A common finding

in time series regression however is that the residuals are correlated with

their own lagged or previous values. Since this correlation is sequential in

time, it is then called serial correlation.

ARMA models are

linear in both mean and variance while the GARCH models are either linear in

mean but non linear in the variance or nonlinear in both mean and variance.

a. ARMA Models

The serial

correlations for the models which are linear in both mean and variance can take

the form of autoregression (AR), Moving Average (MA) or a combination of the

two (ARMA).

These models are

said to be stationary. Stationarity here implies that a time series has memory

depicted in the lags of the error/residual terms or past observations of the

variable under consideration.

·

In an AR model, the variable in the

current period is also related to the variable’s lag and is given by;

Where

is represent the order of the model and

is the disturbance term

with N(0,?2).

·

In an MA model, the variable

is affected by the

previous prediction errors

and is models as;

Where

is represent the order of the model

previous prediction

error term for

and

is the disturbance term

with N(0,?2).

·

The autoregressive and moving average

specifications can be combined to form an ARMA(p,q) specification:

b. ARCH and

GARCH models

For most financial data the Classical Linear

Regression Model assumption that the Variance is homoskedastic is often

violated. If the

variance of the errors is not constant, this would be known as heteroscedasticity.

Autoregression Conditional Heteroskedasticity (ARCH) and the Generalized

Autoregression Conditional Heteroskedasticity (GARCH) models are used to model

time-varying volatilities.

Given a stationary mean given by

,

The ARCH Model of the

will be given by;

Where the

Autoregression in the squared residual has an order of

, or

lags.

If

is

a time varying conditional variance with both Autoregression (AR) and Moving

Average (MA), the GARCH can be modeled by;

One

of the advantages of GARCH over ARCH is parsimonious, i.e. less lags are

required to capture the property of time-varying variance in GARCH.

2. Discuss many variations of GARCH and their

relevance to financial modeling.

(10

Marks)

SOLUTION

Many of the variations to the GARCH model have been suggested as

a consequence of perceived problems with standard GARCH(p, q) models

because;

1. The non-negativity conditions may be

violated by the estimated model. The only way to avoid this for sure would be

to place artificial constraints on the model coefficients in order to force

them to be non-negative.

2. GARCH models cannot account for leverage

effects, although they can account for volatility clustering and leptokurtosis

in a series.

3. The model does not allow for any direct

feedback between the conditional variance and the conditional mean.

Variations of the GARCH model remove some of the restrictions or

limitations of the basic model.

The variations to the GARCH Model can either be Univariate or

Multivariate.

1. UNIVARIATE

MODELS

a.

Model relating the return on a security to its time-varying

volatility or risk

These models allows for a direct feedback between the

conditional variance and the conditional mean

i.

The

GARCH in Mean (GARCH-M) model

Most models used in finance suppose that

investors should be rewarded for taking additional risk by obtaining a higher

return. One way to operationalise this concept is to let the return of a

security be partly determined by its risk.

When the

conditional variance enters the mean equation for a GARCH process, the GARCH-in-Mean

or simply the ARCH-M model is derived and is given by:

Where

, k

= 1, . . ., m are

exogenous variables which could include lagged

. In the sense of

asset pricing, if

is

the return on an asset of a firm, then

,k = 1, . . ., m would

generally include the return on the market and possibly other explanatory

variables such as the price earnings ratio and the size.

The parameter

captures the

sensitivity of the return to the time-varying volatility, or in other words,

links the return to a time-varying risk premium.

This model can

be used as an extension of the CAPM to model with an allowance for time-varying

risk to identify the trade-off preferences of investor between risk and return.

b.

Asymmetric

GARCH Models

One of the primary restrictions of GARCH models is that they

enforce a symmetric response of volatility to positive and negative shocks.

This arises since the conditional variance in the GARCH is a function of the square

of the lagged residuals and not their signs.

However, it has been argued that a negative shock to financial

time series is likely to cause volatility to rise by more than a positive shock

of the same magnitude. In the case of equity returns, such asymmetries are

typically attributed to leverage effects, whereby a fall in the value of

a firm’s stock causes the firm’s debt to equity ratio to rise.

Two of the most popular asymmetric GARCH models are the GJR and

the Exponential GARCH (EGARCH) models)

i.

GJR

Model (also Called Threshold GARCH -TGARCH)

The GJR model is a simple extension of GARCH with an additional

term added to account for possible asymmetries. The conditional variance is now

given by;

Where

This additional term,

account for the leverage effect in most

financial data.

ii.

Exponential

GARCH (EGARCH)

The model

captures asymmetric responses of the time-varying variance to shocks and, at

the same time, ensures that the variance is always positive. For a GARCH(1,1)

model, the volatility is given by;

Where

is asymmetric response

parameter or leverage parameter. The sign of

is expected to be

positive in most empirical cases so that a negative shock increases future

volatility or uncertainty while a positive shock eases the effect on future

uncertainty. This is in contrast to the standard GARCH model where shocks of

the same magnitude, positive or negative, have the same effect on future volatility.

An important

advantage of the EGARCH specification is that the conditional variance is

guaranteed to be positive at each point in time because the variance is

expressed in terms of log.

2.

MULTIVARIATE

GARCH MODELS (MGARCH)

The multivariate

time-varying conditional variance problem is given by the model;

Where vectors

for a bivariate model (GARCH(1,1) are;

,

, and

a.

Full

Parametization (VECH) Model

The VECH specification is presented by;

Where

,

,

,

and

are matrices in their

conventional form, and

means the procedure of

conversion of a matrix into a vector.

For a bivariate

GARCH;

b.

Positive

Definite Parametization (BEKK) Model

It takes the

form;

The important

feature of this specification is that it builds in sufficient generality,

allowing the conditional variances and covariances of the time series to

influence each other, and at the same time, does not require estimating a large

number of parameters.

In a bivariate

GARCH process, BEKK model has only 11 parameters compared with 21 parameters in

the VECH representation.

Even more

importantly, the BEKK process guarantees that the covariance matrices are

positive definite under very weak conditions.

In the bivariate

model, BEKK takes the form;

3. What is the stochastic volatility model? Discuss

the similarities and differences between a GARCH-type model and a stochastic

volatility model. (10

Marks)

SOLUTION

Stochastic volatility models are models that are random in time

and are used to model volatilities which are time-varying. For example a random

walk with a drift model given by;

In a stochastic

model, the best guess for the next value of series is the current value plus

some constant, rather than a deterministic mean value.

Stochastic volatility models contain a second error term, which

enters into the conditional variance equation.

Stochastic volatility models are closely related to the

financial theories used in the options pricing literature.

The primary advantage of stochastic volatility models is that

they can be viewed as discrete time approximations to the continuous time

models employed in options pricing frameworks. However, such models are hard to

estimate.

A GARCH model is deterministic trend rather that stochastic and

is a nonrandom function of time. A GARCH(p,q) model can be modeled as;

Stochastic volatility models differ from GARCH principally in

that the conditional variance equation of a GARCH specification is completely

deterministic given all information available up to that of the previous

period. In other words, there is no error term in the variance equation of a

GARCH model, only in the mean equation.

While stochastic volatility models have been widely employed in

the mathematical options pricing literature, they have not been popular in

empirical discrete-time financial applications, probably owing to the complexity

involved in the process of estimating the model parameters.

Both Stochastic and GARCH models are used in modeling

volatility.

4.

Discuss and comment on the new developments in modeling time-varying

volatilities.

(10

Marks)

SOLUTION

a. Adoption

of Multivariate Volatility Models in Finance

Because of the

new variations in the volatility models, most fund managers are now modeling

volatility as a time-varying variable. This is in contrast to for example the

assumptions made in the CAPM and Option pricing models which take volatility as

measured by the standard deviation of returns to be constant.

b.

Time-Varying

Beta Risk

In most

financial models, specifically CAPM model, the beta is always assumed to be

constant over time, thus the use of historical betas. The restriction of

constant beta risk may, however, be unrealistic. For example, beta risks may be

affected by periods of financial crises and economic booms and recessions. GARCH-based

and other stochastic volatility models have been developed to structure the

beta as time-varying which may be given by the function;

Univariate Conditional

covariance models can be used to model the individual variances while

covariances can be modeled using a simple bivariate GARCH model.

c.

Modeling

transmission of volatilities from one global market to another

GARCH Models are

now being extended to study how volatility is transmitted through different

regions of the world (mainly US, UK and Middle East) during the course of a

global financial trading day. The aim is of these GARCH study is to examine

international linkages in volatility between major global financial regions and

investigate in particular two patterns as possible explanation of international

volatility transmission;

Heatwave: These models are based on the premise that volatility

in any one region or market is a function of the previous day’s volatility in

the same region or market.

Meteor Shower: the models are explains that volatility

in one region is driven mainly by volatility in the region immediately

preceding it in terms of calendar times.

d.

Dynamic

Optimal Hence Ratio

Multivariate

GARCH models are now being used to determine the optimal hedge ratio for an

investor who buys or sell futures contracts to hedge against the movement in

the spot prices of an asset. This is achieved by specifying the covariance

between the returns on the futures and the assets, and the variance of the

return on the futures contracts to be time varying