A self-exciting switching jump diffusion: properties, calibration and hitting time

Abstract

A way to model the clustering of jumps in asset prices consists in combining a diffusion process with a jump Hawkes process in the dynamics of the asset prices. This article proposes a new alternative model based on regime switching processes, referred to as a self-exciting switching jump diffusion (SESJD) model. In this model, jumps in the asset prices are synchronized with changes of states of a hidden Markov chain. The matrix of transition probabilities of this chain is designed in order to approximate the dynamics of a Hawkes process. This model presents several advantages compared to other jump clustering models. Firstly, the SESJD model is easy to fit to time series since estimation can be performed with an enhanced Hamilton filter. Secondly, the model explains various forms of option volatility smiles. Thirdly, several properties about the hitting times of the SESJD model can be inferred by using a fluid embedding technique, which leads to closed form expressions for some financial derivatives, like perpetual binary options.

Keywords:

• Self-exciting process
• Switching process
• Jump diffusion

JEL Classification:

• G10
• G13

Acknowledgements

We thank the anonymous referee and the editor for their comments. We also acknowledge Professor Paul Embrechts from ETH Zrich for his helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The very first process, developed by Hawkes (1971), has been used in seismology to model the frequency of earthquakes and aftershocks.

Additional information

Funding

Griselda Deelstra acknowledges support of the FNRS grant PDR ‘Risk management in finance and insurance’ 2016–2018. Donatien Hainaut thanks the Chair Data Analytics and Models for insurance of BNP Paribas Cardif, hosted by ISFA (Université Claude Bernard, Lyon) and managed by the ‘Fondation Du Risque’.