March  2020, 25(3): 1043-1058. doi: 10.3934/dcdsb.2019207

A free boundary problem for defaultable corporate bond with credit rating migration risk and its asymptotic behavior

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

3. 

Department of Applied Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA

* Corresponding author: Jin Liang

Received  August 2018 Published  September 2019

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

In this paper, valuation of a defaultable corporate bond with credit rating migration risk is considered under the structure framework by using a free boundary model. The existence, uniqueness and regularity of the solution are obtained. Furthermore, we analyze the solution's asymptotic behavior and prove that the solution is convergent to an closed form solution. In addition, numerical examples are also shown.

Citation: 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
References:
[1]

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

[2]

E. Briys and F. de Varenne, Valuing risky fixed rate debt: An extension, The Journal of Financial and Quantitative Analysis, 32 (1997), 239-248.  doi: 10.1142/9789814759595_0012.  Google Scholar

[3]

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

[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]

A. Friedman, Variational Principles and Free Boundary Problems, John Wiley & Sons, New York 1982.  Google Scholar

[6]

M. G. Garrori and J. L. Menaldi, Green Functions for Second Order Parabolic Integro-differential Problems, Longman Scientific & Technical, New York, 1992.  Google Scholar

[7]

J. Hall, Options, Futures, and Other Derivatives, Prentice-Hall, Inc., New Jersey, 1989. Google Scholar

[8]

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

[9]

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

[10]

R. Jarrow and S. Turnbull, Pricing derivatives on financial securities subject to credit risk, Journal of Finance, 50 (1995), 53-86.  doi: 10.1142/9789812819222_0017.  Google Scholar

[11]

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

[12]

L. Jiang, Mathematical Modeling and Methods for Option Pricing, World Scientific, Beijing, 2005. doi: 10.1142/5855.  Google Scholar

[13]

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

[14]

D. Lando, Some elements of rating based credit risk modeling, in Advanced Fixed-Income Valuation Tools, Wiley, (2000), 193–215. Google Scholar

[15]

H. E. Leland, Corporate debt value, bond covenants, and optimal capital structure, Journal of Finance, 49 (1994), 1213-1252.   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

J. LiangY. Zhao and X. 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

[20]

F. Longstaff and E. Schwartz, A simple approach to valuing risky fixed and floating rate debt, Journal of Finance, 50 (1995), 789-819.  doi: 10.1142/9789814759595_0011.  Google Scholar

[21]

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

[22]

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

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]

E. Briys and F. de Varenne, Valuing risky fixed rate debt: An extension, The Journal of Financial and Quantitative Analysis, 32 (1997), 239-248.  doi: 10.1142/9789814759595_0012.  Google Scholar

[3]

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

[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]

A. Friedman, Variational Principles and Free Boundary Problems, John Wiley & Sons, New York 1982.  Google Scholar

[6]

M. G. Garrori and J. L. Menaldi, Green Functions for Second Order Parabolic Integro-differential Problems, Longman Scientific & Technical, New York, 1992.  Google Scholar

[7]

J. Hall, Options, Futures, and Other Derivatives, Prentice-Hall, Inc., New Jersey, 1989. Google Scholar

[8]

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

[9]

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

[10]

R. Jarrow and S. Turnbull, Pricing derivatives on financial securities subject to credit risk, Journal of Finance, 50 (1995), 53-86.  doi: 10.1142/9789812819222_0017.  Google Scholar

[11]

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

[12]

L. Jiang, Mathematical Modeling and Methods for Option Pricing, World Scientific, Beijing, 2005. doi: 10.1142/5855.  Google Scholar

[13]

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

[14]

D. Lando, Some elements of rating based credit risk modeling, in Advanced Fixed-Income Valuation Tools, Wiley, (2000), 193–215. Google Scholar

[15]

H. E. Leland, Corporate debt value, bond covenants, and optimal capital structure, Journal of Finance, 49 (1994), 1213-1252.   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

J. LiangY. Zhao and X. 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

[20]

F. Longstaff and E. Schwartz, A simple approach to valuing risky fixed and floating rate debt, Journal of Finance, 50 (1995), 789-819.  doi: 10.1142/9789814759595_0011.  Google Scholar

[21]

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

[22]

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

Figure 1.  value function $ \psi(x, t) $
Figure 2.  free boundary
Figure 3.  asymptotic behavior
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