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We consider a two-factor model for the valuation of a non callabledefaultable bond which pays coupons at certain given dates. The model underconsideration is the Jump to Default Constant Elasticity of Variance (JDCEV)model. The JDCEV model is an improvement of the reduced form approach, whichunifies credit and equity models into a single framework allowing forstochastic and possible negative interest rates. From the mathematical point ofview, the valuation involves two partial differential equation (PDE) problemsfor each coupon. First, we obtain the existence of solution for these PDEproblems. In order to solve them, we propose appropriate numerical schemesbased on a Crank-Nicolson semi-Lagrangian method for time discretizationcombined with bi-quadratic Lagrange finite elements for space discretization.Once the numerical solutions of the PDEs are obtained, a post-processingprocedure is carried out in order to achieve the value of the bond. Thispost-processing includes the computation of an integral term which isapproximated by using the composite trapezoidal rule. Finally, we present somenumerical results for real market bonds issued by different firms in order toillustrate the proper behaviour of the numerical schemes. Moreover, we obtainan agreement between the numerical results from the PDE approach and those onesobtained by applying a Monte Carlo technique and an asymptotic approximationmethod.

PDE models for the valuation of a non callable defaultable coupon bond under an extended JDCEV model