As we have seen in previous chapters, financial time series often appear to be well approximated by random walks. The relationship between random walks and the theory of efficient capital markets was briefly discussed in chapter 1, where it was argued that the random walk assumption that asset price changes are independent is usually too restrictive to be consistent with a reasonably broad class of optimising models: what is in fact required is that a variable related to the asset price is a martingale.

Martingales and random walks are discussed formally in section 1, with tests of the random walk hypothesis being the subject of section 2. The relaxation of the assumption that changes in a time series must be independent and identically distributed allows the possibility of examining non-linear stochastic processes and the remainder of the chapter thus introduces various non-linear models that are now used regularly in the analysis of financial time series. Stochastic variance models are discussed in section 3, ARCH processes in section 4, and bilinear and other nonlinear models in section 5, including artificial neural networks and chaotic models. Finally, section 6 looks at some tests for non-linearity.

Martingales, random walks and non-linearity

A martingale is a stochastic process that is a mathematical model of a ‘fair game’. The term martingale, which also denotes part of a horse's harness or a ships rigging, refers in addition to a gambling system in which every losing bet is doubled, a usage that may be felt to be rather apposite when considering the behaviour of financial data!