# American Institute of Mathematical Sciences

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

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

 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.

Citation: Yinghui Dong, Kam Chuen Yuen, Guojing Wang. Pricing credit derivatives under a correlated regime-switching hazard processes model. Journal of Industrial & 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] M. Davis and V. Lo, Infectious defaults, Quant. Finance, 1 (2001), 382-387. Google Scholar [7] A. Davies, Credit spread modeling with regime-switching techniques, J. Fixed Income, 14 (2004), 36-48. doi: 10.3905/jfi.2004.461450. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer-Verlag: Berlin-Heidelberg-New York, 1995. Google Scholar [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. Google Scholar [16] R. M. Gaspar and T. Schmidt, Credit risk modeling with shot-noise processes, working paper, 2010. Available from: http://ssrn.com/abstract=1588750.Google Scholar [17] K. Giesecke, A simple exponential model for dependent defaults, J. Fixed Income, 13 (2003), 74-83. doi: 10.2139/ssrn.315088. Google Scholar [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. Google Scholar [19] K. Giesecke and L. Goldberg, Sequential defaults and incomplete information, J. Risk, 7 (2004), 1-26. doi: 10.21314/JOR.2004.100. Google Scholar [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. Google Scholar [21] R. Jarrow and F. Yu, Counterparty risk and the pricing of defaultable securities, J. Finan, 56 (2001), 1765-1799. Google Scholar [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.Google Scholar [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. Google Scholar

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##### References:
 [1] C. Alexander and A. Kaeck, Regime dependent determinants of credit default swap spreads, J. Bank. Finan., 32 (2008), 1008-1021. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] M. Davis and V. Lo, Infectious defaults, Quant. Finance, 1 (2001), 382-387. Google Scholar [7] A. Davies, Credit spread modeling with regime-switching techniques, J. Fixed Income, 14 (2004), 36-48. doi: 10.3905/jfi.2004.461450. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer-Verlag: Berlin-Heidelberg-New York, 1995. Google Scholar [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. Google Scholar [16] R. M. Gaspar and T. Schmidt, Credit risk modeling with shot-noise processes, working paper, 2010. Available from: http://ssrn.com/abstract=1588750.Google Scholar [17] K. Giesecke, A simple exponential model for dependent defaults, J. Fixed Income, 13 (2003), 74-83. doi: 10.2139/ssrn.315088. Google Scholar [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. Google Scholar [19] K. Giesecke and L. Goldberg, Sequential defaults and incomplete information, J. Risk, 7 (2004), 1-26. doi: 10.21314/JOR.2004.100. Google Scholar [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. Google Scholar [21] R. Jarrow and F. Yu, Counterparty risk and the pricing of defaultable securities, J. Finan, 56 (2001), 1765-1799. Google Scholar [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.Google Scholar [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. Google Scholar
Term structure of $s_1$ with $a=1$
Impact of $w$ on $s_1$ with $a=1, T=10.$
Impact of $a$ on $s_1$ with $w=0.01, T=10.$
Term structure of $s^1$ with $a=1$
Impact of $w$ on $s^1$ and with $q=0.5, T=10.$
Impact of $a$ on $s^1$ and with $q=0.5, T=10.$
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