Pricing Catastrophe Options with Stochastic Interest Rates and Compound Poisson Losses

In this paper, we present an approach of changing probability measures associated with numeraire changes to the pricing of catastrophe event (CAT) derivatives. We assume that the underlying asset and a discounted zero-coupon bond follow a stochastic process, respectively. We obtain explicit closed form formulae that permit the interest rate to be random. We shall see that sometimes it is convenient to change the numeraire because of modeling considerations as well. Furthermore, we show that, for compound Poisson losses, sometimes a continuum of jump sizes can be replaced by finitely many jump sizes. Therefore, sometimes we can explore further applications of the closed-form formulae beyond the case that the compound Poisson losses are finitely many jump sizes. Finally, numerical experiments demonstrate how financial risks and catastrophic risks affect the price of double trigger put option.