Monte Carlo Simulations of the NASDAQ Composite

Abstract

In previous work it was shown that annual returns for the NASDAQ Composite Index in the 36 year period since its inception in 1971 do not appear to be randomly distributed in the sense that Monte Carlo simulations which sampled a stationary distribution function derived from observed daily returns failed to reproduce the observed annual results. In particular, the data exhibited a considerably higher probability of loss than predicted by the simulations. An attempt was also made to assess whether memory might account for the discrepancy. Calculation of the Hurst exponent via the rescaled range statistic was inconclusive, however, since the values computed for the NASDAQ data differed only slightly from those calculated from a random walk with daily returns drawn from a Gaussian distribution with the same mean and standard deviation. Two innovations are introduced in the present work. First, sampling is accomplished via a non-stationary Gaussian process in which the average daily return and standard deviation are themselves derived from randomly sampling backward facing moving average distributions derived over the 36-year data period. It is shown that the resultant cumulative distribution function (CDF) of annual returns is invariant from that obtained with stationary sampling. Next, a one-parameter memory function is introduced directly in the Monte Carlo simulations, which allows explicit evaluation of memory effects on the computed CDF of annual returns. The results indicate that, as long as the daily returns are drawn from the distribution derived from the observed NASDAQ data, short-term memory has no significant effect on the predicted distribution of annual returns. With sufficient long-term memory, the predicted probability of loss can be shown to match the data but only at the expense of overestimating the probability of gain. Moreover, analysis of streak statistics (the number of consecutive days of increasing or decreasing returns) shows that the results predicted with increased long-term memory are completely inconsistent with the observed data.