Pathwise moderate deviations for option pricing

TL;DR

This paper develops a unified framework for pathwise moderate deviations in financial models, linking small-time, large-time, and tail asymptotics for diffusions and option-related quantities, with practical implications for numerical computations.

Contribution

It introduces a comprehensive approach to pathwise moderate deviations applicable to various financial models and functionals, connecting them with large deviations theory for improved numerical analysis.

Findings

Unified treatment of moderate deviations for financial models

Connections established between moderate and large deviations rate functions

Practical methods for numerical computation of deviations

Abstract

We provide a unifying treatment of pathwise moderate deviations for models commonly used in financial applications, and for related integrated functionals. Suitable scaling allows us to transfer these results into small-time, large-time and tail asymptotics for diffusions, as well as for option prices and realised variances. In passing, we highlight some intuitive relationships between moderate deviations rate functions and their large deviations counterparts; these turn out to be useful for numerical purposes, as large deviations rate functions are often difficult to compute.