# American Institute of Mathematical Sciences

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  March 2020 Early access  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 and 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. [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. [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. [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] A. Friedman, Variational Principles and Free Boundary Problems, John Wiley & Sons, New York 1982. [6] M. G. Garrori and J. L. Menaldi, Green Functions for Second Order Parabolic Integro-differential Problems, Longman Scientific & Technical, New York, 1992. [7] J. Hall, Options, Futures, and Other Derivatives, Prentice-Hall, Inc., New Jersey, 1989. [8] B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Springer, Heidelberg, New York, 2011. doi: 10.1007/978-3-642-18460-4. [9] 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. [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. [11] R. Lando, D. Jarrow and S. Turnbull, A markov model for the term structure of credit risk spreads, Review of Financial studies, 10 (1997), 481-523. [12] L. Jiang, Mathematical Modeling and Methods for Option Pricing, World Scientific, Beijing, 2005. doi: 10.1142/5855. [13] D. Lando, On cox processes and credit-risky securities, Review of Derivatives Research, 2 (1998), 99-120. [14] D. Lando, Some elements of rating based credit risk modeling, in Advanced Fixed-Income Valuation Tools, Wiley, (2000), 193–215. [15] H. E. Leland, Corporate debt value, bond covenants, and optimal capital structure, Journal of Finance, 49 (1994), 1213-1252. [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. [17] 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. [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. [19] J. Liang, Y. 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. [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. [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. [22] 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.

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] 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. [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. [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] A. Friedman, Variational Principles and Free Boundary Problems, John Wiley & Sons, New York 1982. [6] M. G. Garrori and J. L. Menaldi, Green Functions for Second Order Parabolic Integro-differential Problems, Longman Scientific & Technical, New York, 1992. [7] J. Hall, Options, Futures, and Other Derivatives, Prentice-Hall, Inc., New Jersey, 1989. [8] B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Springer, Heidelberg, New York, 2011. doi: 10.1007/978-3-642-18460-4. [9] 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. [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. [11] R. Lando, D. Jarrow and S. Turnbull, A markov model for the term structure of credit risk spreads, Review of Financial studies, 10 (1997), 481-523. [12] L. Jiang, Mathematical Modeling and Methods for Option Pricing, World Scientific, Beijing, 2005. doi: 10.1142/5855. [13] D. Lando, On cox processes and credit-risky securities, Review of Derivatives Research, 2 (1998), 99-120. [14] D. Lando, Some elements of rating based credit risk modeling, in Advanced Fixed-Income Valuation Tools, Wiley, (2000), 193–215. [15] H. E. Leland, Corporate debt value, bond covenants, and optimal capital structure, Journal of Finance, 49 (1994), 1213-1252. [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. [17] 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. [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. [19] J. Liang, Y. 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. [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. [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. [22] 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.
value function $\psi(x, t)$
free boundary
asymptotic behavior

2021 Impact Factor: 1.497