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We investigate scaling and memory effects in return intervals between price volatilities above a certain threshold $q$ for the Japanese stock market using daily and intraday data sets. We find that the distribution of return intervals can be approximated by a scaling function that depends only on the ratio between the return interval $tau$ and its mean $<tau>$. We also find memory effects such that a large (or small) return interval follows a large (or small) interval by investigating the conditional distribution and mean return interval. The results are similar to previous studies of other markets and indicate that similar statistical features appear in different financial markets. We also compare our results between the period before and after the big crash at the end of 1989. We find that scaling and memory effects of the return intervals show similar features although the statistical properties of the returns are different.
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We study the volatility time series of 1137 most traded stocks in the US stock markets for the two-year period 2001-02 and analyze their return intervals $tau$, which are time intervals between volatilities above a given threshold $q$. We explore the probability density function of $tau$, $P_q(tau)$, assuming a stretched exponential function, $P_q(tau) sim e^{-tau^gamma}$. We find that the exponent $gamma$ depends on the threshold in the range between $q=1$ and 6 standard deviations of the volatility. This finding supports the multiscaling nature of the return interval distribution. To better understand the multiscaling origin, we study how $gamma$ depends on four essential factors, capitalization, risk, number of trades and return. We show that $gamma$ depends on the capitalization, risk and return but almost does not depend on the number of trades. This suggests that $gamma$ relates to the portfolio selection but not on the market activity. To further characterize the multiscaling of individual stocks, we fit the moments of $tau$, $mu_m equiv <(tau/<tau>)^m>^{1/m}$, in the range of $10 < <tau> le 100$ by a power-law, $mu_m sim <tau>^delta$. The exponent $delta$ is found also to depend on the capitalization, risk and return but not on the number of trades, and its tendency is opposite to that of $gamma$. Moreover, we show that $delta$ decreases with $gamma$ approximately by a linear relation. The return intervals demonstrate the temporal structure of volatilities and our findings suggest that their multiscaling features may be helpful for portfolio optimization.
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.
Bid-ask spread is taken as an important measure of the financial market liquidity. In this article, we study the dynamics of the spread return and the spread volatility of four liquid stocks in the Chinese stock market, including the memory effect and the multifractal nature. By investigating the autocorrelation function and the Detrended Fluctuation Analysis (DFA), we find that the spread return is lack of long-range memory, while the spread volatility is long-range time correlated. Moreover, by applying the Multifractal Detrended Fluctuation Analysis (MF-DFA), the spread return is observed to possess a strong multifractality, which is similar to the dynamics of a variety of financial quantities. Differently from the spread return, the spread volatility exhibits a weak multifractal nature.
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.
In informationally efficient financial markets, option prices and this implied volatility should immediately be adjusted to new information that arrives along with a jump in underlyings return, whereas gradual changes in implied volatility would indicate market inefficiency. Using minute-by-minute data on S&P 500 index options, we provide evidence regarding delayed and gradual movements in implied volatility after the arrival of return jumps. These movements are directed and persistent, especially in the case of negative return jumps. Our results are significant when the implied volatilities are extracted from at-the-money options and out-of-the-money puts, while the implied volatility obtained from out-of-the-money calls converges to its new level immediately rather than gradually. Thus, our analysis reveals that the implied volatility smile is adjusted to jumps in underlyings return asymmetrically. Finally, it would be possible to have statistical arbitrage in zero-transaction-cost option markets, but under actual option price spreads, our results do not imply abnormal option returns.
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