An Actuarial Bridge to Option Pricing

Actuaries measure, model, and manage risks. Risk associated with the investment function is a major uncertainty faced by many insurance companies. Actuaries should have knowledge of the asset side of the balance sheet of an insurance company and how it relates to the liability side. Such knowledge includes the operation of financial markets, the instruments available to the insurance companies, the options imbedded in these instruments, and the methods of pricing such options and derivative securities. In this paper we study the pricing of financial options and derivative securities, and their synthetic replication by means of the primitive securities. We show that a time-honored concept in actuarial science, the Esscher transform, is an efficient tool for pricing many options and contingent claims if the logarithms of the prices of the primitive securities are certain stochastic processes with stationary and independent increments. The Swedish actuary F. Esscher (1932) suggested that the Edgeworth approximation (a refinement of the normal approximation) yields better results, if it is applied to a modification or transformation of the original distribution of aggregate claims. Here, this Esscher transform is defined more generally as a change of measure for certain stochastic processes. An Esscher transform of such a 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. A derivative security or contingent claim is valued as the expectation, with respect to this equivalent martingale measure, of the discounted payoffs. Although there may be more than one equivalent martingale measure, in general, the risk-neutral Esscher measure provides a unique and transparent answer, which can be justified if there is a representative investor maximizing his or her expected utility. The option price is unique whenever a self-financing replicating portfolio can be constructed. This is the case in the multidimensional geometric Brownian motion model and also in the multidimensional geometric shifted compound Poisson process model. The latter is at the same time simpler (in view of its sample paths) and richer (the former can be retrieved as a limit). The Esscher method can be extended to pricing the derivative securities of (possibly) dividend-paying stocks. We show that, in the case of a multidimensional geometric Brownian motion model, the price of a European option does not depend on the interest rate, provided that the payoff is a homogeneous function of degree one with respect to the stock prices. Moreover, with the aid of Esscher transforms, a change of numeraire can be discussed in a concise way.