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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

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.

Citation: Yuan Wu, Jin Liang. Free boundaries of credit rating migration in switching macro regions. Mathematical Control and 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. 

[2]

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

[3]

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

[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. 

[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.

[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.

[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.

[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.

[9]

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

[10]

B. HuJ. 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.

[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.

[12]

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

[13]

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

[14]

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

[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. 

[16]

J. LiangY. 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.

[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. 

[18]

J. LiangY. 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.

[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.

[20]

L. ThomasD. 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. 

[21]

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

[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.

[23] Q. YeZ. LiM. Wang and Y. Wu, Introduction of Reaction-Defusion Equation, 2 edition, Science Press, Beijing, 2011. 

show all references

References:
[1]

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

[2]

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

[3]

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

[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. 

[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.

[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.

[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.

[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.

[9]

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

[10]

B. HuJ. 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.

[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.

[12]

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

[13]

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

[14]

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

[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. 

[16]

J. LiangY. 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.

[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. 

[18]

J. LiangY. 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.

[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.

[20]

L. ThomasD. 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. 

[21]

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

[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.

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