This paper derives a framework for arbitrage models in markets with frictions. It generalizes the existence of a valuation operator to such markets. As in perfect markets, the valuation operator is a linear operator and its existence is implied by the noarbitrage condition. In imperfect markets the valuation operator is individual-specific and depends on the agent's position in the market. The methodology employed in the paper is duality in convex programming.
IN SHARP CONTRAST to the vast body of literature on perfect markets, little isknown about markets with frictions. Of special interest are the implications of no arbitrage opportunities in such markets. Models based on arbitrage arguments have considerably improved our understanding of pricing and valuation of risky and riskless securities, e.g., bond pricing, option pricing, and the effects of capital structure.The importance of the valuation operator in a perfect market is explained in Ross [8]. "Arbitrage theory can be used to value ... assets in a simple and straightforward fashion that makes use only of information available in the market." Its crucial role in perfect markets suggests that it will be important in markets with frictions.In a market with frictions, the valuation operator is individual-specific. It may depend on the position of the individual in the market. But the valuation operator of any individual can be used to value assets. However, the implications of noarbitrage arguments to economies with frictions have received little attention.In a perfect market, these implications are derived from either Farkas' Lemma, or from duality in linear programming, as in Ross [7]. In a market with frictions these arguments would not work, unless it is assumed that the frictions are linear. (See Garman and Ohlson [2]).Recently Prisman [6] showed how the theory of duality in convex programming can be used to build arbitrage models in markets with frictions. That paper refers only to riskless bonds and focuses on an estimator of the term structure which minimizes the maximum arbitrage profit in markets with frictions. However, the analysis applies also to risky securities if time and the number of time periods * Arizona State University. The author acknowledges Stephen Ross, the participants of the ORSA/ TIMS meeting (November 1985) for helpful comments, and Jaime Dermody for helping improve the exposition of the paper. This paper was inspired from a paper by Garman and Ohlson [2], in particular from the footnote, "A treatment of the relaxed assumption (linearity of transaction costs) would lead into nonlinear duality theory. . . ". The research was supported in part by the Council 100 grant from Arizona State University. 545 546 The Journal of Finance supluT dq(u) I u} = q*(d). Hence, d admits no arbitrage if q * (d) = 0. For every d, uTdq (u) evaluated at zero is -q (0). Consequently, for every d, we have q* (d) + q (0) > 0. Thus, if q (0) = 0, then for every d, q*(d) 2 0. Q.E.D. LEMMA 2. The problem DF possesses an optimal solution (i.e., the infimum ...