The Small-Maturity Heston Forward Smile

TL;DR

This paper analyzes the small-maturity behavior of forward-start options and the implied volatility smile in the Heston model, revealing explosive out-of-the-money smiles and singularities due to the variance process.

Contribution

It provides a detailed asymptotic expansion for the forward smile in the Heston model and highlights the impact of the variance process's square-root behavior on these asymptotics.

Findings

Out-of-the-money forward smile explodes as maturity approaches zero.

High-order asymptotic expansion characterizes the rate of explosion.

Certain parameter configurations lead to singularities in the asymptotics.

Abstract

In this paper we investigate the asymptotics of forward-start options and the forward implied volatility smile in the Heston model as the maturity approaches zero. We prove that the forward smile for out-of-the-money options explodes and compute a closed-form high-order expansion detailing the rate of the explosion. Furthermore the result shows that the square-root behaviour of the variance process induces a singularity such that for certain parameter configurations one cannot obtain high-order out-of-the-money forward smile asymptotics. In the at-the-money case a separate model-independent analysis shows that the small-maturity limit is well defined for any Ito diffusion. The proofs rely on the theory of sharp large deviations (and refinements) and incidentally we provide an example of degenerate large deviations behaviour.