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Journal Articles Mathematics and Financial Economics Year : 2021

A Gamma Ornstein–Uhlenbeck model driven by a Hawkes process

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Guillaume Bernis
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  • PersonId : 1101932
Riccardo Brignone
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  • PersonId : 1101933
Carlo Sgarra
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  • PersonId : 1101934

Abstract

Abstract We propose an extension of the $$\Gamma $$ Γ -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 $$\Gamma $$ Γ -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 $$\Gamma $$ Γ -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 $$\Gamma $$ Γ -OU Barndorff-Nielsen and Shephard model.
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Dates and versions

hal-03258786 , version 1 (11-06-2021)

Licence

Attribution - CC BY 4.0

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Guillaume Bernis, Riccardo Brignone, Simone Scotti, Carlo Sgarra. A Gamma Ornstein–Uhlenbeck model driven by a Hawkes process. Mathematics and Financial Economics, 2021, ⟨10.1007/s11579-021-00295-0⟩. ⟨hal-03258786⟩
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