American Institute of Mathematical Sciences

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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

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 & 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, (2002).   Google Scholar

[2]

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

[3]

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

[4]

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

[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.  doi: 10.1111/j.1467-9965.2009.00375.x.  Google Scholar

[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.  doi: 10.1016/j.insmatheco.2009.12.004.  Google Scholar

[7]

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

[8]

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

[9]

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

[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.  doi: 10.1016/0167-6687(91)90023-Q.  Google Scholar

[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.  doi: 10.1016/S0167-6687(98)00014-6.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[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, (2003).   Google Scholar

[17]

J. Z. Huang and H. Zhou, Specification analysis of structual credit risk models,, in, (2008).   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

D. Lando, "Credit Risk Modeling: Theory and Applications,", Princeton Series in Finance, (2004).   Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

S. M. Ross, "Stochastic Processes," Second edition,, Wiley Series in Probability and Statistics: Probability and Statistics, (1996).   Google Scholar

[31]

J. Ruf, "Structural Default Models with Jumps,", Ph.D thesis, (2006).   Google Scholar

[32]

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

[33]

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

[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.  doi: 10.1016/j.cam.2009.09.014.  Google Scholar

[35]

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

show all references

References:
[1]

T. Bielecki and M. Rutkowski, "Credit Risk: Modeling, Valuation and Hedging,", Springer Finance, (2002).   Google Scholar

[2]

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

[3]

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

[4]

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

[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.  doi: 10.1111/j.1467-9965.2009.00375.x.  Google Scholar

[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.  doi: 10.1016/j.insmatheco.2009.12.004.  Google Scholar

[7]

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

[8]

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

[9]

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

[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.  doi: 10.1016/0167-6687(91)90023-Q.  Google Scholar

[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.  doi: 10.1016/S0167-6687(98)00014-6.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[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, (2003).   Google Scholar

[17]

J. Z. Huang and H. Zhou, Specification analysis of structual credit risk models,, in, (2008).   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

D. Lando, "Credit Risk Modeling: Theory and Applications,", Princeton Series in Finance, (2004).   Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

S. M. Ross, "Stochastic Processes," Second edition,, Wiley Series in Probability and Statistics: Probability and Statistics, (1996).   Google Scholar

[31]

J. Ruf, "Structural Default Models with Jumps,", Ph.D thesis, (2006).   Google Scholar

[32]

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

[33]

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

[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.  doi: 10.1016/j.cam.2009.09.014.  Google Scholar

[35]

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

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