# An Algorithmic Look at Financial Volatility
## 5 Algorithmic Randomness and GARCH Simulations
### The Universal Distribution and Volatility Autocorrelation  
If GARCH simulated data cannot be transformed into universally distributed trading weeks, can our reported algorithmic structure result in volatility autocorrelation? A negative answer to this question would indicate a total irrelevance of the universal distribution to VC's explanation.  
In this section, we show that algorithmic structure reported in our study could be responsible for volatility autocorrelation, and volatility autocorrelation could be solely another way to present the universal distribution of binary weeks.  
To do this, we start by simulating 20,000 successive binary weeks. As each week consists of 5 trading days, this will lead to 100,000 daily returns. The binary weeks' generation depends on the following distribution law:  
Figure 3: The frequency of binary trading weeks simulated from GARCH (1,1).  
Like what we exposed in Table 2, we distinguish three classes of binary weeks: (I) the most frequent class consisting of 2 possible binary strings, (II) the less frequent one consisting of 10 binary weeks, (III) and the least frequent group with the others. The probability associated with each binary week is calculated by taking the average frequency of their corresponding group in Table 2. For example, we attribute the same probability to\"00000\" and \"11111\" that is fixed at \((0.07072+0.06692)/2=0.0692\). In the same way, we set the probability of the 10 less frequent strings to be \((0.03304+0.0368+0.03348+0.0389+0.03568+0.03518+0.03436+0.03192+0.0358+0.032)/10=0.0352\).  
We make 20,000 random draws following the distribution described in Table 3. The 20,000 \(\times\) 5 = 100,000 binary trading days obtained like this will be transformed into a volatility series following three steps:  
* Firstly, we simulate a 100,000 daily returns from an \(i.i.d.N(0.02,0.2)\). We take the absolute value of daily returns to make a \(i.i.d\) volatility series. Figure 4a gives the histogram of the simulated returns. Figure 4b illustrates the independence of successive volatilities on reporting their PACF. Remark here that no significant autocorrelation can be noted in the PACF.
* Then, all simulated volatilities are divided into 2 subsets: those under the sample median and those above it. We keep exactly the same emerging order in the 2 volatility subsets.
* Always keeping the same appearing order, \"0\"s (resp. \"1\"s) in the binary volatility series take volatility values from the under (resp. above) median subset. This obtained real-number volatility series is subjected to autocorrelation tests. Figure 4c,d export the ACF and PACF of the universally distributed volatility sequence. They expose a significant autocorrelation of successive volatilities. This result indicates that the universal distribution of binary weeks could account for the volatility autocorrelation reported by econometric tools. This simulation has been repeated 10 times, and at each time, we get similar results.  
Figure 4: The universal distribution results in autocorrelation of successive volatilities. (**a**) Histogram of the 100,000 simulated returns; (**b**) PACF of the simulated volatility series; (**c**) ACF of universally distributed data; (**d**) PACF of universally distributed data.  
In this section, we attempt to set out the relation between [PERSON]'s universal distribution and the volatility autocorrelation captured by econometric tests. This relation has been investigated in 2 directions: (1) GARCH simulated autocorrelation cannot be the origin of the universal distribution of binary weeks, (2) universal distributed binary weeks do lead to autocorrelations. This result is somewhat expected by the researchers: the universal distribution highlights the origin of VC, while autocorrelation tests is, for us, solely a description of the stylized fact. Algorithmic structure reported in our study gives a more fundamental explanation of VC.