The Theory and Practice of Investment Management: Asset Allocation, Valuation, Portfolio Construction, and Strategies, Second Edition

ARITHMETIC RANDOM WALKS

Instead of assuming that at each step the asset price can only move up or down by a certain multiple with a given probability, we could assume that the price moves by an amount that follows a normal distribution with mean μ and standard deviation σ. In other words, the price for each period is determined from the price of the previous period by the equation

where is a normal random variable with mean 0 and standard deviation σ. We will also assume that the random variable , describing the change in the price in one time period is independent of the random variables describing the change in the price in any other time period.⁸⁰ A sequence of independent and identically distributed (IID) random variables ,... with zero mean and finite variance σ⁷⁸ is sometimes referred to as white noise.

EXHIBIT 6.3 Binomial Distribution

EXHIBIT 6.4 Standard Normal Distribution

The movement of the price expressed through the equation above is called an arithmetic random ...