On pricing of corporate securities in the case of jump-diffusion

Abstract

Structural models of credit risk are known to present vanishing spreads at very short maturities. This shortcoming, which is due to the diffusive behavior assumed for asset values, can be circumvented by considering discontinuities of the jump type in their evolution over time. In this paper, we extend the pricing model for corporate bond and determine the default probability in jump-diffusion model to address this issue. To make the problem clearly, we first investigate the case that the firm value follows a geometric Brownian motion under similar assumptions to those in Black and Scholes (1973), Briys and de Varenne (1997), i.e, the default barrier is KD (t, T) and the recovery rate is (1 − ω), where D (t, T) is the price of zero coupon default free bond and ω is a constant (0 < ω ≤ 1). By changing the numeraire, we obtain the closed-form solution for both the price of bond and default probability. Further, we consider the case of jump-diffusion and suppose that a firm will go bankruptcy if its value V t ≤ KD (t, T) and at the same time, the bondholder will receive $(1 - \omega )\tfrac{{V_t }} {K} $ . By introducing the Green function of PDE with absorbing boundary and converting the problem to an II-type Volterra integral equation, we get the closed-form expressions in series form for bond price and corresponding default probability. Numerical results are presented to show the impact of different parameters to credit spread of bond.