Stephen A. Clark

The valuation problem in arbitrage price theory

Abstract Suppose a continuous, strictly positive, linear price functional p is given on a subspace M of marketed claims. The valuation problem consists of verifying whether or notthere exists a continuous, strictly positive, linear extension of p from M to the entire contingent claims space X . We solve this problem when X belongs to a large class of Banach lattices including the classical Banach spaces, and also simplify some analogous results found in the literature for other types of financial models.

An extension theorem for rational choice functions

A choice function is strictly rational whenever it can be rationalized by a preference relation in a manner such that alternatives in a choice set are strictly preferred to alternatives in the corresponding rejection set. We demonstrate an Extension Theorem which asserts that a preference relation strictly rationalizes the choice function if and only if the choice function satisfies the Weak Axiom of Revealed Preference and the preference relation is an extension of the revealed weak preference relation. Then we consider various applications to rational choice theory.

Arbitrage approximation theory

Abstract This essay concerns three principles for eliminating arbitrage opportunities: (i) No Arbitrage (NA), (ii) No Approximate Arbitrage (NAA), and (iii) No Free Lunches (NFL). Assuming there is NAA, the space of contingent claims priced by arbitrage is characterized both analytically and geometrically. If the market is incomplete, then this set is typically a superset of the marketed claims space. Furthermore, it is shown that there exists a unique valuation operator if and only if there are NFL and every contingent claim is priced by arbitrage. Consequently, it is possible that every contingent claim is priced by arbitrage in an incomplete market.

Indecisive choice theory

Abstract A choice function is ‘decisive’ provided that it selects a non-empty choice set from every constraint set in its domain. If these choice sets are sometimes empty, then the choice function is ‘indecisive’. This paper takes the position that economic rationality and indecisiveness are compatible ideas, and generalizes some familiar methods from revealed preference theory.

The random utility model with an infinite choice space

SummaryThis essay presents a measure-theoretic version of the random utility model with no substantive restrictions upon the choice space. The analysis is based upon DeFinettis Coherency Axiom, which characterizes a set function as a finitely additive probability measure. The central result is the equivalence of the random utility maximization hypothesis and the coherency of the choice probabilities over all allowable constraint sets.

Revealed preference and linear utility

We study a Linear Axiom of Revealed Preference (LARP) that characterizes the consistency of a choice function with respect to a preference order satisfying the independence axiom. In addition, LARP characterizes lexicographic linear utility rationality when the choice space is a convex subset of a finite-dimensional real vector space, and LARP characterizes linear utility rationality when the choice space corresponds to a finite choice experiment.

An infinite-dimensional LP duality theorem

This paper constructs an infinite-dimensional version of the Duality Theorem for a Linear Program (LP). The algebraic dual LP is replaced with a new program called the topological dual LP that closes the range of the adjoint operator. Under some mild nondegeneracy conditions involving strict positivity, the new Duality Theorem asserts that the optimal value of the primal LP equals the optimal value of the topological dual LP. Some applications to mathematical finance are also included.

Revealed Preference and Expected Utility

This essay gives necessary and sufficient conditions for recovering expected utility from choice behavior in several popular models of uncertainty. In particular, these techniques handle a finite state model; a model for which the choice space consists of probability densities and the expected utility representation requires bounded, measurable utility; and a model for which the choice space consists of Borel probability measures and the expected utility representation requires bounded, continuous utility. The key result is the identification of the continuity condition necessary for the revelation of linear utility.

The representative agent model of probabilistic social choice

Abstract Suppose a random preference relation is defined on a finite set of mutually exclusive actions. This paper proposes a method for aggregating the random preference relation into a determinate preference relation on random actions. If we interpret the random preference relation as the distribution of tastes for a population of rational decision-makers, then the determinate preference relation can be regarded as the tastes of a representative agent, whose optimal behavior is equivalent to the collective behavior of the underlying population. We analyze this aggregation procedure from the viewpoint of normative social choice theory, suggesting a probabilistic resolution to the dilemma surrounding the impossibility theorems.

A concept of stochastic transitivity for the random utility model

Abstract This essay presents several major results concerning the random utility model. First, if a stochastic choice function is ex post rational with respect to a random utility function, then it is also ex ante rational and the ex ante revealed weak preference relation is transitive. Second, ex ante transitive rationality characterizes ex ante rationality with respect to a determinate utility function defined upon lotteries. Third, the additive determinate utility model is characterized in terms of acyclic choice probabilities, and the Luce strict utility model is verified to satisfy this condition.