Option Pricing by Esscher Transforms

The Esscher transform is a time-honored tool in actuarial science. This paper shows that the Esscher transform is also an efficient technique for valuing derivative securities if the logarithms of the prices of the primary securities are governed by certain stochastic processes with stationary and independent increments. This family of processes includes the Weiner process, the Poisson process, the gamma process and the inverse Gaussian process. An Esscher transform of such a stock price process induces an equivalent probability measure on the process. The Esscher parameter or parameter vector is determined so that the discounted price of each primitive security is a martingale under the new probability measure. The price of any derivative security is simply calculated as the expectation, with respect to the equivalent martingale measure, of the discounted payoffs. Straightforward consequences of the method of The Esscher transforms include, among others, the celebrated Black-Scholes option-pricing formula, the binomial option-pricing formula, and the formulas for pricing options on the maximum and minimum of multiple risky assets. Tables of numerical values for the prices of certain European call options (calculated according to four different models for stock-price movements) are also provided.