Most structural models for credit pricing assume geometric Brownian motion to describe the firm asset value. However, the underlying lognormal distribution does not match empirical distributions, typically skewed and leptokurtic. Moreover, defaults are usually driven by shocks, which are not captured by the continuous paths of Brownian motion. We assume that the asset price process is driven by a pure-jump Lévy process and default is triggered by the crossing of a preset barrier. Our model incorporates asymmetry, fat-tail behavior, jumps and instantaneous defaults. Under this model we price credit default swaps, detailing the calculations for the variance gamma process.