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

Diserahkan: 12 Oktober 2019/Diterima: 23 Disember 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

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*Pengarang untuk surat-menyurat; email: rafidah@ukm.edu.my