Affine forward variance models

TL;DR

This paper introduces affine forward variance (AFV) models, unifying the Heston and rough Heston models, and extends to affine forward order flow intensity (AFI) models driven by jumps, with convergence results linking AFI and AFV models.

Contribution

The paper develops the AFV model class, characterizes it via convolution Riccati equations, and introduces AFI models with jump processes, establishing their connection to AFV models.

Findings

AFV models include Heston and rough Heston as special cases.

AFI models satisfy a generalized convolution Riccati equation.

High-frequency limits of AFI models converge to AFV models.

Abstract

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.