Title: Affine forward variance 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.