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

KEY POINTS

• Models of asset dynamics include trees (such as binomial trees) and random walks (such as arithmetic, geometric, and mean-reverting random walks). Such models are called discrete when the changes in the asset price are assumed to happen at discrete time increments. When the length of the time increment is assumed infinitely small, we refer to them as stochastic processes in continuous time.

• The arithmetic random walk is an additive model for asset prices—at every time period, the new price is determined by the price at the previous time period plus a deterministic drift term and a random shock that is distributed as a normal random variable with mean equal to zero and a standard deviation proportional to the square root of the length of the time period. The probability distribution of future asset prices conditional on a known current price is normal.

• The arithmetic random walk model is analytically tractable and convenient; however, it has some undesirable features such as a nonzero probability that the asset price will become negative.

• The geometric random walk is a multiplicative model for asset prices—at every time period, the new price is determined by the price at the previous time period multiplied by a deterministic drift term and a random shock that is distributed as a lognormal random variable. The volatility of the process grows with the square root of the elapsed amount of time. The probability distribution of future asset prices conditional on ...