5.4 Geometric Random Walks

The essential idea underlying the random walk for real processes is the assumption of mutually independent increments of the order of magnitude for each point of time. However, economic time series in particular do not satisfy the latter assumption. Seasonal fluctuations of monthly sales figures for example are in absolute terms significantly greater if the yearly average sales figure is high. By contrast, the relative or percentage changes are stable over time and do not depend on the current level of $X_t$. Analogously to the random walk with i.i.d. absolute increments $Z_t = X_t - X_{t-1}$, a geometric random walk $\{ X_t; \, t \ge 0 \}$ is assumed to have i.i.d. relative increments

$$R_t = \frac{X_t}{X_{t-1}} \, , \quad t = 1, 2, \ldots \, .$$

For example, a geometric binomial random walk is given by

$$X_t = R_t \cdot X_{t-1} = X_0 \cdot \Pi^t_{k=1} \, R_k$$ (5.5)

where $X_0, R_1, R_2, \ldots$ are mutually independent and for $u > 1, \, d < 1\ :$

$$\P(R_k = u) = p\ , \, \, \P(R_k = d) = 1-p\ .$$

Given the independence assumption and $\mathop{\text{\rm\sf E}}[R_k] = (u-d) p+d$ it follows from equation (4.5) that $\mathop{\text{\rm\sf E}}[X_t]$ increases or decreases exponentially as the case may be $\mathop{\text{\rm\sf E}}[R_k] > 1$ or $\mathop{\text{\rm\sf E}} [R_k] < 1$:

$$\mathop{\text{\rm\sf E}}[X_t] = \mathop{\text{\rm\sf E}}[X_0] \cd... ...\rm\sf E}}[R_1])^t = \mathop{\text{\rm\sf E}}[X_0] \cdot \{ (u - d) p+d \}^t .$$

If $\mathop{\text{\rm\sf E}}[R_k] = 1$ the process is on average stable, which is the case for

$$p = \frac{1-d}{u-d} \, .$$

For a recombining process, i.e.  $d = \frac{1}{u},$ this relationship simplifies to

$$p = \frac{1}{u+1} \, .$$

Taking logarithms in equation (4.5) yields:

$$\ln X_t = \ln X_0 + \sum^t_{k=1} \, \ln R_k\ .$$

The process $\tilde{X}_t = \ln X_t$ is itself an ordinary binomial process with starting value $\ln X_0$ and increments $Z_k = \ln R_k$ for which hold:

$$\P(Z_k = \ln u) = p, \quad \P(Z_k = \ln d) = 1-p\ .$$

For $t$ large, $\tilde{X}_t$ is approximately normally distributed, i.e.  $X_t = \exp (\tilde{X}_t)$ is approximately lognormally distributed.