GARCH(1,1) models

1 Ruprecht-Karls-Universit at HeidelbergFakult at f ur Mathematik und InformatikBachelorarbeitzur Erlangung des akademischen GradesBachelor of Science (B. Sc.) garch (1,1) modelsvorgelegt vonBrandon Williams15. Juli 2011 Betreuung: Prof. Dr. Rainer DahlhausAbstraktIn dieser Bachelorarbeit werden garch (1,1)-Modelle zur Analyse finanzieller Zeitreihen unter-sucht. Dabei werden zuerst hinreichende und notwendige Bedingungen daf ur gegeben, dass solcheProzesse uberhaupt station ar werden k onnen. Danach werden asymptotische Ergebnisse uber rel-evante Sch atzer hergeleitet und parametrische Tests entwickelt.

2 Die Methoden werden am Endedurch ein Datenbeispiel this thesis, garch (1,1)- models for the analysis of financial time series are investigated. First,sufficient and necessary conditions will be given for the process to have a stationary , asymptotic results for relevant estimators will be derived and used to develop parametrictests. Finally, the methods will be illustrated with an empirical Introduction22 Stationarity43 A central limit theorem94 Parameter estimation185 Tests226 Variants of the garch (1,1) model267 garch (1,1) in continuous time278 Example with MATLAB349 Discussion3911 IntroductionModelling financial time series is a major application and area of research in probability theory andstatistics.

3 One of the challenges particular to this field is the presence of heteroskedastic effects,meaning that the volatility of the considered process is generally not constant. Here the volatilityis the square root of the conditional variance of the log return process given its previous is, ifPtis the time series evaluated at timet, one defines the log returnsXt= logPt+1 logPtand the volatility t, where 2t= Var[X2t|Ft 1]andFt 1is the -algebra generated byX0,..,Xt 1. Heuristically, it makes sense that the volatilityof such processes should change over time, due to any number of economic and political factors,and this is one of the well known stylized facts of mathematical presence of heteroskedasticity is ignored in some financial models such as the Black-Scholesmodel, which is widely used to determine the fair pricing of European-style options.

4 While this leadsto an elegent closed-form formula, it makes assumptions about the distribution and stationarityof the underlying process which are unrealistic in general. Another commonly used homoskedas-tic model is the Ornstein-Uhlenbeck process, which is used in finance to model interest rates andcredit markets. This application is known as the Vasicek model and suffers from the homoskedasticassumption as (autoregressive conditional heteroskedasticity) models were introduced by Robert Englein a 1982 paper to account for this behavior.

5 Here the conditional variance process is given an au-toregressive structure and the log returns are modelled as a white noise multiplied by the volatility:Xt=et t 2t= + 1X2t 1+..+ pX2t p,whereet(the innovations ) are with expectation 0 and variance 1 and are assumed indepen-dent from kfor allk t. The lag lengthp 0 is part of the model specification and may bedetermined using the Box-Pierce or similar tests for autocorrelation significance, where the casep= 0 corresponds to a white noise process. To ensure that 2tremains positive, , i 0 Bollerslev (1986) extended the ARCH model to allow 2tto have an additional autoregres-sive structure within itself.

6 The garch (p,q) (generalized ARCH) model is given byXt=et t 2t= + 1X2t 1+..+ pX2t p+ 1 2t 1+..+ q 2t model, in particular the simpler garch (1,1) model, has become widely used in financialtime series modelling and is implemented in most statistics and econometric software (1,1) models are favored over other stochastic volatility models by many economists due2to their relatively simple implementation: since they are given by stochastic difference equationsin discrete time, the likelihood function is easier to handle than continuous-time models , and sincefinancial data is generally gathered at discrete , there are also improvements to be made on the standard garch model.

7 A notableproblem is the inability to react differently to positive and negative innovations, where in reality,volatility tends to increase more after a large negative shock than an equally large positive is known as the leverage effect and possible solutions to this problem are discussed further insection loss of generality, the timetwill be assumed in the following sections to take valuesin eitherN0or StationarityThe first task is to determine suitable parameter sets for the model.

8 In the introduction, weconsidered that , , 0 is necessary to ensure that the conditional variance 2tremains non-negative at all timest. It is also important to find parameters , , which ensure that 2thas finiteexpected value or higher moments. Another consideration which will be important when studyingthe asymptotic properties of garch models is whether 2tconverges to a stationary , we will see that these conditions translate to rather severe restrictions on the choiceof 1.: A processXtis called stationary (strictly stationary), if for all timest1.

9 ,tn,h Z:FX(xt1+h,..,xtn+h) =FX(xt1,..xtn)whereFX(xt1,..,xtn) is the joint cumulative distribution function ofXt1,.., >0and , 0. Then the garch (1,1) equations have a stationary solutionif and only ifE[log( e2t+ )]<0. In this case the solution is uniquely given by 2t= (1 + j=1j i=1( e2t i+ )). the equation 2t= + ( e2t 1+ ) 2t 1, by repeated use on t 1, etc. we arrive at theequation 2t= (1 +k j=1j i=1( e2t i+ )) + (k+1 i=1( e2t i+ )) 2t k 1,which is valid for allk N. In particular, 2t (1 +k j=1j i=1( e2t i+ )),since , 0.

10 Assume that 2tis a stationary solution and thatE[log( e2t+ )] 0. We havelogE[j i=1( e2t i+ )] E[logj i=1( e2t i+ )] =j i=1E[log( e2t i+ )]and therefore, ifE[log( e2t+ )]>0, then the product ji=1( e2t i+ ) diverges by the stronglaw of large numbers. In the case thatE[log( e2t+ )] = 0, then ji=1log( e2t i+ ) is a randomwalk process so thatlim supj j i=1log( e2t i+ ) = that in both cases we havelim supj j i=1( e2t i+ ) = all terms are negative we then have 2t lim supj j i=1( e2t i+ ) = is impossible; therefore,E[log( e2t+ )]<0 is necessary for the existence of a stationarysolution.