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

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

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

Yinghui Dong ^1, and Guojing Wang ^1,

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

Received October 2010 Revised October 2011 Published April 2012

Abstract
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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.

Keywords: Dependent jump-diffusion processes, default time, defaultable bond., default probability.

Mathematics Subject Classification: Primary: 49K15, 44A10; Secondary: 47D0.

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 "14^th 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 "14^th 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.

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