School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University

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.