Jump Diffusion in Credit Default Modeling

Security Prices experience random fluctuations.
Sometimes the magnitude of the change is so large
within a very short period of time that
Brownian motion fails to model such behavior. 
In a 1973 pioneering paper by
Merton, Poisson jump diffusion was introduced to model
discontinuous stock price movements and an option
pricing formula similar to that of Black-Scholes was 
obtained. Here we apply the same jump diffusion to
the credit default index model (structural model) to
address the concerns of short time behavior not well
resolved in a Gaussian diffusion model. With the
introduction of jump diffusion to the model,
a partial integro-differential equation with a free 
boundary is derived, where the extra non-local boundary 
condition is imposed to match the market information. 
We will show that the introduction of jumps indeed
improves the behavior of the model by numerical
results. Furthermore, it is possible to use this approach
to bridge the gap between two existing types of credit
default models.