Asymptotic formulae for implied volatility in the Heston model

TL;DR

This paper derives an approximate explicit formula for implied volatility in the Heston model, valid for large maturities, using saddlepoint methods and properties of holomorphic functions.

Contribution

It provides the first explicit large-maturity asymptotic formula for implied volatility in the Heston model expressed with elementary functions.

Findings

The formula accurately approximates implied volatility for large maturities.

The approach simplifies complex calculations of implied volatility in the Heston model.

The method leverages saddlepoint techniques and holomorphic function properties.

Abstract

In this paper we prove an approximate formula expressed in terms of elementary functions for the implied volatility in the Heston model. The formula consists of the constant and first order terms in the large maturity expansion of the implied volatility function. The proof is based on saddlepoint methods and classical properties of holomorphic functions.