Long-term and blow-up behaviors of exponential moments in multi-dimensional affine diffusions

TL;DR

This paper analyzes the conditions under which exponential moments of multi-dimensional affine diffusions remain finite or explode, providing insights into their long-term behavior and implications for financial modeling.

Contribution

It fully characterizes the regions of exponents for exponential moments in affine processes and computes their growth and explosion rates.

Findings

Identifies regions where exponential moments do not explode

Calculates long-term growth rates of moments

Determines explosion rates for exponential moments

Abstract

This paper considers multi-dimensional affine processes with continuous sample paths. By analyzing the Riccati system, which is associated with affine processes via the transform formula, we fully characterize the regions of exponents in which exponential moments of a given process do not explode at any time or explode at a given time. In these two cases, we also compute the long-term growth rate and the explosion rate for exponential moments. These results provide a handle to study implied volatility asymptotics in models where returns of stock prices are described by affine processes whose exponential moments do not have an explicit formula.