Affine Forward Variance Models

Finance and Stochastics, Forthcoming

30 Pages Posted: 28 Jan 2018 Last revised: 12 Jun 2019

See all articles by Jim Gatheral

Jim Gatheral

CUNY Baruch College

Martin Keller-Ressel

Dresden University of Technology - Department of Mathematics

Date Written: January 19, 2018


We introduce the class of affine forward variance (AFV) models of which both the conventional Heston model and the rough Heston model are special cases. We show that AFV models can be characterized by the affine form of their cumulant generating function, which can be obtained as solution of a convolution Riccati equation. We further introduce the class of affine forward order flow intensity (AFI) models, which are structurally similar to AFV models, but driven by jump processes, and which include Hawkes-type models. We show that the cumulant generating function of an AFI model satisfies a generalized convolution Riccati equation and that a high-frequency limit of AFI models converges in distribution to the AFV model.

Keywords: Rough Volatility, Affine Process, Stochastic Volatility, Hawkes Process, Forward Variance

JEL Classification: G13, C02

Suggested Citation

Gatheral, Jim and Keller-Ressel, Martin, Affine Forward Variance Models (January 19, 2018). Finance and Stochastics, Forthcoming. Available at SSRN: or

Jim Gatheral

CUNY Baruch College ( email )

Department of Mathematics
One Bernard Baruch Way
New York, NY 10010
United States

Martin Keller-Ressel (Contact Author)

Dresden University of Technology - Department of Mathematics ( email )

Zellescher Weg 12-14
Willers-Bau C 112
Dresden, 01062

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