Stochastic time changes in catastrophe option pricing

Abstract

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 Only CAT options and combinations of options such as call spreads are traded today, and the ISO index has been replaced by the PCS index.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: 1. (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); 2. (b) the decision of a financial versus actuarial approach to the valuation; 3. (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.