# An Algorithmic Look at Financial Volatility
## 3 [PERSON]'s Universal Distribution
* Whereas, on admitting the simple pattern: \"100 times 0\", that takes only 9 characters to describe, one has a chance of \(2^{-9}\) to generate \(s\) randomly, since a program coding this pattern contains at least 9 characters (Here, we made an approximation to make our example more intuitive. Actually, the binary length of the rule \"100 times 0\", is slightly more that 9 bits, taking the true binary length to be \(m\), the probability of generating \(s\) randomly is \(2^{-m}\)). As \(2^{-100}<<2^{-9}\), \(s\) should be attributed to the second case. Remark here that the shorter is a string's generating rule, the more likely it can be generated by a randomly chosen program. Thus, to measure the likelihood of generating a binary string \(s\) by pure chance, one needs the length of its describing rule, noted by \(m\). As \(s\) could be generated by more than one program, we take the shortest one to calculate \(m\). The length of the shortest program generating \(s\) is defined as the Kolmogorov complexity of \(s\)[23]. For more details on this concept see Figure 1.  
Reference [15; 17] make a perfect link between the probability of generating \(s\) by pure chance and its Kolmogorov complexity. If all finite binary strings can be considered as the output of computer programs, the probability to observe \(s\) is given by Equation (1):  
\[P(s)\propto 2^{-K(s)} \tag{1}\]  
Here, \(P(s)\) is the probability to observe \(s\), and \(K(s)\) its Kolmogorov complexity. \(P(s)\) is called the semimeasure universal for the class of enumerable semimeasures. Like [24] we'll just call it the universal distribution. The universal distribution proposes the _a priori_ probability of a binary string \(s\), when the sample is generated by randomly picked out programs. This _a priori_ probability should not be attributed to any other theoretical assumption. For financial researchers, it is another \"natural\" distribution, like the normal or uniform one, that delivers no information on the validity of economic theories. To make the universal distribution operational, one needs the Kolmogorov complexity of \(s\). While, despite its general acceptation, Kolmogorov complexity of a finite string cannot be calculated directly  
Figure 1: On the definition of Kolmogorov complexity. (Because of the famous halting problem [25], for more detail see [26]). Compression tools can be used to estimate the Kolmogorov complexity of long finite strings. However, as these tools need long data to be efficient, the Kolmogorov complexity of a short string has been difficult to estimate for a long time. Reference [27] make a progress in this direction. These authors code all existing 5-state programs, and estimate the appearing frequency of each finite string at the output, i.e., \(P(s)\) in Equation (1). According to \(P(s)\), reference [27] give a complexity ranking of all binary strings under 7 bits. This ranking is confirmed by 5-state programs in [20]. Based on this ranking, one can verify the validity of the universal distribution with data from physical world. Reference [28] made a first application of the universal distribution in Psychology. To sum up, there are two types of randomness: the normal randomness defined in a mathematical framework and the algorithmic one coming from computer science. Both describe the \"natural\" state of physical world and neither should be attributed to economic assumptions. In the next section, we will conduct an empirical study to verify the distribution of financial volatilities. We will show to which degree daily volatilities can be ranked by their complexity. This result explains, at least to some extent, the general presence of VC in financial time series.