Convergence of Heston to SVI

TL;DR

This paper proves that the SVI implied volatility parameterization and the large-time asymptotic of the Heston model are algebraically equivalent, simplifying the understanding of implied volatility behaviors.

Contribution

It establishes an algebraic equivalence between SVI and Heston asymptotics, confirming a conjecture and providing a simpler expression for implied volatility.

Findings

SVI and Heston asymptotics are algebraically equivalent

Simplified expression for Heston implied volatility at large times

Enhanced interpretation of SVI parameters

Abstract

In this short note, we prove by an appropriate change of variables that the SVI implied volatility parameterization presented in Gatheral's book and the large-time asymptotic of the Heston implied volatility agree algebraically, thus confirming a conjecture from Gatheral as well as providing a simpler expression for the asymptotic implied volatility in the Heston model. We show how this result can help in interpreting SVI parameters.