Pricing the Zero-Coupon Bond and its Fair Premium Under a Structural Credit Risk Model with Jumps

Abstract

In this paper we consider a structural form credit risk model with jumps. We investigate the credit spread, the price, and the fair premium of the zero-coupon bond for the proposed model. The price and the fair premium of the bond are associated with the Laplace transform of default time and the firm's expected present market value at default. We give sufficient conditions under which the Laplace transform and the expected present market value of a firm at default are twice continuously differentiable. We derive closedform expressions for them when the jumps have a hyperexponential distribution. Using the closed-form expressions, we obtain numerical solutions for the default probability, the credit spread, and the fair premium of the bond.

Nonlinear boundary crossing probabilities of Brownian motion with random jumps

“…In a remarkable work, Kou and Wang (2003) obtained the explicit form of the Laplace transform of the FPT for a Brownian motion (BM) with double-exponentially distributed jumps crossing a constant boundary. Dong et al (2011) obtained similar results for hyper-exponential jumps. Perry et al (2004) established the integral equation for the FPT density for BM with general jump sizes.…”

“…Some recent applications can be found in e.g. Cont and Tankov (2004), Kou and Wang (2004), Jiang and Pistorius (2008), Cai et al (2009), Chiarella and Ziogas (2009), Kudryavtsev and Levendorskii (2009), Bakshi and Panayotov (2010), Jeannin and Pistorisus (2010), Chi and Lin (2011), and Dong et al (2011). Given the important role of the FPT in many applications, however, the computation of the FPT densities for jump-diffusion processes turns out to be very challenging and so far very few exact solutions exist.…”

The First Passage Time Problem for Mixed-Exponential Jump Processes with Applications in Insurance and Finance

“…In this section, we give a simple application on the price of the zero-coupon bond under a structural credit risk model with jumps. As in Dong et al [18], we assume that the total market value of a firm under the pricing probability measure P is given by…”

“…Given T > 0 and a short constant rate of interest r > 0, Dong et al (2011) shown that the Laplace transform of the fair price B(0, T ) of a defaultable zero-coupon bound at time 0 with maturity T is given bŷ…”

Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level

“…Numerical results. In this section we present some numerical results for the value of DFP at time 0 by inverting (3.28) via the Gaver-Stehfest algorithm, which is used inKou and Wang (2003)[18] and Dong et al (2011)[4]. For the details of the implementation of the Gaver-Stehfest algorithm, we refer to Section 5 in[18] or Section 4 in[4].For all the computations, the values of certain parameters are held fixed except otherwise indicated: we take T = 5, F 0 = 100, K 0 = 70, a 11 = a 22 = −0.1, r 1 = 0.02, r 2 = 0.05, σ 1 = 0.6, σ 2 = 0.3, λ 11 = λ 21 = 2, λ 12 = λ 22 = 1, m = 1, α 11 = β 11 = 25, α 12 = β 12 = 50, β 01 = 30, β 02 = 60…”