# Density symmetries for a class of 2-D diffusions with applications to finance

## TL;DR

This paper extends symmetry results for 2-D diffusion densities with boundary explosions, enabling better analysis of forward equations and applications in financial models like option pricing.

## Contribution

It generalizes classical symmetry results to two-dimensional diffusions with boundary explosions, facilitating analysis of forward equations in finance.

## Key findings

- Extended symmetry results for 2-D diffusions with boundary explosions
- Reduced boundary condition problems to backward equations without explosions
- Applied symmetry to improve option pricing models in finance

## Abstract

We study densities of two-dimensional diffusion processes with one non-negative component. For such diffusions, the density may explode at the boundary, thus making a precise specification of the boundary condition in the corresponding forward Kolmogorov equation problematic. We overcome this by extending a classical symmetry result for densities of one-dimensional diffusions to our case, thereby reducing the study of forward equations with exploding boundary data to the study of a related backward equation with non-exploding boundary data. We also discuss important applications of this symmetry for option pricing in stochastic volatility models and in stochastic short rate models.