Efficiency and Arbitrage in Financial Markets

International Research Journal of Finance and Economics ISSN 1450-2887 Issue 19 (2008) © EuroJournals Publishing, Inc. 2008 http://www.eurojournals.com/finance.htm Efficiency and Arbitrage in Financial Markets Michael Dothan Atkinson Graduate School of Management, Willamette University Salem, OR 97301, USA E-mail: mdothan@willamette.edu Tel: +011-503-370-6440; Fax: +011-503-370-3011 Abstract The usual description of market efficiency has two parts: (1) prices fully reflect all available information, and (2) there are no trading strategies that produce positive, expected, risk-adjusted excess returns. This paper argues that the two parts should be formalized independently of each other and proves that, in the context of a multiperiod market model that is much more general than the CAPM, the first part implies the second part. In other words, if prices fully reflect all available information, then it is generally true that there are no trading strategies that produce positive, expected, risk-adjusted excess returns. Keywords: Information and market efficiency, asset pricing, trading strategies, limits to arbitrage, σ -martingales 1. Introduction One of the most influential ideas of modern finance is the Efficient Market Hypothesis (EMH), the notion that prices in financial markets fully reflect all available information and that there are no trading strategies that produce positive, expected, risk-adjusted excess returns. This, still arguable, situation follows from the view that, in an intensely competitive financial market, the response of investors to new information is rapid and rational, bidding prices up or down until they eliminate any advantage to trading on the new information. Fama (1970) defines market efficiency verbally as circumstances in which prices fully reflect all available information, and formally as a martingale property of prices. Leroy (1973) and Lucas (1978) were among the first to clarify that, with risk aversion, only discounted, risk-adjusted, prices could have the martingale property. The martingale property, and the weaker supermartingale property of discounted, risk-adjusted prices, reflect the non-existence of trading strategies that produce positive, expected, risk-adjusted excess returns. Other authors have used either the verbal or the formal part as the primary definition of market efficiency. For example, Lo (1974), and Chordia et al. (2007) define market efficiency as circumstances where prices fully reflect all available information, whereas Malkiel (2003) uses the non-existence of trading strategies that produce positive, expected, risk-adjusted excess returns as the definition of market efficiency. To simplify the language, we will use the term “profitable trading strategies” to describe trading strategies that produce positive, expected, risk-adjusted excess returns. Empirical tests of market efficiency aim to confirm or refute the existence of profitable trading strategies. Much of the testing has been done with event studies that examine the reaction of stock prices to new information. International Research Journal of Finance and Economics - Issue 19 (2008) 103 To find evidence against market efficiency, researchers look for profitable trading strategies that are based either on corporate announcements or past performance of the stock. For example, Jegadeesh and Titman (1993) show that their strategy of buying well performing stocks and selling poorly performing stocks earned a positive, expected, risk-adjusted excess returns. Similarly, Michaely et al. (1995) describe profitable trading strategies based on corporate announcements of dividend payments and omissions. In most such studies, authors use the Capital Asset Pricing Model (CAPM) to adjust returns for risk. The connection between prices that fully reflect all available information and the absence of profitable strategies is intuitively appealing, but not trivial. Formalizing the verbal statement that in an efficient market prices fully reflect all available information as a martingale property of discounted, risk-adjusted prices, avoids the question of existence of profitable trading strategies by assuming, in the very definition, that such strategies do not exist. The problem addressed in this paper is the absence of a general proof that prices which fully reflect all available information preclude the existence of profitable trading strategies. Although this implication is intuitively appealing, such a proof, in a multiperiod model that is much more general than the CAPM, settles this question, contributes to our understanding of the meaning of market efficiency, and, as it turns out, is revealing in terms of the required assumptions about the fine properties of the information structure of investors, and the restrictions we have to impose on their trading strategies. To present a general proof that prices which fully reflect all available information imply the non-existence of profitable trading strategies, this paper proposes to separate the two parts of the definition of market efficiency, and to formalize the first, verbal part, that prices fully reflect all available information, as the Markov property of prices. Then, in the context of a very general market model, we prove that the Markov property, a restriction to tame trading strategies, and a somewhat stronger no-arbitrage condition, together imply that there are no trading strategies which produce positive, expected, risk-adjusted excess returns. The next section describes the customary general market model. Section 3 presents the sense in which the intuition that prices fully reflect all available information is captured by the Markov property of prices. Section 4 describes a proof of non-existence of profitable trading strategies. Section 5 concludes the paper. 2. General Market Model The elements of the general market model are information, prices, and trading strategies. Formally, the market model includes a probability space (Ω,F,P) , an information structure F = (Ft ) 0≤t≤T , a d dimensional process X 0 of prices, and a set Θ of d -dimensional trading strategies. The information structure F = (Ft ) 0≤t≤T is an increasing family of σ -fields such that Ft contains the decidable events at time t . As usual, the price process X 0 is assumed to be adapted to the information structure F , meaning that price history up to time t is part of information Ft at time t . To focus ideas, assume that security 1 is a locally riskless discount bond, or accumulation 0 account, in the sense that the price X1,t is a continuous, positive, increasing process with finite variation on compacts. It is customary to refer to X t = Xt 0 0 X 1,t as discounted prices. From now on, we will deal exclusively with discounted prices. The discounted price process X is assumed to be a d -dimensional semimartingale. This t assumption helps ensure that the integral ∫ θ 's dX s , representing gain from trade, exists. Trading strategies θ are restricted to be predictable, X -integrable, self-financing, and tame. 0 104 International Research Journal of Finance and Economics - Issue 19 (2008) A trading strategy is called self-financing if θ ' t X t = θ ' 0 X 0 + ∫ θ ' dX t 0 s s for every 0 ≤ t ≤ T . Because self-financing trading strategies involve only trading and do not involve additions or withdrawals of capital, the resulting value process is initial value plus the gain process. A tame strategy is such that the discounted gain process ∫ θ' 0 t s dX s is uniformly bounded from below. The restriction to tame strategies excludes schemes such as ``doubling on loss'' that guarantee an ultimate gain at the cost of unbounded intermediate losses, and reflects “limits to arbitrage” due to risk aversion and institutional constraints, as suggested by DeLong et al. (1990) and Shleifer and Vishny (1997). 3. Prices Reflect Information Fama (1970) refers to the martingale property of prices as the mathematical characterization of the EMH. Leroy (1973) and Lucas (1978) were among the first to clarify that only discounted, riskadjusted, prices can have the martingale property. A definition based on the martingale property of discounted, risk-adjusted prices goes directly to the notion that there are no profitable trading strategies. Such a definition does not capture the more basic idea that prices fully reflect all available information, it only captures the implication of that idea for the profitability of trading on information. Therefore, this paper proposes to formalize the notion that prices fully reflect all available information as the Markov property of prices. The reasoning that leads to this definition is that if the information Ft is fully reflected in prices X t , then the conditional probability of any relevant future event is the same whether conditioned on the information Ft or on current prices X t . Relevant future events are those determined by the future evolution of prices. Information that is irrelevant to the future evolution of prices need not be reflected in current prices even in an efficient * market. Therefore, thinking of t as the present time, denote by Ft the smallest σ -field that makes measurable all future discounted prices, that is, the random variables {X u t ≤ u ≤ T}. The intuitive notion that prices fully reflect the information structure F = (Ft )0≤t≤T is then the requirement that the * discounted price process X t be Markov, that is, for any A ∈ Ft and any 0 ≤ t ≤ T P (A Ft ) = P (A X t ) This definition generalizes the earlier association in the literature of the EMH with random walks because a process with stationary independent increments is Markov. For further discussion of this association see, for example, Malkiel (2003). The importance of the Markov property of prices is that it makes the information structure F right-continuous, that is, Ft + = Ft for every 0 ≤ t ≤ T . Intuitively, that means that new information at time t arrives precisely at time t and not an instant after t . We need the right continuity of the information structure to apply the fundamental theorem of asset pricing in this model in the proof that profitable trading strategies do not exist. Furthermore, by itself, the assumption that prices fully reflect all available information is not sufficient to infer that there are no profitable trading strategies. To deliver that result requires a ``no free lunch with vanishing risk'' condition that allows the use of the general fundamental theorem of asset pricing in this market model. The “no free lunch with vanishing risk” condition is a somewhat stronger no-arbitrage condition due to Kreps (1981). A “free lunch with vanishing risk” is a self-financing trading strategy that can be approximated by a sequence of tame self-financing trading strategies that converge to an arbitrage strategy. Formally, it is a self-financing trading strategy with zero initial cost and nonzero, nonnegative terminal value θT ' X T such that there is a sequence of tame self-financing trading strategies θ (n ) with zero initial cost such that the terminal values θ (n ) T ' X T converge to θT ' X T and the [ ] International Research Journal of Finance and Economics - Issue 19 (2008) 105 negative parts of terminal values θ (n ) T ' X T converge uniformly to zero in the norm of L∞ . The price system satisfies a “no free lunch with vanishing risk” condition if such trading strategies do not exist. The fundamental theorem of asset pricing that we use characterizes discounted, risk-adjusted prices as σ -martingales. A σ -martingale is a semimartingale that has a representation as an integral with respect to a local martingale. Formally, a d -dimensional semimartingale X is a σ -martingale if there exist a d -dimensional local martingale M and a d -dimensional predictable process φ such that each φ (i) is M (i) -integrable and for each 0 ≤ i ≤ d X (i)t = X (i)0 + {[ ] } − ∫ φ ( ) dM ( ) i s i 0 t s 4. No Profitable Strategies This section demonstrates that if prices fully reflect all available information and satisfy the strong noarbitrage condition then there are no admissible trading strategies that produce positive, expected, riskadjusted excess returns. Formally: Proposition. If discounted prices X have the Markov property and satisfy the “no free lunch with vanishing risk” condition, then the discounted value process is a supermartingale. Proof. Because discounted prices X have the Markov property, it follows from Theorem 4 in Chapter 2, Section 2.3 of Chung (1982) that the information structure of discounted prices is right-continuous: Ft + = Ft for every 0 ≤ t ≤ T . The right continuity of the information structure and the ``no free lunch with vanishing risk'' condition allow us to use the fundamental theorem of asset pricing of Delbaen and Schachermayer (1998) to conclude that there exists a probability measure Q , equivalent to P , that makes X a σ -martingale. Intuitively, the probability measure Q adjusts prices for risk. Next, according to Proposition 6.42 in Chapter 3, Section 6 of Jacod and Shiryaev (2003), the class of σ -martingales is stable under stochastic integration, that is, if X is a d -dimensional σ martingale and θ is a d -dimensional, predictable, X -integrable process then the integral ∫ θ ' dX s 0 t s is also a σ -martingale. Therefore, for any tame self-financing trading strategy θ , the discounted value process Vt = V0 + ∫ θ ' dX s 0 t s is a σ -martingale with respect to the probability measure Q and is uniformly bounded from below. Furthermore, it follows from Proposition 6.35 in Chapter 3, Section 6 of Jacod and Shiryaev (2003) that a σ -martingale that is uniformly bounded from below is a local martingale that is uniformly bounded from below. Applying Fatou's lemma we get that the discounted value process Vt satisfies for any 0 ≤ s ≤ t ≤ T E Q (Vt | Fs ) ≤ Vs Therefore, the discounted value process is a Q-supermartingale. ◊ The supermartingale property of the discounted value process means that, after adjustment for risk and the time value of money, conditional expected returns are nonpositive. Therefore, after adjustment for risk and the time value of money, it is possible, on average, to break even or have a trading loss, but not a trading gain. The fact alone that prices fully reflect all available information does not guarantee the nonexistence of profitable trading strategies. For example, consider the following Markov semimartingale prices ⎧⎡ ⎫ ⎤ 1 X t = exp⎨⎢ A − diag(BB')⎥t + BW t ⎬ ⎦ 2 ⎩⎣ ⎭ 106 International Research Journal of Finance and Economics - Issue 19 (2008) where A is a constant d -dimensional vector, B is a constant d × d -dimensional matrix, and W t is a d dimensional standard Wiener process. If the covariance matrix BB' is singular, then, depending on the vector A , these prices may violate the simple “no arbitrage” condition and, therefore, also the “no free lunch with vanishing risk” condition. For market efficiency, we need both the Markov property of prices and the absence of arbitrage opportunities. 5. Conclusion We have separated the definition of market efficiency into two formal parts, the Markov property of prices which captures the notion that prices fully reflect all available information, and the supermartingale property of discounted, risk-adjusted prices, which captures the notion that there are no admissible trading strategies which produce positive, expected, risk-adjusted excess returns. With a restriction to tame trading strategies and the additional assumption that prices satisfy the “no free lunch with vanishing risk” condition, we showed that the assumption that prices have the Markov property implies that discounted, risk-adjusted value processes are supermartingales, that is, there are no trading strategies that produce positive, expected, risk-adjusted excess returns. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] Chordia, T., R. Roll, and A. 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