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July  2017, 13(3): 1395-1415. doi: 10.3934/jimo.2016079

Pricing credit derivatives under a correlated regime-switching hazard processes model

Yinghui Dong 1,, , Kam Chuen Yuen 2, and  Guojing Wang 3,

1. 

Department of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China
2. 

Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong, China
3. 

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

* Corresponding author:Yinghui Dong

Received  February 2016 Published  October 2016

Fund Project: The authors thank the anonymous referees for valuable comments to improve the earlier version of the paper. The first author is supported by the NSF of Jiangsu Province (Grant No. BK20130260), the NNSF of China (Grant No. 11301369) and Qing Lan Project. The second author is supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7057/13P), and the CAE 2013 research grant from the Society of Actuaries -any opinions, finding, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the SOA. The third author is supported by the NNSF of China (Grant No. 11371274).

In this paper, we study the valuation of a single-name credit default swap and a $k$th-to-default basket swap under a correlated regime-switching hazard processes model. We assume that the defaults of all the names are driven by a Markov chain describing the macro-economic conditions and some shock events modelled by a multivariate regime-switching shot noise process. Based on some expressions for the joint Laplace transform of the regime-switching shot noise processes, we give explicit formulas for the spread of a CDS contract and the $k$th-to-default basket swap.

Keywords: Hazard process, Markov chain, kth-to-default basket swap, multivariate regime-switching shot noise process.
Mathematics Subject Classification: Primary: 91B25, 91B70; Secondary: 60J27.
Citation: Yinghui Dong, Kam Chuen Yuen, Guojing Wang. Pricing credit derivatives under a correlated regime-switching hazard processes model. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1395-1415. doi: 10.3934/jimo.2016079
References:
[1]

C. Alexander and A. Kaeck, Regime dependent determinants of credit default swap spreads, J. Bank. Finan., 32 (2008), 1008-1021. 

[2]

T. Bielecki, S. Crépey, M. Jeanblanc and B. Zargari, Valuation and hedging of CDS counterparty exposure in a Markov copula model, Int. J. Theor. Appl. Finance, 15 (2012), 1250004, 39 pp. doi: 10.1142/S0219024911006498.

[3]

D. Brigo, A. Pallavicini and R. Torresetti, Credit models and the crisis: Default cluster dynamics and the generalized Poisson loss model, J. Credit Risk, 6 (2010), 39-81. 

[4]

J. Buffington and R. J. Elliott, American options with regime switching, Int. J. Theor. Appl. Finance, 5 (2002), 497-514.  doi: 10.1142/S0219024902001523.

[5]

A. Dassios and J. Jang, Pricing of catastrophe reinsurance & derivatives using the Cox process with shot noise intensity, Financ. Stoch, 7 (2003), 73-95.  doi: 10.1007/s007800200079.

[6]

M. Davis and V. Lo, Infectious defaults, Quant. Finance, 1 (2001), 382-387. 

[7]

A. Davies, Credit spread modeling with regime-switching techniques, J. Fixed Income, 14 (2004), 36-48.  doi: 10.3905/jfi.2004.461450.

[8]

G. Di Graziano and L. C. G. Rogers, A dynamic approach to the modelling of correlation credit derivatives using Markov chains, Int. J. Theor. Appl. Finance, 12 (2009), 45-62.  doi: 10.1142/S0219024909005142.

[9]

X. W. Ding, K. Giesecke and P. I. Tomecek, Time-changed birth processes and multiname credit derivatives, Oper. Res., 57 (2009), 990-1005.  doi: 10.1287/opre.1080.0652.

[10]

Y. H. Dong, K. C. Yuen and C. F. Wu, Unilateral counterparty risk valuation of CDS using a regime-switching intensity model, Stat. Probabil. Lett., 85 (2014), 25-35.  doi: 10.1016/j.spl.2013.11.001.

[11]

Y. H. Dong, K. C. Yuen, G. J. Wang and C. F. Wu. A reduced-form model for correlated defaults with regime-switching shot noise intensities, A reduced-form model for correlated defaults with regime-switching shot noise intensities, Methodol. Comput. Appl. Probab., 18 (2016), 459-486.  doi: 10.1007/s11009-014-9431-6.

[12]

D. Duffie and N. Gârleanu, Risk and valuation of collateralized debt obligations, Financ. Anal. J., 57 (2001), 41-59.  doi: 10.2469/faj.v57.n1.2418.

[13]

D. Duffie, D. Filipovic and W. Schachermayer, Affine processes and applications in finance, Ann. Appl. Probab., 13 (2003), 984-1053.  doi: 10.1214/aoap/1060202833.

[14]

R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer-Verlag: Berlin-Heidelberg-New York, 1995.

[15]

R. J. Elliott and T. K. Siu, Default times in a continuous-time Markovian regime switching model, Stoch. Anal. Appl., 29 (2011), 824-837.  doi: 10.1080/07362994.2011.598792.

[16]

R. M. Gaspar and T. Schmidt, Credit risk modeling with shot-noise processes, working paper, 2010. Available from: http://ssrn.com/abstract=1588750.

[17]

K. Giesecke, A simple exponential model for dependent defaults, J. Fixed Income, 13 (2003), 74-83.  doi: 10.2139/ssrn.315088.

[18]

K. Giesecke, F. A. Longstaff, S. Schaefer and I. Ilya Strebulaev, Corporate bond default risk: A 150-year perspective, J. Financ. Econ., 102 (2011), 233-250.  doi: 10.1016/j.jfineco.2011.01.011.

[19]

K. Giesecke and L. Goldberg, Sequential defaults and incomplete information, J. Risk, 7 (2004), 1-26.  doi: 10.21314/JOR.2004.100.

[20]

J. Hull and A. White, Valuation of a CDO and a nth to default CDS without Monte Carlo simulation, J. Derivatives, 12 (2004), 8-23. 

[21]

R. Jarrow and F. Yu, Counterparty risk and the pricing of defaultable securities, J. Finan, 56 (2001), 1765-1799. 

[22]

P. Schonbucher and D. Schubert, Copula dependent default risk in intensity models, Working Paper. Department of Statistics, Bonn University, 2001, Available from: http://ssrn.com/abstract=301968.

[23]

Y. Shen and T. K. Siu, Longevity bond Pricing under stochastic interest rate and mortality with regime switching, Insur. Math. Econ., 52 (2013), 114-123.  doi: 10.1016/j.insmatheco.2012.11.006.

show all references

References:
[1]

C. Alexander and A. Kaeck, Regime dependent determinants of credit default swap spreads, J. Bank. Finan., 32 (2008), 1008-1021. 

[2]

T. Bielecki, S. Crépey, M. Jeanblanc and B. Zargari, Valuation and hedging of CDS counterparty exposure in a Markov copula model, Int. J. Theor. Appl. Finance, 15 (2012), 1250004, 39 pp. doi: 10.1142/S0219024911006498.

[3]

D. Brigo, A. Pallavicini and R. Torresetti, Credit models and the crisis: Default cluster dynamics and the generalized Poisson loss model, J. Credit Risk, 6 (2010), 39-81. 

[4]

J. Buffington and R. J. Elliott, American options with regime switching, Int. J. Theor. Appl. Finance, 5 (2002), 497-514.  doi: 10.1142/S0219024902001523.

[5]

A. Dassios and J. Jang, Pricing of catastrophe reinsurance & derivatives using the Cox process with shot noise intensity, Financ. Stoch, 7 (2003), 73-95.  doi: 10.1007/s007800200079.

[6]

M. Davis and V. Lo, Infectious defaults, Quant. Finance, 1 (2001), 382-387. 

[7]

A. Davies, Credit spread modeling with regime-switching techniques, J. Fixed Income, 14 (2004), 36-48.  doi: 10.3905/jfi.2004.461450.

[8]

G. Di Graziano and L. C. G. Rogers, A dynamic approach to the modelling of correlation credit derivatives using Markov chains, Int. J. Theor. Appl. Finance, 12 (2009), 45-62.  doi: 10.1142/S0219024909005142.

[9]

X. W. Ding, K. Giesecke and P. I. Tomecek, Time-changed birth processes and multiname credit derivatives, Oper. Res., 57 (2009), 990-1005.  doi: 10.1287/opre.1080.0652.

[10]

Y. H. Dong, K. C. Yuen and C. F. Wu, Unilateral counterparty risk valuation of CDS using a regime-switching intensity model, Stat. Probabil. Lett., 85 (2014), 25-35.  doi: 10.1016/j.spl.2013.11.001.

[11]

Y. H. Dong, K. C. Yuen, G. J. Wang and C. F. Wu. A reduced-form model for correlated defaults with regime-switching shot noise intensities, A reduced-form model for correlated defaults with regime-switching shot noise intensities, Methodol. Comput. Appl. Probab., 18 (2016), 459-486.  doi: 10.1007/s11009-014-9431-6.

[12]

D. Duffie and N. Gârleanu, Risk and valuation of collateralized debt obligations, Financ. Anal. J., 57 (2001), 41-59.  doi: 10.2469/faj.v57.n1.2418.

[13]

D. Duffie, D. Filipovic and W. Schachermayer, Affine processes and applications in finance, Ann. Appl. Probab., 13 (2003), 984-1053.  doi: 10.1214/aoap/1060202833.

[14]

R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer-Verlag: Berlin-Heidelberg-New York, 1995.

[15]

R. J. Elliott and T. K. Siu, Default times in a continuous-time Markovian regime switching model, Stoch. Anal. Appl., 29 (2011), 824-837.  doi: 10.1080/07362994.2011.598792.

[16]

R. M. Gaspar and T. Schmidt, Credit risk modeling with shot-noise processes, working paper, 2010. Available from: http://ssrn.com/abstract=1588750.

[17]

K. Giesecke, A simple exponential model for dependent defaults, J. Fixed Income, 13 (2003), 74-83.  doi: 10.2139/ssrn.315088.

[18]

K. Giesecke, F. A. Longstaff, S. Schaefer and I. Ilya Strebulaev, Corporate bond default risk: A 150-year perspective, J. Financ. Econ., 102 (2011), 233-250.  doi: 10.1016/j.jfineco.2011.01.011.

[19]

K. Giesecke and L. Goldberg, Sequential defaults and incomplete information, J. Risk, 7 (2004), 1-26.  doi: 10.21314/JOR.2004.100.

[20]

J. Hull and A. White, Valuation of a CDO and a nth to default CDS without Monte Carlo simulation, J. Derivatives, 12 (2004), 8-23. 

[21]

R. Jarrow and F. Yu, Counterparty risk and the pricing of defaultable securities, J. Finan, 56 (2001), 1765-1799. 

[22]

P. Schonbucher and D. Schubert, Copula dependent default risk in intensity models, Working Paper. Department of Statistics, Bonn University, 2001, Available from: http://ssrn.com/abstract=301968.

[23]

Y. Shen and T. K. Siu, Longevity bond Pricing under stochastic interest rate and mortality with regime switching, Insur. Math. Econ., 52 (2013), 114-123.  doi: 10.1016/j.insmatheco.2012.11.006.

Figure 1.  Term structure of $s_1$ with $a=1$
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Figure 2.  Impact of $w$ on $s_1$ with $a=1, T=10.$
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Figure 3.  Impact of $a$ on $s_1$ with $w=0.01, T=10.$
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Figure 4.  Term structure of $s^1$ with $a=1$
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Figure 5.  Impact of $w$ on $s^1$ and with $q=0.5, T=10.$
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Figure 6.  Impact of $a$ on $s^1$ and with $q=0.5, T=10.$
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