We provide an efficient and unbiased Monte Carlo simulation for the computation of bond prices in a structural default model with jumps. The algorithm requires the evaluation of integrals with the density of the first-passage time of a Brownian bridge as the integrand. Metwally and Atiya suggest an approximation of these integrals. We improve this approximation in terms of precision. We show, from a modeling point of view, that a structural model with jumps is able to endogenously generate stochastic recovery rates. It is well known that allowing a sudden default by a jump results in a positive limit of credit spreads at the short end of the term structure. We provide an explicit formula for this limit, depending only on the Lévy measure of the logarithm of the firm-value process, the recovery rate and the distance to default.