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

ADVANCED RANDOM WALK MODELS

The models we described so far provide building blocks for representing the asset price dynamics. However, observed real-world asset price dynamics has features that cannot be incorporated in these basic models. For example, asset prices exhibit correlation—both with each other, and with themselves over time. Their volatility typically cannot be assumed constant. This section reviews several techniques for making asset price models more realistic depending on observed price behavior.

Correlated Random Walks

So far, we have discussed models for asset prices that assume that the dynamic processes for the prices of different assets evolve independently of each other. This is an unrealistic assumption—it is expected that market conditions and other factors will have an impact on the prices of groups of assets simultaneously. For example, it is likely that stock prices for companies in the oil industry will generally move together, as will stock prices for companies in the telecommunications industry.

The argument that asset prices are codependent has theoretical and empirical foundations as well. If asset prices were independent random walks, then large portfolios would be fully diversified, have no variability, and therefore be completely deterministic. Empirically, this is not the case. Even large aggregates of stock prices, such as the S&P 500, exhibit random behavior.

If we make the assumption that log returns are jointly normally distributed, ...