Informal Assessment of the Model of Stock Prices as a Geometric Random Walk.

  A common starting point for many theories in economics and finance is that the price of the financial commodity (e.g. stock price) is a geometric Gaussian random walk. That is, if the stock price at time t is x_t, then this is a realization of a stochastic process X_t where $\log X_t$ = Z_t is a Gaussian random walk, so  

$$\log X_t \; = \; Z_t \; = \; Z_{t-1} + a_t \; = \; Z_0 \, + \, \sum_{j=1}^t a_j ,$$ (1)

where the a_t's are a $N(\mu,\sigma^2)$ white noise and Z_t-1 is independent of all future a_t's. For instance, the so-called Black-Scholes theory (which resulted in Nobel Prizes for its inventors) was based on an assumption which implies the geometric random walk (they assumed a geometric Brownian motion for the continuous time process, which means that if it is sampled at discrete times one will see a geometric random walk with Gaussian steps). Here, we check that assumption with some informal data analyses.

The time series we will look at is the Dow-Jones index on 292 trading days ending 26 August 1994 which I got from an web site that seems to be defunct now (these things come and go so quickly).

In Section 2, we read these data into Splus and plot the series, logged series, differenced logged series, and the sample autocorrelation function. This provides reasonable evidence for the hypothesis.

In Section 3, we perform a little test of the geometric random walk model by constructing simulated data that would have the same distribution as the original data if the model is correct. We then present 20 plots: 19 are with simulated data and the 20th is of the real data. The real series is in a random place in the 20 plots. We challenge the reader to find the real series from among the plots.

In Section 4 we compute the periodogram (sample spectrum) and do a formal test for white noise based on the analysis of variance (ANOVA) of this sample spectrum. We also test for the presence of a day of week effect which would correspond to a period of 5 in the data.