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摘錄自我博士論文的一節，改天再來闡述我的看法。

The Fractal Market Hypothesis

Fractal market analysis is a pioneering approach to the study of financial markets, and discovers and characterizes the order hidden within seemingly random financial markets, and determines the probability of future events. Blackledge (2010) claimed the Fractal Market Hypothesis (FMH) and used a financial risk assessment model based on Lévy statistics and considers a financial forecasting system that uses a solution to a non-stationary fractional diffusion equation. This study provided a solution to securing an investment portfolio and showed that there is a quantitative relationship between Lévy's characteristic function and a random scaling fractal signal obtained through a Green's function solution to the fractional diffusion equation. Blackledge used the hypothesis in the application of the hypothesis by predicting the volatility associated with foreign exchange markets and the ABX index.

Blackledge assumed that financial signals are fractal signals to be a classification of the Fractal Market Hypothesis (FMH) based on Lévy statistics in contrast to the Efficient Market Hypothesis that is based on Gaussian statistics, and predicated on the assumption that financial time series with a one-dimensional model for fractional diffusion. Blackledge provided connectivity between Lévy distributed processes used to derive the fractional diffusion equation and random scaling fractal signals in terms of a Green's function solution to this equation, and gauged the likely future behavior of the signal under the FMH.

Walter (1999) examined the Lévy-stability-under-addition of the French MATIF notional contract on ten-year government bonds and presents the connection between the stable distributions and the fractal structure of markets. Walter verified the existence of an underlying fractal structure governing the price variations on different time intervals, indicated that the fractality of the market is associated with the Lévy-stability-under-addition property by rescaling space and time.

These empirical studies suggested that there was a stronger degree of long-range dependence in the returns of various markets. However, the fractal approach of the market hypothesis need further more studies and evidences.

Relevant studies of financial market structure

Since the New York stock market crash on October 19, 1987, several studies have attempted to identify reasons for the crash. In addition to traditional statistical methods, certain studies have used chaos theory to determine the reasons.

Whether the market variability due to the random walk or a deterministic process is still controversy in recent years. Although some think that is sufficient to provide evidence to chaos system. Chaos theory has recently been widely applied in finance to explain market anomalies. Table 2-1 shows empirical evidences of chaotic behavior in a wide range of financial asset pricing.

Table 2-1. Summary of nonlinearity and chaos studies

Peters (1989) firstly used the R/S analysis to examine the relative returns of stocks and bonds and both the existence of a fractal structure. The study focused on the average monthly return of the S&P500 from 1950 to 1988, as well as the bond interest rates over three decades. The results showed the values ​​greater than 0.5 and the S&P500 and the return of public debt were not linear with a random walk hypothesis.

Peters (1991a, 1991b) used correlation dimension analysis and R/S analysis to examine major stock markets worldwide. He found that the H value was greater than 0.5 in the main national markets, such as in Germany (0.72), Japan (0.68), the United Kingdom (0.68), which was consistent with the assumptions of chaos theory. In addition, the analysis of correlation dimension also indicated that the S&P500 index converged to 2.33, the United Kingdom index converged to 2.94, the German index converged to 2.41, and the Japan index converged to 3.05. The results also indicated that the market assumed a fractal structure.

Peters (1991b) estimated the Lyapunov exponent for four major capital markets, and the results were all greater than zero; for example, the S&P500 was 0.0241, the British index was 0.028, the Japanese index was 0.0228, and the German index was 0.0168. Therefore, he concluded that the chaotic system was widespread in these capital markets.

However, Hsieh (1991) examined the stock returns from 1963 to 1987, and indicated that the stock returns did not meet the independent and identified distribution (iid) hypothesis and that the data exists dependency. Hsieh attributed the rejection of the iid hypothesis to non-stationarity, chaos and variance heterogeneity, and indicated that the iid hypothesis was rejected not because of unstable factors or the market to be a nonlinear dynamical system. The time series exhibited heterogeneity of variance, and the EGARCH model was used as a forecasting tool. Although Hsieh admitted EGARCH model to describe the price behavior of the stock market is still valid, and perhaps more complex ARCH-type models could be more effectively. Yang and Brorsen (1993) also showed that the Brock residuals test did not support the chaos hypothesis, whereas approximately half of the residuals through the GARCH (1, 1) model were inconsistent with the iid hypothesis, yet consistent with the chaotic phenomena.

DeCoster, Labys, and Mitchell (1992) found that the price behavior had a clear structure, and not just simply reflected the conditional characteristics of the heterogeneity of variance. If the chaotic phenomenon was a null hypothesis, the study could not reject the null hypothesis. Meanwhile, the study also rejected the linear-structure-plus-noise structure and required further research before accepting or rejecting chaos theory.

Moreover, Harris and Kucukozmen (2001), Belaire-French, Contreras, Tordera-Lledo (2002) and Strozzi (2002) found evidence of nonlinear capital asset pricing and indicated that traditional financial theory may have mistaken chaotic behavior for a random walk. The application of chaos theory to financial theory has the potential to substantially influence the future development of the field (Mouck, 1998)

Numerous studies have also indicated that statistical tests established in autocovariance and specific density distribution cannot distinguish whether the information belongs to the stochastic process or the deterministic process (Brock and Dechert, 1986).

The following conclusions are based on our reviews of related empirical studies.

(1) In the capital market, because of the variance of heterogeneity, the fixed variance model cannot fully reflect the true risk; therefore, a simple two-factor model of the mean-variance does not apply.

(2) Although the ARCH-type model has a good daily predictive ability for numerous, its residuals are not entirely consistent with the iid hypothesis, the price behavior does not simply reflect the heterogeneity and its residual sequence implies a specific structure.

(3) The efficient market hypothesis is unsuitable for the current financial environment, and chaos may be superior to explain the true situation of the market.

(4) The economic factors and past prices affect the behavior of the market, and chaotic nonlinear dynamic systems and fractals can integrate fundamental analysis and technical analysis.

(5) The current statistical testing methods are inadequate for distinguishing random processes and determining the differences of the processes.