We have talked too much now, and think that the random walk hypothesis is the most important hypothesis in mathematical finance. It connects the idea of efficient market with Brownian motion in physics, and a set of random mathematical methods derived from it has become the cornerstone of constructing mathematical finance. (Its research mechanism has been widely used in stock research) The proposal of random walk model is closely related to the change pattern of securities prices. The earliest work to analyze the rate of return by statistical method was published by Louis Bachelet in 1900. He applied the method of analyzing gambling to stocks, bonds, futures and options. In Bachelier's paper, his pioneering contribution lies in realizing that the random walking process is Brownian motion. 1953, when British statistician Kendall applied time series analysis to study the fluctuation of stock price and tried to draw a pattern of stock price fluctuation, he came to an amazing conclusion: there is no rule in stock price, which is like "a drunk is walking, almost like the magic of opportunity still gives a random number every week, and adds it to the current price to determine the price for the next week." That is, the stock price follows the law of random walk.
There are two kinds of random walking models, and their mathematical expressions are:
Y t =Y t- 1 +e t ①
Y t =α+Y t- 1 +e t ②
Among them:
Y t is a time series (expressed by stock price or natural logarithm of stock price);
E t is a random term, e (e t) = 0; var(e t)=σ2；
α is a constant term.
Model ① is called "zero drift random walk model", that is, the stock price of the day changes randomly on the basis of the previous day's price. The stock price difference is all contained in the random term e t.
Model ② is called "random walk model of alpha drift", that is, the stock price of the day carries out fixed alpha drift on the basis of the previous day's price, and then changes randomly. The stock price difference includes two parts, one part is a fixed change α, and the other part is a random term E T. ..
From the above random walk model, we can see that the time series of securities prices will be random and will not show some observable or statistically determined trend. That is, the change of securities price is unpredictable, which is precisely the central idea of the law of securities price change revealed by the random walk model. Then, what is the relationship between the change pattern of securities prices determined under the random walk model and the efficiency of the capital market? The random fluctuation of securities prices is not only the evidence of market irrationality, but also the result of many rational investors developing relevant information and reflecting it. In fact, if the change of securities price is predictable, it really shows the inefficiency and irrationality of the market. In other words, if the stock market is efficient, the stock price should really conform to the random walk model.
T)=0, which is a necessary condition for an independent stochastic process. However, when H≠ 1/2, no matter what the value of t is, C(t)≠0. This feature of fractional Brownian motion leads to state persistence or inverse state persistence.
When H> is at 1/2, there is state persistence, that is, there is an upward (or downward) trend before a certain time t, suggesting that there is an upward (or downward) trend after the time t; On the other hand, when H.
In addition, using R/S analysis, we can determine two important aspects of information, the Hurst index H and the average period length. The existence of period has an important influence on further discussion and analysis. When H≠ 1 /2, the probability distribution is abnormal; When 1/2
It is worth pointing out that R/S analysis is a very effective tool, and it is not necessary to assume that the potential distribution is Gaussian. H= 1/2 does not mean that the time series is a Gaussian random walk, but only means that there is no long-term memory. If random walk is no longer applicable, many quantitative analysis methods will lose their effectiveness, especially CAPM and the concept of measuring risk with variance or fluctuation.
From the above discussion, the following basic conclusions are drawn:
1. For the efficient market hypothesis, α must always be equal to 2; For fractal market analysis, α can vary from 1 to 2. This is the main difference between efficient market hypothesis and fractal market analysis in understanding market characteristics. It is precisely because the fractal nature of α fully reflects the characteristics of the market itself.
2. Fractal market analysis does not have to rely on independent, normal or finite variance assumptions.
3. Using R/S analysis method, we can determine two important aspects of information, Hurst index H and average period length.
4. The public's response to information is nonlinear, so biased random walk is the normal state of the market, which is expressed as fractional Brownian motion.
5. The degree of deviation from random walk depends on the index H. ..
On the basis of discussing the emergence and development of EMH, this paper discusses the fractal market analysis method of market characteristics and its market characteristics from the fractal point of view, and summarizes the capital market theory. It is considered that the market is fractal and obeys fractional Brownian motion, that is, biased random walk, and its research method can be R/S analysis. The public's response to information is nonlinear, which shows that the digestion and absorption of information are inconsistent, leading to deviation from random walk and showing the normality of the market.