Credit risk modeling using time-changed Brownian motion

TL;DR

This paper introduces a novel approach to credit risk modeling by using time-changed Brownian motions with jumps and stochastic volatility, enabling efficient computation of default probabilities and realistic credit derivative pricing.

Contribution

It proposes a new first passage time concept for time-changed processes, facilitating realistic and computationally efficient credit risk models with multifirm dependencies.

Findings

Efficient computation of default probabilities using the new first passage time.

Realistic spread curves for defaultable bonds and credit default swaps.

Multifirm models with dependent time changes allowing for rich dynamics.

Abstract

Motivated by the interplay between structural and reduced form credit models, we propose to model the firm value process as a time-changed Brownian motion that may include jumps and stochastic volatility effects, and to study the first passage problem for such processes. We are lead to consider modifying the standard first passage problem for stochastic processes to capitalize on this time change structure and find that the distribution functions of such "first passage times of the second kind" are efficiently computable in a wide range of useful examples. Thus this new notion of first passage can be used to define the time of default in generalized structural credit models. Formulas for defaultable bonds and credit default swaps are given that are both efficiently computable and lead to realistic spread curves. Finally, we show that by treating joint firm value processes as dependent time changes of independent Brownian motions, one can obtain multifirm credit models with rich and plausible dynamics and enjoying the possibility of efficient valuation of portfolio credit derivatives.