We have seen that there is strong empirical and theoretical evidence supporting the conclusion that the volatility of log price changes of a financial asset is a time-dependent stochastic process. In this chapter we discuss an approach for describing stochastic processes characterized by a time-dependent variance (volatility), the ARCH processes introduced by Engle in 1982 [50]. ARCH models have been applied to several different areas of economics. Examples include (i) means and variances of inflation in the UK, (ii) stock returns, (iii) interest rates, and (iv) foreign exchange rates. ARCH models are widely studied in economics and finance and the literature is huge. They can also be very attractive for describing physical systems.

ARCH models are simple models able to describe a stochastic process which is locally nonstationary but asymptotically stationary. This implies that the parameters controlling the conditional probability density function ft(x) at time t are fluctuating. However, such a ‘local’ time dependence does not prevent the stochastic process from having a well defined asymptotic pdf P(x).

ARCH processes are empirically motivated discrete-time stochastic models for which the variance at time t depends, conditionally, on some past values of the square value of the random signal itself. ARCH processes define classes of stochastic models because each specific model is characterized by a given number of control parameters and by a specific form of the pdf, called the conditional pdf, of the process generating the random variable at time t.

In this chapter we present some widely used ARCH processes. We focus our attention on the shape of the asymptotic probability density function and on the scaling properties observed.