Dirac Processes and Default Risk

TL;DR

This paper introduces Dirac processes using delta functions for pricing financial derivatives, expanding modeling capabilities and addressing high implied volatility issues in CDS swaptions.

Contribution

It presents Dirac processes as a novel tool for short-rate pricing, enhancing expressivity and simplifying jump modeling in financial derivatives.

Findings

Dirac processes enable high implied volatility in CDS swaptions.

They expand the modeling capabilities of short-rate approaches.

Jumps become redundant in the presence of Dirac processes.

Abstract

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.