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The characteristic function of rough Heston models
Author(s) -
El Euch Omar,
Rosenbaum Mathieu
Publication year - 2019
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/mafi.12173
Subject(s) - heston model , fractional brownian motion , stochastic volatility , mathematics , hurst exponent , brownian motion , volatility (finance) , statistical physics , riccati equation , mathematical analysis , sabr volatility model , econometrics , differential equation , physics , statistics
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.