Essay on Chaos Theory and Neural Networks

Introduction

This essay aims at reviewing the literature on and discussing two important new theoretical concepts recently proposed for investment analysis and portfolio management in capital markets. The first concept deals with the non-linear nature of actual security returns distribution which not only behaves lognormally but their variance distribution also has a fat tail and high peak, or leptokurtosis. This behavior of security returns contradicts the random walk hypothesis of efficient capital markets which assumes symmetric normality and finite variance. Actual security returns somehow follow other kinds of cross-sectional distributions called fractal distributions whose time-series are characterized by deterministic chaos. The second concept represents the attempts to model non-linearity in price and returns movements through the use of neural networks which are the high-speed artificial intelligence system capable of processing a large amount of market information simultaneously. Not only are these networks able to simulate complicated non-linear relationships among market factors but also learn to mimic the actual market behavior in order to predict the eventual results. Research in this area will enable financial economists to conduct capital market experiments in which their new financial/investment models can be tested without relying on the empirical data, which could be contaminated by undesirable factors.

Part one concentrates on the first concept which includes the discussion of fractal markets hypothesis, the chaos theory, and their conclusion. Neural networks concepts and their applications in finance and investment are discussed in their totality in Part two.

Part One
Fractal Markets Hypothesis and The Chaos Theory

Fractal Markets Hypothesis Back to Top

The traditional models in modern investment theory are based on the prescriptive assumptions that capital markets are both allocatively and informationally efficient following the welfare economics concept of Pareto optimum and efficient markets hypothesis (EMH), and that behavior of security prices and returns is governed only by the two-fund separation of risk-free and risky but equilibrium market portfolios. In contrast, the fractal markets hypothesis (FMH) is based on a premise different from the EMH that capital markets exist and are functioning for liquidity reasons, not for achieving efficiency.

According to Peters (1991) who synthesizes theories relevant to the non-linear dynamic paradigms, FMH is designed to demonstrate that actual capital markets follow the biased random walk along a long-memory process, i.e., security prices are partially deterministic and not totally stochastic. More specifically, the short-term price behavior is characterized by a non-linear stochastic process, while the long-term behavior follows a non-linear chaotic process. As a result, the traditional definition of risk is no longer limited to either total risk (s) or market risk (b) which are finite, but also include the velocity of price changes. As the assumption of finite variance is relaxed, the analysis based on normal distribution is no longer insightful. Since the time-series of security prices are not totally random and their returns variance are not finite, it can be concluded that actual capital markets are fractal instead of efficient as being perceived for the last forty years through the eyes of modern investment theorists.

Fractal Dimensions and The Biased Random Walks Back to Top

The fractal dimensions are non-Euclidean, non-integer or fractional spaces with gaps and holes within them. The fractal object or time series fills its space unevenly and non-randomly because its parts are correlated and interdependent. The volatility of stock returns can be measured by the standard deviation only if the dispersion of returns is random, or independently and identically distributed (IID). Since the empirical evidence has shown that the distribution of stock returns is not IID, it can be concluded that changes in stock returns are somehow serially correlated which falls into a time-series category of fractal dimensions.

In the actual capital markets, most investors wait for more information to confirm their beliefs and do not react until a trend is clearly established. However, the amount of information needed for trend-setting varies unevenly. This accumulation of information causes investors to have biased expectations about the price trend, thereby resulting in a biased random walk, or fractal time series. The biased random walk patterns were observed by Hurst (1951) and applied to stock prices time series by Mandelbrot (1964, 1966, 1971).

The Hurst Exponent

Hurst was interested in measuring the range of fluctuation of the reservoir around its average level over time. The range of this fluctuation would change in response to the passage of time, i.e., the time-varying variance or volatility. If the series were random, the volatility would increase with the square root of time (T^½). This volatility is standardized or rescaled by divided by the standard deviation of the observations, resulting in the so-called rescaled range analysis The rescaled range analysis (R/S analysis) is given by:

R/S = (aN)^H

where: R/S = rescaled range; a = constant; N = number of observations; H = Hurst exponent

The Hurst exponent (H) should be equal to 0.5 for the series to be a random walk. If H differs from 0.5, it means that the probability distribution is not normally distributed. If 0.5<H<1, then the series is fractal. Fractal time series behave differently from random walk. Most of natural phenomena follow a biased random walk or a trend with noise. The distribution of the biased random walks is called the fractal distributions.

Fractal Distributions

Fractal distributions are also known as Pareto, Pareto-Levy, or Stable Paretian distributions. Levy (1925) is the first to derive the properties of this distribution. His derivation was based upon Pareto's (1897) work on welfare economics. Pareto finds that income has an approximately lognormal distribution, except for 3% of the upper-class income-earners. For this 3% portion, income follows an inverse-power law, which results in a fat tail. Pareto attributes this fat tail to the fact that the upper-income group can lever its wealth more effectively than average individuals to create more wealth and achieve even higher income level. The feedback loops within this top 3% group also causes the tail to be fatter.

Leptokurtic Distributions

The characteristic shape of fractal distributions is leptokurtosis and given by:
log f(t) = idt - l|t|a[1+b(t/|t|)tan(ap/2)]

where: d = location parameter of the mean.; l = scale parameter to adjust the difference between daily and weekly data.; b = skewness ranging from -1 to +1; if b = 0, it is symmetric.; a = peakedness and fatness of the tail ranging from 0 to 2 inclusive.

If a = 2, b = 0, l = 1, and d = 1, the equation becomes the normal distribution function.

The difference between EMH and FMH is that the former assumes that a must be equal to 2 while the latter says that a can range between 1 and 2. However, changing the value of a changes the characteristics of the time series substantially. Pareto distributions are fractal because they are statistically self-similar with respect to time. If a distribution of daily prices has a mean µ and a = a, the five-day price distribution would have a mean of 5µ and still have a = a. Adjusting for time scale, the time-series' probability distribution still has the same shape and is said to be scale invariant. If a = 2, and the distribution is a special case of the fractal distributions. When a ¹ 2, the characteristics of the distribution change.

The distribution of stock prices is assumed in modern investment theory to be continuous. However, in a fractal distribution, large changes occur through a small number of large changes. Large price changes can be discontinuous and abrupt. A fractal distribution for the stock market would explain why the Great Depression and the stock market crash of October 1987 had occurred. In those markets, lack of liquidity caused abrupt and discontinuous pricing.

Infinite Variance Distributions

Mandelbrot (1964,1966) argues against finite variance distributions using advanced mathematics. He also employs the R/S analysis to supplement his argument but lacks empirical proof to support. The absence of empirical evidence of infinite variance had brought Mandelbrot's proposition for FMH to a halt. The concept of infinite variance distributions ceases its development in modern finance theory because their implications are mathematically complicated and untidy. EMH is neater and easier for academicians to conceptualize and model. Defining risk only as the variance of stock returns simplifies both theoretical works and empirical studies. However, recent empirical puzzles have pointed out that variance could not capture all of the market risks. Market anomalies become the evidence that the distribution of stock returns is not normal such as the calendar effect, the small-stock effect, the high B/P effect, etc. Trading strategies based on these anomalies earn abnormal returns without increasing risk.

Fractal Time Series

Fractal time series are characterized by the long-memory processes. They possess cycles and trends and result in a non-linear dynamic system, or deterministic chaos. Information is not immediately reflected in stock prices, but is shown as a bias in returns. This bias stays indefinitely, although the system can lose memory of initial conditions. Each increment of time is correlated with all increments that follow it. All six-month periods are correlated with all subsequent six-month periods. All two-year periods are correlated with all subsequent two-year periods. Information biases the system until unexpected economic events or shocks change them. This biased random walk describes capital markets better than the IID random walk does.

Non-linear Dynamic Systems Back to Top

Non-linear dynamic systems is based on the transition from stability to turbulence. Yet, this common event of transition form a stable state to a turbulent state cannot be modeled by standard Newtonian physics. Newtonian physics assumes that the cause-effect relationships between variables are linear, all systems seek an equilibrium state, and nature is orderly. However, actual nature witnesses many non-linear relationships. Initial conditions could affect eventual results a great deal. This effect is referred to as sensitive dependence on initial conditions and becomes the important characteristic of dynamic systems. A dynamical system is inherently unpredictable in the long run. Two features of dynamic systems should be noted. First, they are feedback systems which repeat and transform themselves infinitely. Second, there exists a certain critical level at which the systems cannot tolerate and therefore collapse. Hence, the dynamic system is a non-linear feedback system with critical level. In essence, the determinants of chaotic dynamic systems include sensitive dependence on initial conditions, critical levels, and fractal dimensions.

Conclusion on FMH and the Chaos Theory Back to Top

The review and discussion of Peters's literature provide insights about how to bring in highly advanced mathematical concepts of fractal distributions, infinite variances, and chaotic non-linear dynamic systems into the modeling and empirical tests of capital markets. This leads to the reconceptualization of finance and investment theory which provides added benefit and offers new tools to the current views of modern paradigms. Hence, FMH and chaos theory should be seen as supporting EMH rather than opposing it in a more positively-oriented way.

Part Two
Neural Networks System and Financial Applications

Fundamentals of Neural Networks Back to Top

Neural networks (NN) have tremendous impacts on today's information processing industries including financial services. They are the artificial intelligence-based simulation models that emulate a biological neural network system such as the human brain and capable of parallel processing, quick retrieval and feedback of large quantity of data, and learning to recognize historical patterns of information. These capabilities create a breakthrough opportunity in the analysis of non-linear behavior of stock prices and returns consistent with the theoretical approach of fractal markets hypothesis and the chaotic dynamic systems of fractal time-series.

Medsker, Turban, and Trippi (1993) characterize NN's capabilities to perform different non-linear emulations including processing, summation, transformation, and learning functions. The system functions by recognizing the inputs' weights relative to their importance, transforming their non-linear into linear relationships, repeatedly adjusting these weights, comparing the results with the objective functions, and producing the outputs. Other unique characteristics of NN are fault tolerance, generalizability, and adaptability. Fault tolerance means that a damage to a few links in NN will not halt the system. Generalization is the ability to generate a reasonable response from an unstructured or unknown inputs, while adaptability provides the NN with the ability to learn in new environments. Yet, high costs, limited scope, lack of supporting facilities, lengthy training-learning period, and lack of guarantee for an optimal solution from repeated processing of the same inputs become the barriers for the applications of NN.

The most remarkable feature of the NN is their training-learning function. NN can be trained to learn from the algorithms the analysts have constructed. There are two kinds of algorithm based on input format from which the NN can learn: 1) the discrete-value input, and 2) the continuous-value input. The learning patterns can be either supervised or unsupervised depending on the algorithms and the system's architectures. The NN will learn through several training iterations to adjust the weights of the inputs to produce the desirable outputs. This capability is proven to be very useful in the study of stock price movements because NN can be trained to recognize, transform, and make adjustment to the weights of interrelated factors in the non-linear stock returns generating functions.

Neural Networks Application in Stock Returns Prediction Back to Top

Shifting away from standard econometric techniques which are based on linear modeling, Lo and MacKinlay (1988) attempt to use NN technology to investigate the complex non-linear relationships between stock prices and various economic variables such as interest rates, unemployment rates, stock market cycles and trading volume, macroeconomic policies and activities, and individual- and firm-specific characteristics. Beside non-linear modeling, NN are also trained to uncover systematic patterns in analysts' earnings estimates and to develop a performance evaluation method for traders. It is expected from applying NN to model stock price behavior that the sources of returns predictability and the patterns of influence of aggregate economic activity on the capital markets can be determined in the context of the tradeoffs between risk and expected returns. Once these expectations are realized, the efficient markets hypothesis (EMH) based on the random-walk model can be refuted with higher confidence.

Neural Networks for Other Financial Applications Back to Top

Based on the seminal works of Lo and MacKinlay (1986, 1988), there have been other NN- and non-NN-based computational tools developed to model the non-linear time-series relationships among various economic and financial variables. Among them, four prominent ones are worth mentioning: 1) the artificial life model at Santa Fe Institute, 2) the non-traditional financial analysis model at MIT, 3) the modular neural networks at Fujitsu and Nikko Securities, and 4) the hybrid neural network forecaster at Chase Manhattan Bank.

Artificial Life Model

Artificial life model (ALM) is developed by Langton (1989) for simulation project called Swarm Project experimented by the Santa Fe Institute (SFI) jointly with the Los Alamos National Laboratory. It is based on the assumption that most of the complex systems that occur in nature share a common architecture and have the transition patterns called swarm embedded in such an architecture. Swarm project is being used to model social behavior in the economic systems, or what SFI refers to as community intelligence. In terms of capital markets analysis, the focus is on the ability to emulate market as a swarm of economic agents. The objective is to bring behavioral aspects of these economic agents into the ALM by creating a flexible and powerful framework to simulate the complex behavior arising in environments consisting of many interactive elements. For transitory patterns or time-series studies, the ALM is used to examine systems without actually constructing them since it can learn to project the future realities. The simulation results are realistic because they account for the crucial factors that characterize an actual environment.

Non-Traditional Financial Analysis Model

Non-traditional financial analysis (NTFA) has been carried out at MIT using the NN-based simulators and focusing on the time-series studies of security prices. The objective is to analyze capital markets data in order to establish which factors influence a given phenomenon in such markets. The main assumption of NTFA is that capital markets do not behave according to the models implied by the modern investment theory. Its basic hypothesis is that while expected returns are lognormally distributed, returns volatility exhibits non-linear dependencies to initial conditions and autocorrelated errors with long-term memory. Among the modeling input assumptions for NN simulators are leptokurtic distributions, infinite variance distributions, cross-correlation of actual returns, and conditional volatilities. The findings based on NTFA indicate that 1) effective risk management tools must account for volatility dynamics using continuously updated feedback and associated forecasts, 2) volatility models for forecasting should exploit chaotic features of price movements, and 3) dynamic non-linear semi-stochastic models provide the better forecasts than static linear stochastic models. Of importance to them is the acknowledgment of asymmetric nature of market information and price change discontinuities.

Modular Neural Networks

Modular neural networks (MNN) is jointly developed by two Japanese firms - Fujitsu Computer and Nikko Securities - to predict stock price movements in the Tokyo Stock Exchange and to verify whether it provides better results than other non-NN models. Within the system, many MNN modules learn to recognize the weights between technical and fundamental data to be used for the market timing strategy. The goal of MNN is to predict the best market-timing trading strategy within one month using six input indices: 1) returns vector curve, 2) volume turnover, 3) interest rate, 4) exchange rate, 5) Dow Jones average, and 6) others market indices. The so-called TOPIX, or the weighted average of market prices of all stocks listed on the fist section of the Tokyo Stock Exchange, is used as the inputs' weights. While the technical and economic indices are converted into a cross-sectional pattern for MNN's input, the outputs are used in the simulation for market timing. It is concluded that using the price forecast outputs from MNN to derive trading strategy improves average returns significantly. However, a system in which training data is generated in combination with a statistical method should be further developed and tested in actual trading environment. This system requires more unsupervised training in order to simulate actual data automatically.

Hybrid Neural Networks Forecaster

Hybrid neural network forecaster (HNNF) is developed by Inductive Inference Inc. based on its NN system called ADAM for use in the loan and credit assessment by Chase Manhattan Bank. ADAM is a module formulation of predictive patterns using statistical techniques to extract a collection of Boolean formulae from historical data and capture the rules most significant in determining the corporate customer's creditworthiness. HNNF in turn learns and forecasts those patterns provided by ADAM based on two inputs: 1) historical financial statement data on good, critical, and charged-off customers, and 2) industry norms calculated using financial statement data from companies in specific industries from COMPUSTAT database. The accuracy of HNNF depends on the patterns of the two inputs. Chase Manhattan Bank finds that ADMA is able to select good patterns for HNNF to maximize its precision and minimize bias effectively.
Conclusion on Neural Networks and Capital Markets Back to Top

NN technology is among many other advanced statistical techniques which are being developed to capture complex and non-linear relationships between variables. The development of these advanced techniques may not necessarily be fine-tuned with that of NN, but they all attempt to overcome the computational difficulty that prevent the financial economists to come up with better economic models. As computational power increases and is more cost-effective, the advancement in these NN and non-NN areas will become very fast.

There are two motives for the study of NN applications in capital markets, as suggested by Lo (1988). First, there is much misunderstanding about NN that they can be applied without taking into account the fundamental economic structure that drives the capital markets. Both NN and other traditional statistical methods such as multivariate time-series model, vector autoregressions and non-parametric regression, should be constantly and rigorously compared to see if NN indeed provide any improvements in predictive power. Second, NN which are capable of parallel processing have the ability to process relatively simple tasks and do many of these tasks simultaneously. This feature best suits the financial analysis and capital markets applications since there are large collections of observations which require this kind of parallel processing ability. Moreover, the ability to learn to recognize past patterns and memorize them is very beneficial in modeling the non-linear relationships among those observations. Essentially, NN serve as indispensable analytical tools for fractal market hypothesis and chaotic time-series to be tested in practice, just like modern investment paradigms, such as EMH and CAPM, have conventional statistical tools to support their assumptions and underlie their models.

* Worapot Ongkrutaraksa is a lecturer in Finance and Strategic Management at Maejo University's Faculty of Agricultural Business, Chiang Mai, Thailand. He used to conduct his post-graduate research in financial economics at Kent State University and international political economy at Harvard University through the Fulbright sponsorship between 1995 and 1998.

E mail: worapot@iname.com

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