  In this paper we provide an expansion formula for Hawkes processes which
involves the addition of jumps at deterministic times to the Hawkes process in
the spirit of the well-known integration by parts formula (or more precisely
the Mecke formula) for Poisson functional. Our approach allows us to provide an
expansion of the premium of a class of cyber insurance derivatives (such as
reinsurance contracts including generalized Stop-Loss contracts) or risk
management instruments (like Expected Shortfall) in terms of so-called shifted
Hawkes processes. From the actuarial point of view, these processes can be seen
as "stressed" scenarios. Our expansion formula for Hawkes processes enables us
to provide lower and upper bounds on the premium (or the risk evaluation) of
such cyber contracts and to quantify the surplus of premium compared to the
standard modeling with a homogenous Poisson process.
