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A Computational View of Market Efficiency

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A Computational View of Market Efficiency Quantitative Finance, Vol. 11, No. 7, July 2011, 1043–1050 A computational view of market efficiency JASMINA HASANHODZICy, ANDREW W. LOz and EMANUELE VIOLA*x yAlphaSimplex Group, LLC, One Cambridge Center, Cambridge, MA 02142, USA zMIT Sloan School of Management, 50 Memorial Drive, E52–454, Cambridge, MA 02142, USA xCollege of Computer and Information Science, Northeastern University, 440 Huntington Ave., #246WVH, Boston, MA 02115, USA (Received 25 April 2010; in final form 16 November 2010) We study market efficiency from a computational viewpoint. Borrowing from theoretical computer science, we define a market to be efficient with respect to resources S (e.g., time, memory) if no strategy using resources S can make a profit. As a first step, we consider memory-m strategies whose action at time t depends only on the m previous observations at times t À m, ..., t À 1. We introduce and study a simple model of market evolution, where strategies impact the market by their decision to buy or sell. We show that the effect of optimal strategies using memory m can lead to ‘market conditions’ that were not present initially, such as (1) market spikes and (2) the possibility for a strategy using memory m04m to make a bigger profit than was initially possible. We suggest ours as a framework to rationalize the technological arms race of quantitative trading firms. Keywords: Agent based modelling; Bound rationality; Complexity in finance; Behavioral finance 1. Introduction ‘frictionless’ markets and costless trading, then prices fully reflect all available information. Therefore, no Market efficiency—the idea that ‘‘prices fully reflect all profits can be garnered from information-based trading available information’’—is one of the most important because such profits must have already been captured. concepts in economics. A large number of articles have In mathematical terms, prices follow martingales. been devoted to its formulation, statistical implementa- This stands in sharp contrast to finance practitioners tion, and refutation since Fama (1965a,b, 1970) and who attempt to forecast future prices based on past prices, Samuelson (1965) first argued that price changes must be and, perhaps surprisingly, some of them do appear unforecastable if they fully incorporate the information to make consistent profits that cannot be attributed and expectations of all market participants. The more to chance alone. efficient the market, the more random the sequence of price changes generated by it, and the most efficient market of all is one in which price changes are completely 1.1. Our contribution random and unpredictable. In this paper we suggest that a reinterpretation of market According to the proponents of market efficiency, this efficiency in computational terms might be the key to randomness is a direct result of many active market reconciling this theory with the possibility of making participants attempting to profit from their information. profits based on past prices alone. We believe that it does Driven by profit opportunities, legions of investors not make sense to talk about market efficiency without pounce on even the smallest informational advantages taking into account that market participants have at their disposal, and in doing so, they incorporate their bounded resources. In other words, instead of saying information into market prices and quickly eliminate the that a market is ‘efficient’ we should say, borrowing from profit opportunities that first motivated their trades. theoretical computer science, that a market is efficient If this occurs instantaneously, as in an idealized world of with respect to resources S, e.g. time, memory, etc., if no *Corresponding author. Email: [email protected] Jasmina Hasanhodzic is now at Boston University Department of Economics, 270 Bay State Road, Boston, MA 02215, USA. Quantitative Finance ISSN 1469–7688 print/ISSN 1469–7696 online ß 2011 Taylor & Francis http://www.informaworld.com DOI: 10.1080/14697688.2010.541487 1044 J. Hasanhodzic et al. strategy using resources S can generate a substantial profit. presidential cycle. Although one can consider more Similarly, we cannot say that investors act optimally given complicated dependencies, working with these finite all the available information, but rather they act opti- patterns keeps the mathematics simple and is sufficient mally within their resources. This allows for markets to be for the points made below. efficient for some investors, but not for others; for Now we consider a new agent A who is allowed to trade example, a computationally powerful hedge fund may the good daily, with no transaction costs. Intuitively, extract profits from a market that looks very efficient agent A can profit whenever she can predict the sign of from the point of view of a day-trader who has less the next return. However, the key point is that A is resources at his disposal—arguably the status quo. computationally bounded: her prediction can only depend As is well-known, suggestions in this same spirit have on the previous m returns. Continuing the above example already been made in the literature. For example, Simon and setting memory m ¼ 2, we see that upon seeing (1955) argued that agents are not rational but boundedly returns (À1, 1), A’s best strategy is to guess that the next rational, which can be interpreted in modern terms as return will be negative: this will be correct for two out of bounded in computational resources. Many other works the three occurrences of (À1, 1) (in each occurrence of the in this direction are discussed in section 1.2. The main pattern). difference between this line of research and ours is: while We now consider a model of market evolution where it appears that most previous works use sophisticated A’s strategy impacts the market by pushing the return continuous-time models, in particular explicitly address- closer to 0 whenever it correctly forecasts the sign of the ing the market-making mechanism (which sets the price return—thereby exploiting the existing possibility given agents’ actions), our model is simple, discrete, and of profit—or pushing it away from 0 when the forecast abstracts from the market-making mechanism. is incorrect—thereby creating a new possibility for profit. In other words, the evolved market is the difference between the original market and the strategy guess (we 1.1.1. Our model and results. We consider an economy think of the guess in {À1, 0, 1}). For example, figure 2 where there exists only one good. The daily returns (price shows how the most profitable strategy using memory differences) of this good follow a pattern, e.g. m ¼ 2 evolves the market given by the pattern ð1, 1, À 1, 1, À 1, 1, 2, À 1, 1, À 1, 1, [À2, 2, À2, 2, À2, 2, 3] (similar to the previous example). À 1, 1, 2, À 1, 1, À 1, 1, À 1, 1, 2, ...Þ, The main question addressed in this work is: what do markets evolved by memory-bounded strategies look like? which we write as [À1, 1, À1, 1, À1, 1, 2] (see figure 1). We show that the effect of optimal strategies using A good whose returns follow a pattern may arise from a memory m can lead to ‘market conditions’ that were not variety of causes ranging from biological to political; present initially, such as (1) market spikes and (2) the think of seasonal cycles for commodities, or the 4-year possibility that a strategy using memory m04m can make 2 2 1 1 0 0 −1 −1 2 4 6 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Figure 1. The market corresponding to a market pattern. A computational view of market efficiency 1045 larger profits than previously possible. By market information and marginal-utility-weighted prices follow spikes (1), we mean that some returns will grow much martingales. Market efficiency has been extended in many larger, an effect already anticipated in figure 2. For point other directions, but the general thrust is the same: (2), we consider a new agent B that has memory m0 that is individual investors form expectations rationally, markets larger than that of A. We let B trade on the market evolved aggregate information efficiently, and equilibrium prices by A. We show that, for some initial market, B may make incorporate all available information instantaneously. See more profit than what would have been possible if A had Lo (1997, 2007) for a more detailed summary of the not evolved the market, or even if another agent with large market efficiency literature in economics and finance. memory m0 had evolved the market instead of A. Thus, it is There are two branches of the market efficiency precisely the presence of low-memory agents (A) that literature that are particularly relevant for our paper: allows high-memory agents (B) to make large profits. the asymmetric information literature, and the literature Regarding the framework for (2), we stress that we only on asset bubbles and crashes. In Black’s (1986) presiden- consider agents trading sequentially, not simultaneously: tial address to the American Finance Association, he after A evolves the market, it becomes incorporated into a argued that financial market prices were subject to ‘noise’, new market, on which B may trade. While a natural which could temporarily create inefficiencies that would direction is extending our model to simultaneous agents, ultimately be eliminated through intelligent investors we argue that this sequentiality is not unrealistic. competing against each other to generate profitable Nowadays, some popular strategies A are updated only trades. Since then, many authors have modeled financial once a month. Within any such month, a higher- markets by hypothesizing two types of traders—informed frequency strategy B can indeed trade in the market and uninformed—where informed traders have private evolved by A without A making any adjustment.
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