Stochastic volatility (SF) is offered as a more complex alternative to the Generalized Auto-Regressive Conditional Hydroelectricity (GARTH) framework in modeling dementedness (conditional) return volatility. The main difference is that in SF, conditional volatility is only partially determined by the information up to time t. An empirical test of this model is carried out using the Kalmia filter and results summarized in a separate MS Excel file. 1 Chaos and Non-Linear Dynamics 1. Words like random and chaos are often used to represent activity in a closed or limited system. The decay of radioactive atoms is, for example, random when looking at a few atoms. But it is often the case that in larger aggregates, these apparently random actions ‘average’ into a very precisely predictable outcome. And so, while at the smallest scale, we cannot use cause and effect reasoning, on a larger sale, we can do that with great reliability.

The case of the formation of the solar system is an excellent example of a situation where, while there are many dimensions of random actions: random particle temperatures, random particle motion – we can treat the entire system as a very stable and predictable distribution of mass and energy. Theory. This is not, as one might assume, about the behavior of random activities, but the recognition that there are more sophisticated predictable patterns that we ad once assumed to be random: things like turbulence and so forth.

Along with the ‘recognition’ is a comprehensive mathematical structure to deal with these patterns. The Random Walk Hypothesis The Random Walk Hypothesis states that stock market prices evolve according to a random walk and that endeavourers to predict future movements will be fruitless. There are two forms: Narrow Version – It asserts that the movements of a stock or the market as a whole cannot be predicted from past behavior (Wallach, 1968). This would suggest that an investor cannot beat the market.

Broad Version – It expands on the narrow version, claiming that “in a hallucinogenic market, all known information has already been discounted” (Wallach, 1968). Similar to the Efficient Market Hypothesis, if new information becomes available, any past information becomes irrelevant; as it has already caused any market movements that it was capable of. The broad version of the Random Walk Hypothesis assumes that all information is reflected immediately in the prices of stocks. There are two flaws with the fundamental assumptions in this theory. ) The assumption that all investors have the same access to the same information is rely not the case. Reliable and detailed information is usually obtained from a paid service, or through the employment of analysts. In addition, all investors do not act upon the information at the same time may not hold. Investors receiving information after it becomes public may still act on it and this would affect prices. B) The effect of information obtained at time t may not be fully understood without information at time t + 1. Thus, the past information can still have an effect on the market prices.

Random vs.. Unpredictable The best way to distinguish these two concepts is by illustration of their application ) Unpredictable: The weather is unpredictable meaning that it would not be possible to gauge at this moment what the weather will be in six hours’ time. Taking into account more information, like for a meteorologist, the odds of being correct would increase but it is not guaranteed to be correct. Also, as we extend the timeline, the likelihood of being correct would decrease. B) Random: The weather is not random however.

If, at midday today, it is seven degrees and raining, it is fairly certain that it will not be twenty four degrees and sunny six hours later. This is because there are deterministic relationships at play here. There are many variables that affect the weather, many of which may not be taken into consideration when making predictions about future weather conditions. The Distinction: Unpredictable events or systems can be described as those that we are unable to forecast, or are only able to partially forecast, due to a lack of information whereas random systems are systems in which no deterministic relationship exists.

Chaos Theory Chaos is a non-linear deterministic process, which looks random (Whish, 1991). There are several characteristics of chaos which are listed below. 1) Sensitive Dependence on Initial Conditions The chaotic nature of a system’s evolution arises from this. In a standard statistically modeled system, one expects that, if the independent variable is altered by some proportion, then there will be a similar or predictable change in the dependent variable. Thus they facilitate prediction of events.

However, in a chaotic system, an infinitesimally small change in the initial conditions can cause the model to evolve in a completely different fashion. This phenomenon was discovered by a meteorologist, Edward Lorenz, while he was running a weather-stripping model in 1961, he wished to re-examine a certain portion of the exults and, in the interest of expedience, he used the data from a read-out which he had obtained previously for the beginning of that sequence, rather than re-running the entire model.

The system evolved in a completely different fashion from his earlier models. The reason, he discovered, was that during the initial run, the computer had used figures to six decimal places but he had only printed out figures to three decimal places. A change of Just over a thousandth of a significant figure completely altered the model. This has 2 very obvious implications for economics or finance systems if they are indeed found o be chaotic. 2) Apparent Randomness Disguising Deterministic Relationships The best explanation of this concept is through example of a roulette wheel.

The outcome is believed to be random and certainly seems that way on first observation. However, the result has several influencing factors: the speed and number of rotations of the uses to throw the ball and others. So what seems random is in fact deterministic. 3) Strange Attractors An economic or financial system that is chaotic can be modeled. A crucial aspect of the theory is the point at which order arises from disorder. While the positions of ATA at a specific time cannot necessarily be predicted, quite accurate models of the overall behavior of the system can be created.

An attractor is the equilibrium level of a system, but should not be confused with an econometric equilibrium, which is a narrow form of an attractor. An attractor is the level or value a system attempts to regain after external effects have abated (Peters, 1991). A strange (or chaotic) attractor, with a non-integer dimension, is present in a system that tends towards a set of possible values. The possible values are infinite in number but limited in range. Chaotic attractors are not periodic, I. . They do not have any repetition regardless of the length of the timeline (Peters, 1991). ) Fractal Dimension The most basic way to understand fractal dimension is as a measure of how chaotic a system is; the closer to the higher integer the dimension is between, the more chaotic the system. On a more complex scale, fractal dimension is a statistical quality, giving a measure of how completely a fractal fills space. Mathematical Abstraction A tent map is a piecewise linear, one-dimensional map on the interval [0,1] exhibiting chaotic dynamics: The tent map is an example of a chaotic map or system. Piecewise functions are those with different sub-functions for different ranges, say x = O and x > 0. Analyzing chaotic behavior is important because it has potential to explain fluctuations in the economy and financial markets, which appear to be random, but may actually not follow a random process. Hence, there is need to test for the presence of chaos. We are interested in low complexity chaos. Detecting chaos needs infinite amounts of data, otherwise deterministic chaos may not be distinguished from randomness. Low complexity chaotic processes have short term predictability, UT this is only executed through non-linear models.

Testing for low-complexity chaos: Method 1; Observing Chaotic Maps One way to identify chaos is through observing chaotic maps, as they do not fill up’ enough space in high dimension. Plotted on 1 dimension, a chaotic process (say a dimensions starting from 2, the chaotic data leave large ‘holes’, unlike the random data. More complex chaotic data need analysis using higher dimensions. Using the chaos map graphical method for this is not practical, hence an alternative method was developed; correlation dimension. This was by Grabbers and Procaine (1983).

Method 2; Correlation Dimension It is a 4-step method. Step 1: Remove autocorrelation, if present. It can affect some chaos tests. (Typical procedure: Run an auto regression to filter the data, determining lag length through either Awake or Schwartz information criterion). Step 2: Form n histories from the filtered data, denoted as follows: An n history is a point in n-dimensional space. N is called the embedding dimension. Step 3: Calculate the correlation integral, CNN (E) The correlation integral is the mean probability that the states at two different times are close, that is Step 4: Calculate the slope of the graph of log CNN (E) versus log E for small values of E. More precisely, we are calculating the quantity: The correlation dimension, v, is a measure of the dimensionality of the space occupied by a set of random points, or how much space is filled up’ by a string of data. If van does not increase with n, the data is consistent with chaotic behavior. In fact, the Grabbers-procaine correlation dimension is defined as: This 4-step process is straight-forward in principle, but a number of issues arise in practical applications on financial and economic data which a listed below. One problem is lack of sufficient data points. Scientists use 100,000 or more data points, even when detecting low dimensional chaotic systems. This is because one cannot use a finite amount of data to test for an infinite correlation dimension. However, financial economists have significantly fewer points, with the largest having about 2000 observations. 2. Another problem arising from the use of small data sets is that van is biased downwards, hence biasing results in favor of finding chaos, even if there is none. 3.

Also, the correlation dimension method is a graphical one, hence cannot quantify the accuracy of the correlation dimension. A statistical test is needed for this, and Brock, Etcher and Chainman (1987) proposed one. 1. 5. 1 The BEDS statistic If *x t ; t = 1, , T+ is a random sample of lid observations, then: Win,t (E) is the BEDS Statistic. It has a limiting standard normal distribution. Is an estimate of the asymptotic standard error of: does not imply lid. Since the BEDS statistic is a new procedure, it is useful to study its finite sample distribution. This can be done through use of Monte Carlo simulations. 1. Empirical Examples Demonstrating Random Market Hypothesis Failures ) Real Exchange Rates Choc (1999) examines whether the Random Walk Hypothesis is observed for real exchange rates. He uses the log-differenced US real monthly exchange rates and certain other major currencies. In his paper he sets out a null hypothesis that a random walk is observed. The alternative hypothesis is that there is serial correlation present. Several tests were run for each currency, and the results were mixed. In his conclusions, he states that “for the full sample, the null is rejected at conventional significance levels for Japan, Switzerland and Britain”.

Here the Random Walk Hypothesis is rejected, with the possibility arising of the presence of a non-linear system. B) A Chaotic Attractor for the S&P 500 Peters (1991) examined the S&P 500 index in order to ascertain whether or not there was a chaotic attractor present. The dynamic observable used was the log-linear deflated S&P 500. The results, again, help to refute the Random Walk Hypothesis. Firstly, he found that the fractal dimension of the trended S&P 500 is approximately 2. 33 (incidentally, this is the same fractal dimension as cauliflower).

If the data were completely random, the dimension would have been an integer. Random data, as stated above, fills any space available to it. He states that the attractor is “chaotic”, with a positive Lollipop exponent which measures the loss in predictive power experienced by non-linear systems over time, by measuring the divergence of nearby trajectories over time. A positive exponent indicates expansion while a negative one indicates contraction. The positive Lollipop exponent indicates that the system is subject to sensitive dependence on initial conditions.

When using Lollipop exponents as a measure of divergence from initial trajectories, it is moon to simply use the largest one, Maximal Lollipop exponent, MEL. C) Returns on T-Bill Rates Larkin (1991) rightly asserts that if the past interest rates affect the future evolution of interest rates, then the Random Market Hypothesis is false. If this turns out to be the case, then a genuine and Justified use of investment tools and strategies can be generate a profit. According to Larkin (1991), both fundamentals and technical analysis can be used to determine future interest rates.

Moreover, the relationships are nonlinear in nature. This also gives further credence to the idea that markets are Hattie in nature. He concludes that there is a non-linear structure in the series for T-bill rates and that this non-linear structure, while not explicitly guaranteeing mathematical chaos, does allow for the possibility of it arising, under certain market conditions. Conclusions and Remarks Three important conclusions are drawn. First, the Random Walk Hypothesis is not correct in its narrow or broad form as is shown by the empirical data.

Second, it is possible that chaos theory could be used to describe some markets but is not to be adopted as the complete alternative. Third, where the situation arises that a system s not chaotic, it may very well be nonlinear, and so still requires that we do not assume randomness. Sorts noted that there seems to be little correlation between fundamentals and stock prices. Chaos Theory urges an approach that is based entirely on market prices. It notes that there is no distinct proportional relationship between inputs into the market (such as earnings) and the output (the price).

The chaos thinkers conclude that markets have to be studied as chaotic, nonlinear systems. 2 Stochastic Volatility 2. 1 Volatility describes the variability of a financial time series, that is, the magnitude and peed of the time series’ fluctuations. It conveys the uncertainty in which financial decision making is accomplished. Volatility is often expressed as the standard deviation of asset returns and, more generally, when returns are assumed to be non- normal, as the scale of the return distribution. 2 In financial modeling, volatility is a forward-looking concept.

It is the variance of the yet unrealized asset return conditional on all relevant, available information. Denote by kit-l the set of information available up to time t – 1. This information set includes, or example, past asset returns and information about past trading volume. The volatility at time t is given by TTL-l = vary(art Kit-l )= E ((art – putt-l ) 1Volatility clustering is an important phenomenon that is characteristic of the dynamics of asset returns. Mandelbrot (1963) was one of the first to note that “large changes [in asset prices] tend to be followed by large changes Ђ??of either sign-??and small changes tend to be followed by small changes. ” In other words, volatility clustering describes the tendency of asset returns to alternate between periods of high volatility and low volatility. The periods of high volatility see large magnitudes of asset returns (both positive and negative), while in periods of low volatility the market is “calm” and returns do not fluctuate much.

Clearly, this stylized fact about financial time series contradicts the efficient market hypothesis. In an efficient market, investors would react immediately to the arrival of new information so that its effect is quickly dissipated; changes in asset returns are independent through time. Two other empirically observed features of returns are that returns exhibit keenness and heavier tails (higher kurtosis) than suggested by the normal distribution and that volatility displays an asymmetric behavior to positive and negative return shocks-??it tends to be higher when the market falls than when it rises.

Volatility models attempt to explain these stylized facts about asset returns. A stochastic process is assumed for the dynamics of the normal mean/variance. When only the normal variance is varied, the resulting distribution is called a scale mixture of normal, whereas when both the mean and variance are varied, the result is a location-scale mixture of normal. 8 Since the volatility (and the expected return) today depends on the volatility (and the expected return) yesterday, it is clear that today’s asset return is not independent from yesterdays asset return. Therefore, we can write an expression describing the evolution of returns through time incorporating the time-varying conditional volatility. In general, although asset returns can be thought of as evolving in a mutinous fashion, the retransmitting process is often modeled in the discrete time domain.