# An Algorithmic Look at Financial Volatility
## 4 The Universal Distribution of Stock Volatility  
In this section, we use daily returns of common stocks observed from international exchanges to show how well the universal distribution fits volatility data. Table 1 gives a presentation of our sample. To show the wide presence of the universal distribution, we sampled a price-driven market (NYSE) as well as order-driven ones. Our sample covers more than 20 years of transactions and have been observed from 3 variant continents. For each exchange, we investigate the components of its market index. The composition of these indexes is fixed on 31 March 2014. All data has been furnished by the commercialized platform IDOS. For each individual stock, daily returns are calculated by the difference of logarithmic Closes, and volatility is the absolute value of successive returns. Volatilities calculated like this are noted in real numbers. However, the universal distribution is solely relevant to binary strings. To reveal algorithmic structures in financial volatilities, it is necessary to make some data transformation. In our work, this data transformation is conducted asset by asset. For each individual stock, every trading week is transformed into a 5-bit binary string following 3 steps:  
* To transform a real number series into a binary one, we take daily volatilities of each stock, and set those above (resp. under) the sample median to be \"1\" (resp. \"0\"). Thus, in every binary series, we observe as much \"0\" (representing small volatilities) as \"1\" (representing large ones). This property is appreciated in our study since it allows a direct comparison between universal and uniform distributions. * Then, every binary sequence is divided into trading weeks according to civil agenda. When a civil week contains no-trading days, all trading data during this week is deleted from the sample. * The above-described two steps are repeated with each individual stock. The frequency of all possible binary trading weeks is calculated for the 5 exchanges in our sample. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Market** & **Stocks** & **Sample Period** \\ \hline NYSE & Dow Jones Industrial Average (30 components) & 1 January 1985-31 March 2014 \\ Nasdaq & Nasdaq 100 (100 components) & 1 January 1985-31 March 2014 \\ Euronext paris & CAC 40 (40 components) & 1 January 1985-31 March 2014 \\ LSE & FTSE 100 (100 components) & 1 January 1985-31 March 2014 \\ HSE & Hingsheng (50 components) & 1 January 1985-31 March 2014 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Data presentation. Table 2 exposes the numeric results, and Figure 2 gives an intuitive presentation of them. Please note that the 32 possible binary trading weeks can be divided into 3 classes, according to their appearing frequency:  
1. \"00000\" and \"11111\" are most frequent, since they are the simplest 5-bit strings according to [27]. Their average frequency is 0.07072 and 0.06692 respectively for the 5 sampled exchanges. 2. \"00001\", \"01111\", \"10000\", \"11110\", \"00010\", \"01000\", \"10111\", \"11101\", \"00100\" and \"11011\" constitute the second class, with their average frequency ranging from 0.03192 to 0.03890. 3. The others are even less frequent, with their average frequency varying from 0.02244 to 0.02898, what is interesting to underline here, it that they happen to be the most complex 5-bit strings up to [27]. Remark that these three classes can be observed from individual exchanges as well as the averaged data. In Figure 2, \(X\)-axis represents all possible binary weeks ranked by their Kolmogorov complexity. Colorful continual lines give the frequency evolution of individual exchanges. The dotted grey curve describes their average. The dotted orange one is what one should expect from Independent and Identically Distributed random variables.