Statistical Properties of MRW Process

In order to demonstrate distinctive feature of MRW process, one can compare its realization with realization of geometric Brownian motion (original random walk model of Bachelier), which sample increments and path are shown in Figs. 1 and 2. Sample realizations of increments and path of MRW are shown in Figs. 3 and 4.

Comparing Fig. 3 with Fig. 1, one can notice significant differences in the way, which each process goes. When dynamics of increments of random walk (Fig. 1) are very regular and one can not observe large deviations from the mean value, the dynamics of MRW (Fig. 3) is much more intermittent, one can easily spot volatility clustering and large excursions (extreme events).

 o О І I I , , , I ‘ 0 2000 4000 6000 8000 n  Fig. 3 Increments of MRW process for A2 = 0.06, a = 7.5 • 10~5 and L = 1024

Presence of the heavy tails of pdf for MRW can be shown more clearly with the ranking plot (see Fig. 5) for various aggregation level. The interval of scales 10 3 ІДгХдг [t] ^ 1 illustrates the tails of pdf which decay much slower than for

the normal distribution that is also presented on the plot for comparison. In other words, the probability of observing extremely large increment (return) for MRW is much larger than for the normal distribution where the probability of observing a value larger than three-four standard deviations is essentially zero.

Figure 5 also illustrates another stylized fact, namely—aggregational gaussianity. One can see from the Fig. 5, that slope of the tail line tends to the slope of the tail line for normally distributed data, when aggregation level (which is defined as a number of consecutive increments of initial MRW process that are summed to obtain single

increments  increment of aggregated process) rises. For instance, for aggregation level equals 4096 tail of the distribution converges to Gaussian distribution.

Volatility clustering, that one can observe in Fig. 3 is a result of the presence of long memory in volatility. In order to quantify it we have considered four different measures of the volatility. The first one is the simplest squared values of returns   (increments). Second is the definition of volatility as a standard deviation of returns in a rolling window of size nt:

Third is the widely-used volatility estimator as a Exponentially-Weighted Moving Average (EWMA), which can be defined as

oi = ^ Астг2_! + (1 — A) rf2_ і, (18)

where A є [0,1] is the rate of decay of the exponential weight within time window. Finally, we have also considered Muller estimator of the volatility (Muller 2000) which is similar to the EWMA, but involves recursion both of lagged and current squared returns:

On M = f 1 M + (u — M) rl_і + (1 — u) rl; (19)

where a = (ln—ln_i)/r is the rate of decay of the exponential weight; m = e_a is an exponential weight itself and u = (1 — M)/a. Autocorrelation functions computed for above estimators of volatility are presented in Fig. 6. As one can see from the Fig. 6, autocorrelation of all proxies of volatility is signif­icantly non zero in a very wide range (of the lags up to 1000 and more). Compared to Fig. 7 Log-log plot of moments of increments (4) calculated using the MRW sample of length 217 for A2 = 0.06, ct = 7.5 • 10_5, L = 2048 and q = 1,2, 3, 4, 5. Dashed lines correspond to linear fit of dependency (4)

autocorrelation for squared returns, the rate of decay of autocorrelations for standard deviation, EWMA and Muller estimator decay much slower due to the fact, that above the three estimators perform recursive procedures for volatility computation. These recursion-based estimators capture the features of volatility behavior better than squared returns.

In order to illustrate the scale invariance in simulated MRW sample, one have to consider moments of increments of the realization (4). As described above, the presence of scale invariance is qualified with the power law behavior of the the moments (4). As one can see from the Fig. 7 this holds for the analyzed MRW process, as the absolute moments Mq (l) for all q has linear or close to linear (for q = 5) form in log-log scale, which tells about the presence of power law dependency in the ordinary scale.