The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 346 Wednesday, February 4, 2004 12:59 PM

346 The Mathematics of Financial Modeling and Investment Management

The ARCH/GARCH Family of Models

The ARCH models were proposed by Engle^31 as a model of inflation.

The empirical fact behind ARCH models is the clustering of volatility

observed in many economic and financial series. If instantaneous volatil-

ity is defined as a hidden variable in a price model and estimated as the

variance of returns over relatively long periods, one finds periods of

high volatility followed by periods of low volatility and vice versa.

Note that a new strain of econometric literature deals with instanta-

neous volatility as an observed variable. The observability of volatility

is made possible by the availability of high frequency data. In this case,

there is a variety of models for the volatility process, in particular long-

memory fractional models.^32 We maintain the classical definition of vol-

atility as a hidden variable.

Engle proposed a model in the spirit of state-space modeling where

volatility is modeled by an autoregressive process and then injected mul-

tiplicatively in the price process. More precisely, the simplest ARCH

model is defined as follows:

xt βλxt – 1

2

= + zt

In the above equation, x is the process variable and the terms z form

an IID sequence. The ARCH model was extended by Bollerslev,^33 who

proposed the GARCH family of models. In the GARCH models, volatil-

ity is modeled as a more general ARMA process and then treated as

before:

xt = σtzt

p q

σ^22

t = β+ ∑λixti– + ∑δiσtj–

i = 1 j = 1

The key ingredients of ARCH modeling are an ARMA process for vol-

atility and a regressive process where volatility multiplies a white-noise

(^31) R.F. Engle, “Autoregressive Conditional Heteroscedasticity with Estimates of the
Variance of United Kingdom Inflation,” Econometrica 50 (July 1982), pp. 987–
1007.
(^32) T.G. Andersen, T. Bollerslev, F.X. Diebold, and P. Labys, “Modeling and Fore -
casting Realized Volatility,” Econometrica 71, 2003, pp. 529–626.
(^33) T. Bollerslev, “Generalized Autoregressive Conditional Heteroscedasticity,” Jour -
nal of Econometrics 31 (1986), pp. 307–327.