo method for American options. Internat. J. Theoret. Appl. Finance 7, 591–614.CrossRefGoogle Scholar
Neveu, J. (1975). Discrete-Parameter Martingales. North-Holland, Amsterdam.Google Scholar
Puterman, M. L. (1994). Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley, New York.Google Scholar
Rogers, L. C. G. (2002). Monte Carlo valuation of American options. Math. Finance 12, 271–286.Google Scholar
Schoenmakers, J. (2005). Robust Libor Modelling and Pricing of Derivative Products. Chapman and Hall/CRC Press, Boca Raton, FL.CrossRefGoogle Scholar