Transition probability of Brownian motion in the octant and its application to default modeling

TL;DR

This paper derives a semi-analytic formula for the transition probability of 3D Brownian motion in the octant, with applications to default probability calculations in credit risk modeling.

Contribution

It introduces a novel semi-analytic solution for the transition probability in a constrained Brownian motion and applies it to credit risk assessment.

Findings

Derived a semi-analytic formula for transition probabilities.

Developed computational methods for solving the eigenvalue problem.

Applied results to default probability and credit valuation adjustments.

Abstract

We derive a semi-analytic formula for the transition probability of three-dimensional Brownian motion in the positive octant with absorption at the boundaries. Separation of variables in spherical coordinates leads to an eigenvalue problem for the resulting boundary value problem in the two angular components. The main theoretical result is a solution to the original problem expressed as an expansion into special functions and an eigenvalue which has to be chosen to allow a matching of the boundary condition. We discuss and test several computational methods to solve a finite-dimensional approximation to this nonlinear eigenvalue problem. Finally, we apply our results to the computation of default probabilities and credit valuation adjustments in a structural credit model with mutual liabilities.