Saddle-Point Approach to Large-Time Volatility Smile

TL;DR

This paper develops a saddle-point method to analytically derive large-time implied volatility smiles for a broad class of arbitrage-free models, extending theoretical understanding and practical modeling capabilities in financial mathematics.

Contribution

It introduces a theoretical framework for large-time volatility smiles using saddle-point equations, applicable to a wide range of models with Lévy-type scaling.

Findings

Analytical expressions for large-time volatility smiles.

Applicability to Lévy-type models with scaling behavior.

Extension of SVI-like parametrizations to large times.

Abstract

We extend upon the saddle-point equation presented in [1] to derive large-time model-implied volatility smiles, providing its theoretical foundation and studying its applications in classical models. As long as characteristic function fulfills a L\'evy-type scaling behavior in large time, the approach allows us to study analytically the large-time smile behaviors under specific models, and moreover, to reach a very wide class of arbitrage-free model-inspired parametrizations, in the same manner as stochastic-volatility-inspired (SVI).