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Crossing Probabilities for Diffusion Processes with Piecewise Continuous Boundaries

Author

Listed:
  • Liqun Wang

    (University of Manitoba)

  • Klaus Pötzelberger

    (University of Economics and Business Administration Vienna)

Abstract

We propose an approach to compute the boundary crossing probabilities for a class of diffusion processes which can be expressed as piecewise monotone (not necessarily one-to-one) functionals of a standard Brownian motion. This class includes many interesting processes in real applications, e.g., Ornstein–Uhlenbeck, growth processes and geometric Brownian motion with time dependent drift. This method applies to both one-sided and two-sided general nonlinear boundaries, which may be discontinuous. Using this approach explicit formulas for boundary crossing probabilities for certain nonlinear boundaries are obtained, which are useful in evaluation and comparison of various computational algorithms. Moreover, numerical computation can be easily done by Monte Carlo integration and the approximation errors for general boundaries are automatically calculated. Some numerical examples are presented.

Suggested Citation

  • Liqun Wang & Klaus Pötzelberger, 2007. "Crossing Probabilities for Diffusion Processes with Piecewise Continuous Boundaries," Methodology and Computing in Applied Probability, Springer, vol. 9(1), pages 21-40, March.
    • Handle: RePEc:spr:metcap:v:9:y:2007:i:1:d:10.1007_s11009-006-9002-6
    DOI: 10.1007/s11009-006-9002-6
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    References listed on IDEAS

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    Cited by:

    1. Lee, Hangsuck & Ha, Hongjun & Kong, Byungdoo, 2024. "Pricing first-touch digitals with a multi-step double boundary and American barrier options," Finance Research Letters, Elsevier, vol. 59(C).
    2. Zhiyong Jin & Liqun Wang, 2017. "First Passage Time for Brownian Motion and Piecewise Linear Boundaries," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 237-253, March.
    3. Mario Abundo, 2010. "On the First Hitting Time of a One-dimensional Diffusion and a Compound Poisson Process," Methodology and Computing in Applied Probability, Springer, vol. 12(3), pages 473-490, September.
    4. Lee, Taeho, 2023. "Exact simulation for the first hitting time of Brownian motion and Brownian bridge," Statistics & Probability Letters, Elsevier, vol. 193(C).
    5. Hangsuck Lee & Hongjun Ha & Minha Lee, 2022. "Piecewise linear double barrier options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(1), pages 125-151, January.
    6. Qinglai Dong & Lirong Cui, 2019. "First Hitting Time Distributions for Brownian Motion and Regions with Piecewise Linear Boundaries," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 1-23, March.
    7. Hangsuck Lee & Hongjun Ha & Minha Lee, 2022. "Piecewise linear boundary crossing probabilities, barrier options, and variable annuities," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(12), pages 2248-2272, December.
    8. Andrew N. Downes & Konstantin Borovkov, 2008. "First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes," Methodology and Computing in Applied Probability, Springer, vol. 10(4), pages 621-644, December.

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