Stochastic Time Changes in Catastrophe Option Pricing

Catastrophe insurance derivatives (Futures and options) were | introduced in December 1992 by the Chicago Board of Trade in order | to offer insurers new ways of hedging their underwriting risk. Only | CAT options and combinations of options such as call spreads are | traded today, and the ISO index has been replaced by the PCS index. | Otherwise, the economic goal of these instruments continues to be | for insurers an alternative to reinsurance and for portfolio | managers a new class of assets to invest in. The pricing | methodology of these derivatives relies on some crucial elements: | (a) the choice of the stochastic modelling of the aggregate | reported claim index dynamics (since the terminal value of this | index defines the pay-off of the CAT options; (b) the decision of a | financial versus actuarial approach to the valuation; (c)the number | of sources of randomness in the model and the determination of a | "martingale measure" for insurance and reinsurance instruments. We | represent in this paper the dynamics of the aggregate claim index | by the sum of a geometric Brownian motion which accounts for the | randomness in the reporting of the claims and a Poisson process | which accounts for the occurrence of catastrophes (only | catastrophic claims are incorporated in the index). Geman (1994) | and Cummins and Geman (1995) took this modelling for the | instantaneous claim process. Our choice here is closer to the | classical actuarial representation while preserving the | quasi-completeness of insurance derivative markets obtained by | applying the Delbaen and Haezendonck (1989) methodology to the | class of layers of reinsurance replicating the call spreads. | Moreover, we obtain semi-analytical solutions for the CAT options | and call spreads by extending to the jump-diffusion case the method | of the Laplace transform and stochastic time changes introduced in | Geman and Yor (1993, 1996) in order to price financial | path-dependent options through the properties of excursion theory.