A Bivariate Compound Dynamic Contagion Process for Cyber Insurance

TL;DR

This paper introduces a novel bivariate compound dynamic contagion process to model cyber insurance losses, capturing contagious, catastrophic, and joint cyber attack risks with analytical and simulation tools.

Contribution

It develops a new bivariate process combining external shocks and self-excited jumps, with analytical properties and applications to cyber insurance premium calculation.

Findings

The process models aggregate cyber losses effectively.

Analytical expressions for moments and Laplace transforms are derived.

Simulation algorithms enable practical risk assessment.

Abstract

As corporates and governments become more digital, they become vulnerable to various forms of cyber attack. Cyber insurance products have been used as risk management tools, yet their pricing does not reflect actual risk, including that of multiple, catastrophic and contagious losses. For the modelling of aggregate losses from cyber events, in this paper we introduce a bivariate compound dynamic contagion process, where the bivariate dynamic contagion process is a point process that includes both externally excited joint jumps, which are distributed according to a shot noise Cox process and two separate self-excited jumps, which are distributed according to the branching structure of a Hawkes process with an exponential fertility rate, respectively. We analyse the theoretical distributional properties for these processes systematically, based on the piecewise deterministic Markov process developed by Davis (1984) and the univariate dynamic contagion process theory developed by Dassios and Zhao (2011). The analytic expression of the Laplace transform of the compound process and its moments are presented, which have the potential to be applicable to a variety of problems in credit, insurance, market and other operational risks. As an application of this process, we provide insurance premium calculations based on its moments. Numerical examples show that this compound process can be used for the modelling of aggregate losses from cyber events. We also provide the simulation algorithm for statistical analysis, further business applications and research.