A common findingin time series regression however is that the residuals are correlated withtheir own lagged or previous values. Since this correlation is sequential intime, it is then called serial correlation. ARMA models arelinear in both mean and variance while the GARCH models are either linear inmean but non linear in the variance or nonlinear in both mean and variance.a. ARMA ModelsThe serialcorrelations for the models which are linear in both mean and variance can takethe form of autoregression (AR), Moving Average (MA) or a combination of thetwo (ARMA).These models aresaid to be stationary.

Stationarity here implies that a time series has memorydepicted in the lags of the error/residual terms or past observations of thevariable under consideration. · In an AR model, the variable in thecurrent 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 termwith N(0,?2).· In an MA model, the variable is affected by theprevious prediction errors and is models as; Where is represent the order of the model previous predictionerror term for and is the disturbance termwith N(0,?2).· The autoregressive and moving averagespecifications can be combined to form an ARMA(p,q) specification: b. ARCH andGARCH models For most financial data the Classical LinearRegression Model assumption that the Variance is homoskedastic is oftenviolated. If thevariance of the errors is not constant, this would be known as heteroscedasticity.

Autoregression Conditional Heteroskedasticity (ARCH) and the GeneralizedAutoregression Conditional Heteroskedasticity (GARCH) models are used to modeltime-varying volatilities.Given a stationary mean given by ,The ARCH Model of the will be given by; Where theAutoregression in the squared residual has an order of , or lags.If isa time varying conditional variance with both Autoregression (AR) and MovingAverage (MA), the GARCH can be modeled by; Oneof the advantages of GARCH over ARCH is parsimonious, i.e.

less lags arerequired to capture the property of time-varying variance in GARCH. 2. Discuss many variations of GARCH and theirrelevance to financial modeling. (10Marks)SOLUTION Many of the variations to the GARCH model have been suggested asa consequence of perceived problems with standard GARCH(p, q) modelsbecause;1. The non-negativity conditions may beviolated by the estimated model.

The only way to avoid this for sure would beto place artificial constraints on the model coefficients in order to forcethem to be non-negative. 2. GARCH models cannot account for leverageeffects, although they can account for volatility clustering and leptokurtosisin a series. 3. The model does not allow for any directfeedback between the conditional variance and the conditional mean.Variations of the GARCH model remove some of the restrictions orlimitations of the basic model.The variations to the GARCH Model can either be Univariate orMultivariate.1.

UNIVARIATEMODELS a. Model relating the return on a security to its time-varyingvolatility or riskThese models allows for a direct feedback between theconditional variance and the conditional meani. TheGARCH in Mean (GARCH-M) modelMost models used in finance suppose thatinvestors should be rewarded for taking additional risk by obtaining a higherreturn. One way to operationalise this concept is to let the return of asecurity be partly determined by its risk.

When theconditional variance enters the mean equation for a GARCH process, the GARCH-in-Meanor simply the ARCH-M model is derived and is given by: Where , k= 1, . . .

, m areexogenous variables which could include lagged . In the sense ofasset pricing, if isthe return on an asset of a firm, then ,k = 1, . . ., m wouldgenerally include the return on the market and possibly other explanatoryvariables such as the price earnings ratio and the size.

The parameter captures thesensitivity of the return to the time-varying volatility, or in other words,links the return to a time-varying risk premium.This model canbe used as an extension of the CAPM to model with an allowance for time-varyingrisk to identify the trade-off preferences of investor between risk and return.b. AsymmetricGARCH ModelsOne of the primary restrictions of GARCH models is that theyenforce a symmetric response of volatility to positive and negative shocks.This arises since the conditional variance in the GARCH is a function of the squareof the lagged residuals and not their signs. However, it has been argued that a negative shock to financialtime series is likely to cause volatility to rise by more than a positive shockof the same magnitude.

In the case of equity returns, such asymmetries aretypically attributed to leverage effects, whereby a fall in the value ofa firm’s stock causes the firm’s debt to equity ratio to rise. Two of the most popular asymmetric GARCH models are the GJR andthe Exponential GARCH (EGARCH) models)i. GJRModel (also Called Threshold GARCH -TGARCH)The GJR model is a simple extension of GARCH with an additionalterm added to account for possible asymmetries.

The conditional variance is nowgiven by; Where This additional term, account for the leverage effect in mostfinancial data.ii. ExponentialGARCH (EGARCH)The modelcaptures asymmetric responses of the time-varying variance to shocks and, atthe same time, ensures that the variance is always positive. For a GARCH(1,1)model, the volatility is given by; Where is asymmetric responseparameter or leverage parameter. The sign of is expected to bepositive in most empirical cases so that a negative shock increases futurevolatility or uncertainty while a positive shock eases the effect on futureuncertainty.

This is in contrast to the standard GARCH model where shocks ofthe same magnitude, positive or negative, have the same effect on future volatility.An importantadvantage of the EGARCH specification is that the conditional variance isguaranteed to be positive at each point in time because the variance isexpressed in terms of log.2. MULTIVARIATEGARCH MODELS (MGARCH)The multivariatetime-varying conditional variance problem is given by the model; Where vectorsfor a bivariate model (GARCH(1,1) are; , , and a.

FullParametization (VECH) ModelThe VECH specification is presented by; Where , , , and are matrices in theirconventional form, and means the procedure ofconversion of a matrix into a vector.For a bivariateGARCH; b. PositiveDefinite Parametization (BEKK) ModelIt takes theform; The importantfeature of this specification is that it builds in sufficient generality,allowing the conditional variances and covariances of the time series toinfluence each other, and at the same time, does not require estimating a largenumber of parameters.

In a bivariateGARCH process, BEKK model has only 11 parameters compared with 21 parameters inthe VECH representation. Even moreimportantly, the BEKK process guarantees that the covariance matrices arepositive definite under very weak conditions.In the bivariatemodel, BEKK takes the form; 3.

What is the stochastic volatility model? Discussthe similarities and differences between a GARCH-type model and a stochasticvolatility model. (10Marks) SOLUTION Stochastic volatility models are models that are random in timeand are used to model volatilities which are time-varying. For example a randomwalk with a drift model given by; In a stochasticmodel, the best guess for the next value of series is the current value plussome constant, rather than a deterministic mean value. Stochastic volatility models contain a second error term, whichenters into the conditional variance equation.Stochastic volatility models are closely related to thefinancial theories used in the options pricing literature.The primary advantage of stochastic volatility models is thatthey can be viewed as discrete time approximations to the continuous timemodels employed in options pricing frameworks.

However, such models are hard toestimate. A GARCH model is deterministic trend rather that stochastic andis a nonrandom function of time. A GARCH(p,q) model can be modeled as; Stochastic volatility models differ from GARCH principally inthat the conditional variance equation of a GARCH specification is completelydeterministic given all information available up to that of the previousperiod. In other words, there is no error term in the variance equation of aGARCH model, only in the mean equation.While stochastic volatility models have been widely employed inthe mathematical options pricing literature, they have not been popular inempirical discrete-time financial applications, probably owing to the complexityinvolved in the process of estimating the model parameters.Both Stochastic and GARCH models are used in modelingvolatility. 4.Discuss and comment on the new developments in modeling time-varyingvolatilities.

(10Marks) SOLUTION a. Adoptionof Multivariate Volatility Models in FinanceBecause of thenew variations in the volatility models, most fund managers are now modelingvolatility as a time-varying variable. This is in contrast to for example theassumptions made in the CAPM and Option pricing models which take volatility asmeasured by the standard deviation of returns to be constant.

b. Time-VaryingBeta RiskIn mostfinancial models, specifically CAPM model, the beta is always assumed to beconstant over time, thus the use of historical betas. The restriction ofconstant beta risk may, however, be unrealistic. For example, beta risks may beaffected by periods of financial crises and economic booms and recessions. GARCH-basedand other stochastic volatility models have been developed to structure thebeta as time-varying which may be given by the function; Univariate Conditionalcovariance models can be used to model the individual variances whilecovariances can be modeled using a simple bivariate GARCH model. c.

Modelingtransmission of volatilities from one global market to anotherGARCH Models arenow being extended to study how volatility is transmitted through differentregions of the world (mainly US, UK and Middle East) during the course of aglobal financial trading day. The aim is of these GARCH study is to examineinternational linkages in volatility between major global financial regions andinvestigate in particular two patterns as possible explanation of internationalvolatility transmission;Heatwave: These models are based on the premise that volatilityin any one region or market is a function of the previous day’s volatility inthe same region or market.Meteor Shower: the models are explains that volatilityin one region is driven mainly by volatility in the region immediatelypreceding it in terms of calendar times.d.

DynamicOptimal Hence Ratio MultivariateGARCH models are now being used to determine the optimal hedge ratio for aninvestor who buys or sell futures contracts to hedge against the movement inthe spot prices of an asset. This is achieved by specifying the covariancebetween the returns on the futures and the assets, and the variance of thereturn on the futures contracts to be time varying