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Credit Risk Modeling Using Time-Changed Brownian Motion

Author

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  • T. R. HURD

    (Department of Mathematics and Statistics, McMaster University, Hamilton ON L8S 4K1, Canada)

Abstract

Motivated by the interplay between structural and reduced form credit models, we propose to model the firm value process as a time-changed Brownian motion that may include jumps and stochastic volatility effects, and to study the first passage problem for such processes. We are lead to consider modifying the standard first passage problem for stochastic processes to capitalize on this time change structure and find that the distribution functions of such "first passage times of the second kind" are efficiently computable in a wide range of useful examples. Thus this new notion of first passage can be used to define the time of default in generalized structural credit models. Formulas for defaultable bonds and credit default swaps are given that are both efficiently computable and lead to realistic spread curves. Finally, we show that by treating joint firm value processes as dependent time changes of independent Brownian motions, one can obtain multifirm credit models with rich and plausible dynamics and enjoying the possibility of efficient valuation of portfolio credit derivatives.

Suggested Citation

  • T. R. Hurd, 2009. "Credit Risk Modeling Using Time-Changed Brownian Motion," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(08), pages 1213-1230.
    • Handle: RePEc:wsi:ijtafx:v:12:y:2009:i:08:n:s0219024909005646
    DOI: 10.1142/S0219024909005646
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    Citations

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    Cited by:

    1. Thomas R. Hurd & Zhuowei Zhou, 2011. "Two-factor capital structure models for equity and credit," Papers 1110.5846, arXiv.org.
    2. Marwa Belhaj Salem & Mitra Fouladirad & Estelle Deloux, 2021. "Prognostic and Classification of Dynamic Degradation in a Mechanical System Using Variance Gamma Process," Mathematics, MDPI, vol. 9(3), pages 1-25, January.
    3. Flavia Barsotti, 2012. "Optimal Capital Structure with Endogenous Default and Volatility Risk," Working Papers - Mathematical Economics 2012-02, Universita' degli Studi di Firenze, Dipartimento di Scienze per l'Economia e l'Impresa.
    4. T. R. Hurd & Zhuowei Zhou, 2011. "Statistical Inference for Time-changed Brownian Motion Credit Risk Models," Papers 1102.2412, arXiv.org.
    5. Mario Abundo, 2018. "The Randomized First-Hitting Problem of Continuously Time-Changed Brownian Motion," Mathematics, MDPI, vol. 6(6), pages 1-10, May.
    6. Marcos Escobar & Peter Hieber & Matthias Scherer, 2014. "Efficiently pricing double barrier derivatives in stochastic volatility models," Review of Derivatives Research, Springer, vol. 17(2), pages 191-216, July.
    7. Son-Nan Chen & Pao-Peng Hsu & Chang-Yi Li, 2016. "Pricing credit-risky bonds and spread options modelling credit-spread term structures with two-dimensional Markov-modulated jump-diffusion," Quantitative Finance, Taylor & Francis Journals, vol. 16(4), pages 573-592, April.
    8. Salem, Marwa Belhaj & Fouladirad, Mitra & Deloux, Estelle, 2022. "Variance Gamma process as degradation model for prognosis and imperfect maintenance of centrifugal pumps," Reliability Engineering and System Safety, Elsevier, vol. 223(C).
    9. Fernández Lexuri & Hieber Peter & Scherer Matthias, 2013. "Double-barrier first-passage times of jump-diffusion processes," Monte Carlo Methods and Applications, De Gruyter, vol. 19(2), pages 107-141, July.
    10. Hieber, Peter & Scherer, Matthias, 2012. "A note on first-passage times of continuously time-changed Brownian motion," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 165-172.
    11. Zhang, Yuxin & Brockett, Patrick, 2020. "Modeling stochastic mortality for joint lives through subordinators," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 166-172.
    12. Masaaki Kijima & Chi Chung Siu, 2014. "Credit-Equity Modeling Under A Latent Lévy Firm Process," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(03), pages 1-41.
    13. Omar, Arti & Prasanna, P. Krishna, 2021. "Asymmetric effects of noise in Merton default risk model: Evidence from emerging Asia," Pacific-Basin Finance Journal, Elsevier, vol. 65(C).

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