4.1 Standard Model of Asset Price Behavior and Reality
Quantitative understanding of asset risk and return for investment, trading, and risk management involves asset price dynamics, the behavior of asset prices over time, discussed in this chapter, and equilibrium asset prices, the configuration of asset prices at a point in time, which we discuss in the next.
The geometric Brownian motion model, a continuous stochastic process based on the discrete random walk, is the baseline for asset price dynamics in financial modeling. In it, asset prices have a tendency to move up or down at random. Proportional changes in the prices i.e., logarithmic returns, are normally distributed. The martingale property states that successive and nonoverlapping changes in an asset price are independent of one another and of its initial position ; knowing what the price was last week adds no useful information to knowing what it is now. It can include a drift, or tendency to move up or down gradually over time with certainty, overlaid on the short-term fluctuations. Figure 4.1 displays simulated paths of a discrete process similar to a geometric Brownian motion.2
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; knowing what the price was last week adds no useful information to knowing what it is now. It can include a drift, or tendency to move up or down gradually over time with certainty, overlaid on the short-term fluctuations. Figure 4.1 displays simulated paths of a discrete process similar to a geometric Brownian motion.2