ARCH AND GARCHVAIBHAV GUPTAMIB, DOC, DSE, DU

REFS RISK AND VOLATILITY: ECONOMETRIC MODELSAND FINANCIAL PRACTICENobel Lecture, December 8, 2003Robert F. Engle III 

INTRODUCTION Advantage of knowing about risks: You can change

your behaviour to avoid it. To avoid all risks would be impossible: no flying, no

driving, no walking, eating, drinking, no to sunshine. To some who are obsessed with a early morning bath, NO BATH as well.

There are some risks we choose to take because the benefits exceed the costs.

Optimal behaviour takes risks that are worthwhile. This is central paradigm to finance. Thus we optimize our behaviour, in particular our

portfolio, to maximize rewards and minimize risks.

SIMPLE CONCEPT OF RISK CAN MEAN A LOT OF NOBEL CITATIONS Markowitz (1952) and Tobin (1958)

associated risk with the variance in the value of a portfolio: From the avoidance of risk they derived optimizing portfolio and banking behaviour. (Nobel Prize 1981)

Sharpe (1964) developed the implications when all investors follow the same objectives with the same information. This theory is called the Capital Asset Pricing Model or CAPM, CAPM, and shows that there is a natural relation between expected returns and variance. (Nobel Prize 1990)

Black and Scholes (1972) and Merton (1973) developed a model to evaluate the pricing of options. (1997 Nobel Prize)

Typically the square root of the variance, called the volatility, was reported. They immediately recognized that the volatilities were changing over time.

A simple approach, sometimes called historical volatility, was and widely used. (sample standard deviations over a short period)

What is the right period: Too long: Not relevant for today Too short: Very noisy Furthermore, it is volatility over a future

period that should be considered as risk, forecast also needed in the measure of today.

Theory of dynamic volatility is needed: ARCH

Until the early 80s econometrics had focused almost solely on modelling the means of series, i.e. their actual values. Recently however we have focused increasingly on the importance of volatility, its determinates and its effects on mean values.

  A key distinction is between the conditional and

unconditional variance.  the unconditional variance is just the standard

measure of the variance 

var(x) =E(x -E(x))2

the conditional variance is the measure of our uncertainty about a variable given a model and an information set. 

cond var(x) =E(x-E(x| ))2 this is the true measure of uncertainty

mean

variance

Conditional variance

Stylised Facts of asset returns

i) Thick tails, they tend to be leptokurtic

ii)Volatility clustering, Mandelbrot, ‘large changes tend to be followed by large changes of either sign’

iii)Leverage Effects, refers to the tendency for changes in stock prices to be negatively correlated with changes in volatility.

iv)Non-trading period effects. when a market is closed information seems to accumulate at a different rate to when it is open. eg stock price volatility on Monday is not three times the volatility on Tuesday.

v) Forcastable events, volatility is high at regular times such as news announcements or other expected events, or even at certain times of day, eg less volatile in the early afternoon.

vi) Co-movements in volatility. There is considerable evidence that volatility is positively correlated across assets in a market and even across markets

Engle(1982) ARCH Model 

Auto-Regressive Conditional Heteroscedasticity

2

1

22

2

)(

),0(~

L

NXY

q

iitit

ttttt

22

tttdefine

tt

L 22 )(

an AR(q) model for squared innovations.

note as we are dealing with a variance

ialli 00

even though the errors may be serially uncorrelated they are not independent, there will be volatility clustering and fat tails.

if the standardised residuals

tttz /

are normal then the fourth moment for an ARCH(1) is

13)31/()1(3)(/)( 222224 ifEE tt

VOLATILITY Volatility – conditional variance of the

process Don’t observe this quantity directly (only one

observation at each time point) Common features

Serially uncorrelated but a depended process Stationary Clusters of low and high volatility Tends to evolve over time with jumps being rare Asymmetric as a function of market increases or

market decreases

THE BASIC MODELS Consider a process r(t) where

)()()( tattr

))1(|)(()( tFtrEt

p

j

q

kkj ktajtrt

1 10 )()()(

))1(|)(()(2 tFtrVart

Conditional mean evolves as an ARMA process

How does the conditional variance evolve?

MODELING THE VOLATILITY Evolution of the conditional variance follows

to basic sets of models The evolution is set by a fixed equation (ARCH,

GARCH,…) The evolution is driven by a stochastic equation

(stochastic volatility models). Notation:

a(t)=shock or mean-corrected return; is the positive square root of the volatility

)(t

ARCH MODEL We have the general format as before The equation defining the evolution of the

volatility (conditional variance) is an AR(m) process.

)()()( ttta

)()1()( 2210

2 mtatat m

Why would this model yield “volatility

clustering”?

BASIC PROPERTIES ARCH(1)

0)))(()([

))]1(|)()(([))]1(|)(([)]([

tEtEtFttEE

tFtaEEtaE

Unconditional mean is 0.

BASIC PROPERTIES, ARCH(1)

)1/()]([)]([

)]1([)]1([

)]1([

)]([0

)]1(|)(([))]1(|)(([)]([

10

10

10

210

210

2

taVartaVartaVartaE

taE

tE

tFtaVarEtFtaEVartaVar

Unconditional variance

What constraint does this

put on 1?

BASIC PROPERTIES OF ARCH 01<1 Higher order moments lead to additional constraints

on the parameters Finite positive (always the case) fourth moments

requires0 1

2<1/3 Moment conditions get more difficult as the order

increases – see general framework of equation 3.6 Note – in general the kurtosis for a(t) is greater than 3

even if the ARCH model is built from normal random variates.

Thus the tails are heavier and you expect more “outliers” than “normal”.

ARCH ESTIMATION, MODEL FITTING AND FORECASTING MLE for normal and t-dist ’s is given on pages 88 and

89. The full likelihood is very difficult and thus the

conditional likelihood is most generally used. The conditional likelihood ignores the component of

the likelihood that involves unobserved values (in other words, obs 1 through m)

MLE for joint estimation of parameters and degree of the t-distribution is given.

Model selection Fit ARMA model to mean structure Review PACF to identify order of ARCH Check the standardized residuals – should be WN

Forecasting – identical to AR forecasting but we forecast volatility first and then forecast the process.

GARCH MODEL Generalize the ARCH model by including an

MA component in the model for the volatility or the conditional variance.

m

j

s

kkj ktjtat

ttta

1 1

220

2 )()()(

)()()(

m

j

s

kkj ktjtat

ttta

1 1

220

2 )()()(

)()()(

Proceed as before – using all you learned from ARMA models.