The Heston Riemannian distance function

TL;DR

This paper derives explicit formulas for the Riemannian distance function in the Heston model using geometric and analytical methods, linking it to sub-Riemannian geometry and large deviation principles.

Contribution

It introduces a novel approach to compute the Heston distance function by connecting geometric geodesics with sub-Riemannian analysis and large deviations.

Findings

Explicit formulas for the Heston Riemannian distance function.

Connection established between Heston and Grushin plane geometries.

Partial large deviation principle proved for the Grushin plane.

Abstract

The Heston model is a popular stock price model with stochastic volatility that has found numerous applications in practice. In the present paper, we study the Riemannian distance function associated with the Heston model and obtain explicit formulas for this function using geometrical and analytical methods. Geometrical approach is based on the study of the Heston geodesics, while the analytical approach exploits the links between the Heston distance function and the sub-Riemannian distance function in the Grushin plane. For the Grushin plane, we establish an explicit formula for the Legendre-Fenchel transform of the limiting cumulant generating function and prove a partial large deviation principle that is true only inside a special set.