• Previous Article
    On the exact controllability and the stabilization for the Benney-Luke equation
  • MCRF Home
  • This Issue
  • Next Article
    Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries
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  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 & 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. CholleteA. 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. 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.  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. JarrowD. 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. 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.  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. 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.  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. 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.   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. YeZ. LiM. 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. CholleteA. 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. 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.  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. JarrowD. 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. 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.  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. 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.  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. 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.   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. YeZ. LiM. Wang and Y. Wu, Introduction of Reaction-Defusion Equation, 2 edition, Science Press, Beijing, 2011.   Google Scholar
Figure 1.  Value function in different regimes
Figure 2.  Free boundary
Figure 3.  Differences between two cases
[1]

Yuan Wu, Jin Liang, Bei Hu. A free boundary problem for defaultable corporate bond with credit rating migration risk and its asymptotic behavior. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1043-1058. doi: 10.3934/dcdsb.2019207

[2]

Prasenjit Pramanik, Sarama Malik Das, Manas Kumar Maiti. Note on : Supply chain inventory model for deteriorating items with maximum lifetime and partial trade credit to credit risk customers. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1289-1315. doi: 10.3934/jimo.2018096

[3]

Yinghui Dong, Kam Chuen Yuen, Guojing Wang. Pricing credit derivatives under a correlated regime-switching hazard processes model. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1395-1415. doi: 10.3934/jimo.2016079

[4]

Puspita Mahata, Gour Chandra Mahata. Two-echelon trade credit with default risk in an EOQ model for deteriorating items under dynamic demand. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020138

[5]

Yazhe Li, Tony Bellotti, Niall Adams. Issues using logistic regression with class imbalance, with a case study from credit risk modelling. Foundations of Data Science, 2019, 1 (4) : 389-417. doi: 10.3934/fods.2019016

[6]

Honglin Yang, Heping Dai, Hong Wan, Lingling Chu. Optimal credit periods under two-level trade credit. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1753-1767. doi: 10.3934/jimo.2019027

[7]

Chonghu Guan, Fahuai Yi, Xiaoshan Chen. A fully nonlinear free boundary problem arising from optimal dividend and risk control model. Mathematical Control & Related Fields, 2019, 9 (3) : 425-452. doi: 10.3934/mcrf.2019020

[8]

Jui-Jung Liao, Wei-Chun Lee, Kuo-Nan Huang, Yung-Fu Huang. Optimal ordering policy for a two-warehouse inventory model use of two-level trade credit. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1661-1683. doi: 10.3934/jimo.2017012

[9]

Pin-Shou Ting. The EPQ model with deteriorating items under two levels of trade credit in a supply chain system. Journal of Industrial & Management Optimization, 2015, 11 (2) : 479-492. doi: 10.3934/jimo.2015.11.479

[10]

Sankar Kumar Roy, Magfura Pervin, Gerhard Wilhelm Weber. Imperfection with inspection policy and variable demand under trade-credit: A deteriorating inventory model. Numerical Algebra, Control & Optimization, 2020, 10 (1) : 45-74. doi: 10.3934/naco.2019032

[11]

Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. An integrated inventory model with variable holding cost under two levels of trade-credit policy. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 169-191. doi: 10.3934/naco.2018010

[12]

Sankar Kumar Roy, Magfura Pervin, Gerhard Wilhelm Weber. A two-warehouse probabilistic model with price discount on backorders under two levels of trade-credit policy. Journal of Industrial & Management Optimization, 2020, 16 (2) : 553-578. doi: 10.3934/jimo.2018167

[13]

Kun-Jen Chung, Pin-Shou Ting. The inventory model under supplier's partial trade credit policy in a supply chain system. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1175-1183. doi: 10.3934/jimo.2015.11.1175

[14]

Yizhao Qin, Yuxia Guo, Peng-Fei Yao. Energy decay and global smooth solutions for a free boundary fluid-nonlinear elastic structure interface model with boundary dissipation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1555-1593. doi: 10.3934/dcds.2020086

[15]

Siyu Liu, Haomin Huang, Mingxin Wang. A free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1649-1670. doi: 10.3934/dcdsb.2019245

[16]

Lijun Bo. Portfolio optimization of credit swap under funding costs. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 12-. doi: 10.1186/s41546-017-0023-6

[17]

Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1345-1373. doi: 10.3934/jimo.2018098

[18]

Qiang Lin, Ying Peng, Ying Hu. Supplier financing service decisions for a capital-constrained supply chain: Trade credit vs. combined credit financing. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1731-1752. doi: 10.3934/jimo.2019026

[19]

W.C. Ip, H. Wong, Jiazhu Pan, Keke Yuan. Estimating value-at-risk for chinese stock market by switching regime ARCH model. Journal of Industrial & Management Optimization, 2006, 2 (2) : 145-163. doi: 10.3934/jimo.2006.2.145

[20]

Claudio Albanese, Simone Caenazzo, Stéphane Crépey. Credit, funding, margin, and capital valuation adjustments for bilateral portfolios. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 7-. doi: 10.1186/s41546-017-0019-2

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (89)
  • HTML views (511)
  • Cited by (0)

Other articles
by authors

[Back to Top]