  We consider a two-factor model for the valuation of a non callable
defaultable bond which pays coupons at certain given dates. The model under
consideration is the Jump to Default Constant Elasticity of Variance (JDCEV)
model. The JDCEV model is an improvement of the reduced form approach, which
unifies credit and equity models into a single framework allowing for
stochastic and possible negative interest rates. From the mathematical point of
view, the valuation involves two partial differential equation (PDE) problems
for each coupon. First, we obtain the existence of solution for these PDE
problems. In order to solve them, we propose appropriate numerical schemes
based on a Crank-Nicolson semi-Lagrangian method for time discretization
combined with bi-quadratic Lagrange finite elements for space discretization.
Once the numerical solutions of the PDEs are obtained, a post-processing
procedure 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. Moreover, we obtain
an agreement between the numerical results from the PDE approach and those ones
obtained by applying a Monte Carlo technique and an asymptotic approximation
method.
