Time-scale effects on the gain-loss asymmetry in stock indices

The gain-loss asymmetry, observed in the inverse statistics of stock indices is present for logarithmic return levels that are over $2\%$, and it is the result of the non-Pearson-type autocorrelations in the index. These non-Pearson-type correlations can be viewed also as functionally dependent daily volatilities, extending for a finite time interval. A generalized time-window shuffling method is used to show the existence of such autocorrelations. Their characteristic time scale proves to be smaller (less than 25 trading days) than what was previously believed. It is also found that this characteristic time scale has decreased with the appearance of program trading in the stock market transactions. Connections with the leverage effect are also established.

Figure 1

(a) The investment horizon distributions of the original DJIA index. (b) The inverse statistics for the shuffled version of the index (is symmetric). The superimposed curves with the same color correspond to different permutations. The return level considered for the inverse statistics is five times larger than the daily volatility of returns: $\rho =5\sigma \approx 5\%$.

Figure 2

(a) The original time series of daily returns. (b) The time series of the randomly permuted daily returns considering time windows of $T=500$ trading days. We illustrate the original and new positions of two selected time windows.

Figure 3

(a) The investment horizon distributions of the original DJIA index. (b) The inverse statistics for the shuffled version of the index using time windows of $T=25$ trading days. The superimposed curves with the same color correspond to different permutations. The return levels considered for the inverse statistics are five times larger than the volatility of returns: $\left|\rho \right|=5\sigma \approx 5\%$.

Figure 4

(a) Positions of the ${\tau }_{\left|\rho \right|}^{*}\left(T\right)$ (dashed green) and ${\tau }_{-\left|\rho \right|}^{*}\left(T\right)$ (solid red) maxima after shuffling with $T$ time window for $\left|\rho \right|=5\sigma$ logarithmic return level, in comparison to the fully shuffled case, ${\tau }_{\rho }^{*}\left(1\right)=14$ days (solid black, horizontal). The dotted black and grey horizontal lines are corresponding to ${\tau }_{\left|\rho \right|}^{*}=24$ and ${\tau }_{-\left|\rho \right|}^{*}=11$ days, respectively, and are shown to guide the eyes. The inset shows the $w_{\pm }(T,1)$ dissimilarity parameters. (b) The corresponding asymmetry level $w\left(T\right)$ for different $n_p$ permutations, using $\mathrm{\Delta }{\tau }_{\rho }^{*}\left(\infty \right)=13$ days. The dashed vertical line denotes the $T=25$ shuffle window size; see also Fig. 3.

Figure 5

The scaling of the ${\tau }_{\rho }^{*}$ optimal investment horizons for positive (dashed green curves) and negative (solid red curves) $\rho$ return levels, using the original DJIA index (a) and the fully shuffled ($T=1$) index (b) on log-log scales. The dotted black and gray lines indicate the slopes fitted for $\left|\rho \right|>3\sigma$ return levels. The $\left|\rho \right|>3\sigma$ limit is illustrated by the thin vertical line. Scaling exponents are given in the legend. We also present a comparative plot with an inset showing the region of small return levels using linear scales on the axis (c): $\left|\rho \right|\in [\sigma ,8\sigma ]$.

Figure 6

Asymmetry remaining after shuffling with $T$ time window for different $\rho$ return levels in terms of the $\sigma$ volatility ($n_p=1000$).

Figure 7

The asymmetry measure of the DJIA index: $|1-w(T\left)\right|$ curves on a log-normal scale for different $\rho$ return levels in terms of the $\sigma$ volatility. Non-Pearson-type correlations decreasing as $e^{-T/\theta }$ for $T<30$ days ($T_{\infty }=1000$).

Figure 8

(a) The asymmetry parameter of the DJIA index as a function of the $T$ time window for human trading periods (1928–1980) and for program trading (1980–2011). (b) The characteristic times ${\theta }_h$ and ${\theta }_p$ measured from the exponential fits corresponding to human and program trading, respectively. For both pictures the volatility of returns is chosen to be five times the daily volatility: $\left|\rho \right|=5\sigma \approx 5\%$ and $T_{\infty }=1000$.

Figure 9

Normalized probability distributions of the positive and negative daily returns, $r_1\left(t\right)$, of the DJIA and the STT index. (a) Log-log scale, with the dotted gray and black lines indicating two power-law fits for the tail with exponents ${\beta }_1$ and ${\beta }_2$, respectively. Both distributions show a similar fat tail. (b) Log-$x^2$ scale, in comparison with two Gaussian distribution functions $\mathcal{N}(0,{\sigma }_1^2)$ and $\mathcal{N}(0,{\sigma }_2^2)$, respectively.

Figure 10

The scaling of the ${\tau }_{\rho }^{*}$ optimal investment horizons for positive (dashed green curve) and negative (solid red curve) $\rho$ return levels, using an artificial index with daily returns having a fat-tailed Student's T distribution (STT). Please note the logarithmic scales. The dotted black and gray lines indicate the slopes fitted for $\left|\rho \right|>3\sigma$ return levels, denoted by the thin vertical line. The scaling exponents are given in the legend.

Figure 11

The asymmetry parameter as a function of the $T$ time window for the DJIA, S&P500, and NASDAQ100 indices. The volatility of returns are chosen to be five times of the daily volatilities of the respective indices: $\left|\rho \right|=5\sigma$. The curves start to saturate in the same interval: $T\in [10,30]$ trading days, indicated by dashed vertical lines.

Figure 12

Normalized probability distributions of daily returns, $r_1\left(t\right)$, of the DJIA index for the human and program trading periods.