10 - Diffusion and volatility

Summary

Many observed stochastic processes may produce occasional very extreme values or periods with exceptional amounts of variability. Both of these phenomena are known as volatility. (In econometrics and finance, this term often has the more restricted meaning of models with changing variance over time.)

Two common approaches to modelling volatile phenomena are

1. (i) to use distributions with heavy tails and

2. (ii) to allow the dispersion to depend on some measure of the previous variability about the location regression function.

We already have met some volatility models related to the second approach in Sections 7.3 and 7.4. Here, I shall consider both approaches. However, I shall first look at an important class of processes in which the dispersion increases in a deterministic way with time: diffusion processes.

Because, in most non-normal models, the variance is a function of the mean, a location parameter changing over time implies that the variability, or volatility, is also changing. Thus, the distinctiveness of specifically modelling the volatility is most relevant for members of the location-scale family, such as the normal, Cauchy, Laplace, and Student t distributions, where the location and dispersion can vary independently of each other.

Wiener diffusion process

If the variance of a stochastic process increases systematically over time, a diffusion process may provide an appropriate model. One question will be whether the variance can increase without limit or slowly approaches some constant maximum. I shall consider the first possibility in this section.

Theory

Brownian motion may be one of the most famous stochastic processes, perhaps because of its connection with the movement of small particles floating on water.