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Spectral Expansions for Credit Risk Modelling with Occupation Times

Author

Listed:
  • Giuseppe Campolieti

    (Department of Mathematics, Faculty of Science, Wilfrid Laurier University, 75 University Ave. West, Waterloo, ON N2L 3C5, Canada)

  • Hiromichi Kato

    (Department of Mathematics, Faculty of Science, Wilfrid Laurier University, 75 University Ave. West, Waterloo, ON N2L 3C5, Canada)

  • Roman N. Makarov

    (Department of Mathematics, Faculty of Science, Wilfrid Laurier University, 75 University Ave. West, Waterloo, ON N2L 3C5, Canada)

Abstract

We study two credit risk models with occupation time and liquidation barriers: the structural model and the hybrid model with hazard rate. The defaults within the models are characterized in accordance with Chapter 7 (a liquidation process) and Chapter 11 (a reorganization process) of the U.S. Bankruptcy Code. The models assume that credit events trigger as soon as the occupation time (the cumulative time the firm’s value process spends below some threshold level) exceeds the grace period (time allowance). The hazard rate model extends the structural occupation time models and presumes that other random factors may also lead to credit events. Both approaches allow the firm to fulfill its obligations during the grace period. We derive new closed-from pricing formulas for credit derivatives containing the (risk-neutral) probability of defaults and credit default swap (CDS) spreads as special cases, which are derived analytically via a spectral expansion methodology. Our method works for any solvable diffusion, such as the geometric Brownian motion (GBM) and several state-dependent volatility processes, including the constant elasticity of variance (CEV) model. It allows us to write the pricing formulas explicitly as infinite series that converges rapidly. We then calibrate our models (assuming that GBM governs the firm’s value) to market CDS spreads from the Total Energy company. Our calibration results show that the computations are fast, and the fit is near-perfect.

Suggested Citation

  • Giuseppe Campolieti & Hiromichi Kato & Roman N. Makarov, 2022. "Spectral Expansions for Credit Risk Modelling with Occupation Times," Risks, MDPI, vol. 10(12), pages 1-20, November.
    • Handle: RePEc:gam:jrisks:v:10:y:2022:i:12:p:228-:d:989220
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    References listed on IDEAS

    as
    1. Giuseppe Campolieti & Roman N. Makarov & Karl Wouterloot, 2013. "Pricing Step Options under the CEV and other Solvable Diffusion Models," Papers 1302.3771, arXiv.org.
    2. Roman N. Makarov, 2016. "Modeling liquidation risk with occupation times," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(04), pages 1-11, December.
    3. Aurélien Alfonsi & Jérôme Lelong, 2012. "A Closed-Form Extension To The Black-Cox Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(08), pages 1-30.
    4. Mark Broadie & Mikhail Chernov & Suresh Sundaresan, 2007. "Optimal Debt and Equity Values in the Presence of Chapter 7 and Chapter 11," Journal of Finance, American Finance Association, vol. 62(3), pages 1341-1377, June.
    5. Merton, Robert C, 1974. "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates," Journal of Finance, American Finance Association, vol. 29(2), pages 449-470, May.
    6. Makarov, R. & Metzler, A. & Ni, Z., 2015. "Modelling default risk with occupation times," Finance Research Letters, Elsevier, vol. 13(C), pages 54-65.
    7. Galai, Dan & Raviv, Alon & Wiener, Zvi, 2007. "Liquidation triggers and the valuation of equity and debt," Journal of Banking & Finance, Elsevier, vol. 31(12), pages 3604-3620, December.
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    10. Bin Li & Qihe Tang & Lihe Wang & Xiaowen Zhou, 2014. "Liquidation risk in the presence of Chapters 7 and 11 of the US bankruptcy code," Journal of Financial Engineering (JFE), World Scientific Publishing Co. Pte. Ltd., vol. 1(03), pages 1-19.
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