Exchange Options when One Underlying Price Can Jump

Many problems in life insurance and finance can be described in terms of exchange options. These contracts give their holders the right to exchange an asset against another one at some specified later date. Exchange options were introduced in the classical diffusive framework where an explicit formula can be obtained for the price. This article extends this framework by taking jumps into account. In the particular case where one asset follows a jump diffusion model, the present authors present two alternative approaches for the pricing of these exchange options. The first one is a complete probabilistic approach where a quasi-closed form formula can be obtained. The second one is based on the generalized Fourier transform approach. With the latter, this article gives a general methodology for pricing exchange options when one underlying can jump. This methodology can then be used in many areas such as the study of guaranteed funds in life insurance.