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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 |
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.
Mathematics Subject Classification:
Primary: 35R35, 35K40; Secondary: 35Q91.
Citation:
Yuan Wu, Jin Liang. Free boundaries of credit rating migration in switching macro regions. Mathematical Control and Related Fields,
2020, 10
(2)
: 257-274.
doi: 10.3934/mcrf.2019038
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References:
| [1] |
F. Black and J. C. Cox,
Some effects of bond indenture provisions, Journal of Finance, 31 (1976), 351-367.
|
| [2] |
N. P. B. Bollen,
Valuing options in regime-switching models, Journal of Derivatives, 6 (1998), 38-49.
doi: 10.3905/jod.1998.408011. |
| [3] |
L. Chollete, A. Heinen and A. Valdesogo,
Modeling international financial returns with a multivariate regime switching copula, Journal of Financial Econometrics, 7 (2008), 437-480.
|
| [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] |
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. |
| [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. |
| [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. |
| [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. |
| [9] |
B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Springer, Heidelberg, New York, 2011.
doi: 10.1007/978-3-642-18460-4. |
| [10] |
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. |
| [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. |
| [12] |
R. Jarrow, D. Lando and S. Turnbull,
A Markov model for the term structure of credit risk spreads, Review of Financial studies, 10 (1997), 481-523.
|
| [13] |
L. S. Jiang, Mathematical Modeling and Methods for Option Pricing, World Scientific Publishing Co., Inc., River Edge, NJ, 2005.
doi: 10.1142/5855. |
| [14] |
D. Lando,
On cox processes and credit-risky securities, Review of Derivatives Research, 2 (1998), 99-120.
|
| [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.
|
| [16] |
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. |
| [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.
|
| [18] |
J. Liang, Y. 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. |
| [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. |
| [20] |
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.
|
| [21] |
W. Weron,
Modelling electricity prices: Jump diffusion and regime switching, Physica A Statistical Mechanics and Its Applications, 336 (2004), 39-48.
|
| [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. |
| [23] |
Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Defusion Equation, 2 edition, Science Press, Beijing, 2011.
|
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] |
N. P. B. Bollen,
Valuing options in regime-switching models, Journal of Derivatives, 6 (1998), 38-49.
doi: 10.3905/jod.1998.408011. |
| [3] |
L. Chollete, A. Heinen and A. Valdesogo,
Modeling international financial returns with a multivariate regime switching copula, Journal of Financial Econometrics, 7 (2008), 437-480.
|
| [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] |
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. |
| [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. |
| [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. |
| [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. |
| [9] |
B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Springer, Heidelberg, New York, 2011.
doi: 10.1007/978-3-642-18460-4. |
| [10] |
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. |
| [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. |
| [12] |
R. Jarrow, D. Lando and S. Turnbull,
A Markov model for the term structure of credit risk spreads, Review of Financial studies, 10 (1997), 481-523.
|
| [13] |
L. S. Jiang, Mathematical Modeling and Methods for Option Pricing, World Scientific Publishing Co., Inc., River Edge, NJ, 2005.
doi: 10.1142/5855. |
| [14] |
D. Lando,
On cox processes and credit-risky securities, Review of Derivatives Research, 2 (1998), 99-120.
|
| [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.
|
| [16] |
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. |
| [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.
|
| [18] |
J. Liang, Y. 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. |
| [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. |
| [20] |
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.
|
| [21] |
W. Weron,
Modelling electricity prices: Jump diffusion and regime switching, Physica A Statistical Mechanics and Its Applications, 336 (2004), 39-48.
|
| [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. |
| [23] |
Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Defusion Equation, 2 edition, Science Press, Beijing, 2011.
|
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