  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.

Mathematical Finance