A new look at the Heston characteristic function

TL;DR

This paper introduces a new, more transparent expression for the Heston model's characteristic function, enabling precise analysis of its properties, domain, and moments, with applications to volatility smile and log-spot distribution approximation.

Contribution

It provides a novel explicit form of the Heston characteristic function, clarifies its analytic properties, and offers new tools for analyzing moments and distributions in the model.

Findings

Exact domain of the moment generating function is determined.

Factorization of the moment generating function into Bessel type factors.

An approximating sequence for the law of log-spot is constructed.

Abstract

A new expression for the characteristic function of log-spot in Heston model is presented. This expression more clearly exhibits its properties as an analytic characteristic function and allows us to compute the exact domain of the moment generating function. This result is then applied to the volatility smile at extreme strikes and to the control of the moments of spot. We also give a factorization of the moment generating function as product of Bessel type factors, and an approximating sequence to the law of log-spot is deduced.