Why financial markets makes unexpected "black swan" is always present? Maybe we used the wrong distribution assumption!
The normal distribution is a statistic often used in financial transactions distributional assumptions. This distribution assumption is based on the "Central Limit Theorem" on. The content of the theorem is to assume that we extract from any population sample size of n samples, and when n is large enough, the sample means sampling distribution approximated with mean [mu], and variance σ ^ a bell-type 2 / n is normal distribution.
Most characteristic is the normal data set in the middle on both sides at least partially dispersed. But to achieve these results there is an implied condition that these sample results with each other should be independent of each other.
Independent of each other mean, the previous sample results should not affect the outcome of the next sample. The most representative independent sampling process is lost coin, regardless of the results of the last throw of the coin is positive or negative, it will not affect the possibility of the next throw of the coin. So long as the result of lost coins more than enough, we can see the results of an approximate normal distribution.
In real life, if you do not involve a cross-section of a time series of a large sample, we can be considered in line with the normal distribution of "independence" criteria. For example, we measure all the A-share closing price closing one day, you will find that their distribution is in line with the normal distribution: the price of most stocks concentrated in the vicinity of 10-30 yuan, located in the lower part of the small or 2-3 yuan 100 yuan higher range.
However, if we measure the variables are not independent of each other, then the normal distribution would not be established, but will become exponential or power-law distribution. Both the distribution is a concave crescent: the first half of the fluctuation range higher but small sample size, low fluctuation range of the second half but samples are available.
Features power-law distribution different from the exponential distribution is more evenly distributed around its numerical rate of decline more "slow."
In real life, whenever and human activities related variables in time series there is a certain correlation. For example, a rise in stock prices and the previous day's often the day after there is a strong correlation. Thus, the stock price change is largely in line with a power-law distribution.
It features power-law distribution, In a word is 20/80 laws. A stock price performance in a period of time inside, often completed within 20% of the range rose or fell. The remaining 80 percent of the time inside, it is often just doing random sideways.
In addition to stocks, the power-law distribution is also widely distributed in the language (20% of the words occupy the frequency of occurrence of 80%), distribution of wealth (20% of the population control 80% of wealth), and network traffic (20% of the sites occupy 80% CTR )etc.
There is a power law distribution, allows us to predict the time series in the variable distribution of changes, there must be a greater error tolerance range. This is because the "fat tail" Phenomena power-law distribution is more significant: Due to the presence of mutual influence between the variables, resulting in more extreme cases Rongyifasheng. Rising stock price will continue to rise, while oversold stock continued to fall.
If we follow the normal distribution is estimated to predict, then 95% of the stock price movement may be concentrated in the mean plus or minus 1.64 standard deviations. However, due to the fact that changes in stock prices follow a power law distribution, 95% of the stock price movement may be extended to the mean plus or minus two or three standard deviations. Therefore, based on the normal distribution of mean plus or minus 1.64 standard deviation set "standard forecast", may actually cause investors to sell too low or too high to buy, assumed the additional transactionrisk.
One reason there is most ironic, probably because more and more people use this hypothesis "the independence of each other is distributed transaction event" to guide trading, leading to the independence of convergence between different varieties with the logic of the transaction rather it is gone! This also explains why that never happened more and more on the history of the "small probability" event will appear on the financial markets in recent years, such as flash prices collapse 30%, the repo rate soared 10-fold and so on. This is because if we assume a power-law distribution to analyze fluctuations, these events have always been normal probability distribution. Leading to the prediction error, not the market, but the use of hypothetical trader itself. Or, out of greed and the endless pursuit of profit and the perils of humanity.
Source: quantitative investment club
Further Reading:
1. a quantitative strategist Confessions (Good text strongly recommended)
2. classic quantitative trading strategies available in the market are here! (Source)
3. futures / stock data Daquan query (History / real-time / Tick / finance, etc.)
5. From the high-frequency trading to quantify, can not read five books
