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Application: The Square Root of Time Rule for the Simple Wiener Process

The Wiener process follows 0(0, 1). When the coefficient ct — o is constant, the variable is again stationary. Consider any variable that has a constant variance per unit of time, with independent random increments at each time point. This could be the case for a stock return, notably. Under such a process, the uncertainty measured by the standard deviations increases as the square root of time.

Let S represent an asset return with zero drift. The market efficiency hypothesis postulates that all prices reflect all information available at the initial date. Consider two consecutive time intervals of duration 1 (unit of time). Two consecutive random changes between two consecutive time points are independent. Assume that the volatility does not depend on time. The random term over two time intervals has mean zero since it sums up two random variations at each interval with zero mean. The variance over two time intervals is the sum of the variances, or 2c2, since variances of independent variables are additive. Hence the standard deviation over the two time intervals is cW2. This is the rationale for so-called "square root of time rule for uncertainty," which says, loosely speaking, that the uncertainty measured by the standard deviation of returns increases like the square root of time. Generalizing to several time points, up to T, the variance of the summation of the diffusion terms is the summation of the variances of each change.

The standard deviation unit has to be consistent with the time unit. If t — 1 year, then o is the annual standard deviation, and so on. Under these assumptions, the process is "stationary," meaning that it remains the same though time.

The Generalized Wiener Process

The generalized Wiener process includes a drift and a constant standard deviation per unit of time. This simple process considers that the drift, a, and the coefficient of the innovation term, b, are constant. The process has now the form:

Taking the expectation, E[dx] — a dt, is the constant drift. The incremental change of value is constant per unit of time. The constant drift represents a trend. The standard deviation is bc^lt. The time path is a steady trend with random variations at each time point around this trend.

The discrete equivalent of this equation serves for simulating the time path of the process. The horizon Tis, divided into n small intervals of same length At — Tin. The continuous equation takes a discretized form such that all small variations are related by the same relation:

The random innovation term is C£^At. A time path for S can be simulated by incrementing the variable by a At at each step and adding a random number, normally distributed with mean zero and standard deviation cr/Af.

 
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