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June  2020, 10(2): 257-274. doi: 10.3934/mcrf.2019038

Free boundaries of credit rating migration in switching macro regions

Yuan Wu 1, and  Jin Liang 2,,

1. 

School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China
2. 

School of Mathematical Science, Tongji University, Shanghai 200092, China

* Corresponding author: Jin Liang

Received  September 2018 Revised  March 2019 Published  June 2020 Early access  August 2019

Fund Project: The second author is supported by National Natural Science Foundation of China (No. 11671301)

In this paper, under the structure framework, a valuation model for a corporate bond with credit rating migration risk and in macro regime switch is established. The model turns to a free boundary problem in a partial differential equation (PDE) system. By PDE techniques, the existence, uniqueness and regularity of the solution are obtained. Furthermore, numerical examples are also presented.

Keywords: Credit rating migration, free boundary, regime switch, structure model, credit risk.
Mathematics Subject Classification: Primary: 35R35, 35K40; Secondary: 35Q91.
Citation: Yuan Wu, Jin Liang. Free boundaries of credit rating migration in switching macro regions. Mathematical Control & Related Fields, 2020, 10 (2) : 257-274. doi: 10.3934/mcrf.2019038
References:
[1]

F. Black and J. C. Cox, Some effects of bond indenture provisions, Journal of Finance, 31 (1976), 351-367.   Google Scholar

[2]

N. P. B. Bollen, Valuing options in regime-switching models, Journal of Derivatives, 6 (1998), 38-49.  doi: 10.3905/jod.1998.408011.  Google Scholar

[3]

L. Chollete, A. Heinen and A. Valdesogo, Modeling international financial returns with a multivariate regime switching copula, Journal of Financial Econometrics, 7 (2008), 437-480.   Google Scholar

[4]

S. Das and P. Tufano, Pricing credit-sensitive debt when interest rates, credit ratings, and credit spreads are stochastic, Journal of Financial Engineering, 5 (1996), 161-198.   Google Scholar

[5]

F. X. Diebold, J. Lee and G. C. Weinbach, Regime switching with time-varying transition probabilities, in Nonstationary Time Series Analysis and Cointegration (ed. C. Hargreaves), Oxford University Press, (1993), 283–302. Google Scholar

[6]

D. Duffe and K. J. Singleton, Modeling term structures of defaultable bonds, The Review of Financial Studies, 12 (1999), 687-720.  doi: 10.1093/rfs/12.4.687.  Google Scholar

[7]

J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57 (1989), 357-384.  doi: 10.2307/1912559.  Google Scholar

[8]

M. R. Hardy, A regime-switching model of long-term stock returns, North American Actuarial Journal, 5 (2001), 41-53.  doi: 10.1080/10920277.2001.10595984.  Google Scholar

[9]

B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Springer, Heidelberg, New York, 2011. doi: 10.1007/978-3-642-18460-4.  Google Scholar

[10]

B. Hu, J. Liang and Y. Wu, A free boundary problem for corporate bond with credit rating migration, Journal of Mathematical Analysis and Applications, 428 (2015), 896-909.  doi: 10.1016/j.jmaa.2015.03.040.  Google Scholar

[11]

R. A. Jarrow and S. M. Turnbull, Pricing derivatives on financial securities subject to credit risk, Financial Derivatives Pricing, (2008), 377–409. doi: 10.1142/9789812819222_0017.  Google Scholar

[12]

R. Jarrow, D. Lando and S. Turnbull, A Markov model for the term structure of credit risk spreads, Review of Financial studies, 10 (1997), 481-523.   Google Scholar

[13]

L. S. Jiang, Mathematical Modeling and Methods for Option Pricing, World Scientific Publishing Co., Inc., River Edge, NJ, 2005. doi: 10.1142/5855.  Google Scholar

[14]

D. Lando, On cox processes and credit-risky securities, Review of Derivatives Research, 2 (1998), 99-120.   Google Scholar

[15]

H. Leland and B.K. Toft, Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads, Journal of Finance, 51 (1996), 987-1019.   Google Scholar

[16]

J. Liang, Y. Wu and B. Hu, Asymptotic traveling wave solution for a credit rating migration problem, Journal of Differential Equations, 261 (2016), 1017-1045.  doi: 10.1016/j.jde.2016.03.032.  Google Scholar

[17]

J. Liang and C. K. Zeng, Corporate bonds pricing under credit rating migration and structure framework, Applied Mathematics A Journal of Chinese Universities, 30 (2015), 61-70.   Google Scholar

[18]

J. Liang, Y. J. Zhao and X. D. Zhang, Utility indifference valuation of corporate bond with credit rating migration by structure approach, Economic Modelling, 54 (2016), 339-346.  doi: 10.1016/j.econmod.2015.12.002.  Google Scholar

[19]

R. C. Merton, On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance, 29 (1974), 449-470.  doi: 10.1142/9789814759588_0003.  Google Scholar

[20]

L. Thomas, D. Allen and N. Morkel-Kingsbury, A hidden Markov chain model for the term structure of bond credit risk spreads, International Review of Financial Analysis, 11 (2002), 311-329.   Google Scholar

[21]

W. Weron, Modelling electricity prices: Jump diffusion and regime switching, Physica A Statistical Mechanics and Its Applications, 336 (2004), 39-48.   Google Scholar

[22]

Y. Wu and J. Liang, A new model and its numerical method to identify multi credit migration boundaries, International Journal of Computer Mathematics, 95 (2018), 1688-1702.  doi: 10.1080/00207160.2017.1329529.  Google Scholar

[23] Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Defusion Equation, 2 edition, Science Press, Beijing, 2011.   Google Scholar

show all references

References:
[1]

F. Black and J. C. Cox, Some effects of bond indenture provisions, Journal of Finance, 31 (1976), 351-367.   Google Scholar

[2]

N. P. B. Bollen, Valuing options in regime-switching models, Journal of Derivatives, 6 (1998), 38-49.  doi: 10.3905/jod.1998.408011.  Google Scholar

[3]

L. Chollete, A. Heinen and A. Valdesogo, Modeling international financial returns with a multivariate regime switching copula, Journal of Financial Econometrics, 7 (2008), 437-480.   Google Scholar

[4]

S. Das and P. Tufano, Pricing credit-sensitive debt when interest rates, credit ratings, and credit spreads are stochastic, Journal of Financial Engineering, 5 (1996), 161-198.   Google Scholar

[5]

F. X. Diebold, J. Lee and G. C. Weinbach, Regime switching with time-varying transition probabilities, in Nonstationary Time Series Analysis and Cointegration (ed. C. Hargreaves), Oxford University Press, (1993), 283–302. Google Scholar

[6]

D. Duffe and K. J. Singleton, Modeling term structures of defaultable bonds, The Review of Financial Studies, 12 (1999), 687-720.  doi: 10.1093/rfs/12.4.687.  Google Scholar

[7]

J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57 (1989), 357-384.  doi: 10.2307/1912559.  Google Scholar

[8]

M. R. Hardy, A regime-switching model of long-term stock returns, North American Actuarial Journal, 5 (2001), 41-53.  doi: 10.1080/10920277.2001.10595984.  Google Scholar

[9]

B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Springer, Heidelberg, New York, 2011. doi: 10.1007/978-3-642-18460-4.  Google Scholar

[10]

B. Hu, J. Liang and Y. Wu, A free boundary problem for corporate bond with credit rating migration, Journal of Mathematical Analysis and Applications, 428 (2015), 896-909.  doi: 10.1016/j.jmaa.2015.03.040.  Google Scholar

[11]

R. A. Jarrow and S. M. Turnbull, Pricing derivatives on financial securities subject to credit risk, Financial Derivatives Pricing, (2008), 377–409. doi: 10.1142/9789812819222_0017.  Google Scholar

[12]

R. Jarrow, D. Lando and S. Turnbull, A Markov model for the term structure of credit risk spreads, Review of Financial studies, 10 (1997), 481-523.   Google Scholar

[13]

L. S. Jiang, Mathematical Modeling and Methods for Option Pricing, World Scientific Publishing Co., Inc., River Edge, NJ, 2005. doi: 10.1142/5855.  Google Scholar

[14]

D. Lando, On cox processes and credit-risky securities, Review of Derivatives Research, 2 (1998), 99-120.   Google Scholar

[15]

H. Leland and B.K. Toft, Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads, Journal of Finance, 51 (1996), 987-1019.   Google Scholar

[16]

J. Liang, Y. Wu and B. Hu, Asymptotic traveling wave solution for a credit rating migration problem, Journal of Differential Equations, 261 (2016), 1017-1045.  doi: 10.1016/j.jde.2016.03.032.  Google Scholar

[17]

J. Liang and C. K. Zeng, Corporate bonds pricing under credit rating migration and structure framework, Applied Mathematics A Journal of Chinese Universities, 30 (2015), 61-70.   Google Scholar

[18]

J. Liang, Y. J. Zhao and X. D. Zhang, Utility indifference valuation of corporate bond with credit rating migration by structure approach, Economic Modelling, 54 (2016), 339-346.  doi: 10.1016/j.econmod.2015.12.002.  Google Scholar

[19]

R. C. Merton, On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance, 29 (1974), 449-470.  doi: 10.1142/9789814759588_0003.  Google Scholar

[20]

L. Thomas, D. Allen and N. Morkel-Kingsbury, A hidden Markov chain model for the term structure of bond credit risk spreads, International Review of Financial Analysis, 11 (2002), 311-329.   Google Scholar

[21]

W. Weron, Modelling electricity prices: Jump diffusion and regime switching, Physica A Statistical Mechanics and Its Applications, 336 (2004), 39-48.   Google Scholar

[22]

Y. Wu and J. Liang, A new model and its numerical method to identify multi credit migration boundaries, International Journal of Computer Mathematics, 95 (2018), 1688-1702.  doi: 10.1080/00207160.2017.1329529.  Google Scholar

[23] Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Defusion Equation, 2 edition, Science Press, Beijing, 2011.   Google Scholar
Figure 1.  Value function in different regimes
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Figure 2.  Free boundary
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Figure 3.  Differences between two cases
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