 Abstract
 Introduction
 Literature review
 Theory foundation
 Experiments on the Chinese stock market
 Conclusion
 References
Abstract
As the cornerstone of modern financial theory, the Effective Market Hypothesis (EMH) has been increasingly questioned in recent years. The fractal theory is originally used to study the coastline and it has been introduced into the research of the stock market, and has attracted more and more attention from researchers. Since the 20th century, there have been many theoretical and empirical studies on the fractal characteristics of the stock market. This paper uses the fractal market hypothesis model to verify its effectiveness, and finds that the daily return rate of Chinese stock market shows a characteristic of peaks and tails with a biased random walk. Through R/S analysis and calculation of V statistics, we estimate the cycle of the three major indexes of the Ashare market. Comparing the trend of the V statistics and the Hurst index, we estimate that the average cycle of the three major indexes are 220 trading days. On this basis, We constructed the local Hurst index to predict the inflection point of market prices, and found that the local Hurst index showed a strong priori when predicting the future trend of the market. In the period before the market is about to reverse, the Hurst index has been less than 0.5, giving a clear signal that the market is about to reverse.
Keywords: Fractal, Stock market, Hurst index, Inflection point
Introduction
As we all know, many phenomena in nature are complex, such as the distribution of nebulae on large scales, the meandering shapes of mountains, geological rock layers, the spread of lightning, the distribution network of rivers, coastlines, and even animal blood vessels, etc, which present strange characteristics. These salient features cannot be described in traditional geometric terms because they have too many irregularities. In Euclidean geometry, the topological dimension of a shape is constant, and this constant topological dimension cannot describe the abovementioned irregular objects. The most typical example is the measurement of coastline length. The concept of fractal was first proposed by (Mandelbrot, 1967) when studying the length of the British coastline. The coastline as a curve is characterized by extremely irregular and extremely smooth, showing extremely meandering and complicated changes, after which he founded fractal geometry. Based on this, a science for studying the properties and applications of fractals has been formed, called fractal theory. He believes that fractals have the following characteristics: first, they have a fine structure, that is, they can present more fine details at any small scale; second, fractal objects have a wide range of scale changes; Third, fractals are selfsimilar, that is, parts are connected to the whole in some way.
For more than half a century, the efficient market theory has been the cornerstone of the establishment and research of modern financial theory, and has been dominating financial economics. The theory of random walk (Baehelier, 1900) believes that capital market price movements are independent, random, and unpredictable; the efficient market hypothesis (Malkiel and Fama, 1970) believes that if capital market prices can reflect all relevant information in a timely, rapid, and accurate manner (Historical information, public information, and inside information), the market is efficient. These two theories implicitly assume that the capital market has such statistics: the first is that the price series are independent of each other or at most shortterm related; the second is that the price series follow a normal distribution; the third is that the price series are linear time series. However, does the actual capital market really meet this statistical characteristic? The answer is no. There are many blind spots in the effective market theory’s interpretation of the actual operation of the financial market. The US stock market disaster that occurred on October 19, 1987, and the financial crisis in 1997 and 2008 were all unexplainable. With the development of the stock market, the understanding of the laws of the stock market has deepened. The internal and external factors of the financial market interact each other. This market is essentially a complex dynamic system composed of multiple factors, making it difficult to understand and characterize the operating laws of the financial market. After entering the 1980s, the efficient market hypothesis encountered a considerable challenge and was helpless with a large number of anomalies. Since (Mandelbrot, 1967) put forward the concept of fractal, the fractal market hypothesis has gradually emerged, which has challenged the efficient market hypothesis. The fractal market hypothesis believes that stock prices will follow certain rules instead of random walks, and has longterm memory. It is possible to predict price trends by analyzing historical data. The efficient market hypothesis has gradually proved to be nonexistent in the real market, but the fractal market hypothesis can well explained many phenomena that the effective market hypothesis cannot explain.
Literature review
(Malkiel and Fama, 1970) proposed the efficient market theory. Fama pointed out that investors in the market are rational, the prices of securities are reasonable, and the prices of securities reflect all the information in the market.
The R/S analysis method (rescale range analysis) occupies a fundamental position in fractal analysis. The R/S analysis method started from a paper published by (Hurst, 1951), and first proposed an important R/S analysis method to explore the characteristics of fractal structures. Later (Peters, 1991) used the R/S analysis method to analyze the market of the United States, the United Kingdom. The research on financial markets including Germany and Japan shows that the financial markets of these countries show significant fractal structural characteristics. At present, many literatures have used fractal theory to study financial markets. (Lebaron and Scheinkman, 1989) has studied the sequence of US stock returns and found that there is chaos in the fractal dimension between 5 and 6. (Peters, 1991) verified the normality of asset price changes in the capital market by applying the R/S analysis method, and confirmed that the asset price or asset income series conformed to the fractal Brownian motion or biased random walk. (Nwarocki, 1995) studied the famous S&P300 stock index and found that the average period of the stock index is five years. (Byers et al., 1996) studied exchange rates using fractal theory.
(Xu and Lu, 1999) used R/S analysis method to analyze Shanghai Composite Index and Shenzhen Composite Index before October 1998 in China. The Hurst value of the Shanghai Composite Index is 0.661 and the average cycle length of the Shanghai Composite Index is about 195 days. The Hurst index value of the Shenzhen Composite Index is 0.643, and both markets have fractal characteristics.
(Huang, 2004) estimated the scale function, generalized Hurst index, and multifractal spectrum of the Shanghai Composite Index’s daily closing index from December 19, 1990 to September 8, 2003, and confirmed that the Shanghai securities market has obvious multifractal structure characteristics.
(Huang, 2005) used the R/S method to conduct an empirical analysis of the fractal structure of the Chinese stock market. The results show that the H index of the daily and weekly return series of the Shanghai Composite Index and the Shenzhen Composite Index are significantly greater than 0.5, so there are strong state persistence, with fractal characteristics.
(He, 2007) used the R/S nonlinear estimation method to conduct empirical research on the long memory problem in China’s stock market return series. It is found that the longmemory nature of China’s stock market returns generally exists. Only some stocks do not have longmemory nature and the Shenzhen stock market has stronger longmemory nature than the Shanghai stock market.
(Zhao, 2009) tried to use the GARCH model and the ARIMA model to make shortterm predictions of stock price volatility trends. Firstly, the 7day moving average method was used to perform appropriate noise reduction on the daily closing price data of the Shanghai Composite Index, and then ARIMA modeling and GARCH modeling analysis were performed on this trend series, and shortterm trend prediction was performed using the established model. It is found that there are better shortterm prediction effects.
(Wen et al., 2012) used the R/S analysis method to calculate the Hurst index of the Shanghai logarithmic return series, which was about 0.6298. The results showed that the Shanghai stock market showed a fractal characteristic and a long memory cycle, about one and a half years. Through phase space reconstruction, the Shanghai stock market attractor dimension has converged to 1.335, which indicates that the Shanghai stock market has chaotic characteristics. At least two variables are needed to build a dynamic system for the Shanghai stock market. The results of the principal component analysis support the conclusion that the Shanghai stock market has chaotic characteristics. The fractal and chaotic characteristics of the Shanghai stock market reveal the nonlinear characteristics of the Chinese stock market, and the nonlinear perspective is more conducive to the formulation of countermeasures for the development of the Chinese stock market.
(Li, 2017) conducted research on long memory based on fractal theory. The research object is the long memory of time series of financial market returns. He introduced and used empirical research on long memory in financial markets using methods such as rescaling extreme range, detrending analysis, detrending movement analysis, and generalized Hurst index to measure long memory and related statistics.
(Zhao, 2013) conducted research on the chaotic phenomenon of fractal dimensions in financial markets, using the theory and estimation methods of correlation dimension and Hurst index to study the structural characteristics of Shanghai stock market. Based on the empirical analysis of the chaos and fractal characteristics of the Shanghai stock market, combined with actual data, the Hurst index value was estimated. Based on the characteristics of the Hurst index value, it was concluded that the Shanghai stock market has a fractal structure and exists in the phase space of the system Nonlinear system of chaotic attractors.
Most researchers focus on analyzing the fractal phenomenon of the stock market using the traditional R/S method. Researchers use data from various stock markets to test it. In terms of data selection and use, each researcher uses data different intervals, different time scales, and different securities market structures have different research results, and different researchers have different research results on the same securities market. And there are few researches on the causes of fractal characteristics in financial markets. Most articles only use data to conduct empirical analysis and then obtain empirical results. There is still a lot of space in the application of fractal theory in the stock market.
Theory foundation
Capital market efficiency
Capital market efficiency refers to the effectiveness of the capital market in optimizing the allocation of resources to capital. The specific meaning is that capital market efficiency has two meanings: first, to what extent can the capital market provide capital resources to the demanders of capital resources to reduce the transaction costs in the process; At what level can the effective supply of capital resources be provided to society? If the capital market is in a state of higher efficiency, it will allocate limited capital resources to higherquality capital resource demanders (that is, companies or industries that require capital resources), thereby maximizing the value of the entire market.
Efficient market hypothesis
Foundation of efficient market hypothesis
(Malkiel and Fama, 1970) proposed the efficient market theory. Fama pointed out that investors in the market are rational, the prices of securities are reasonable, and the prices of securities reflect all the information in the market. Under effective market conditions, information and securities analysis cannot obtain additional returns. It is assumed that firstly the price series are independent of each other or at most shortterm related; the second is that the price series follow a normal distribution; the third is that the price series is linear time series. The efficiency of capital market is closely related to many theories in the field of financial economy, such as value evaluation theory, portfolio theory, and capital structure theory. The theoretical premise includes that the capital market is effective. Therefore, the efficient market hypothesis is of great theoretical significance in the field of financial economy.
An efficient market is a market in which all information is quickly understood by market participants and immediately reflected in market prices. This theory assumes that investors participating in the market are sufficiently rational to respond quickly and reasonably to all market information. The efficient market hypothesis assumes that in a society full of information exchange and information competition, a specific piece of information can be quickly known to investors in the stock market. Subsequently, the competition in the stock market will drive the stock price to fully and timely reflect the group of information, so that the transactions performed by the group of information do not have abnormal returns, and can only earn a riskadjusted average market return. As long as the market price of securities can fully and timely reflect all valuable information, and the market price represents the true value of securities, such a market is called an effective market.
For example, LazyT Oil Company has just discovered oil in the Gulf of Alaska. It was announced at 11.30 am on Tuesday. When will the price of LazyT Oil company’s stock rise? Efficiency market theory believes that this news will be immediately reflected in prices. Market participants will immediately respond and raise the price of LacyT’s stock to the expected level. In short, at every point in time, the market has digested all the latest news available and included it in stock or grain prices or other speculative prices. This means that if you see a big frost in Florida from the newspaper, don’t think you can make a fortune by buying frozen orange juice futures during lunch breaks; while reporting the news, even before that, the price of oranges juice has risen.
Efficient market theory holds that market prices already contain all available information. It is impossible to make money by looking at past information or past forms of price changes.
Validity hypothesis of efficient market
The assumption of market effectiveness is based on a perfect market:
 There is no friction in the entire market, that is, there are no transaction costs and indicators; all assets are completely divisible and tradable; there are no predetermined regulations.
 the entire market is fully competitive and all market participants are price takers.
 The cost of information is zero.
 All market participants receive information at the same time. All market participants are rational and seek to maximize utility.
In real life, these assumptions are difficult to hold. Investors must consider the following costs when investing:
 Transaction cost; every time an investor makes a transaction, he must pay a certain fee to the broker.
 Tax: Investors must pay a certain percentage of tax based on the amount or income of each transaction.
 Investors must also pay certain fees in order to mine various information.
 Opportunity cost, including time, effort, etc.
The division of capital market effectiveness
The above description of efficient markets is an ideal state, and in the real world, the effectiveness of capital markets varies widely. Therefore, in order to facilitate academic research and guide investment practices, financial economists divide capital markets into three categories: weakly efficient markets, semistrongly efficient markets, and strong efficient markets. Strictly speaking, in addition to the aforementioned three types of markets, there are also invalid markets. However, after hundreds of years of development of the capital market, the capital market in an ineffective state hardly exists in the world.
 Weak effective market: In a weak effective market, asset prices already reflect all historical information related to asset prices (including all historical transaction price, volume information, and all other historical public information), that is, That said, all historical information related to asset prices is already in the price.
 Semistrong effective market: In the semistrong effective market, asset prices have reflected all public information related to asset prices (including all historical transaction price information, all other historical public information, and all current public information). Information), which means that all public information related to the price of an asset is already in the price.
 Strong effective market: In the strong effective market, asset prices have reflected all information related to asset prices (including all historical transaction price information, all other historical public information, all current public information, and all inside information), which means that all information related to the price of an asset is already in the price.
Challenges of the efficient market hypothesis
As a basic theory in financial economics, the efficient market hypothesis is widely accepted, but it is not a perfect theory.
The US stock market disaster that occurred on October 19, 1987, and the financial crisis of 1997 and 2008 were all explained by efficient market theory. The efficient market hypothesis does not explain the common anomalies in capital markets such as the crash and the bubble caused by herding behavior (Herding Behavior, Grossman and Stiglitz (1976), Robert Shiller (2004)). Anomalies).
The research results of some researchers point out that the return sequence of financial markets is not completely unpredictable. Historical historical price data has a certain ability to predict future prices. Current prices do not fully reflect all information in the open market. Effective market patterns are inconsistent. A widespread anomaly in efficient market theory is that the probability distribution of the stock market’s rate of return is not a normal distribution originally assumed, but has an obvious peak and fat tails. (Leptokurtic).
As a complex and open system, the operation of the capital market is affected by various factors such as internal mechanisms and external interference, and is not as orderly and hierarchical as described in the efficient market hypothesis. In other words, the actual capital market cannot meet the two basic prerequisites of complete rationality and complete information necessary for the efficient market hypothesis model. With the development of the stock market, people’s understanding of the laws of the stock market has deepened, and the internal and external factors of the financial market interact. This market is essentially a complex dynamic system composed of multiple factors, making it difficult to understand and characterize the operating laws of the financial market. These findings strongly challenge the efficient market hypothesis.
Brown motion
Brownian motion refers to a kind of random walk without correlation, which meets the statistical selfsimilarity, that is, it has the characteristics of random fractal, but its time function (moving trajectory) is selfaffine. It has the following main characteristics: the movement of particles is composed of translation and its transfer, which appears very irregular and its trajectories are almost tangent everywhere; the movements of particles are obviously independent of each other, even when the particles approach each other to a distance smaller than their diameter. This is also the case; the smaller the particles or the lower the viscosity of the liquid or the higher the temperature, the more active the movement of the particles; the composition and density of the particles have no effect on their movement; the movement of the particles never stops.
The original Brownian motion (BM) was proposed by Robert Brown in 1827. It refers to the random movement of suspended particles in a liquid. It was not until 1877 that J. Del Soso made a correct qualitative analysis: Motion is actually caused by an imbalanced collision of surrounding liquid molecules. In 1905, A. Einstein made a physical analysis of this “irregular motion”, and became the pioneer of Brownian kinetic theory, and first proposed a mathematical model of Brownian motion. In 1923, Norbert Wiener proposed the definition of measures and integrals on Brownian motion space, thus forming the concept of Wiener space, and made a strict mathematical definition of Brownian motion. According to this definition, Brown Motion is an independent incremental process, a stochastic process with continuous time parameters and continuous state space.
The mathematical description of Brownian motion is as follows:
 $B(t)$ is an independent incremental process,for any $t>s$, $B(t)B(s)$is independent of previous processes $\mathrm{B}(\mathrm{u}): 0<=\mathrm{u}<=\mathrm{s}$.
 $B(t)$ has a normal distribution increment： $B(t)B(s)$ satisfy the normal distribution with Expectation $0$ and variance $ts$.
 $B(t), t>=0 $ is continuous function of $t$.
Application of Brownian motion in financial markets
Linking the Brownian motion with stock price behavior, and then establishing a mathematical model of the Wiener process is an important financial innovation of this century and occupies an important position in modern financial mathematics. So far, the general view still holds that the stock market is fluctuating randomly. Random fluctuations are the most fundamental characteristic of the stock market and the normal state of the stock market. In many classic financial pricing models and analysis methods, stock prices are assumed to be an independent random Brownian motion:
The Brownian motion assumption is the core assumption of modern capital market theory. Modern capital market theory believes that securities and futures prices have random characteristics. The socalled randomness here refers to the memoryless nature of the data, that is, past data does not constitute a basis for predicting future data. No surprisingly similar iterations will occur. The mathematical definition of random phenomenon is: the results of individual experiments appear uncertain; the results of statistical experiments in a large number of repeated experiments. The Wiener process of Brownian motion, which describes one of the stock price behavior models, is a special form of Markov random processes; Markov processes are a special type of random processes. Stochastic process is a probability model based on probability space. It is considered to be the dynamics of probability theory, that is, its research object is a random phenomenon that evolves with time. So random behavior is a statistically regular behavior. Stock price behavior models are usually expressed using the wellknown Wiener process. It is tempting to assume that the stock price follows a generalized Wiener process, that is, it has a constant rate of expected drift and variance. The Wiener process shows that only the current value of the variable is related to future predictions, and the past history of the variables and how the variables have evolved from the past to the present are not related to future predictions. The Markov nature of stock prices is consistent with the weak form of market efficiency, that is, the current price of a stock already contains all the information, including of course all past price records. But when people started to use fractal theory to study financial markets, they found that its operation does not follow Brownian motion, but obeys the more general fractional Brownian motion.
Rescale range analysis
(R/S) analysis method is a British hydrologist H.E. Hurst, when studying the relationship between the water flow and storage capacity of the Nile Reservoir, found in a series of empirical studies that natural phenomena follow the biased random walk. That is, a trend is added with noise, and a rescaled range analysis (R/S analysis) is proposed, and based on this, an R/S analysis method is proposed to establish the Hurst index (H).
Suppose we have a set of independent random variables with a mean of 0 and a variance of 1 that appear sequentially in time. The time range of this time series within a certain time span T is linearly related to the $\frac{1}{2} $ power of $ T $. The familiar Brownian motion increments satisfy this property (the increments are independent of each other). However, Hurst found that for many time series in nature, their range of variation within the time span $ T $ is not directly proportional to the $\frac{1}{2} $ power of $ T $, but rather than $ \frac{1}{2} $ is proportional to higher powers, which indicates that the values of the time series are not independent but affect each other, that is, the autocorrelation coefficient of the time series is not 0. We say that such time series have long memories.
Long memory corresponds to shortterm dependency. For a time series with shortterm correlation, its autocorrelation coefficient quickly decays to 0 or exponentially with the increase of the lag. For time series with long memory, its autocorrelation coefficient decays more slowly. This definition indicates that if the decay rate of the autocorrelation function of a stationary time series obeys the power law decay (that is, slower than the exponential decay), then the time series has long memory. The Hurst exponent H is used to characterize this long memory; it is used to measure how the fluctuation range of a time series changes over time,that is
\(\mathrm{E}\left[\frac{R(n)}{S(n)}\right]=A n^{H} \text { as } n \rightarrow \infty\) where $ n $ is the length of time series, $ R(n) $ is the interval of time series； $ S(n) $ standard deviation of time series. Using $ R(n) $ to standalize $ S(n) $ , we obtain $ \frac{R(n)}{S(n)} $, it is called rescaled range； $ A $ is constant, $ H $ is Hurst index.
This method is unbiased in a general sense, and is generally used as an index for judging whether time series data follows a random walk or a biased random walk process. The R/S analysis method is an effective tool to test that the economic time series obeys the assumption of independent random distribution or nonlinear characteristics, and the method does not involve the difference between stationary and nonstationary processes, and does not need to set up a specific model. It is ideal test method, we use the R/S analysis method to distinguish between stochastic and nonstochastic systems, and find the continuation of the system trend, cycle。
The algorithm is as follows:
Step 1: Assume a time series $x_{t=1}^{M}$ with length $M$ , divide it into $\frac{M}{N}$ subintervals with length $N$
Step 2: For each subinterval $u$, calculate mean:$e_{u} = \frac{1}{N} \sum_{i=1}^{N} x_{i+(u1) \times N},(u=1,2, \cdots,\frac{M}{N})$
Step 3: For each subinterval $u$, calculate $y_{u, i}=x_{i+(u1) \times_{N}}e_{u}$ $(i=1,2, \cdots, N)$, let $z_{u, i}=\sum_{i=1}^{N} y_{u, i}$ is cumulative rate of return of $u$ $(u=1,2, \cdots,\frac{M}{N})$
Step 4: Calculate $R_{u}=\max {1 \leqslant i \leqslant N} z{u, i}\min {1 \leqslant i \leqslant N} z{u,i}$ on each subinterval $u$, $S_u$ is standard variance of each subinterval ;
Step 5: Calculate $R_{u} / S_{u}$, for $\frac{M}{N}$ number of interval, we can get $\frac{M}{N}$ number of $R_{u} / S_{u}$ values. $(u=1,2, \cdots,\frac{M}{N})$ Calculate the mean of $R_{N} / S_{N}$ with the length of $N$, that is the R/S value.
Step 6: Taking the logarithm on both sides,$R_{N} / S_{N}=b N^{H}$, then $\log \left(R_{N} / S_{N}\right)=H \log N+\log b$ , $b$ is constant, $H$ is Hurst index;
Step 7: For different $N$, repeat Step1 and Step6 ,we can get a series of $R_{N} / S_{N}$ , using ols method of calculate $H$.
Hurst index H is used to measure correlation and trend strength:

When $ H = 0.5 $, it indicates that the time series is white noise and no memory, which can be described by the random walk process;

When $ 0.5 <H <1 $, it indicates that the time series is black noise, that is, the time series is persistent, which implies a longterm memory time series. If $H = 1$, the time series is a straight line, and history can fully predict the future;

When $ 0 <H <0.5 $, it indicates that the time series is pink noise, that is, the time series has antipersistence, which implies that the time series has a mean recovery process;
Applying R/S analysis to financial markets can determine whether the yield sequence is memorable and whether the memory is persistent or antipersistent.
Fractional Brown Motion
Mandelbrot pointed out that the return distribution of the market does not follow a normal distribution as predicted by the random walk theory, and the price sequence is not a random walk in the strict sense. The return distribution of the financial market has a characteristic of peak and fat tails distribution. He was also the first to apply fractal mathematics to financial markets and found that the price sequence of financial markets was not subject to the Brownian motion in the strict sense.
The world is nonlinear, and most of the universe is not an orderly, linear, stable, and balanced, but a chaotic, nonlinear, unstable and fluctuating boiling world. In other words, the universe is full of fractals. There is a type of random process in a large number of natural and social phenomena such as price fluctuations in the stock market, fluctuations in heart rate and brain waves, noise in electronic components, natural landforms, etc. They have the following characteristics: in the time or space domain it has selfsimilarity and longterm correlation. In the frequency domain, its power spectral density basically conforms to the polynomial attenuation rule of $1 / \mathrm{f}^{\gamma}$ in a certain frequency range. So it is called $\frac{1}{f}$ family of stochastic processes. When modeling such processes, the commonly used ARMA method is only suitable for processes where the related structure decays exponentially, and its effect is not good. Therefore, people are constantly looking for various models to simulate such random processes. Although the Hurst index is available, we still do not have a model for analyzing such time series. so the theory of fractional Brownian motion was proposed.
Fractional Brownian Motion (also known as Fractal Brownian Motion) is born out of standard Brownian motion. FBM is a continuous random process $B_{H}(t)$ defined in the time domain, which satisfies:

for any $t$ and $\Delta t>0$, the expectation of $B_{H}(t+\Delta t)B_{H}(t)$ is $0$.

for any $t$ and $s$ , the covariance function is \(\mathrm{E}\left[B_{H}(t) B_{H}(s)\right]=\frac{1}{2}\left(t^{2 H}+s^{2 H}ts^{2 H}\right)\)
where $H$ is hurst index, the core properties of FBM are its incremental smoothness, selfsimilarity, and autocorrelation (except $ H = 0.5 $; when $ H = 0.5 $, the FBM changes to standard Brownian motion).
The earliest proposed concept of fractal market is \citep {peters1991fractal}. He introduced the fractal theory into the economic and financial system based on Mandelbrot, and explicitly proposed the fractal market hypothesis. He believes that the Fractional Brownian motion can more accurately describe the fluctuation characteristics of financial market. The fractal market hypothesis is a specific application of fractal theory in the research of capital market theory. From a nonlinear point of view, Peters proposed FMH (Fractal Market Hypothesis) \citep {peters1994fractal}, which looks less perfect but is more in line with reality. It supports this new hypothesis with a lot of empirical research evidence, which explains the anomalies that the efficient market hypothesis and the existing capital market theory cannot explain.
The correlation coefficient between the increments over two time spans is defined as $ C $, which can be derived as $ C = 2^{2 H1} 1 $, which shows that $ C $ is independent of $T$, the increment is selfstable.
But in reality, it is easy to see that the stock price is not independent and random, but has a memorable trend change. The price changes of the yesterday is likely to affect today’s trend.
For stock price series, on the scales of monthly, weekly, daily, hourly, and minute ranges, stock prices have obvious selfsimilar structures, that is, the frequency of time series of stock prices is increased, and the microstructure of the stock market can be observed. The highfrequency financial data of the hour ranges, the minute ranges, and even the second, the stock price has a more obvious selfsimilar structure. Edgar Peter proved that the stock market is a nonlinear selforganizing fractal organization. The change in asset prices is not a simple random walk. The stock price will repeat its behavior on various scales.
Experiments on the Chinese stock market
The Necessity of Hurst Index Fractal Study in Ashare Market
We used the logarithmic rate of return of the Ashare market from January 5, 2009 to December 31, 2019 as the research sample. The survey objects were Shanghai Stock Index, Shenzhen Component Index, CSI 300, and the R/S analysis method was used to examine the fractal nature of the stock market.
There are some reasons that the Ashare market does not meet the efficient market hypothesis and presents a fractal structure: First, the Ashare market participants do not meet the conditions of a rational economic person. Many individual investors in the Ashare market are still immature and cannot respond to specific information in a timely and correct manner. Secondly, the imperfect Ashare market information disclosure system leads to untimely and insufficient information disclosure of listed companies.
In addition, the price under the daily limit in the Ashare market cannot fully reflect the wishes of traders, that is, the socalled equilibrium price is not a market equilibrium performance. Based on the above reasons and data results, it can be known that the Ashare market does not meet the conditions of an effective market, which provides a rational and necessary basis for us to use the Hurst index to study the fractal market structure of the Ashare market.
The distribution of returns of three index we mentioned are as follows:
We see that the daily returns of the Ashare market follows Leptokurtic distribution, which is consistent with the assumption of the fractal market hypothesis.
Regular average cycle length of A shares market
First we use the R/S method to examine the average cycle of the index. When $H> 0.5$, the Hurst index measures the memory of the market, that is, the future market trend will continue the previous market trend. The larger the Hurst value, the stronger the memory of the current market. When the length of the segmented subsequence exceeds the length of the cyclic period, the Hurst exponent will become smaller. Then, the slope of the straight line obtained by least square fitting from $ln(R/S)_{n}$ and $ln (n)$ should become smaller. That is to say, when the linear relationship between the two is broken , it is the average cycle of the market.
We calculate R/S values and Hurst index of the three indexes and plot the figures as follows:
From the figure above, we can see that the linear relationship between the two is broken at $ log (n) $. To determine the length of the market cycle more accurately, we examine
\[V_ {n} = (R / S) _ {n} / \sqrt{n}\]If $R/S_{n}$ is the same order as $\sqrt{n}$, then this proportional value should be a straight line with the change of n. In other words, the V statistic corresponding to the Gaussian distribution is a horizontal line in the coordinate system with $Log(n)$ as the abscissa. When a time series is memorable $(H> 0.5)$, the growth rate of $R/S_{n}$ value is higher than $Log (n)$, then the corresponding straight line should be upward; otherwise, when $H <0.5$, the corresponding The straight line should be downward. Then, in the coordinate system with the $V$ statistic as the ordinate and $Log (n)$ as the abscissa, the length of the cycle corresponds to the point in time when the V graph trend changes, that is, when the market memory begins to disappear.
The corresponding trend graph of $V$ values with n changes can be seen that the average market cycles determined are consistent. The average cycle of the three indexes are about 220 trading days, which means that the system loses the memory after 220 trading days. The memory of initial conditions, or the impact of an event on the system, can last an average of 220 trading days.
Construction of Local Hurst index
The Hurst index is a characteristic quantity to measure the memorability of the market. Most studies only regard it as a concept describing the overall market. By calculating the static Hurst index, it is predicted that the future market trend will continue the previous trend or will occur. However, in a dynamic market that changes in real time, in order to capture changes in the strength of the market’s memory in different periods, we built a local Hurst index.
A time series of length ${x_t}$, the local Hurst exponent at the ith time point can be calculated based on the subsequence between time points $(in + 1, i)$, and then $H_i$ At this point, you can get a local Hurst exponent sequence, where $(in + 1, i)$ is called the time window.
The local Hurst index can be used to reflect the sentiment of the stock market in a certain period of time, and determine whether the market will show an upward trend, a downward trend, or a random trend in the future. We expect to use the local Hurst index to study this trend. Because the size of the local Hurst index is closely related to the choice of the time window length, the choice of the length of the time window is very important. Most studies point out that the length of the time window should be the average cycle period, but on the other hand, from the perspective of economics, in order to eliminate the market impact caused by the supply and demand cycle, the length of the time window is preferably not more than the length of trading year. Based on the above reasons, we chose 220 trading days as the local Hurst The length of the exponential time window. Theoretically, when the Hurst index is 0.5, the market is in a random state and lacks direction selectivity. However, the local Hurst index of the stock market is usually higher than 0.5. Therefore, in order to more realistically capture the time of market direction conversion and determine the trading strategy, we indicate the expected Hurst index for a specific time window length. The algorithm of the Hurst exponent is similar to the algorithm of the Hurst index,we use the following formula proposed by Peters:
\[E(R / S)_{n}=[(n0.5) / n] \times[n \times(\pi / 2)]^{0.5} \times \sum_{r=1}^{n1} \sqrt{(\mathrm{n}r) / r}\]Where $n$ is the length of each subsequence.
We can continue the analysis through the following figures:
From the figure above, we can see that at most points the local Hurst index is above 0.5, and most of them are above the expected value of the Hurst index, which indicates that the future market trend and the previous market performance are positive in most times. Relatedly, the market exhibits longterm memory at most points in time. In addition, we note that market reversal occurs mostly when the local Hurst index is below the expected value of the Hurst index.
Inflection point
Then we can build a method to find the inflection point accordingly, and mark the points with a Hurst index less than 0.5 with red points.
Through this visualization method, we can clearly find that in the ten years from January 5, 2009 to December 31, 2019, the accuracy rate of finding inflection points of the Shanghai Composite Index is 85%, the accuracy rate of the Shenzhen Composite Index is 88%, and the accuracy rate of the CSI300 is 85%, we carefully calculated that when the red points began to appear densely, the subsequent stock price trend would reverse, because when $ H <0.5$, the time series would show the characteristics of mean recovery, that is, antipersistence. However, when a sparse red points appears, the subsequent inflection point trend cannot be successfully predicted. What’more, the trend after the reversal is incomplete, that is, the trend after the inflection point is not continuous, the end of the trend after the inflection point cannot be predicted, and it needs to be used in conjunction with other methods. After all, the prediction accuracy of this method is acceptable. It is applicable to all three major indexes, indicating its universality. And it can also be applied in the latest year 2019, indicating that it has not expired and there is still worth continuing this research.
Conclusion
The use of the fractal market hypothesis model in empirical research on the Ashare market has its rationality and necessity. Due to the special equity structure, information disclosure system, and investor composition of the Ashare market, the Ashare market does not meet the prerequisites of the efficient market hypothesis, which makes the daily return rate of the market show a sharp and thick tail with a random walk. The fractal market hypothesis takes market liquidity and investment duration into model considerations, and describes the market status more closely to reality.
The Hurst indexes of the three major indexes are 0.58, 0.61, and 0.61, all of which show longterm memory characteristics, and the Shenzhen market is more memorable than the Shanghai market. The average memory length of the two cities is 220 trading days by R/S analysis.
The local Hurst index shows a strong priori when predicting the mediumterm and longterm trends of market. When the hurst index reaches 0.5, it indicates that the randomness of market starts, the stock index trend weakens, and the turning begins. And the stock index should have a real trend. If there is no trend, the index cannot be used to carry out trend prediction.
There is currently no prediction on shortterm trends using local Hurst index, and it is impossible to predict a stock market crash. This needs to be used in conjunction with other methods, which is also the direction of our future research.
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