Asymptotic formulae for implied volatility in the Heston model

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.Comment: Presentation in Section 2 has been improved. Theorem 3.1 has been slightly generalised. Figures 2 and 3 now include the at-the-money point