Sains Malaysiana 49(4)(2020): 941-952

http://dx.doi.org/10.17576/jsm-2020-4904-23

Defaultable Bond Pricing under the Jump Diffusion Model with Copula Dependence Structure

(Penentuan Harga Bon Boleh Mungkir di Bawah Model Resapan Lompat dengan Struktur Kebersandaran Kopula)

SITI NORAFIDAH MOHD RAMLI^1* & JIWOOK JANG²

¹Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia

²Department of Actuarial Studies and Business Analytics, Macquarie Business School, Macquarie University, North Ryde NSW 2109 Sydney, Australia

Received: 12 October 2019/Accepted: 23 December 2019

ABSTRACT

We study the pricing of a defaultable bond under various dependence structure captured by copulas. For that purpose, we use a bivariate jump-diffusion process to represent a bond issuer's default intensity and the market short rate of interest. We assume that each jump of both variables occur simultaneously, and that their sizes are dependent. For these simultaneous jumps and their sizes, a homogeneous Poisson process and three copulas, which are a Farlie-Gumbel-Morgenstern copula, a Gaussian copula, and a Student t-copula are used, respectively. We use the joint Laplace transform of the integrated risk processes to obtain the expression of the defaultable bond price with copula-dependent jump sizes. Assuming exponential marginal distributions, we compute the zero coupon defaultable bond prices and their yields using the three copulas to illustrate the bond. We found that the bond price values are the lowest under the Student-t copula, suggesting that a dependence structure under the Student-t copula could be a suitable candidate to depict a riskier environment. Additionally, the hypothetical term structure of interest rates under the risky environment are also upward sloping, albeit with yields greater than 100%, reflecting a higher compensation required by investors to lend funds for a longer period when the financial market is volatile.

Keywords: Bivariate jump-diffusion model; credit risk; default intensity; short rate; zero coupon bond

ABSTRAK

Kertas ini mengkaji penentuan harga bon boleh mungkir dengan kadar faedah pendek dan nilai keamatan ingkar penerbit bon, dengan struktur kebersandaran yang diwakili oleh kopula. Untuk tujuan itu, proses resapan-lompat bivariat digunakan untuk mewakili proses keamatan ingkar penerbit bon dan kadar faedah pendek pasaran. Setiap lompatan oleh kedua-dua pemboleh ubah diandaikan berlaku serentak, dan saiznya adalah bersandaran antara satu sama lain. Bagi mewakili proses lompatan serentak dan struktur kebersandaran saiznya, proses Poisson yang homogen dan tiga kopula, iaitu kopula Farlie-Gumbel-Morgenstern, Gaussian, dan student-t digunakan. Transformasi Laplace tercantum bagi proses risiko bersepadu digunakan untuk mendapatkan persamaan harga bon boleh mungkir dengan saiz lompatan faktor yang bersandar dengan struktur kopula. Harga bon boleh mungkir tanpa kupon dan kadar hasilnya dihitung di bawah tiga jenis kopula dengan taburan marginal eksponen untuk mewakili kebersandaran antara kedua-dua faktor. Kajian mendapati bahawa nilai harga bon adalah yang paling rendah apabila faktor kebersandaran digambarkan oleh kopula student-t, yang menunjukkan bahawa struktur kebersandaran di bawah kopula student-t adalah lebih sesuai untuk menggambarkan persekitaran yang berisiko berbanding kopula FGM dan Gaussian. Di samping itu, walaupun struktur masa kadar faedah bagi jangka panjang di bawah persekitaran yang berisiko juga menunjukkan pola menaik, kadar hasil yang melebihi 100%, mencerminkan situasi bahawa pelabur memerlukan pampasan yang lebih tinggi bagi aktiviti meminjamkan dana untuk tempoh yang lebih lama apabila situasi pasaran kewangan adalah tidak menentu.

Kata kunci: Bon sifar kupon; kadar keamatan mungkir; kadar pendek; model resapan-lompat bivariat; risiko kredit

REFERENCES

Black, F. & Cox, J.C. 1976. Valuing corporate securities: Some effects of bond indenture provisions. Journal of Finance 31(2): 351-367.

Brigo, D. & Chourdakis, K. 2009. Counterparty risk for credit default swap-Impact of spread volatility and default correlation. International Journal of Theoretical and Applied Finance 12(7): 1007-1026.

Cox, J.C., Ingersoll, J.E. & Ross, S.A. 1985. A theory of the term structure of interest rates. Econometrica 53(2): 385-408.

Donelly, C. & Embrechts, P. 2010. The devil is in the tails: Actuarial mathematics and the subprime mortgage crisis. ASTIN Bulletin 40(1): 1-33.

Duffie, D. & Singleton, K. 1999. Modelling term structures of defaultable bonds. The Review of Financial Studies 12(4): 687-720.

Giesecke, K. 2004. Correlated default within complete information. Journal of Banking and Finance 28: 1521-1545.

Herbertsson, A., Jang, J. & Schmidt, T. 2011. Pricing basket default swaps in a tractable shot noise model. Statistics & Probability Letters 81(8): 1196-1207.

Hull, J. & White, A. 1995. The impact of default risk on the prices of options and other derivative securities. Journal of Banking & Finance 19(2): 299-322.

Hyong-Chol, O. & Ning, W. 2005. Analytical pricing of defaultable bond with stochastic default  intensity-The case with exogenous default recovery. Working Paper, Department of Applied Mathematics, Tong-ji University, Shanghai.

Ignatieva, K. & Platen, E. 2010. Modelling co-movements and tail dependency in the international stock market via copulae. Asia-Pacific Financial Market 7(3): 261-304.

Jang, J. 2007. Jump-diffusion process and their applications in insurance and finance, Insurance: Mathematics and Economics 41(1): 62-70.

Jang, J. & Mohd Ramli, S.N. 2018. Hierarchical Markov model in life insurance and social benefit schemes. Risks 6(3): 63. DOI: 10.3390/risks6030063.

Jarrow, R.A. & Protter, P. 2004. Structural versus reduced form models: A new information based perspective. Journal of Investment Management 2(2): 110.

Jarrow, R.A. & Yu, F. 2001. Counterparty risk and the pricing of defaultable securities. Journal of Finance 56(5): 1765-1800.

Jarrow, R. & Turnbull, S. 1995. Pricing derivatives on financial securities subject to credit risk. Journal of Finance 50(1): 53-85.

Jarrow, R., Lando, D. & Turnbull, S.M. 1997. A Markov Model for the term structure of credit risk spreads. The Review of Financial Studies 10(2): 481-523.

Jouanin, J.F., Rapuch, G., Riboulet, G. & Roncalli, T. 2001. Modelling dependence for credit derivatives with copulas. Working Paper, Groupe de Recherche Opérationnelle, Crédit Lyonnais, France. http://dx.doi.org/10.2139/ssrn.1032561.

Kijima, M. 2000. Valuation of a credit swap of the basket type. Review of Derivatives Research 4(1): 81-97.

Lando, D. 1998. On cox processes and credit risky securities. Review of Derivatives Research 2(2): 99-120.

Li, D.X. 2000. On default correlation: A copula function approach. Journal of Fixed Income 9(4): 43-54.

Longstaff, S. & Schwartz, E. 1995. A simple approach to valuing risky fixed and floating rate debt. Journal of Finance 50(3): 789-821.

Lund, S., Manyika, J., Mehta, A. & Goldshtein, D. 2018. A decade after the global financial crisis: What has (and hasn’t) changed? McKinsey Global Institute (Unpublished).

Ma, Y.K. & Kim, J.H. 2010. Pricing the credit default swap rate for jump-diffusion default intensity processes. Quantitative Finance 10(8): 809-818.

Merton, R. 1974. On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance 29(2): 449-470.

Mohd Ramli, S.N. & Jang, J. 2015. A multivariate jump-diffusion process for counterparty risk in CDS rates. Journal of the Korean Society for Industrial and Applied Mathematics 19: 2345.

Mohd Ramli, S.N. & Jang, J. 2014. Neumann series on the recursive moments of copula-dependent aggregate discounted claims. Risks 2(2): 195-210.

Nelsen, R.B. 2006. An Introduction to Copula. New York: Springer

Ruf, J. & Scherer, M. 2011. Pricing corporate bonds in an arbitrary jump-diffusion model based on an improved Brownian-bridge algorithm. Journal of Computational Finance 14(3): 127-144.

Salmon, F. 2009. Recipe for disaster: The formula that killed Wall Street. Wired Magazine. February 23, 2009.

Schmidt, T. 2006. Coping with Copula. In Copulas: From Theory to Application in Finance. London: Risk Books. pp.1-23.

Schonbucher, P.J. & Schubert, D. 2001. Copula-dependent default risk in intensity models. Working Paper, Department of Statistics, Bonn University.

Shamiri, A., Hamzah, N. & Pirmoradian, A. 2011. Tail dependence estimate in financial market risk management: Clayton-Gumbel copula approach. Sains Malaysiana 40(8): 927-935.

Stewart, I. 2012. The mathematical equation that caused the banks to crash. The Guardian. February 12, 2012.

*Corresponding author; email: rafidah@ukm.edu.my