Alternate entropy measure for assessing volatility in financial markets

We propose two alternate information theoretical approaches to assess non-Gaussian fluctuations in the return dynamics of financial markets. Specifically, we use superinformation, which is a measure of the disorder of the entropy of time series. We argue on theoretical grounds on its usefulness and show that it can be applied effectively for analyzing returns. A study of stock market data for over five years has been carried out using this approach. We show how superinformation helps to identify and classify important signals in the time series. The financial crisis of 2008 comes out very clearly in the superinformation plots. In addition, we introduce the super mutual information. Distinct super mutual information signatures are observed that might be used to mitigate idiosyncratic risk. The universality of our approach has been tested by carrying out the analysis for the 100 stocks listed in S&P100 index. The average superinformation values for the S&P100 stocks correlates very well with the VIX.

Figure 1

Superinformation for daily “returns” $(+1,0,-1)$ as given in Eq. (9) for (a) Citigroup, (b) Exxon, (c) Immunomedics, and (d) Lehman Brothers. The $y$ axis represents the $r$%.

Figure 2

Superinformation for daily “returns” $(+1,0,-1)$ as given in Eq. (10) for (a) Citigroup, (b) Exxon, (c) Immunomedics, and (d) Lehman Brothers. The $y$ axis represents the $r$%.

Figure 3

Superinformation for daily “returns” $(+1,0,-1)$ as given in Eq. (10) for 100 stocks listed in S&P100. The $y$ axis represents 100 different stocks. The stocks are sorted according to the occurrence of their peak value of $H_s$.

Figure 4

Super mutual information ($K=1)$ for daily “returns” $(+1,0,-1)$ as given in Eq. (9) for (a) Citigroup, (b) Exxon, (c) Immunomedics, and (d) Lehman Brothers. The $y$ axis represents the $r$%.

Figure 5

(a) The time series for VIX and the average superinformation for S&P100 (scaled by arbitrary factor). The correlation coefficient = 0.819 with $p$ value of 2.2 ×10${}^{-16}$. The median of the superinformation values correlates significantly with a correlation coefficient of 0.812. (b) The corresponding scatter plot.

Figure 6

Linear regression analysis on the superinformation values of S&P100 stocks. Note that the normalized coefficients are both positive and negative. The companies corresponding to some of the peaks are marked. The $p$ value is 3.4×10${}^{-15}$.

Figure 7

Plot of normalized GARCH (1,1) volatility measure, ${\sigma }_{\mathrm{av}}$ (dotted line), and SIV (continuous line) for $r=1$% for (a) Exxon and (b) Immunomedics.

Figure 8

(a) Superinformation for daily “returns” $(+1,0,-1)$ as given in Eq. (10) for Apple Inc. (b) Plot of normalized GARCH (1,1) volatility measure, ${\sigma }_{\mathrm{av}}$ (dotted line) and SIV (continuous line) for $r=1$%.