• Previous Article
    Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions
  • DCDS-B Home
  • This Issue
  • Next Article
    Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions
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, 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
[1]

Yuan Wu, Jin Liang. Free boundaries of credit rating migration in switching macro regions. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019038

[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]

Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045

[4]

Jian Luo, Xueqi Yang, Ye Tian, Wenwen Yu. Corporate and personal credit scoring via fuzzy non-kernel SVM with fuzzy within-class scatter. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-14. doi: 10.3934/jimo.2019078

[5]

Xiaofeng Ren. Shell structure as solution to a free boundary problem from block copolymer morphology. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 979-1003. doi: 10.3934/dcds.2009.24.979

[6]

Honglin Yang, Heping Dai, Hong Wan, Lingling Chu. Optimal credit periods under two-level trade credit. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-15. doi: 10.3934/jimo.2019027

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[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, 2017, 13 (5) : 1-26. doi: 10.3934/jimo.2018167

[13]

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, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019032

[14]

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

[15]

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

[16]

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, 2017, 13 (5) : 1-22. doi: 10.3934/jimo.2019026

[17]

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, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019245

[18]

Toyohiko Aiki. On the existence of a weak solution to a free boundary problem for a model of a shape memory alloy spring. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 1-13. doi: 10.3934/dcdss.2012.5.1

[19]

Jian Yang. Asymptotic behavior of solutions for competitive models with a free boundary. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3253-3276. doi: 10.3934/dcds.2015.35.3253

[20]

Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355

2018 Impact Factor: 1.008

Article outline

Figures and Tables

[Back to Top]