Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T17:12:50.001Z Has data issue: false hasContentIssue false
  • English
  • Français

Russian options with a finite time horizon

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Erik Ekström
Erik Ekström*
Affiliation:
Uppsala University
*
∗ Postal address: Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden. Email address: ekstrom@math.uu.se
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the Russian option with a finite time horizon in the standard Black–Scholes model. The value of the option is shown to be a solution of a certain parabolic free boundary problem, and the optimal stopping boundary is shown to be continuous. Moreover, the asymptotic behavior of the optimal stopping boundary near expiration is studied.

Keywords

Optimal stoppingmaximum processoptimal stopping boundaryoptions

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 313 - 326
Copyright
Copyright © Applied Probability Trust 2004 

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

Barles, G., Burdeau, J., Romano, M., and Samsoen, N. (1995). Critical stock price near expiration. Math. Finance 5, 77–95.CrossRefGoogle Scholar
Barles, G., Daher, C., and Romano, M. (1994). Optimal control on the L ∞ norm of a diffusion process. SIAM J. Control Optimization 32, 612–634.Google Scholar
Bather, J., Chernoff, H., and Petkau, J. (1989). The effect of truncation on a sequential test for the drift of Brownian motion. Sequential Anal. 8, 169–190.Google Scholar
Chernoff, H. (1972). Sequential Analysis and Optimal Design (Conf. Board Math. Sci. Regional Conf. Ser. Appl. Math. 8). Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Duffie, D., and Harrison, M. J. (1993). Arbitrage pricing of Russian options and perpetual lookback options. Ann. Appl. Prob. 3, 641–651.Google Scholar
Duistermaat, J. J., Kyprianou, A. E., and van Schaik, K. (2003). Finite expiry Russian options. Preprint.Google Scholar
El Karoui, N. (1981). Les aspects probabilistes du contrôle stochastique. In Ninth Saint Flour Probability Summer School (Lecture Notes Math. 876). Springer, Berlin, pp. 73–238.Google Scholar
Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Graversen, S. E., and Peskir, G. (1997). On the Russian option: the expected waiting time. Theory Prob. Appl. 42, 416–425.Google Scholar
Guo, X., and Shepp, L. (2001). Some optimal stopping problems with nontrivial boundaries for pricing exotic options. J. Appl. Prob. 38, 647–658.CrossRefGoogle Scholar
Karatzas, I., and Shreve, S. (1998). Methods of Mathematical Finance. Springer, Berlin.Google Scholar
Karatzas, I., and Shreve, S. (2000). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin.Google Scholar
Lamberton, D. (1995). Critical price for an American option near maturity. In Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993) (Progr. Prob. 36), Birkhäuser, Basel, pp. 353–358.Google Scholar
Pedersen, J. (2000). Discounted optimal stopping problems for the maximum process. J. Appl. Prob. 37, 972–983.Google Scholar
Peskir, G. (2003). The Russian option: finite horizon. To appear in Finance Stoch.Google Scholar
Shepp, L., and Shiryaev, A. N. (1993). The Russian option: reduced regret. Ann. Appl. Prob. 3, 631–640.Google Scholar
Shepp, L., and Shiryaev, A. N. (1994). A new look at pricing of the ‘Russian option’. Theory Prob. Appl. 39, 103–119.Google Scholar