The Theory of Good-Deal Pricing in Financial Markets

Abstract

The term ‘no-good-deal pricing’ in this paper encompasses pricing techniques base on the absence of attractive investment opportunities — good deals — in equilibrium. We borrowed the term from [8] who pioneered the calculation of price bands conditional on the absence of high Sharpe Ratios. Alternative methodologies for calculating tighter-than-no-arbitrage price bounds have been suggested by [4], [6], [12]. The theory presented here shows that any of these techniques can be seen as a generalization of no-arbitrage pricing. The common structure is provided by the Extension and Pricing Theorems, already well known from no-arbitrage pricing, see [15]. We derive these theorems in no-good-deal framework and establish general properties of no-good-deal prices. These abstract results are then applied to no-goos-deal bounds determined by von Neumann-Morgenstern preferences in a finite state model¹. One important result is that no-good-deal bouns generated by an unbounded utility function are always strictly tighter than the no-arbitrage bounds. The same is not true for bounded utility functions. For smooth utility functions we show that one will obtain the no-arbitrage and the representative agent equilibrium as the two opposite ends of a spectrum of no-good-deal equilibrium restrictions indexed by the maximum attainable certainty equivalent gains.