Large deviations for the extended Heston model: the large-time case

TL;DR

This paper analyzes the large-time behavior of affine stochastic volatility models, deriving a formula for the implied volatility smile at large maturities and identifying conditions for convergence to the SVI parameterization.

Contribution

It provides a closed-form expression for the large-maturity implied volatility smile and establishes the necessary and sufficient condition for convergence to the SVI model.

Findings

Derived a closed-form formula for large-maturity implied volatility

Identified pathological behaviors in the asymptotic smile

Established the necessary and sufficient condition for SVI convergence

Abstract

We study here the large-time behaviour of all continuous affine stochastic volatility models (in the sense of Keller-Ressel) and deduce a closed-form formula for the large-maturity implied volatility smile. Based on refinements of the Gartner-Ellis theorem on the real line, our proof reveals pathological behaviours of the asymptotic smile. In particular, we show that the condition assumed in Gatheral and Jacquier under which the Heston implied volatility converges to the SVI parameterisation is necessary and sufficient.