Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-19T22:27:06.487Z Has data issue: false hasContentIssue false
  • English
  • Français

An explicit solution to an optimal stopping problem with regime switching

Part of: Stochastic processes Stochastic systems and control Sequential methods

Published online by Cambridge University Press:  14 July 2016

Xin Guo
Xin Guo*
Affiliation:
IBM Research Division and University of Alberta
*
∗ Postal address: T. J. Watson Research Center, P.O. Box 218, IBM, Yorktown Heights, NY 10598, USA. Email address: xinguo@us.ibm.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate an optimal stopping time problem which arises from pricing Russian options (i.e. perpetual look-back options) on a stock whose price fluctuations are modelled by adjoining a hidden Markov process to the classical Black-Scholes geometric Brownian motion model. By extending the technique of smooth fit to allow jump discontinuities, we obtain an explicit closed-form solution. It gives a non-standard application of the well-known smooth fit principle where the optimal strategy involves jumping over the optimal boundary and by an arbitrary overshoot. Based on the optimal stopping analysis, an arbitrage-free price for Russian options under the hidden Markov model is derived.

Keywords

hidden Markov processmartingaleoptimal stoppingregime switchingRussian options

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 62L10: Sequential analysis 93E20: Optimal stochastic control
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 2 , June 2001 , pp. 464 - 481
Copyright
Copyright © Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chernoff, H. (1961). Sequential tests for the mean of a normal distribution. In Proc. 4th. Berkeley Symp. Statist. Prob. Vol. 1. University of California Press, Berkeley, pp. 79–91.Google Scholar
Delbaen, F., and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520.Google Scholar
Duffie, D., and Harrison, M. (1993). Arbitrage pricing of perpetual lookback options. Ann. Appl. Prob. 3, 641–651.CrossRefGoogle Scholar
Elliott, R. (1982). Stochastic calculus and applications. Springer, Berlin.Google Scholar
Gerber, H. U., and Shiu, E. S. W. (1996). Martingale approach to pricing perpetual American options on two stocks. Math. Finance 6, 303–322.CrossRefGoogle Scholar
Guo, X. (2001). Information and option pricing. Quant. Finance 1, 38–44.CrossRefGoogle Scholar
Harrison, M., and Kreps, D. (1979). Martingales and arbitrage in multiperiod securities markets. J. Econom. Theory 20, 381–408.Google Scholar
Harrison, M., and Pliska, S. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stoch. Proc. Appl. 11, 215–260.CrossRefGoogle Scholar
Karatzas, I., and Shreve, S. (1998). Brownian Motion and Stochastic Calculus. Springer, Berlin.CrossRefGoogle Scholar
Revuz, D., and Yor, M. (1991). Continuous Martingale and Brownian Motion. Springer, Berlin.CrossRefGoogle Scholar
Shepp, L. A. (1999). A model for stock price fluctuations based on information. Preprint, Rutgers University.Google Scholar
Shepp, L. A., and Shiryaev, A. N. (1993). The Russian option: reduced regret. Ann. Appl. Prob. 3, 631–640.CrossRefGoogle Scholar
Shepp, L. A., and Shiryaev, A. N. (1994). A new look at pricing of the ‘Russian options’. Theory Prob. Appl. 39, 103–119.Google Scholar
Van Moerbeke, P. L. J. (1976). On optimal stopping and free boundary problems. Arch. Rational Mechanics Anal. 60, 101–148.CrossRefGoogle Scholar