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On the first passage problem for correlated Brownian motion

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  • Metzler, Adam

Abstract

Suppose that X=(X1,X2) is two-dimensional correlated Brownian motion. Let [tau]i denote the first passage time of Xi to a fixed level, and [tau] the minimum of [tau]1,[tau]2. When X has zero drift, several distributions of interest are available in closed form, including the joint density of the passage times and the distribution of X([tau]). Unfortunately these published formulae contain errors, and the corresponding distributions in the presence of drift are not expressible in closed form. The purpose of this paper is to address these issues by presenting corrected formulae and outlining a Monte Carlo algorithm for approximating quantities of interest in the presence of drift.

Suggested Citation

  • Metzler, Adam, 2010. "On the first passage problem for correlated Brownian motion," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 277-284, March.
    • Handle: RePEc:eee:stapro:v:80:y:2010:i:5-6:p:277-284
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    References listed on IDEAS

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    1. Mark B. Wise & Vineer Bhansali, 2008. "Correlated Random Walks and the Joint Survival Probability," Papers 0812.2000, arXiv.org.
    2. Joonghee Huh & Adam Kolkiewicz, 2008. "Computation of Multivariate Barrier Crossing Probability and its Applications in Credit Risk Models," North American Actuarial Journal, Taylor & Francis Journals, vol. 12(3), pages 263-291.
    3. Zhou, Chunsheng, 2001. "An Analysis of Default Correlations and Multiple Defaults," The Review of Financial Studies, Society for Financial Studies, vol. 14(2), pages 555-576.
    4. Christophette Blanchet-Scalliet & Fr'ed'eric Patras, 2008. "Counterparty risk valuation for CDS," Papers 0807.0309, arXiv.org.
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    Cited by:

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    3. Andrey Itkin & Alexander Lipton, 2017. "Structural default model with mutual obligations," Review of Derivatives Research, Springer, vol. 20(1), pages 15-46, April.
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    6. Wai-Ki Ching & Jia-Wen Gu & Harry Zheng, 2014. "On Correlated Defaults and Incomplete Information," Papers 1409.1393, arXiv.org, revised Jan 2016.
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    13. James Brugler & Oliver Linton, 2014. "Circuit Breakers on the London Stock Exchange: Do they improve subsequent market quality?," Cambridge Working Papers in Economics 1453, Faculty of Economics, University of Cambridge.
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    16. Weiping Li, 2016. "Probability of Default and Default Correlations," JRFM, MDPI, vol. 9(3), pages 1-19, July.

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