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Register a new userStock market volatility: An approach based on Tsallis entropy
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One of the major issues studied in finance that has always intrigued, both scholars and practitioners, and to which no unified theory has yet been discovered, is the reason why prices move over time. Since there are several well-known traditional techniques in the literature to measure stock market volatility, a central point in this debate that constitutes the actual scope of this paper is to compare this common approach in which we discuss such popular techniques as the standard deviation and an innovative methodology based on Econophysics. In our study, we use the concept of Tsallis entropy to capture the nature of volatility. More precisely, what we want to find out is if Tsallis entropy is able to detect volatility in stock market indexes and to compare its values with the ones obtained from the standard deviation. Also, we shall mention that one of the advantages of this new methodology is its ability to capture nonlinear dynamics. For our purpose, we shall basically focus on the behaviour of stock market indexes and consider the CAC 40, MIB 30, NIKKEI 225, PSI 20, IBEX 35, FTSE 100 and SP 500 for a comparative analysis between the approaches mentioned above.
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We investigate the probability distribution of the volatility return intervals $tau$ for the Chinese stock market. We rescale both the probability distribution $P_{q}(tau)$ and the volatility return intervals $tau$ as $P_{q}(tau)=1/bar{tau} f(tau/bar{tau})$ to obtain a uniform scaling curve for different threshold value $q$. The scaling curve can be well fitted by the stretched exponential function $f(x) sim e^{-alpha x^{gamma}}$, which suggests memory exists in $tau$. To demonstrate the memory effect, we investigate the conditional probability distribution $P_{q} (tau|tau_{0})$, the mean conditional interval $<tau|tau_{0}>$ and the cumulative probability distribution of the cluster size of $tau$. The results show clear clustering effect. We further investigate the persistence probability distribution $P_{pm}(t)$ and find that $P_{-}(t)$ decays by a power law with the exponent far different from the value 0.5 for the random walk, which further confirms long memory exists in $tau$. The scaling and long memory effect of $tau$ for the Chinese stock market are similar to those obtained from the United States and the Japanese financial markets.
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The distribution of the return intervals $tau$ between volatilities above a threshold $q$ for financial records has been approximated by a scaling behavior. To explore how accurate is the scaling and therefore understand the underlined non-linear mechanism, we investigate intraday datasets of 500 stocks which consist of the Standard & Poors 500 index. We show that the cumulative distribution of return intervals has systematic deviations from scaling. We support this finding by studying the m-th moment $mu_m equiv <(tau/<tau>)^m>^{1/m}$, which show a certain trend with the mean interval $<tau>$. We generate surrogate records using the Schreiber method, and find that their cumulative distributions almost collapse to a single curve and moments are almost constant for most range of $<tau>$. Those substantial differences suggest that non-linear correlations in the original volatility sequence account for the deviations from a single scaling law. We also find that the original and surrogate records exhibit slight tendencies for short and long $<tau>$, due to the discreteness and finite size effects of the records respectively. To avoid as possible those effects for testing the multiscaling behavior, we investigate the moments in the range $10<<tau>leq100$, and find the exponent $alpha$ from the power law fitting $mu_msim<tau>^alpha$ has a narrow distribution around $alpha eq0$ which depend on m for the 500 stocks. The distribution of $alpha$ for the surrogate records are very narrow and centered around $alpha=0$. This suggests that the return interval distribution exhibit multiscaling behavior due to the non-linear correlations in the original volatility.
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