On refined volatility smile expansion in the Heston model

Abstract

It is known that Heston's stochastic volatility model exhibits moment explosion, and that the critical moment s ₊ can be obtained by solving (numerically) a simple equation. This yields a leading-order expansion for the implied volatility at large strikes: σ_BS(k, T)² T ∼ Ψ(s ₊ − 1) × k (Roger Lee's moment formula). Motivated by recent ‘tail-wing’ refinements of this moment formula, we first derive a novel tail expansion for the Heston density, sharpening previous work of Drăgulescu and Yakovenko [Quant. Finance, 2002, 2(6), 443–453], and then show the validity of a refined expansion of the type σ_BS(k, T)² T = (β₁ k ^1/2 + β₂ + ···)², where all constants are explicitly known as functions of s ₊, the Heston model parameters, the spot vol and maturity T. In the case of the ‘zero-correlation’ Heston model, such an expansion was derived by Gulisashvili and Stein [Appl. Math. Optim., 2010, 61(3), 287–315]. Our methods and results may prove useful beyond the Heston model: the entire quantitative analysis is based on affine principles and at no point do we need knowledge of the (explicit, but cumbersome) closed-form expression of the Fourier transform of log S _T (equivalently the Mellin transform of S _T ). What matters is that these transforms satisfy ordinary differential equations of the Riccati type. Secondly, our analysis reveals a new parameter (the ‘critical slope’), defined in a model-free manner, which drives the second- and higher-order terms in tail and implied volatility expansions.

Keywords:

• Volatility smile fitting
• Stochastic volatility
• Differential equations
• Derivatives pricing

Acknowledgements

P.F. and S.S. (affiliated to TU Berlin while this work was started) acknowledge support by MATHEON. S.G. was partially supported by the Austrian Federal Financing Agency and the Christian-Doppler-Gesellschaft.

Notes

∥From a private communication with a derivative trader at a major investment bank.

†Exceptions include Fahrner (2007) and Keller-Ressel (2011).

‡Such higher-order Euler estimates are studied in—and form the foundation of—rough path theory; see Friz and Victoir (2008) and Chapter 3, 10 in Friz and Victoir (2010).

†We thank Roger Lee for helpful comments on this numerical evaluation.