Dirac Processes and Default Risk

We introduce Dirac processes, using Dirac delta functions, for short-rate-type pricing of financial derivatives. Dirac processes add spikes to the existing building blocks of diffusions and jumps. Dirac processes are Generalized Processes, which have not been used directly before because the dollar value of non-Real numbers is meaningless. However, short-rate pricing is based on integrals so Dirac processes are natural. This integration directly implies that jumps are redundant whilst Dirac processes expand expressivity of short-rate approaches. Practically, we demonstrate that Dirac processes enable high implied volatility for CDS swaptions that has been otherwise problematic in hazard rate setups.