The characteristic function of rough Heston models

TL;DR

This paper derives the characteristic function of rough Heston models, revealing a fractional Riccati equation structure that parallels the classical Heston model, aiding in derivatives pricing under rough volatility.

Contribution

It introduces a novel method linking Hawkes processes to fractional volatility models to compute the characteristic function in rough Heston models.

Findings

Characteristic function expressed via a fractional Riccati equation

Establishes a connection between Hawkes processes and rough volatility models

Provides a new tool for derivatives pricing in rough volatility context

Abstract

It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to reproduce the behavior of both historical and implied volatilities. However, due to the non-Markovian nature of the fractional Brownian motion, they raise new issues when it comes to derivatives pricing. Using an original link between nearly unstable Hawkes processes and fractional volatility models, we compute the characteristic function of the log-price in rough Heston models. In the classical Heston model, the characteristic function is expressed in terms of the solution of a Riccati equation. Here we show that rough Heston models exhibit quite a similar structure, the Riccati equation being replaced by a fractional Riccati equation.