Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T13:11:27.382Z Has data issue: false hasContentIssue false
  • English
  • Français

Explicit Bounds for Approximation Rates of Boundary Crossing Probabilities for the Wiener Process

Part of: Probabilistic methods, simulation and stochastic differential equations Markov processes Stochastic processes Mathematical economics

Published online by Cambridge University Press:  14 July 2016

K. Borovkov  and
A. Novikov
K. Borovkov*
Affiliation:
University of Melbourne
A. Novikov*
Affiliation:
University of Technology, Sydney
*
∗Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville 3010, Australia. Email address: kostya@ms.unimelb.edu.au
∗∗Postal address: Department of Mathematical Sciences, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give explicit upper bounds for convergence rates when approximating both one- and two-sided general curvilinear boundary crossing probabilities for the Wiener process by similar probabilities for close boundaries of simpler form, for which computation of the boundary crossing probabilities is feasible. In particular, we partially generalize and improve results obtained by Pötzelberger and Wang in the case when the approximating boundaries are piecewise linear. Applications to barrier option pricing are also discussed.

Keywords

Wiener processboundary crossing probabilitybarrier option

MSC classification

Secondary: 60J65: Brownian motion 65C50: Other computational problems in probability 91B70: Stochastic models 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 1 , March 2005 , pp. 82 - 92
Copyright
© Applied Probability Trust 2005 

References

Acton, F. S. (1990). Numerical Methods that Work. Mathematical Association of America, Washington, DC.CrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.CrossRefGoogle Scholar
Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory. Springer, New York.CrossRefGoogle Scholar
Borovkov, K. A. (1982). Limit theorems for some types of random walks. , Steklov Mathematical Institute, Moscow.Google Scholar
Daniels, H. E. (1969). The minimum of a stationary Markov process superimposed on a U-shaped trend. J. Appl. Prob. 6, 399–408.CrossRefGoogle Scholar
Daniels, H. E. (1996). Approximating the first crossing-time density for a curved boundary. Bernoulli 2, 133–143.CrossRefGoogle Scholar
Di Nardo, E., Nobile, A. G., Pirozzi, E. and Ricciardi, L. M. (2001). A computational approach to first-passage-time problems for Gauss–Markov processes. Adv. Appl. Prob. 33, 453–482.CrossRefGoogle Scholar
Durbin, J. (1992). The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Prob. 29, 291–304.CrossRefGoogle Scholar
Lerche, H. R. (1986). Boundary Crossing of Brownian Motion (Lecture Notes in Statistics 40). Springer, Berlin.Google Scholar
Nagaev, S. V. (1970). The rate of convergence in a certain boundary value problem. I. Theory Prob. Appl. 15, 179–199.Google Scholar
Nagaev, S. V. (1970). The rate of convergence in a certain boundary value problem. II. Theory Prob. Appl. 15, 419–444.Google Scholar
Novikov, A. A., Frishling, V. and Kordzakhia, N. (1999). Approximations of boundary crossing probabilities for a Brownian motion. J. Appl. Prob. 36, 1019–1030.CrossRefGoogle Scholar
Novikov, A. A., Frishling, V. and Kordzakhia, N. (2003). Time-dependent barrier options and boundary crossing probabilities. Georgian Math. J. 10, 325–334.Google Scholar
Pötzelberger, K. and Wang, L. (2001). Boundary crossing probability for Brownian motion. J. Appl. Prob. 38, 152–164.CrossRefGoogle Scholar
Roberts, G. O. and Shortland, C. F. (1997). Pricing barrier options with time-dependent coefficients. Math. Finance 7, 83–93.CrossRefGoogle Scholar
Sacerdote, L. and Tomassetti, F. (1996). On evaluations and asymptotic approximations of first-passage-time probabilities. Adv. Appl. Prob. 28, 270–284.Google Scholar
Sahanenko, A. I. (1974). On the rate of convergence in a boundary value problem. Theory Prob. Appl. 19, 416–421.Google Scholar
Shikin, E. V. and Plis, A. I. (1995). Handbook on Splines for the User. CRC Press, Boca Raton, FL.Google Scholar
Shiryaev, A. N. (1999). Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific, Singapore.CrossRefGoogle Scholar
Wang, L. and Pötzelberger, K. (1997). Boundary crossing probability for Brownian motion and general boundaries. J. Appl. Prob. 34, 54–65.CrossRefGoogle Scholar
Wilmott, P., Dewynne, J. and Howison, S. (1995). Option Pricing: Mathematical Models and Computation. Cambridge Financial Press.Google Scholar