Pricing of Extension of European Exchange Option under Esscher Transforms

This paper studies the price of extension of the European exchange option (including generalized exchange option; compound exchange option; barrier exchange option; traffic-light option) with the geometric Brownian motion. Firstly, the reflection principle and property of the Browian motion with drift are given; Secondly, the definitions and properties of the Esscher transform of multidimensional processes with stationary and independent increments and two-dimensional Browian motion with drift are given by borrowing from the idea of Gerber and Shiu (1994); Finally, using related theory of Esscher transform, pricing formulas of extension of several European exchange options are obtained when the price of the underlying asset follows the geometric Brownian motion.