vector is announced• All agents trade at exactly the same prices• Final prices and allocations are completely determined from agent preferencesand endowments• Agent behavior is very simple, involving nothing more than truthful reporting ofdemands at announced prices.Unfortunately, the job of the Walrasian auctioneer as the central entity to computethe equilibrium price is extremely hard. Equilibrium computation requires a mechanism for converging to a fixed point in a finite length of time, using a boundedamount of resources.Definition 5.6. A fixed point of a function is a point that is mapped to itself by thefunction. That is to say, x is a fixed point of the function f , if and only if f (x) = x.Without such a mechanism there is little reason to believe that a fixed point wouldever be observed. Fixed-point theorems were introduced into economic theory byvon Neuman in his work on the input–output model [409, 410]. Since then, manydomains of economic theory have come to depend on fixed point theorems to provethe existence of equilibria, notably general equilibrium theory, but also Nash equilibria in game theory [308]. For a recent perspective on this, see [153]. The Brouwerfixed point theorem was one of the early achievements of algebraic topology, andprovided the basis of more general fixed point theorems, such as Brouwer and Kakutani’s fixed point theorem.Theorem 5.1. Let Bn be the n-dimensional closed unit ball and let f : Bn −→ Bn bea continuous function. Then f has a fixed point: for some x ∈ Bn , f (x) = x.The proof of Brouwer’s fixed point theorem assumes that either a triangle is beingmapped onto itself, or another two-dimensional space is being mapped onto a triangle. Then a series of triangulations of decreasing size is constructed, and Sperner’sLemma is used to help conclude that the convergence of the series proves the existence of a fixed point.Lemma 5.1. (Sperner’s Lemma). Suppose the interior of a triangle is triangulated(that is, divided up internally into small triangles). The vertices of the triangle arecolored red, green, and blue, respectively. All other vertices, where lines meet insideor around the outside edges of the triangle, are also colored red, green, or blue withthe restriction that no edge of the main triangle contains all three colors. Let TC
5.2 Calculating Fixed Point Market Equilibrium
33
be the number of small triangles whose vertices are colored red, green, and bluein clockwise order; and let TA be the number of small triangles whose vertices arecolored red, green, and blue in anticlockwise order. Then|TC − TA | = 1
(5.1)
In particular, the total number of small red–green–blue triangles must be odd, andis certainly never zero.One famous real-world example of this theorem is stirring a glass of water. Whenthe molecules stop moving, there will be one molecule that is in the same place itwas before the stirring began. Another example involves two identical pieces of paper. If one of the pieces is crumpled and set on top of the other, there will be onepoint on that paper that lies directly above the corresponding point of the uncrumple paper.Algorithms for computing Brouwer fixed points fall into a complexity class thatmakes them among the hardest problems in all of computer science. For instance, thelower bound for worst-case computation of Brouwer fixed points is exponential inthe dimension of the problem [192] where the dimension is the size of the commodity space in the Arrow–Debreu version of general economic equilibrium [13]. Furthermore, it has recently been shown that the computational complexity of Brouwerand Kakutani fixed points are closely related to the complexity of the parity argument, the connection between the two being Sperner’s Lemma [319]. The constructive problem arising from the application of Sperner’s Lemma to the Brouwer andKakutani fixed points of the Walrasian equilibrium model is that there are no polynomial time algorithms for the general case with nonlinear utility functions. Suchresults can only be obtained for linear utility functions.
5.2.2 Computational Complexity of Decentralized EquilibriumIt has been argued that the Walrasian model of exchange is problematic for severalreasons. Recent results on the computational complexity of Brouwer and Kakutanifixed points suggest that real markets cannot possibly operate according to the Walrasian model. Furthermore, the Walrasian market model has no empirical underpinnings (e.g., [185], p. 55) and therefore does not present a reasonable picture of howan exchange economy works. The aggregation of information into the net demandfunctions of the Walrasian model has long been known to be excessive. Hayek [186]notesWe cannot expect that this problem will be solved by first communicating all this information to a central board which, after integrating all knowledge, issues its orders [. . . ]. Theproblem is to show how a solution is produced by interaction of people each of whom haspartial knowledge.
In addition, there are a variety of non-Walrasian exchange mechanisms that yieldequilibrium allocations that are Pareto optimal. These mechanisms are radically
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5 Views from Complexity Theory
more decentralized than the Walrasian one, with its single uniform price vector,and display a more realistic picture of real economic processes. Rust ([351]) recommends using massively parallel computers and decentralized algorithms that allowcompetitive equilibria to arise as emergent computations.The reason why large scale computable general equilibrium problems are difficult foreconomists to solve is that they are using the wrong hardware and software. Economistsshould design their computations to mimic the real economy, using massively parallel computers and decentralized algorithms that allow competitive equilibria to arise as “emergentcomputations” [. . . ]. The most promising way for economists to avoid the computationalburdens associated with solving realistic large scale general equilibrium models is to adoptan “agent-based” modeling strategy where equilibrium prices and quantities emerge endogenously from the decentralized interactions of agents.
By using a decentralized k-lateral exchange model with local price formation mechanisms which studies the performance of market systems as a function of their scale,i.e., the number of agents in the marketplace and the number of commodities beingtraded, [19] analytically derives complexity properties for economies as large as amillion agents and 20,000 commodities per agent. These properties are better (i.e.,less complex) than those of Walrasian exchange models. Due to the fact that polynomial complexity is a prerequisite of real world equilibrium price formation mechanisms, [19] decentralized systems are the more plausible computational device.
5.2.3 Adaptive/Inductive Learning of Rational ExpectationsEquilibriaA major part of the coordination processes in markets relies on the identification andcalculation of fixed point mappings of global signals such as equilibrium prices. Byforming rational expectations of fixed points of market equilibrium prices, economicagents are meant to coordinate their activity and iron out inconsistent expectations.Spear [374] was the first to show that the problem of identifying a set of fixed pointsfor market equilibrium price functions is algorithmically unsolvable.The absence of a unique decision procedure or a deductive means by which toselect appropriate forecast models/functions has been intuitively justified as arisingfrom the self-referential character of the problem by [16]:the expectational models investors choose affect the price sequence, so that [. . . ] their [. . . ]very choices of models affect their data and so their choices of model.
Arthur et al. [16] refer to the above self-referential structure of the problem as causing it to be ill-defined. Likewise, in the El Farol game, [14] gives the classic prototype of a problem with a contrarian structure for which there is no deductive meansof arriving at a solution. These problems are well understood to be algorithmicallyunsolvable [259, 260, 374, 407].
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Due to the absence of a unique decision procedure or a deductive means by whichto select appropriate forecast models/functions, it is questionable whether rationalagents exist at all. The best an agent can do is to find approximations to optimal solutions. Given a limited amount of time, some algorithms will be able to find betterapproximations than others. Given the same algorithm, the faster a computer is themore potential that it can find better solutions. Therefore, when serious computationis involved, the actual degree of an agent’s rationality depends on which algorithmshe uses, and what computing power he has access to.However, computational complexity is not the only reason why the rationality assumption is challenged. Challenges also come from cognitive reasoning [8] whichquestions the behavior of optimal human beings and proposes a more realistic notionof rationality. This notion is called bounded rationality [369] and describes the property of an agent that behaves in a manner that is nearly as optimal with respect to itsgoals as its resources will allow. These resources include processing power, algorithm, and time available to the agent. Some of these resources are not always easyto quantify. Therefore, any mathematical model that assumes rationality is subjectto the definition of resources available. By using faster computers and better algorithms within a given time limit, it is therefore possible that an agent can become“more rational.”
5.3 Computational Difficulties with Efficiency5.3.1 Information InterpretationIntellectual incoherency of the rationality concept [304] with the theory of effectivecomputability has also been used as major argument against the notion of marketefficiency. According to [103], even if we assume that all actual market analyseswould have equal merit (whether supported by human or machine), a new marketplayer could always take advantage of the older players unless the older analyseswere not merely the same, but also optimal. The assumption of rationality wouldtherefore imply that market players are equally endowed with intelligence, and thatthis intelligence is nothing but maximal. It may be thought that by use of the mostsophisticated computational analyses, statistical techniques, etc. we may achievesuch optimality. But certainly we do not have such an optimal understanding of themarkets now. It seems hard to believe that market behavior is the uniquely best understood of all subjects of human science. In fact, most scientists would probablytend to think the opposite. Moreover, the computational difficulties of optimally interpreting information prior to acting in the market are severe in principle. Turing’shalting problem tells us that the general case of finding a single computable functionamongst the class of all computable functions is undecidable [398]. In the context ofmarket modeling, this would mean that there is no algorithm which will guarantee
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5 Views from Complexity Theory
the discovery of the best market model within finite time. Thus, not even the mostsophisticated method of inductive and statistical inference can guarantee to find thebest predictor function for any market.Rather than looking for the very best model of market behavior – requiring of usa general search through all computable functions – we might suppose that a lesserconcept of optimality may suffice. That is, by restricting the space of models thecomputational difficulties will be lessened. For example, we could use linear regression using all available explanatory variables. In that case, there is a fairly quickand simple computer program to find the best linear model of market behavior. Atthe very least, we should like to select among the explanatory variables, rejectingthose which are irrelevant (insignificant) to price predictions. But even for functional forms such as linear variable selection with Akaike’s information criterion(AIC) this will be a computationally infeasible, i.e., NP-complete task under theassumption that P = NP holds (see [97]). Intuitively, the inclusion of more complicated functional forms as compared to the linear one (e.g., logistic regression) willrequire more complicated computations. We are therefore left with a choice betweengrossly inadequate models or computational infeasibility. Thus, we are left eitherwith being satisfied with a suboptimal linear model or with being unable (withinfeasible time) to find the optimal nonlinear model.The application of heuristics therefore leads to an imperfect state of nature whererational expectations remain but profit potential arises. As an immediate consequence, agents become opponents which are effectively competing to win. In orderto exploit these profit opportunities, agents develop strategies and distinguish amongthemselves. Thus, it is unlikely that there is any such thing as a typical agent. Furthermore, agents adapt their strategies in the hope of improving performance. Arthur[15] comments on this issue as follows:Economic agents, be they banks, consumers, firms or investors, continually adjust their market moves, buying decisions, prices, and forecasts to the situation these moves or decisionsor prices or forecasts together create. But unlike ions in a spin glass, which always react in asimple way to their local magnetic field, economic “elements” – human agents – react withstrategy and foresight by considering outcomes that might result as a consequence of behavior they might undertake. This adds a layer of complication to economics not experiencedin the natural sciences.
Apart from the inherent, and mostly severe, computational difficulties with the discovery of optimal models (and the verification of optimality), there are also technological difficulties standing in the way of the optimal use of optimal models. Thereason is that any computation employed by the model will be constrained by thespeed and memory constraints imposed by the current level of technology. Therefore, we would not even be able technically to identify the “very best” model ofmarket behavior. Since our analytic techniques are suboptimal and its use is constrained by the sophistication and power of the technology available to us, we cananticipate that the introduction of new generations of computers and other technology will continue to generate new arbitrage opportunities. Statistics not havingbeen “solved,” we can also anticipate that the introduction of new techniques andthe refinement of existing techniques, of statistical analysis, machine learning, etc.
5.3 Computational Difficulties with Efficiency
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will supply new means to take advantage of relevant information as it arises. Thissuggests that arbitrage opportunities exist today and will continue to exist tomorrow.None of which means that the arbitrage opportunities are constant: as old ones areutilized they diminish, while new ones arise.
5.3.2 Computational Complexity of ArbitrageComputational complexity of arbitrage has been studied for various frictional markets [95], including frictional markets in securities [96] and also foreign exchangemarkets [67]. A frictional foreign exchange market is characterized by bid-offerspreads, bound constraints, and integrality constraints. Bid-offer spreads reflect theexistence of transaction costs which may differ among traders. Bound and integrality constraints refer to limited fixed integer traded amounts in multiples of hundreds,thousands, or millions. For such markets, [67] first demonstrated that finding arbitrage opportunities is an undecideable problem:Theorem 5.2. There exists a fixed amount of information ε > 0 available to participating agents such that approximating the optimization version of the arbitrageproblem within a factor of nε is NP-hard.It is only in two special cases that the maximum value of arbitrage revenue can becomputed in polynomial time.Theorem 5.3. There are polynomial time algorithms for the optimization versionof the arbitrage problem if the number of currencies is a constant or the exchangegraph is star-shaped (i.e., there exist only exchanges between one “center” currencyand the other currencies in the market).The results have important consequences towards the fundamental arbitrage-freeassumption which is implied in many theories of finance. The efficient markets hypothesis, for instance, is based on the belief that if an arbitrage opportunity everexisted, it would disappear in an arbitrarily short period of time. However, if generating an arbitrage opportunity is a computationally hard problem, then this assumption may not hold in practice. In order for an agent to take advantage of an arbitrageopportunity, he needs to be equipped with all of the information about an exchangein order to generate arbitraging bids that are guaranteed to be in an optimal allocation. However, agents are often constrained by time and computer power. Thecomputational complexity of detecting arbitrage in the foreign exchange marketstherefore presents one important reason why arbitrageurs often have to use heuristics to support economic decisions.
Chapter 6
Conclusions
It is difficult for any model to describe adequately, and with a firm empirical basis, all features of modern economies that are relevant to determining exchange ratemovements. This reflects in part the difficulty of modeling international financialmarkets and capital flows. Economists have developed a variety of methods to estimate equilibrium exchange rates [271]. The methods differ considerably in theirconstruction and in their estimations of equilibrium values. In some sense, comparing the models is similar to comparing “apples and oranges” because they canradically differ in structure and can even use different measures of the real effective exchange rate. Often, they are attempting to measure entirely different kinds ofequilibrium. That does not mean the models do not provide useful information. Tothe contrary, they provide valuable insights, but one must recognize that they arelimited by the use of somewhat simplified structures, which are often necessary ifthey are to have a reasonable empirical underpinning.Complexity matters in economics and finance and in the special case of foreign exchange markets. It prevents easy solutions, efficient algorithms to solve theproblems and often even efficient algorithms to approximate optimal solutions. Butnevertheless, in real life, decisions have to be made. For instance, despite the complexity, traders in financial firms need to take bets on whether the exchange rateis going to rise or fall within a certain time interval. In addition, risk managers ofnonfinancial firms need to decide on when and how to hedge their foreign exchangeexposures. The computational complexity of a given problem is therefore an argument that helps agents to choose the right tools to support their decisions. Byconsidering above results on the general computational impossibility of detectingforeign exchange market inefficiencies, we know that it is suitable to use heuristics, which are able to handle problems of high dimensionality such as exchangerate forecasting or approximating the optimal solution to decision problems such asexchange rate hedging.
C. Ullrich, Forecasting and Hedging in the Foreign Exchange Markets, Lecture Notesin Economics and Mathematical Systems 623, DOI: 10.1007/978-3-642-00495-7 6,c Springer-Verlag Berlin Heidelberg 2009
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Chapter 7
Introduction
Over the past several decades, researchers and practitioners have used various forecasting methods to study foreign exchange market time series events, thus implicitlychallenging the concepts of informational and speculative efficiency. These forecasting methods largely stemmed from the fields of financial econometrics and machine learning. For example, the 1960s saw the development of a number of largemacroeconometric models purporting to describe the economy using hundreds ofmacroeconomic variables and equations. It was found that although complicatedlinear models can track the data very well over the historical period, they often perform poorly for out-of-sample forecasting [287]. This has often been interpreted thatthe explanatory power of exchange rate models is extremely poor. Nelson [310] discovered that univariate autoregressive moving average (ARMA) models with smallvalues for p and q produce more robust results than the big models. Box [58] developed the autoregressive integrated moving average (ARIMA) methodology forforecasting time series events. The basic idea of ARIMA modeling approaches is theassumption of linearity among the variables. However, there are many time seriesevents for which the assumption of linearity may not hold. Clearly, ARIMA models cannot be effectively used to capture and explain nonlinear relationships. WhenARIMA models are applied to processes that are nonlinear, forecasting errors oftenincrease greatly as the forecasting horizon becomes longer. To improve forecastingnonlinear time series events, researchers have developed alternative modeling approaches. These include nonlinear regression models, the bilinear model [171], thethreshold autoregressive model [392], and the autoregressive heteroskedastic modelby [115]. Although these methods have shown improvement over linear models forsome specific cases, they tend to be application specific, lack generality, and are often harder to implement [424].An alternative strategy is for the computer to attempt to learn the input/outputfunctionality from examples, which is generally referred to as supervised learning,a subdiscipline of the machine learning field of research. Machine learning models are rooted in artificial intelligence which distinguishes itself from economic andeconometric theory by making statistical inferences without any a priori assumptions about the data. Intelligent systems are therefore designed to automaticallyC. Ullrich, Forecasting and Hedging in the Foreign Exchange Markets, Lecture Notesin Economics and Mathematical Systems 623, DOI: 10.1007/978-3-642-00495-7 7,c Springer-Verlag Berlin Heidelberg 2009
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7 Introduction
detect patterns in data, despite complex, nonlinear behavior.1 If the patterns detected are significant, a system can expect to make predictions about new data coming from the same source. An intelligent system can thus acquire generalizationpower by learning something about the source generating the data. Intelligent systems are therefore data driven learning methodologies that seek to approximate optimal solutions of problems of high dimensionality. Since, in contrast, theory drivenapproaches give rise to precise specifications of the required algorithms for solvingsimplified models the search for patterns replaces the search for reasons.Artificial neural networks (ANN) are general-purpose self-learning systems thatgrew out of the cognitive and brain science disciplines for approximating the wayinformation is being processed [196]. Instead of analyzing the vertical relationshipbetween underlying cause and its derived effect, ANN learning models focus onhow effects reproduce themselves horizontally and what this reveals about theirinherent dynamics. Hence, instead of a linear cause–effect relation, ANN modelsare able to exploit nonlinear data relationships by moving the effect in a constantfeedback loop. During the last decade, the application of ANNs as supervised learning methods has exploded in a variety of areas [191, 379, 425]. Within the realmof financial forecasting, ANNs have been used to develop prediction algorithmsfor financial asset prices, such as technical trading rules for stocks and commodities [130, 221, 223, 365, 377, 416, 421]. The effectiveness of ANNs and their performance in comparison to traditional forecasting methods has also been a subject ofmany studies [91, 426]. ANNs have proven to be comprehensive and powerful formodeling nonlinear dependencies in financial markets [336], notably for exchangerates [48, 107, 234, 306]. However, ANN models have been criticized because oftheir black-box nature, excessive training times, danger of overfitting, and the largenumber of parameters required for training. As a result, deciding on the appropriate network involves much trial and error. These shortcomings paired with the logicthat complex real-world problems might require more elaborate solutions led tothe idea of combining ANNs with other technologies to hybrid and modular solutions [9]. For a survey of the application of ANN to forecasting problems in generalsee [424, 426].Support vector machines (SVM) [55, 405] are a new kind of supervised learning system which, based on the laws of statistical learning theory [405], maps theinput dataset via kernel into a high dimensional feature space in order to enable linear data classification and regression. SVM has proven to be a principled and verypowerful method that in the few years since its introduction has already outperformed many other systems in a variety of applications, such as text categorization[207], image processing [328, 333], hand-written digit recognition [247], and bioinformatic problems, for example, protein homology detection [203] and gene expression [64]. Subsequent applications in time series prediction [301] further indicatedthe potential that SVMs have with respect to the economic and financial audience. Inthe special case of predicting Australian foreign exchange rates, [219] showed thatmoving average-trained SVMs have advantages over an ANN based model which1 By patterns we understand any relations, regularities, or structure inherent in a given dataset.
7 Introduction
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was shown to have advantages over ARIMA models [217]. Kamruzzaman et al.[218] had a closer look at SVM regression and investigated how they perform withdifferent standard kernel functions. It was found that Gaussian radial basis and polynomial kernels appeared to be a better choice in forecasting the Australian foreignexchange market than linear or spline kernels. However, although Gaussian kernelsare adequate measures of similarity when the representation dimension of the spaceremains small, they fail to reach their goal in high dimensional spaces [131].This shortcoming has been addressed by [400–402]. They examined the suitability of SVMs, equipped with p-Gaussian, as well as several standard kernel functions and trained with exogenous financial market data in order to forecast EURexchange rate ups and downs. This research will be presented in detail in the following. The task is to examine the potential of SVM models to overcome foreignexchange market efficiency in its weak-form. We therefore analyze the ability ofSVMs to correctly classify daily EUR exchange rate return data. If an SVM modelis able to outperform the nave prediction by exploiting nonlinear patterns in dailyEUR/GBP, EUR/JPY, and EUR/USD exchange rate returns, market efficiency in itsweak form can be rejected. Indeed, it is more useful for traders and risk managers toforecast exchange rate fluctuations rather than their levels. To predict that the levelof the EUR/USD, for instance, is close to the level today is trivial. On the contrary,to determine if the market will rise or fall is much more interesting. Since SVMperformance depends to the most extent on choosing the right kernel, we empirically verify the use of customized p-Gaussians by comparing them with a range ofstandard kernels.The chapter is organized as follows: in the next section, we conduct statisticalanalyses of EUR/GBP, EUR/JPY, and EUR/USD time series. Chapter 9 outlines theprinciples of support vector classification including a description of the binary classification problem which is the theoretic problem under consideration, and the ideaof learning in feature space. Chapter 10 describes the details of the empirical study,such as the procedure for obtaining an explanatory input dataset (description model)as well as the particular SVM model and kernel functions used (forecasting model).Chapter 10 also provides the benchmarks and metrics used for model evaluationalong with the results.
Chapter 8
Statistical Analysis of Daily Exchange Rate Data
8.1 Time Series PredictabilityA time series {yt } is a discrete time continuous state process where the variable yis identified by the value that it takes at time t denoted yt . Time is taken at equallyspaced intervals from −∞ to +∞ and the finite sample size T of data on y is fort = 1, 2, . . . , T . Time series {yt } may emerge from deterministic and/or stochasticinfluences. For example, a time trend yt = t is a very simple deterministic timeseries. If {yt } is generated by a deterministic linear process, it has high predictability, and its future values can be forecasted very well from the past values. A basicstochastic time series is white noise, yt = εt , where εt is an independent and identically distributed (i.i.d.) variable with mean 0 and variance σ 2 for all t, writtenεt ∼ i.i.d.(0, σ 2 ). A special case is Gaussian white noise, where the εt are independent and normally distributed variables with mean 0 and variance σ 2 for all t,written εt ∼ NID(0, σ 2 ). A time series generated by a stochastic process has lowpredictability, and its past values provide only a statistical characterization of thefuture values. Predictability of a time series can therefore be considered as the signal strength of the deterministic component of the time series to the whole timeseries. Usually, a given time series is not simply deterministic or stochastic, butrather some combination of both:yt = α + β t + εt
(8.1)
8.2 Empirical AnalysisThe purpose of this section is to examine the statistical properties of dailyEUR/GBP, EUR/JPY, and EUR/USD exchange rate data from 1 January 1997to 31 August 2003. This is done for two reasons. First, according to above, timeseries analysis gives an understanding on the degree of randomness inhibited in thechosen time interval. The strategy is to build econometric models in order to extractC. Ullrich, Forecasting and Hedging in the Foreign Exchange Markets, Lecture Notesin Economics and Mathematical Systems 623, DOI: 10.1007/978-3-642-00495-7 8,c Springer-Verlag Berlin Heidelberg 2009
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statistical dependencies within a time series that may be based on linear and/ornonlinear relationships. If such dependencies are significant, then the time seriesis not totally random since it contains deterministic components. These may beimportant indicators for the predictability of {yt }. Second, analysis of {yt } is usefulfor building empirical time series models that will serve as benchmarks for ourSVM models. The advantage of our data-driven approach is that it allows variablesto speak for themselves, without the confines of economic or financial theories.The investigation is based on publicly available London daily closing prices asobtained from http://www.oanda.com/. When examining daily data, closing pricesare more relevant than opening prices since they represent the matching of supplyand demand at the end of the trading day. The series for the period from 1997 to1998 were constructed by using the fixed EUR/DEM conversion rate agreed in1998 (1 EUR = 1.95583 DEM), combined with the GBP/DEM, JPY/DEM, andUSD/DEM daily market rates. It is also important to note that we do not include theperiod from 1 September 2003 to 31 December 2004 in our analysis since it will beneeded for out-of-sample forecasting and is not known beforehand.Plots of the three time series are shown in Figs. 8.1–8.3 which demonstrate thedevaluation of the EUR since its introduction in 1998 until the end of 2000.The devaluation had been followed by a rapid upward trend against the threeforeign currencies reaching to the end of the specified time window. A comparison of the magnitude of these upward and downward movements is visualized byFig. 8.4, which shows that among the three exchange rates EUR/JPY has been themost volatile one, and EUR/GBP the least volatile one. The vertical bar drawn at29 August 2003 divides the series into an insample period (left of the bar) and andout-of-sample period (right of the bar).
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