A new numerical method for the valuation of a non callable defaultable coupon bond under an extended JDCEV model

M.C. Calvo-Garrido, S. Diop, A Pascucci, C. Vázquez: In this paper, we consider the numerical solution of a two factor model for the valuation of a non callable defaultable bond which pays coupons at certain given dates. Such a model under consideration is the Jump to Default Constant Elasticity of Variance (JDCEV) model. The stochastic factors are the risk free interest rate and the stock price. Moreover, we take into account the possibility of negative interest rates. From the mathematical point of view, the valuation problem requires the numerical solution of two partial differential equation (PDE) problems for each coupon and with maturity those coupon payment dates. In order to solve these PDE problems, we propose appropriate numerical schemes based on a Crank-Nicolson semi-Lagrangian method for time discretization combined with biquadratic Lagrange finite elements for space discretization. Once the numerical solutions of the PDEs are obtained, a kind of post-processing is carried out in order to achieve the value of the bond. This post-processing includes the computation of an integral term which is approximated by using the composite trapezoidal rule. Finally, we present some numerical results for real market bonds issued by different firms in order to illustrate the proper behaviour of the numerical schemes.