Esscher Transforms Methods Application In Extension Of Euopean Exchange Opition Pricing

Exchange option is an exotic option and gives its holder the right to exchange one asset for another at maturity T moment. Its expanding forms include generalized exchange option, compound exchange option, barrier exchange option, traffic-light option. The Esscher transform is a time-honored tool in acturial science. The Esscher transform is an efficient technique for valuing option if the logarithms of the stock-price process is a stochastic process with independent and stationary increments. An Esscher transform of a stochastic process with independent and stationary increments induces an equivalent probability measure on the process. The Esscher parameter vector is a martingale under the equivalent probability measure. The price of any option is the conditional expection or expection of the discounted payoffs with respect to the equivalent martingale measure.This paper, using Esscher transform, studies the extension of the European exchange option pricing when stock prices follow an geometric Brownian motion. Firstly, reflection principle and property of Browian motion with drift are given; sec-ondly, definitions and properties of the Esscher transform of multidimensional pro-cesses with stationary and independent increments and two-dimensional Browian mo-tion with drift are obtained; finally, using related theory of Esscher transform, pricing formula of extension of several European exchange options is given when stock prices follow geometric Brownian motion. These pricing formulas include generalized ex-change option pricing formula, compound exchange option pricing formula, barrier exchange option pricing formula, traffic-light option pricing formula.