Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues

9

Other financial models: from ARMA to the GARCH family

The previous chapter dealt with stochastic processes, which consist of (returns) models involving a mixture of deterministic and stochastic components. By contrast, the models developed here present three major differences:

• These models are deterministic; since they are aiming to model a non-deterministic variable such as a return, the difference between the model output and the actual observed value is a probabilistic error term.
• By contrast with stochastic processes described by differential equations, these models are built in discrete time, in practice, the periodicity of the modeled return (daily, for example).
• By contrast with usual Markovian stochastic processes, these models incorporate in the general case a limited number of previous return values, so that they are not Markovian.

For a time series of past observations on the variable x up to t − 1, all these processes are of the form

where f(.) is linear.

9.1 THE AUTOREGRESSIVE (AR) PROCESS

Let us consider a series of past returns {r0, …, rt−1} or, in short, {rt}, of 0 mean, such as:

where t is the error term, also called “innovation”, 1 or “white noise”. In practice, ...