Figure 1

Returns, percentiles, and the excursion process. (a) IBM stock prices, every 5 min, during the 252 trading days in 2005. The opening (9:30 to 9:40 AM) and closing (3:50 to 4:00 PM) prices are excluded, leaving 75 prices per day (9:40 AM, 9:45 AM,$...$, 3:50 PM). (b) Intraday 5-min returns for the prices displayed in (a). There are $252\times 74=18648$ data points. (c) Returns, with the 10th and 90th percentiles superimposed. (d) Zoomed portion of (c) with 200 returns. The “excursion process” is the discrete time zero-one process that signals (with ones) returns above or below the selected percentiles.Reuse & Permissions

Figure 2

Geometric (0.2) and empirical waiting times. The empirical waiting-time distribution of 1-min returns of IBM stock in 2005 was compared with the geometric distribution with parameter 0.2. (left) Log plots for the geometric distribution and the empirical waiting-time distribution. The $x$ axis is the waiting times, and the $y$ axis is the log probabilities of the waiting times. (right) $P$-$P$ plots for the geometric distribution vs the empirical waiting-time distribution. The KS distance is the maximum horizontal (maximum vertical) distance between the $P$-$P$ curve [shown in blue (dark gray)] and the diagonal [shown in red (light gray)].Reuse & Permissions

Figure 3

Scale invariance. (top) Empirical waiting-time distributions captured from 30-s, 1-min, and 5-min returns of IBM in 2005. (bottom) $P$-$P$ plots for the three waiting-time distributions taken two at a time and their corresponding Kolmogorov-Smirnov distances.Reuse & Permissions

Figure 4

Histogram of KS distances from 2008. Each panel shows the histogram of Kolmogorov-Smirnov distances between excursion waiting-time distributions at different time scales in 2008 for approximately 300 stocks.Reuse & Permissions

Figure 5

Invariance to stock. Comparisons of excursion waiting-time distributions for 5-min returns between IBM and GPS in (top left) 2005 and (top right) 2008. (bottom) Histograms of KS distributions for all pairs of stocks show a breakdown of invariance across stocks in 2008 as compared to 2005. The table gives a summary of year-by-year comparisons of waiting-time distributions across stocks. With the exception of 2008, waiting times are nearly invariant to stock.Reuse & Permissions

Figure 6

Fractional Brownian motion and excursion waiting times. (left) For each Hurst index $H=0.76,0.77,...,0.89$ we generated 500 FBM samples and extracted 18 648 returns, matching the 18 648 returns in the 5-mine 2005 IBM data. The average KS distances between the FBM excursion waiting times and the empirical IBM waiting times are plotted. The best fit, with a KS of about 0.046, is at $H=0.81$. (right) $P$-$P$ plot of excursion waiting-time distribution for a sample from the best-fitting FBM vs the empirical IBM distribution. FBM overestimates the probabilities of short and very long waiting times.Reuse & Permissions

Figure 7

Exchangeability and time scale. The 5-min returns on IBM stock in 2005 were tested for their departure from exchangeability, as reflected in the excursion waiting-time distribution. (a) $P$-$P$ plot of the excursion waiting-time distribution of the IBM returns vs the geometric distribution (corresponding to the waiting time between successes in a Bernoulli sequence with a probability of 0.2 of success). The distributions would be nearly identical if the returns were exchangeable. (b) Trajectory of IBM prices from 9:45 AM to 3:45 PM, sampled every 5 min, for a randomly selected day in 2005. (c) Same starting price as in (b), but with the twelve 5-min returns in each of the six 1-h intervals randomly and independently permuted. Since the returns within a given hour are exactly preserved, the stock prices in (b) and (c) are the same at 10:45 AM and at each hour thereafter. The dynamics governing the trajectories in (b) and (c) are not apparently different. (d) Distribution of KS distances to the geometric distribution, obtained from 5000 surrogate return sequences corresponding to 5000 random permutations of the 18 648 IBM 5-min returns. The vertical red line marks the KS distance (0.131) of the original sequence of returns. (e) Test for local exchangeability. Surrogates were produced by independently permuting every disjoint 20-min block of four 5-min returns. The distribution of KS distances was again computed from 5000 surrogates. In general, tests employing larger time intervals produce still lower $p$ values. Thus, despite appearances, the evidence strongly points to a highly significant difference between the trajectories in (b) and (c). (f) The ensemble of surrogates derived from permutations of pairs of returns for every 10-min block are indistinguishable from the original sequence with respect to the departure of their excursion waiting-time distributions from geometric.Reuse & Permissions

Figure 8

Implied volatility generates exchangeable returns. Eight days of minute-by-minute 2008 Citigroup stock and put prices (strike price 22.5, maturing on 19 April 2008) were used to calculate the minute-by-minute implied volatility and to generate simulated minute-by-minute returns from a geometric Brownian motion with volatility function ($\sigma ={\sigma }_t$) equal to the implied volatility. (left) Minute-by-minute implied volatility. (middle) The excursion waiting-time distribution of the simulated returns closely resembles the geometric distribution, unlike the real 1-min returns for which the $P$-$P$ plot against the geometric is essentially identical to the one shown in the Fig. 7a (5-min returns of IBM). (right) Simulated returns were not distinguishable from exchangeable returns through the KS statistic, despite substantial fluctuations in the implied volatility at multiple time scales.Reuse & Permissions

Figure 9

Simulated 1-min IBM returns using GARCH. One-minute returns on IBM for all of 2005 were used to fit a GARCH model with autoregressive and moving average terms each of order 10 ($p=q=10$). (top left) Waiting-time distribution between excursions in the simulated returns was a near-perfect match to the empirical distribution. Results of permutation tests for (top right) global exchangeability and (bottom) local exchangeability (4-min intervals, bottom left, and 2-min intervals, bottom right) were essentially identical to the results for the real returns (not shown).Reuse & Permissions

Figure 10

Failure of the GARCH model to match waiting-time scale invariance of real returns. (left) $P$-$P$ plot matching excursion waiting-time distributions for the IBM 1-min to the IBM 5-min returns (2005). Distributions are nearly identical. (right) Same comparison using GARCH-generated 1-min returns, aggregated to make a record of 5-min returns. Although there is an excellent fit to the 1-min data (see Fig. 9), the model fails to scale across different return intervals.Reuse & Permissions

Figure 11

Interval time measured by number of trades. In 2005, there was an average of about 110 trades of IBM stock every 5 min. If market time were measured by the number of trades and were adequate to transform prices into a Lévy process, then returns over 110-trade intervals would be exchangeable. (left) Excursion waiting-time distribution for equal-market-time intervals (110 trades) does not match the geometric distribution. (right) Equal-market-time returns are not exchangeable, as evidenced by the distribution of KS values under permutations. Market time measured by volume instead of trades also fails to render returns exchangeable.Reuse & Permissions