Efficiency analysis, shortage functions, arbitrage, and martingales

Abstract

This paper shows that standard tools of efficiency analysis, directional distance functions, can be used to characterize the investment-returns technology. That ability to characterize the investment-returns technology and fundamental duality relationships imply that directional distance functions can be used to detect the presence of an arbitrage, to value financial assets in the absence of an arbitrage lying in the span of the market and to place bounds on the no-arbitrage values of assets lying outside the span of the market.

Introduction

In a series of influential papers, Luenberger, 1992, Luenberger, 1994a, Luenberger, 1994b introduced the benefit and shortage functions and investigated their role in facilitating the calculation and characterization of Pareto-optimal outcomes. Chambers et al., 1996, Chambers et al., 1998 later showed that versions of these functions provide directional generalizations of the Shephard, 1953, Shephard, 1970 distance functions. Thus, under appropriate disposability assumptions, shortage functions can characterize production technologies, which in turn implies that they provide an apparatus for a profit-based approach to efficiency measurement and for shadow pricing inputs and outputs.

Although their jargons differ, production analysis and modern financial analysis both have deep mathematical roots in convex analysis. In fact, the standard model of a frictionless financial market is a special case of a constant returns to scale production technology. Therefore, tools used in efficiency analysis should be applicable to problems in financial analysis.

This paper does several things. It examines the role that the shortage function plays in characterizing feasible investment-return outcomes. It shows that a version of the shortage function, which we refer to as a directional distance function, can detect the presence or absence of arbitrages. It shows that directional distance functions provide valuation functions and martingale pricing kernels for both complete and incomplete financial markets. In particular, by appropriate choice of numeraire, directional distance functions can provide sublinear valuation functions that correspond with the linear pricing kernel over the span of the market and dominate its no-arbitrage extensions outside the market span. Hence, directional distance functions can price exactly assets lying in the span of the market and provide calculable bounds on the no-arbitrage valuations of financial assets lying outside the market span. Finally, this paper compares the tightness of those bounds on the no-arbitrage valuations derived from using different versions of the directional distance functions.

Before proceeding to the results, it is important to highlight a characteristic of our analysis. An arbitrage in financial analysis is a costless current-period, financial transaction that ensures a strictly positive return in at least one future state of nature. Or alternatively, it is a profitable, current-period, financial transaction that requires no future payouts. Mathematically, it is a solution to a system of linear inequalities. As such, determining its existence or deriving pricing implications from its existence requires no assumptions on decisionmaker preferences or on the underlying state space. Because no assumptions are made on preferences, our results are completely independent of any assumption on the individual decisionmaker’s preferences. In particular, they do not require that preferences be state independent, nor do they require that preferences not be state independent.

That does not mean, however, that our results have no implications once specific assumptions are made. They do. And in our concluding remarks we discuss some of these implications.