Actuarial bridges to dynamic hedging and option pricing

Abstract

We extend the method of Esscher transforms to changing probability measures in a certain class of stochastic processes that model security prices. According to the Fundamental Theorem of Asset Pricing, security prices are calculated as expected discounted values with respect to a (or the) equivalent martingale measure. If the measure is unique, it is obtained by the method of Esscher transforms; if not, the risk-neutral Esscher measure provides a unique and transparent answer, which can be justified if there is a representative investor maximizing his expected utility. We construct self-financing replicating portfolios in the (multidimensional) geometric shifted (compound) Poisson process model, in which the classical (multidimensional) geometric Brownian motion model is a limiting case. With the aid of Esscher transforms, changing numéraire is explained concisely. We also show how certain American type options on two stocks (for example, the perpetual Margrabe option) can be priced. Applying the optional sampling theorem to certain martingales (which resemble the exponential martingale in ruin theory), we obtain several explicit pricing formulas without having to deal with differential equations.