#### Document Type

Article

#### Publication Date

2011

#### Publication Title

Quant. Finance

#### 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* = (β_{1} *k* ^{1/2} + β_{2} + ···)^{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.

#### Suggested Citation

Friz, Peter K.
, Gerhold, Stefan
, Gulisashvili, Archil
, Sturm, Stephan
(2011). On refined volatility smile expansion in the Heston model. *Quant. Finance, 11*(8), 1151-1164.

Retrieved from:
http://digitalcommons.wpi.edu/mathematicalsciences-pubs/71

#### Volume

11

#### Issue

8

#### First Page Number

1151

#### Last Page Number

1164

#### DOI

10.1080/14697688.2010.541486