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Modeling and pricing credit risk with a focus on recovery risk

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  • Liu, Haibo
  • Tang, Qihe

Abstract

Consider a defaultable bond traded in a financial market that is subject to shocks and regime shifts. Its recovery payment has a hybrid structure, comprising two components: one contingent on historical information up to the time of default, and the other an independent variable indexed by the regime at the time of default. The default intensity, interest rate, and reference rate are assumed to be general deterministic functions of certain state variables, while these state variables jointly follow a jump-diffusion process, with drift and volatility coefficients governed by the regime and with jumps induced by shocks. We construct a risk-neutral pricing measure that prices all risk sources in an integrated manner. A rigorous verification of this pricing measure reveals the corresponding time-dependent market prices of these risk sources. The resulting pricing framework is applicable to most defaultable bonds and credit derivatives.

Suggested Citation

  • Liu, Haibo & Tang, Qihe, 2025. "Modeling and pricing credit risk with a focus on recovery risk," Journal of Banking & Finance, Elsevier, vol. 170(C).
    • Handle: RePEc:eee:jbfina:v:170:y:2025:i:c:s0378426624002310
    DOI: 10.1016/j.jbankfin.2024.107317
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    as
    1. Gambetti, Paolo & Gauthier, Geneviève & Vrins, Frédéric, 2019. "Recovery rates: Uncertainty certainly matters," Journal of Banking & Finance, Elsevier, vol. 106(C), pages 371-383.
    2. Duffie, Darrell & Saita, Leandro & Wang, Ke, 2007. "Multi-period corporate default prediction with stochastic covariates," Journal of Financial Economics, Elsevier, vol. 83(3), pages 635-665, March.
    3. Jennie Bai & Pierre Collin-Dufresne & Robert S. Goldstein & Jean Helwege, 2015. "On Bounding Credit-Event Risk Premia," The Review of Financial Studies, Society for Financial Studies, vol. 28(9), pages 2608-2642.
    4. repec:bla:ecnote:v:33:y:2004:i:2:p:183-208 is not listed on IDEAS
    5. Alain Monfort & Fulvio Pegoraro & Jean-Paul Renne & Guillaume Roussellet, 2021. "Affine Modeling of Credit Risk, Pricing of Credit Events, and Contagion," Management Science, INFORMS, vol. 67(6), pages 3674-3693, June.
    6. Gürtler, Marc & Hibbeln, Martin, 2013. "Improvements in loss given default forecasts for bank loans," Journal of Banking & Finance, Elsevier, vol. 37(7), pages 2354-2366.
    7. Andrew Ang & Geert Bekaert & Min Wei, 2008. "The Term Structure of Real Rates and Expected Inflation," Journal of Finance, American Finance Association, vol. 63(2), pages 797-849, April.
    8. Stephan Höcht & Rudi Zagst, 2010. "Pricing credit derivatives under stochastic recovery in a hybrid model," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 26(3), pages 254-276, May.
    9. Duffie, Darrell & Lando, David, 2001. "Term Structures of Credit Spreads with Incomplete Accounting Information," Econometrica, Econometric Society, vol. 69(3), pages 633-664, May.
    10. Robert A. Jarrow & David Lando & Fan Yu, 2008. "Default Risk And Diversification: Theory And Empirical Implications," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 19, pages 455-480, World Scientific Publishing Co. Pte. Ltd..
    11. Naik, Vasanttilak, 1993. "Option Valuation and Hedging Strategies with Jumps in the Volatility of Asset Returns," Journal of Finance, American Finance Association, vol. 48(5), pages 1969-1984, December.
    12. Calabrese, Raffaella & Zenga, Michele, 2010. "Bank loan recovery rates: Measuring and nonparametric density estimation," Journal of Banking & Finance, Elsevier, vol. 34(5), pages 903-911, May.
    13. Liuren Wu & Frank Xiaoling Zhang, 2008. "A No-Arbitrage Analysis of Macroeconomic Determinants of the Credit Spread Term Structure," Management Science, INFORMS, vol. 54(6), pages 1160-1175, June.
    14. Acharya, Viral V. & Bharath, Sreedhar T. & Srinivasan, Anand, 2007. "Does industry-wide distress affect defaulted firms? Evidence from creditor recoveries," Journal of Financial Economics, Elsevier, vol. 85(3), pages 787-821, September.
    15. Peter Christoffersen & Du Du & Redouane Elkamhi, 2017. "Rare Disasters, Credit, and Option Market Puzzles," Management Science, INFORMS, vol. 63(5), pages 1341-1364, May.
    16. Ang, Andrew & Bekaert, Geert, 2002. "Regime Switches in Interest Rates," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(2), pages 163-182, April.
    17. Barthélemy, Jean & Marx, Magali, 2017. "Solving endogenous regime switching models," Journal of Economic Dynamics and Control, Elsevier, vol. 77(C), pages 1-25.
    18. Qiang Dai & Kenneth J. Singleton, 2000. "Specification Analysis of Affine Term Structure Models," Journal of Finance, American Finance Association, vol. 55(5), pages 1943-1978, October.
    19. Junchi Ma & Mobolaji Ogunsolu & Jinniao Qiu & Ayşe Deniz Sezer, 2023. "Credit risk pricing in a consumption‐based equilibrium framework with incomplete accounting information," Mathematical Finance, Wiley Blackwell, vol. 33(3), pages 666-708, July.
    20. Chang, Yoosoon & Choi, Yongok & Park, Joon Y., 2017. "A new approach to model regime switching," Journal of Econometrics, Elsevier, vol. 196(1), pages 127-143.
    21. Joshua D. Coval & Jakub W. Jurek & Erik Stafford, 2009. "Economic Catastrophe Bonds," American Economic Review, American Economic Association, vol. 99(3), pages 628-666, June.
    22. Gregory R. Duffee, 2002. "Term Premia and Interest Rate Forecasts in Affine Models," Journal of Finance, American Finance Association, vol. 57(1), pages 405-443, February.
    23. Joost Driessen, 2005. "Is Default Event Risk Priced in Corporate Bonds?," The Review of Financial Studies, Society for Financial Studies, vol. 18(1), pages 165-195.
    24. Philip Lowe, 2002. "Credit risk measurement and procyclicality," BIS Working Papers 116, Bank for International Settlements.
    25. Ravi Bansal & Hao Zhou, 2002. "Term Structure of Interest Rates with Regime Shifts," Journal of Finance, American Finance Association, vol. 57(5), pages 1997-2043, October.
    26. Edward I. Altman & Brooks Brady & Andrea Resti & Andrea Sironi, 2005. "The Link between Default and Recovery Rates: Theory, Empirical Evidence, and Implications," The Journal of Business, University of Chicago Press, vol. 78(6), pages 2203-2228, November.
    27. Qiang Dai & Kenneth J. Singleton & Wei Yang, 2007. "Regime Shifts in a Dynamic Term Structure Model of U.S. Treasury Bond Yields," The Review of Financial Studies, Society for Financial Studies, vol. 20(5), pages 1669-1706, 2007 12.
    28. Hamilton, James D., 1988. "Rational-expectations econometric analysis of changes in regime : An investigation of the term structure of interest rates," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 385-423.
    29. Li, Junye & Sarno, Lucio & Zinna, Gabriele, 2024. "Risks and risk premia in the US Treasury market," Journal of Economic Dynamics and Control, Elsevier, vol. 158(C).
    30. Tomas Björk & Yuri Kabanov & Wolfgang Runggaldier, 1997. "Bond Market Structure in the Presence of Marked Point Processes," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 211-239, April.
    31. Ang, Andrew & Bekaert, Geert, 2002. "Short rate nonlinearities and regime switches," Journal of Economic Dynamics and Control, Elsevier, vol. 26(7-8), pages 1243-1274, July.
    32. Derrien, François & Kecskés, Ambrus & Mansi, Sattar A., 2016. "Information asymmetry, the cost of debt, and credit events: Evidence from quasi-random analyst disappearances," Journal of Corporate Finance, Elsevier, vol. 39(C), pages 295-311.
    33. Duffie, Darrell & Singleton, Kenneth J, 1999. "Modeling Term Structures of Defaultable Bonds," The Review of Financial Studies, Society for Financial Studies, vol. 12(4), pages 687-720.
    34. Liu, Wenchien & Miu, Peter & Chang, Yuanchen & Ozdemir, Bogie, 2012. "Information asymmetry and bank regulation: Can the spread of debt contracts be explained by recovery rates?," Journal of Financial Intermediation, Elsevier, vol. 21(1), pages 123-150.
    35. Qiang Dai & Kenneth Singleton, 2003. "Term Structure Dynamics in Theory and Reality," The Review of Financial Studies, Society for Financial Studies, vol. 16(3), pages 631-678, July.
    36. Doshi, Hitesh & Elkamhi, Redouane & Ornthanalai, Chayawat, 2018. "The Term Structure of Expected Recovery Rates," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 53(6), pages 2619-2661, December.
    37. Grunert, Jens & Weber, Martin, 2009. "Recovery rates of commercial lending: Empirical evidence for German companies," Journal of Banking & Finance, Elsevier, vol. 33(3), pages 505-513, March.
    38. R. J. Elliott & M. Jeanblanc & M. Yor, 2000. "On Models of Default Risk," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 179-195, April.
    39. Giesecke, Kay, 2004. "Correlated default with incomplete information," Journal of Banking & Finance, Elsevier, vol. 28(7), pages 1521-1545, July.
    40. Alain BÉlanger & Steven E. Shreve & Dennis Wong, 2004. "A General Framework For Pricing Credit Risk," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 317-350, July.
    41. Lu, Chia-Wu & Chen, Tsung-Kang & Liao, Hsien-Hsing, 2010. "Information uncertainty, information asymmetry and corporate bond yield spreads," Journal of Banking & Finance, Elsevier, vol. 34(9), pages 2265-2279, September.
    42. Lindset, Snorre & Lund, Arne-Christian & Persson, Svein-Arne, 2014. "Credit risk and asymmetric information: A simplified approach," Journal of Economic Dynamics and Control, Elsevier, vol. 39(C), pages 98-112.
    43. Candian, Giacomo & Dmitriev, Mikhail, 2020. "Default recovery rates and aggregate fluctuations," Journal of Economic Dynamics and Control, Elsevier, vol. 121(C).
    44. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    45. Kim, Chang-Jin & Piger, Jeremy & Startz, Richard, 2008. "Estimation of Markov regime-switching regression models with endogenous switching," Journal of Econometrics, Elsevier, vol. 143(2), pages 263-273, April.
    46. Hui Chen & Scott Joslin, 2012. "Generalized Transform Analysis of Affine Processes and Applications in Finance," The Review of Financial Studies, Society for Financial Studies, vol. 25(7), pages 2225-2256.
    47. Pham, Huyên, 2010. "Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1795-1820, August.
    48. Duffie, Darrell, 2005. "Credit risk modeling with affine processes," Journal of Banking & Finance, Elsevier, vol. 29(11), pages 2751-2802, November.
    49. Schläfer, Timo & Uhrig-Homburg, Marliese, 2014. "Is recovery risk priced?," Journal of Banking & Finance, Elsevier, vol. 40(C), pages 257-270.
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    More about this item

    Keywords

    Risk-neutral pricing; Default; Shock; Regime shift; Recovery risk; Doubly stochastic Poisson process;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G33 - Financial Economics - - Corporate Finance and Governance - - - Bankruptcy; Liquidation
    • C41 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Duration Analysis; Optimal Timing Strategies

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