Valuation of Risky Assets in Arbitrage Free Economies with Frictions

Abstract

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 ...

Arbitrage, linear programming and martingales¶in securities markets with bid-ask spreads

“…After the early contributions of Garman and Ohlson (1981), Leland (1985), and Prisman (1986), the last decade has witnessed a mounting interest in the effects of introducing bid-ask spreads in the standard no-arbitrage model with frictionless markets. Merton (1990) and Boyle and Vorst (1992) have generalized the valuation-by-replication binomial option pricing model to the case of bid-ask spreads on the stock.…”

Optimal Consumption Portfolio and No-Arbitrage with Nonproportional Transaction Costs

“…Arbitrage has proven to be a very important tool in the study of financial engineering and has received much attention in the literature, see for example, Carassus, Pham, and Touzi (2001), Jouini and Kallal (1995), Hubermann (1982), Prisman (1986), Dermody and Prisman (1993), Wilhelm (1985), Ardalan (1999), Ross (1976), Li and Wang (2001), and Wang and Xia (2002), to name a few. One of the most fundamental results in finance is the equivalence between no-arbitrage condition and the existence of a pricing operator in markets with no transaction costs (see Ross, 1978).…”

Necessary and Sufficient Conditions for Weak No-Arbitrage in Securities Markets with Frictions

“…In literature, there are mainly two types of no-arbitrage: strong no-arbitrage and weak no-arbitrage (see, e.g., Prisman (1986)). The former (alias, arbitrage of the first kind) is the most commonly used concept and is the one we have discussed so far.…”

“…Garman and Ohlson (1981) extended the work of Ross (1978) to markets with proportional transaction costs. Prisman (1986) applied linear programming, non-linear programming and duality in convex programming to the valuation of risky assets with taxation. Ross (1987) extended the martingale analysis of no-arbitrage pricing to worlds with taxation in a one-period setting.…”

“…Kabanov and Stricker (2001) extended the work of Harrison and Kreps (1979) to a currency market with proportional transaction costs in a discrete-time setting. In this paper, we extend the model of Prisman (1986), which included only the friction of taxes, to a more general case of friction including transaction costs, bid-ask spreads and taxes, and establish a set of necessary and sufficient conditions to characterize the weak no-arbitrage.…”