April  2012, 8(2): 391-410. doi: 10.3934/jimo.2012.8.391

The dependence of assets and default threshold with thinning-dependence structure

1. 

Department of Mathematics and Center for Financial Engineering, Soochow University, Suzhou 215006, China, China

Received  October 2010 Revised  October 2011 Published  April 2012

In this paper, we model the value of a firm and a default threshold using two dependent jump-diffusion processes. We give the explicit solutions for the Laplace transform of the first passage time and the expected discounted ratio of the firm value to the default threshold at default, and show the impact of dependent jumps of the firm value and the default threshold on the default probabilities and the spreads of corporate defaultable bonds.
Citation: Yinghui Dong, Guojing Wang. The dependence of assets and default threshold with thinning-dependence structure. Journal of Industrial and Management Optimization, 2012, 8 (2) : 391-410. doi: 10.3934/jimo.2012.8.391
References:
[1]

T. Bielecki and M. Rutkowski, "Credit Risk: Modeling, Valuation and Hedging," Springer Finance, Springer-Verlag, Berlin, 2002.

[2]

F. Black and J. Cox, Valuing corporate securities liabilities: Some effects of bond indenture provisions, J. Finan., 31 (1976), 351-367. doi: 10.2307/2326607.

[3]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654. doi: 10.1086/260062.

[4]

N. Cai, On first passage times of a hyper-exponential jump diffusion process, Oper. Res. Lett., 37 (2009), 127-134. doi: 10.1016/j.orl.2009.01.002.

[5]

N. Chen and S. G. Kou, Credit spreads, optiomal capital structure, and implied volatility with endogenous defaults and jump risk, Math. Financ., 19 (2009), 343-378. doi: 10.1111/j.1467-9965.2009.00375.x.

[6]

Y. C. Chi, Analysis of expected discounted penalty function for a general jump-diffusion risk model and applications in finance, Insurance Math. Econom., 46 (2010), 385-396. doi: 10.1016/j.insmatheco.2009.12.004.

[7]

P. Collin-Dufresne and R. S. Goldstein, Do credit spreads reflect stationary leverage ratios?, J. Finan., 56 (2001), 1929-1957. doi: 10.1111/0022-1082.00395.

[8]

D. Duffie and K. Singleton, Modeling term structure of defaultable bond, Rev. Financ. Stud., 12 (1999), 687-720. doi: 10.1093/rfs/12.4.687.

[9]

D. Duffie and D. Lando, Term structures of credit spreads with incomplete accounting information, Econometrica, 69 (2001), 633-664. doi: 10.1111/1468-0262.00208.

[10]

F. Dufresne and H. U. Gerber, Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance Math. Econom., 10 (1991), 51-59. doi: 10.1016/0167-6687(91)90023-Q.

[11]

H. U. Gerber and B. Landry, On the discounted penalty at ruin in a jump-diffusion and the perpetual put option, Insurance Math. Econom., 22 (1998), 263-276. doi: 10.1016/S0167-6687(98)00014-6.

[12]

H. U. Gerber and E. S. W. Shiu, On the time value of ruin, N. Amer. Actuarial J., 2 (1998), 48-78.

[13]

K. Giesecke and L. Goldberg, Forecasting default in the face of uncertainty, J. Derivatives, 12 (2004), 14-25. doi: 10.3905/jod.2004.434534.

[14]

R. Goldstein, N. Ju and H. Leland, An EBIT-based model of dynamic capital structure, J. Bus., 74 (2001), 483-512. doi: 10.1086/322893.

[15]

B. Hilberink and L. C. G. Rogers, Optimal capital structure and endogenous default, Financ. Stoch, 6 (2002), 237-263. doi: 10.1007/s007800100058.

[16]

J. Z. Huang and M. Huang, How much of the corporate-treasury yield spread is due to credit risk?: A new calibration approach, in "14th Annual Conference on Financial Economics and Accounting," 2003. Available from: http://ssrn.com/abstract=307360.

[17]

J. Z. Huang and H. Zhou, Specification analysis of structual credit risk models, in "Finance and Economics Discussion Series," Penn State and Federal Reserve Board, 2008. Available from: http://ssrn.com/paper=1105640.

[18]

R. Jarrow and S. Turnbull, Pricing derivatives on financial securities subject to default risk, J. Finan., 50 (1995), 53-86. doi: 10.2307/2329239.

[19]

N. Ju and H. Ou-Yang, Capital structure, debt maturity, and stochastic interest rates, J. Bus., 79 (2006), 2469-2502. doi: 10.1086/505241.

[20]

S. G. Kou, A jump-diffusion model for option pricing, Manag. Sci., 48 (2002), 1086-1101. doi: 10.1287/mnsc.48.8.1086.166.

[21]

S. G. Kou and H. Wang, Option pricing under a double exponential jump diffusion model, Manag. Sci., 50 (2004), 1178-1192.

[22]

S. G. Kou and H. Wang, First passage times of a jump diffusion process, Adv. App. Probab., 35 (2003), 504-531.

[23]

D. Lando, "Credit Risk Modeling: Theory and Applications," Princeton Series in Finance, Princeton University Press, Princeton, 2004.

[24]

H. E. Leland, Corporate debt value, bond covenants, and optimal capital structure, J. Finan., 49 (1994), 1213-1252. doi: 10.2307/2329184.

[25]

H. Leland and K. B. Toft, Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads, J. Finan., 51 (1996), 987-1019. doi: 10.2307/2329229.

[26]

F. Longstaff and E. Schwartz, Valuing risky debt: A new approach, J. Finan., 50 (1995), 789-821. doi: 10.2307/2329288.

[27]

D. B. Madan and H. Unal, Pricing the risks of default, Rev. Deriv. Res., 2 (1998), 121-160. doi: 10.1007/BF01531333.

[28]

R. C. Merton, On the pricing of corporate debt: The risky structure of interest rates, J. Finan., 29 (1974), 449-470. doi: 10.2307/2978814.

[29]

C. A. Ramezani and Y. Zeng, Maximum likelihood estimation of the double exponential jump-diffusion process, Ann. Finan., 3 (2007), 487-507. doi: 10.1007/s10436-006-0062-y.

[30]

S. M. Ross, "Stochastic Processes," Second edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1996.

[31]

J. Ruf, "Structural Default Models with Jumps," Ph.D thesis, University of Ulm, 2006.

[32]

T. Schmidt and A. Novikov, A structural model with unobserved default boundary, Appl. Math. Finan., 15 (2008), 183-203. doi: 10.1080/13504860701718281.

[33]

G. J. Wang and K. C. Yuen, On a correlated aggregate claims model with thinning-dependence structure, Insurance Math. Econom., 36 (2005), 456-468. doi: 10.1016/j.insmatheco.2005.04.004.

[34]

Z. M. Zhang, H. Yang and S. M. Li, The perturbed compound Poisson risk model with two-sided jumps, J. Comput. Appl. Math., 233 (2010), 1773-1784. doi: 10.1016/j.cam.2009.09.014.

[35]

C. S. Zhou, The term structure of credit spreads with jump risk, J. Bank. Finan., 25 (2001), 2015-2040. doi: 10.1016/S0378-4266(00)00168-0.

show all references

References:
[1]

T. Bielecki and M. Rutkowski, "Credit Risk: Modeling, Valuation and Hedging," Springer Finance, Springer-Verlag, Berlin, 2002.

[2]

F. Black and J. Cox, Valuing corporate securities liabilities: Some effects of bond indenture provisions, J. Finan., 31 (1976), 351-367. doi: 10.2307/2326607.

[3]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654. doi: 10.1086/260062.

[4]

N. Cai, On first passage times of a hyper-exponential jump diffusion process, Oper. Res. Lett., 37 (2009), 127-134. doi: 10.1016/j.orl.2009.01.002.

[5]

N. Chen and S. G. Kou, Credit spreads, optiomal capital structure, and implied volatility with endogenous defaults and jump risk, Math. Financ., 19 (2009), 343-378. doi: 10.1111/j.1467-9965.2009.00375.x.

[6]

Y. C. Chi, Analysis of expected discounted penalty function for a general jump-diffusion risk model and applications in finance, Insurance Math. Econom., 46 (2010), 385-396. doi: 10.1016/j.insmatheco.2009.12.004.

[7]

P. Collin-Dufresne and R. S. Goldstein, Do credit spreads reflect stationary leverage ratios?, J. Finan., 56 (2001), 1929-1957. doi: 10.1111/0022-1082.00395.

[8]

D. Duffie and K. Singleton, Modeling term structure of defaultable bond, Rev. Financ. Stud., 12 (1999), 687-720. doi: 10.1093/rfs/12.4.687.

[9]

D. Duffie and D. Lando, Term structures of credit spreads with incomplete accounting information, Econometrica, 69 (2001), 633-664. doi: 10.1111/1468-0262.00208.

[10]

F. Dufresne and H. U. Gerber, Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance Math. Econom., 10 (1991), 51-59. doi: 10.1016/0167-6687(91)90023-Q.

[11]

H. U. Gerber and B. Landry, On the discounted penalty at ruin in a jump-diffusion and the perpetual put option, Insurance Math. Econom., 22 (1998), 263-276. doi: 10.1016/S0167-6687(98)00014-6.

[12]

H. U. Gerber and E. S. W. Shiu, On the time value of ruin, N. Amer. Actuarial J., 2 (1998), 48-78.

[13]

K. Giesecke and L. Goldberg, Forecasting default in the face of uncertainty, J. Derivatives, 12 (2004), 14-25. doi: 10.3905/jod.2004.434534.

[14]

R. Goldstein, N. Ju and H. Leland, An EBIT-based model of dynamic capital structure, J. Bus., 74 (2001), 483-512. doi: 10.1086/322893.

[15]

B. Hilberink and L. C. G. Rogers, Optimal capital structure and endogenous default, Financ. Stoch, 6 (2002), 237-263. doi: 10.1007/s007800100058.

[16]

J. Z. Huang and M. Huang, How much of the corporate-treasury yield spread is due to credit risk?: A new calibration approach, in "14th Annual Conference on Financial Economics and Accounting," 2003. Available from: http://ssrn.com/abstract=307360.

[17]

J. Z. Huang and H. Zhou, Specification analysis of structual credit risk models, in "Finance and Economics Discussion Series," Penn State and Federal Reserve Board, 2008. Available from: http://ssrn.com/paper=1105640.

[18]

R. Jarrow and S. Turnbull, Pricing derivatives on financial securities subject to default risk, J. Finan., 50 (1995), 53-86. doi: 10.2307/2329239.

[19]

N. Ju and H. Ou-Yang, Capital structure, debt maturity, and stochastic interest rates, J. Bus., 79 (2006), 2469-2502. doi: 10.1086/505241.

[20]

S. G. Kou, A jump-diffusion model for option pricing, Manag. Sci., 48 (2002), 1086-1101. doi: 10.1287/mnsc.48.8.1086.166.

[21]

S. G. Kou and H. Wang, Option pricing under a double exponential jump diffusion model, Manag. Sci., 50 (2004), 1178-1192.

[22]

S. G. Kou and H. Wang, First passage times of a jump diffusion process, Adv. App. Probab., 35 (2003), 504-531.

[23]

D. Lando, "Credit Risk Modeling: Theory and Applications," Princeton Series in Finance, Princeton University Press, Princeton, 2004.

[24]

H. E. Leland, Corporate debt value, bond covenants, and optimal capital structure, J. Finan., 49 (1994), 1213-1252. doi: 10.2307/2329184.

[25]

H. Leland and K. B. Toft, Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads, J. Finan., 51 (1996), 987-1019. doi: 10.2307/2329229.

[26]

F. Longstaff and E. Schwartz, Valuing risky debt: A new approach, J. Finan., 50 (1995), 789-821. doi: 10.2307/2329288.

[27]

D. B. Madan and H. Unal, Pricing the risks of default, Rev. Deriv. Res., 2 (1998), 121-160. doi: 10.1007/BF01531333.

[28]

R. C. Merton, On the pricing of corporate debt: The risky structure of interest rates, J. Finan., 29 (1974), 449-470. doi: 10.2307/2978814.

[29]

C. A. Ramezani and Y. Zeng, Maximum likelihood estimation of the double exponential jump-diffusion process, Ann. Finan., 3 (2007), 487-507. doi: 10.1007/s10436-006-0062-y.

[30]

S. M. Ross, "Stochastic Processes," Second edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1996.

[31]

J. Ruf, "Structural Default Models with Jumps," Ph.D thesis, University of Ulm, 2006.

[32]

T. Schmidt and A. Novikov, A structural model with unobserved default boundary, Appl. Math. Finan., 15 (2008), 183-203. doi: 10.1080/13504860701718281.

[33]

G. J. Wang and K. C. Yuen, On a correlated aggregate claims model with thinning-dependence structure, Insurance Math. Econom., 36 (2005), 456-468. doi: 10.1016/j.insmatheco.2005.04.004.

[34]

Z. M. Zhang, H. Yang and S. M. Li, The perturbed compound Poisson risk model with two-sided jumps, J. Comput. Appl. Math., 233 (2010), 1773-1784. doi: 10.1016/j.cam.2009.09.014.

[35]

C. S. Zhou, The term structure of credit spreads with jump risk, J. Bank. Finan., 25 (2001), 2015-2040. doi: 10.1016/S0378-4266(00)00168-0.

[1]

Ishak Alia. A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion. Mathematical Control and Related Fields, 2019, 9 (3) : 541-570. doi: 10.3934/mcrf.2019025

[2]

Caibin Zhang, Zhibin Liang, Kam Chuen Yuen. Portfolio optimization for jump-diffusion risky assets with regime switching: A time-consistent approach. Journal of Industrial and Management Optimization, 2022, 18 (1) : 341-366. doi: 10.3934/jimo.2020156

[3]

Xin Zhang, Hui Meng, Jie Xiong, Yang Shen. Robust optimal investment and reinsurance of an insurer under Jump-diffusion models. Mathematical Control and Related Fields, 2019, 9 (1) : 59-76. doi: 10.3934/mcrf.2019003

[4]

Kunyang Song, Yuping Song, Hanchao Wang. Threshold reweighted Nadaraya–Watson estimation of jump-diffusion models. Probability, Uncertainty and Quantitative Risk, 2022, 7 (1) : 31-44. doi: 10.3934/puqr.2022003

[5]

Matteo Ludovico Bedini, Rainer Buckdahn, Hans-Jürgen Engelbert. On the compensator of the default process in an information-based model. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 10-. doi: 10.1186/s41546-017-0017-4

[6]

Charles S. Tapiero, Pierre Vallois. Implied fractional hazard rates and default risk distributions. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 2-. doi: 10.1186/s41546-017-0015-6

[7]

Tao Chen, Wei Liu, Tao Tan, Lijun Wu, Yijun Hu. Optimal reinsurance with default risk: A reinsurer's perspective. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2971-2987. doi: 10.3934/jimo.2020103

[8]

Qiang Du, Zhan Huang, Richard B. Lehoucq. Nonlocal convection-diffusion volume-constrained problems and jump processes. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 373-389. doi: 10.3934/dcdsb.2014.19.373

[9]

Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial and Management Optimization, 2021, 17 (1) : 133-149. doi: 10.3934/jimo.2019103

[10]

Kai Zhang, Xiaoqi Yang, Kok Lay Teo. A power penalty approach to american option pricing with jump diffusion processes. Journal of Industrial and Management Optimization, 2008, 4 (4) : 783-799. doi: 10.3934/jimo.2008.4.783

[11]

Zhuo Jin, George Yin, Hailiang Yang. Numerical methods for dividend optimization using regime-switching jump-diffusion models. Mathematical Control and Related Fields, 2011, 1 (1) : 21-40. doi: 10.3934/mcrf.2011.1.21

[12]

Qing-Qing Yang, Wai-Ki Ching, Wanhua He, Tak-Kuen Siu. Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales. Journal of Industrial and Management Optimization, 2019, 15 (1) : 293-318. doi: 10.3934/jimo.2018044

[13]

Isabelle Kuhwald, Ilya Pavlyukevich. Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3175-3190. doi: 10.3934/dcdsb.2016092

[14]

Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3037-3050. doi: 10.3934/dcdsb.2018298

[15]

Chao Xu, Yinghui Dong, Zhaolu Tian, Guojing Wang. Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2603-2623. doi: 10.3934/jimo.2019072

[16]

Chuancun Yin, Kam Chuen Yuen. Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1247-1262. doi: 10.3934/jimo.2015.11.1247

[17]

Dan Li, Jing'an Cui, Yan Zhang. Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2069-2088. doi: 10.3934/dcdsb.2015.20.2069

[18]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control and Related Fields, 2022, 12 (2) : 371-404. doi: 10.3934/mcrf.2021026

[19]

Wan-Hua He, Chufang Wu, Jia-Wen Gu, Wai-Ki Ching, Chi-Wing Wong. Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2077-2094. doi: 10.3934/jimo.2021057

[20]

Sheng Li, Wei Yuan, Peimin Chen. Optimal control on investment and reinsurance strategies with delay and common shock dependence in a jump-diffusion financial market. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022068

2020 Impact Factor: 1.801

Metrics

  • PDF downloads (87)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]