Search Advanced

AIMS Mathematics

2020, Volume 5, Issue 6: 7746-7775. doi: 10.3934/math.2020495
Previous Article Next Article
Research article

Free boundary problem pricing defaultable corporate bonds with multiple credit rating migration risk and stochastic interest rate

  • 1.
    Guangzhou Institute of International Finance, Guangzhou University, Guangzhou, P. R. China
  • 2.
    Department of Mathematics, City University of Hong Kong, Hong Kong, P. R. China
  • 3.
    School of Mathematics, Sun Yat-sen University, Guangzhou, P. R. China
  • Received: 25 August 2020 Accepted: 20 October 2020 Published: 22 October 2020

  • MSC : 91G40, 91G30, 35R35

  • In this paper, valuation for a defaultable corporate bond subject to multiple credit rating migration risk and stochastic volatility of interest rate is addressed in the structure framework through a free boundary problem, which is derived by a series of transformations. The existence, uniqueness and regularity of solution to the free boundary problem are obtained to verify the rationality of the bond pricing model. Furthermore, we show that the solution of the free boundary problem is convergent to a close form steady status, which may provide some information on the developing characteristics of the bond price. As the coexistence of stochastic interest rate and defaulting boundary, this convergence is achieved through an auxiliary free boundary problem and a Lyapunov argument. Interestingly, the converged steady status can be explicitly solved, which is not the case in the existing literatures on multiple credit rating migration. Finally, we present an explicit formula for valuating this defaultable bond with multiple credit rating migration risk and stochastic interest rate.

    Citation: Zhehao Huang, Yingting Miao, Zhenzhen Wang. Free boundary problem pricing defaultable corporate bonds with multiple credit rating migration risk and stochastic interest rate[J]. AIMS Mathematics, 2020, 5(6): 7746-7775. doi: 10.3934/math.2020495

    Related Papers:

  • In this paper, valuation for a defaultable corporate bond subject to multiple credit rating migration risk and stochastic volatility of interest rate is addressed in the structure framework through a free boundary problem, which is derived by a series of transformations. The existence, uniqueness and regularity of solution to the free boundary problem are obtained to verify the rationality of the bond pricing model. Furthermore, we show that the solution of the free boundary problem is convergent to a close form steady status, which may provide some information on the developing characteristics of the bond price. As the coexistence of stochastic interest rate and defaulting boundary, this convergence is achieved through an auxiliary free boundary problem and a Lyapunov argument. Interestingly, the converged steady status can be explicitly solved, which is not the case in the existing literatures on multiple credit rating migration. Finally, we present an explicit formula for valuating this defaultable bond with multiple credit rating migration risk and stochastic interest rate.


    加载中


    [1] H. Bessembinder, S. E. Jacobsen, W. F. Maxwell, K. Venkataraman, Capital Commitment and Illiquidity in Corporate Bonds, Journal of Finance, 73 (2017), 1615-1661.
    [2] F. Black, J. C. Cox, Valuing Corporate Securities: Some Effects of Bond Indenture Provisions, Journal of Finance, 31 (1976), 351-367.
    [3] E. Briys, F. de Varenne, Valuing risky fixed rate debt: An extension, The Journal of Financial and Quantitative Analysis, 32 (1997), 239-348.
    [4] T. Chellathurai, Probability density of recovery rate given default of a firm's debt and its constituent tranches, International Journal of Theoretical and Applied Finance, 20 (2017), 1750023.
    [5] G. d'Amico, G. Di Biase, J. Janssen, R. Manca, Semi-Markov Migration Models for Credit Risk, 1 Eds., Hoboken: John Wiley & Sons, 2017.
    [6] S. Das, P. Tufano, Pricing credit-sensitive debt when interest rates, credit ratings, and credit spreads are stochastic, Journal of Financial Engineering, 5 (1996), 161-198.
    [7] D. Duffie, K. J. Singleton, Modeling term structures of defaultable bonds, Review of Financial Studies, 12 (1999), 687-720.
    [8] F. Fei, A. Fuertes, E. Kalotychou, Credit Rating Migration Risk and Business Cycles, Journal of Business Finance & Accounting, 39 (2012), 229-263.
    [9] H. Frydman, T. Schuermann, Credit Rating Dynamics and Markov Mixture Models, Journal of Banking and Finance, 32 (2008), 1062-1075.
    [10] M. G. Garrori, J. L. Menaldi, Green Functions for Second Order Parabolic Integro-differential Problems, 1 Eds., New York: Longman Scientific & Technical, 1992.
    [11] B. Hu, J. Liang, Y. Wu, A free boundary problem for corporate bond with credit rating migration, J. Math. Anal. Appl., 428 (2015), 896-909.
    [12] Z. Huang, T. Jiang, Z. Wang, On a multiple credit rating migration model with stochastic interest rate, Math. Method. Appl. Sci., 43 (2020), 7106-7134.
    [13] J. C. Hull, Options, Futures and Other Derivatives, 8 Eds., New York: Pearson International Edition, 2012.
    [14] R. Jarrow, S. Turnbull, Pricing derivatives on financial securities subject to credit risk, Journal of Finance, 50 (1995), 53-86.
    [15] R. A. Jarrow, D. Lando, S. M. Turnbull, A Markov Model for the Term Structure of Credit Risk Spreads, Review of Financial Studies, 10 (1997), 481-523.
    [16] L. Jiang, Mathematical Modeling and Methods of Option Pricing, 8 Eds., New Jersey: World Scientific, 2005.
    [17] D. Lando, On Cox Processes and Credit Risky Securities, Review of Derivatives Research, 2 (1998), 99-120.
    [18] H. E. Leland, K. B. Toft, Optimal Capital Structure, Endogenous Bankruptcy, and the Term Structure of Credit Spreads, Journal of Finance, 51 (1996), 987-1019.
    [19] S. Li, X. Zhao, C. Yin, Z. Huang, Stochastic interest model driven by compound Poisson process and Brownian motion with applications in life contingencies, Quantitative Finance and Economics, 2 (2018), 246-260.
    [20] J. Liang, Y. Zhao, X. Zhang, Utility indifference valuation of corporate bond with credit rating migration by structure approach, Economic Modelling, 54 (2016), 339-346.
    [21] J. Liang, Y. Wu, B. Hu, Asymptotic traveling wave solution for a credit rating migration problem, J. Differ. Equations, 261 (2016), 1017-1045.
    [22] J. Liang, H. M. Yin, X. Chen, Y. Wu, On a corporate bond pricing model with credit rating migration risks and stochastic interest rate, Qualitative Finance and Economics, 1 (2017), 300-319.
    [23] J. Liang, H. Zou, Valuation of Credit Contingent Interest Rate Swap with Credit Rating Migration, Int. J. Comput. Math., (2020), 1-15.
    [24] Y. Liang, C. Xu, An efficient conditional Monte Carlo method for European option pricing with stochastic volatility and stochastic interest rate, Int. J. Comput. Math., 97 (2020), 638-655.
    [25] F. Longstaff, E. Schwartz, A simple approach to valuing risky fixed and floating rate debt, Journal of Finance, 50 (1995), 789-819.
    [26] D. G. McMillan, The information content of the stock and bond return correlation, Quantitative Finance and Economics, 2 (2018), 757-775.
    [27] F. Mehrdoust, A. R. Najafi, A short memory version of the Vasicek model and evaluating European options on zero-coupon bonds, J. Comput. Appl. Math., 375 (2020), 112796.
    [28] R. C. Merton, ON The pricing of corporate debt: the risk structure of interest rates, Journal of Finance, 29 (1974), 449-470.
    [29] H. Pedersen, N. R. Swanson, A survey of dynamic Nelson-Siegel models, diffusion indexes, and big data methods for predicting interest rates, Quantatitive Finance and Economics, 3 (2019), 22-45.
    [30] L. C. Thomas, D. E. Allen, N. Morkelkingsbury, A Hidden Markov Chain Model for the Term Structure of Bond Credit Risk Spreads, International Review of Financial Analysis, 11 (2002), 311-329.
    [31] P. Valta, Strategic Default, Debt Structure, and Stock Returns, Journal of Financial and Quantitative Analysis, 51 (2016), 197-229.
    [32] Z. Wang, Z. Liu, T. Jiang, Z. Huang, Asymptotic traveling wave for a pricing model with multiple credit rating migration risk, Commun. Math. Sci., 17 (2019), 1975-2004.
    [33] Y. Wu, X. Liang, Vasicek model with mixed-exponential jumps and its applications in finance and insurance, Adv. Differ. Equ-NY, 2018 (2018), 138.
    [34] Y. Wu, J. Liang, A new model and its numerical method to identify multi credit migration boundaries, Int. J. Comput. Math., 95 (2018), 1688-1702.
    [35] Y. Wu, J. Liang, B. Hu, A free boundary problem for defaultable corporate bond with credit rating migration risk and its asymptotic behavior, Discrete & Continuous Dynamical System-B, 25 (2020), 1043-1058.
    [36] Y. Wu, J. Liang, Free boundaries of credit rating migration in switching macro regions, Math. Control Relat. F., 10 (2020), 257-274.
    [37] H. M. Yin, J. Liang, Y. Wu, On a New Corporate Bond Pricing Model with Potential Credit Rating Change and Stochastic Interest Rate, Journal of Risk and Financial Management, 11 (2018), 87.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(3220) PDF downloads(126) Cited by(2)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog