A Gamma Ornstein-Uhlenbeck Model Driven by a Hawkes Process

Mathematics and Financial Economics 15, 747–773 (2021). https://doi.org/10.1007/s11579-021-00295-0

30 Pages Posted: 9 May 2019 Last revised: 7 Sep 2021

See all articles by Guillaume Bernis

Guillaume Bernis

Natixis Assurances

Riccardo Brignone

University of Freiburg

Simone Scotti

University of Pisa - Department of Economics and Management

Carlo Sgarra

Politecnico di Milano- Dipartimento di Matematica

Date Written: March 20, 2020

Abstract

We propose an extension of the Gamma-OU Barndorff-Nielsen and Shephard model taking into account jump clustering phenomena.
We assume that the intensity process of the Hawkes driver coincides, up to a constant, with the variance process. By applying the theory of continuous-state branching processes with immigration, we prove existence and uniqueness of strong solutions of the SDE governing the asset price dynamics. We propose a measure change of self-exciting Esscher type in order to describe the relation between the risk-neutral and the historical dynamics, showing that the Gamma-OU Hawkes framework is stable under this probability change. By exploiting the affine features of the model we provide an explicit form for the Laplace transform of the asset log-return, for its quadratic variation and for the ergodic distribution of the variance process.
We show that the proposed model exhibits a larger flexibility in comparison with the Gamma-OU model, in spite of the same number of parameters required.
We calibrate the model on market vanilla option prices via characteristic function inversion techniques, we study the price sensitivities and propose an exact simulation scheme.
The main financial achievement is that implied volatility of options written on VIX is upward shaped due to the self-exciting property of Hawkes processes, in contrast with the usual downward slope exhibited by the Gamma-OU Barndorff-Nielsen and Shephard model.

Keywords: Stochastic volatility, Hawkes processes, Jump clusters, Exponential affine processes, Variance swap, Implied volatility for variance options

JEL Classification: C63, G12, G13

Suggested Citation

Bernis, Guillaume and Brignone, Riccardo and Scotti, Simone and Sgarra, Carlo, A Gamma Ornstein-Uhlenbeck Model Driven by a Hawkes Process (March 20, 2020). Mathematics and Financial Economics 15, 747–773 (2021). https://doi.org/10.1007/s11579-021-00295-0, Available at SSRN: https://ssrn.com/abstract=3370304 or http://dx.doi.org/10.2139/ssrn.3370304

Guillaume Bernis

Natixis Assurances ( email )

59 av Pierre Mendes-France
Paris, 75013
France

Riccardo Brignone

University of Freiburg ( email )

Freiburg, DE
Germany

Simone Scotti

University of Pisa - Department of Economics and Management ( email )

Italy

Carlo Sgarra (Contact Author)

Politecnico di Milano- Dipartimento di Matematica ( email )

Piazza Leonardo da Vinci
Milan, Milano 20100
Italy

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