Financial economics plays a far more prominent role in the training of economists than it did even a few years ago. This change is generally attributed to the parallel transformation in financial markets that has occurred in recent years. Assets worth trillions of dollars are traded daily in markets for derivative securities, such as options and futures, that hardly existed a decade ago. However, the importance of these changes is less obvious than the changes themselves. Insofar as derivative securities can be valued by arbitrage, such securities only duplicate primary securities. For example, to the extent that the assumptions underlying the Black–Scholes model of option pricing (or any of its more recent extensions) are accurate, the entire options market is redundant because by assumption the payoff of an option can be duplicated using stocks and bonds. The same argument applies to other derivative securities markets. Thus it is arguable that the variables that matter most – consumption allocations – are not greatly affected by the change in financial markets. Along these lines one would no more infer the importance of financial markets from their volume of trade than one would make a similar argument for supermarket clerks or bank tellers based on the fact that they handle large quantities of cash.

Introduction

The analytical framework in the classical finance models discussed in this book is largely the same as in general equilibrium theory: agents, acting as price-takers, exchange claims on consumption to maximize their respective utilities. Because the focus in financial economics is somewhat different from that in mainstream economics, we will ask for greater generality in some directions while sacrificing generality in favor of simplification in other directions.

As an example of greater generality, it is assumed that uncertainty will always be explicitly incorporated in the analysis. We do not assert that there is any special merit in doing so; the point is simply that the area of economics that deals with the same concerns as finance but concentrates on production rather than uncertainty has a different name (capital theory). Another example is that markets are incomplete: the Arrow–Debreu assumption of complete markets is an important special case, but in general it will not be assumed that agents can purchase any imaginable payoff pattern on securities markets.

As an example of simplification, it is assumed that only one good is consumed and that there is no production. Again, the specialization to a single-good exchange economy is adopted only to focus attention on the concerns that are distinctive to finance rather than microeconomics, in which it is assumed that there are many goods (some produced), or capital theory, in which production economies are analyzed in an intertemporal setting.

Introduction

Expected utility provides a framework for the analysis of agents' attitudes toward risk. In this chapter we expand on the formal definition of risk aversion presented in Chapter 8 and introduce measures of the intensity of risk aversion such as the Arrow–Pratt measures and risk compensation. The main result of this chapter, the Pratt Theorem, establishes the equivalence of these different measures of risk aversion.

Agents' preferences for risky consumption plans are assumed – except in Section 9.10 – to have a state-independent expected utility representation with continuous von Neumann–Morgenstern utility functions. The consumption plans in the domain of an expected utility function may be defined either narrowly or broadly. The axioms of expected utility imply that any consumption plan can be viewed narrowly as a random variable on the set S of states equipped with an agent's subjective probability measure. Thus, if the objects of choice are specified as the consumption plans that emerge from the axioms of expected utility, they are appropriately defined narrowly as random variables that can take S values with given probabilities. However, the analysis of this chapter applies equally well if consumption plans are broadly interpreted as arbitrary random variables (that is, as random variables with an arbitrary number of realizations and arbitrary probabilities). The choice between these interpretations is a matter of taste. In Section 9.10 we discuss risk aversion for a multiple-prior expected utility.

Introduction

We have thus far limited ourselves to models of two-date security markets in which securities are traded only once before their payoffs are realized. These models are suitable for the introductory study of the risk-return relation for securities and the role of securities in the equilibrium allocation of risk. However, two-date models require the assumption that all uncertainty is resolved at once. It is more realistic to assume that uncertainty is resolved only gradually. As the uncertainty is resolved, agents trade securities again and again. The multidate model of this and the following chapters assumes that there are a finite number of future dates. This specification allows for the gradual resolution of uncertainty and the retrading of securities as new information about security prices and payoffs becomes available.

Uncertainty and Information

In the multidate model, just as in the two-date model, uncertainty is specified by a set of states S. Each of the states is a description of the economic environment for all dates t = 0, 1, …, T. At date 0 agents do not know which state will be realized. But as time passes, they obtain more and more information about the state. At date T they learn the actual state.

Formally, the information of agents at date t is described by a partition Ft of the set of states S (a partition Ft of S is a collection of subsets of S such that each state s belongs to exactly one element of Ft).