Do Financial Markets Exhibit Chaotic Behavior? Evidence from BIST
Knowing of the chaos theory by the economists has caused the understanding of the difficulties of the balance in economy. The applications of the chaos theory related to economy have aimed to overcome these difficulties. Chaotic deterministic models with sensitive dependence on initial conditions provide a powerful tool in understanding the apparently random movements in financial data. The dynamic systems are analyzed by using linear and/or nonlinear methods in the previous studies. Although the linear methods used for stable linear systems, generally fails at the nonlinear analysis, however, they give intuition about the problem. Due to a nonlinear variable in the difference equations describing the dynamic systems, unpredictable dynamics may occur. The chaos theory or nonlinear analysis methods are used to examine such dynamics systems. The chaos that expresses an irregular condition can be characterized by "sensitive dependence on initial conditions" We employ four tests, viz. the BDS test on raw data, the BDS test on pre filtered data, Correlation Dimension test and the Brock's Residual test. The financial markets considered are the stock market, the foreign exchange market. The results from these tests provide very weak evidence for the presence of chaos in Turkish financial markets. BIST, Exchange Rate and Gold Prices. In this study, the methods for the chaotic analysis of the time series, obtained based on the discrete or continuous measurements of a variable are investigated. The chaotic analysis methods have been applied on the time series of various systems. Chaos, is the way a deterministic system can behave in a disordered manner. For example sometimes chaotic situations can be seen in the flow of a liquid passing from a smooth pipe. Once the flow rate of the fluid passes a certain value, eddies are formed and the Newton laws lose their validity. Namely now the flow is chaotic. Although J. Henri Poincare is accepted as the father of chaos concept and theory, the most important contribution for the theory was made by Edward Lorenz who became a meteorology professor in M.I.T. in 1960. Lorenz entered data to his computer in order to prepare a simple weather forecast report and as a result showed the temperature values he found in graphics. Lorenz, restarted the function by increasing the randomly selected temperature values in small amounts that even a very sensitive thermometer cannot detect and found out that totally different functions were formed even though he expected functions would not create any difference in graphics. It was observed that the decrease and increase in graphics in long term caused a pattern like a butterfly. Figure 1: Lorenz Attractions The comment Lorenz made from this result is as follows: As a result of its chaotic behavior a correct and reliable long term weather forecast cannot pass a certain time, for this reason, in a system that shows non periodical behavioristic characteristics is not possible. Lorenz, tried to explain "chaos theory" by putting forth two main features of chaotic systems that look like disordered but have an internal order. After this study of Lorenz, two main characteristic features of chaotic systems that chaos theory tries to explain, looking like disorder from outside but having an internal order are clearly set forth. The technical features of chaotic processes are as follows; 1. Dependency on Beginning Situation If the beginning situation and equation of a deterministic system is known, the subsequent behavior of the system can be determined. In chaotic systems, in order to determine the development of the system throughout the time, it is necessary to know the beginning values with an infinite precision. Since chaotic systems are not linear, the error shall increase exponentially in time. 94 INTERNATIONAL CONFERENCE ON EURASIAN ECONOMIES 2016 In theory, essentially everything occurs according to time, for example; pollen production, population increase, economical changes, world ice mass etc. can be chaotic. Chaotic studies can be seen in fields like physics, chemistry, mathematics, communication, biology, physiology, sociology, economics, history, ecology, astronomy, hydraulics, atmospheric, international relations, solar systems, engineering. Figure 2: Chaotic Corelogram The series in a chaotic structure shows a sensitive dependency on beginning conditions. For this reason, the relation between past and feature values is very weak. As can be seen in the Figure, according to this fact, as the delay increases the series that have autocorrelation values converging to zero are the series that have very weak relation between past and future. Consequently, it can be said that this kind of series have a chaotic structure. 2. The Situation of Being Random. In a more clear statement: In fact many events in the world have a chaotic structure (everything is not linear like the way it is thought to us) like the formation of a snowball, the increase of a cigarette smoke, the development of the stems of a tree, the flight of bevy of birds, the movements of waves in the sea. So these behavior patterns conform with the laws of new science. It is claimed that in the process that cannot be set into the concept of causality of classical science and that can be named as dynamic systems, chaos would destroy determinism and brought a new horizon to the science that stuck. Dynamic Systems and Chaos Theory, is a theory that points out the fact that minor fluctuations made in complex systems which are not linear can lead to big changes in the future. It is a theory that says prediction of events that might happen in the near future is easy, however prediction of the events that might happen in long term can be hard. The most common example that can be given for chaos theory is butterfly affect. The basic propositions of chaos theory can be listed as follows; a. Order creates disorder. b. There is an order in disorder. c. Order comes from disorder. d. In the new system reconciliation and dependency show themselves very shortly after the change. e. The new order reached, develops into a new direction through a process that was organized by itself. 3. Fractal Geometry Mandelbrot defines the "fractal" concept as geometrical shapes that the parts and the total Show similar structural characteristics. [Tosun, 2006] From this point of view, fractal geometry, with its simplest form, can be defined as the geometry of Chaos. The shapes that have fractured dimensions, self similar structures that give the same shape in every scale, the infinite embranchments and the resemblance of these embranchments with the main structure form the base fractal geometry. It is tried to explain the fractal structure and real nature structures with rules. These self recurring structures cannot be understood with Oklid geometry. However, fractal geometry can solve the mystery of these shapes. Trying to simplify these structures and explain complex systems with this simplicity that is reached establish the reason for the research of fractal structures [Tosun, 2006] The basic and simple rule of forming Fractal Shapes, is continuously repeating a rule that is accepted. Namely, if a rule is taken as fixed and continuously applied a fractal shape can be obtained. The fractal structure of WaclawSierpinski that can be seen in Figure 2 can be given as an example for this. Sierpinski forms a new triangle from the median points in an equilateral triangle. By renewing this for every triangle formed here, the triangle shapes in Figure 2 are created [Peters, 1996] SESSION 4A: Finance 95 Figure 3: Sierpinskn's Fraktal Sample Fractal time series for several grounds arises if the display time series in self similar structures. Fractal time series time series are distributed substantially random and deterministic all the features they do not show in certain periods. Therefore, similar to the structure forms the main structure is made by chance in time series. A series of linear fractal dimension is also quite close to 1. Therefore, with high sharpness, the rise and fall of this size is more structure and 2 are both approaching a fractional number. This situation can be explained why the application of chaos theory to Tonis Vaga the financial markets. According to the market in the risks and rewards Vaga has a non volatile linear structure [Vaga, 1991] .Therefore fractal dimension and structure of the system is important to make a good analysis. Because this way, high returns at low risk according to market certain conditions it is possible to achieve. Similarly, investors such as Warren Buffett and George Soros, have declared in a similar way to achieve higher profits than the average over the long term [Tosun, 2006] Attractions briefly are the pictures of the dynamic system. Accordingly, a dynamic movement is analyzed fractal geometry, work is greatly facilitated. For example, while expressing that point strange attractors stood unchanged by time the system is attractive complex mostly shaped in interesting building. Results have chaotic nature of the system; unpredictable function is a structure constituting the time series. This is equivalent to the function that vary randomly increase and decrease. Differently from each other in phase space for each different chaotic fluctuation in the system it consists of different trailers in other words. A major attraction of the chaos arising in the framework of the historical development work can be summarized as follows [Kasap ve Kurt, 2011] Lorenz Attraction Rossler Attraction Henon Attraction Here in the conceptual point of view trailers contain two important points. These are attractive to himself and his form. According to the behavior of a system that is understood by fully engaged. Therefore, it is called a tow truck, but it can be described by fractal geometry attracted to strange attractors. The concept of strange attractors of chaos indispensable wor