Distance to the line in the Heston model

TL;DR

This paper investigates the geometric problem of calculating the distance from a point to a line within the Heston model's Riemannian manifold, providing explicit formulas and applying them to analyze implied volatility at small maturities.

Contribution

It introduces a new coordinate system and formulas for distances to lines in the Heston manifold, simplifying calculations especially for vertical lines and enabling applications to implied volatility.

Findings

Derived explicit formulas for distances to lines in the Heston manifold.

Simplified distance calculations for vertical lines.

Applied formulas to analyze small maturity implied volatility.

Abstract

The main object of study in the paper is the distance from a point to a line in the Riemannian manifold associated with the Heston model. We reduce the problem of computing such a distance to certain minimization problems for functions of one variable over finite intervals. One of the main ideas in this paper is to use a new system of coordinates in the Heston manifold and the level sets associated with this system. In the case of a vertical line, the formulas for the distance to the line are rather simple. For slanted lines, the formulas are more complicated, and a more subtle analysis of the level sets intersecting the given line is needed. We also find simple formulas for the Heston distance from a point to a level set. As a natural application, we use the formulas obtained in the present paper to compute the small maturity limit of the implied volatility in the correlated Heston model.