Towards a Semigroup Pricing Theory

Abstract

In an arbitrage-free economy, there will always exist a set of linear operators which map future contingent dividends of securities into their current prices. It happens that such operators will also form an as a consequence of intertemporal analysis of the no-arbitrage restriction. This paper summarizes some of the major implications of the semigroup properties, but avoids almost all of the technical discussion which underlies them. Instead, several practical examples are presented. Some wellknown continuous-time results are replicated by this alternative method, and certain new developments are explored. IT IS WELL KNOWN that in an arbitrage-free economy, there exists a nonnegative linear operator mapping dividend streams into current prices.1 What is less well known is that these operators, as a consequence of intertemporal no-arbitrage, must additionally possess a semigroup property. This paper is an overview of work in progress which deals with the semigroup features, but which is contained in rather longer and more complicated papers.2 Here I shall attempt to make the main results (and prospective results in some cases) accessible to a broader financial economics readership. At first blush, the terminology itself is intimidating: we shall be dealing with the apparently difficult notions of (1) linear operators; (2) infinitesimal generators; and (3) evolution semigroups. These terms become less daunting when it is realized that almost everyone has encountered a simple example of each via the discounting of future payments. Consider a world of certainty where the discount function between dates t and T is given as tQT. That is, if the current time is t and a future payment is to be received at time T in the amount A,, then the present value of the future payment would be given as Vt = tQ,A,. We see that tQ, may be considered an operator, since it maps future payments into current values; it is linear, since two (contemporaneous) payments may be added together first and then discounted, or discounted individually and added, with the same result; and finally it is nonnegative, since no positive future payment can ever have a present value of less than zero. Thus, the (certainty) discount function is an instance of a nonnegative linear operator. The infinitesimal generator may be thought of as the time derivative of the operator, in our certainty case the rate of change of the discount function. In other words, we