Normal Weighted Inverse Gaussian Distributions and Em Algorithm With Appliactions to Risk Measures and Dependence Modelling

Abstract

High frequencyfinancialdataischaracterizedbynon-normality,asymmetric,leptokurtic and fat-tailedbehaviour.Thenormaldistributionisinadequateincapturingthese characteristics. Tothisend,variousflexibledistributionshavebeenproposed.Inthis Thesis weintroducedanewclassofdistributionsknownasNormalWeightedInverse Gaussian distributions. WeightedInverseGaussiandistributionsarespecialcasesofGeneralizedInverseGaussian (GIG)distributionwhicharerelatedtoInverseGaussian(IG)distribution.Finitemixtures of thesespecialcasesarealsoweightedInverseGaussian(WIG)distributions.Usingthese WIG distributionsasmixingdistributionstotheNormalVarianceMeanMixture(NVMM) weobtainaclassofNormalWeightedInverseGaussian(NWIG)distributions. The propertiesconsideredforthesemodelsaremean,variance,skewnessandkurtosis. For dataanalysisweconsiderthreedatasets:RangeResourceCorporation(RRC),Shares of ChevronCorporation(CVX)ands&p500index.Theperiod3/01/2000to1/07/2013with 702 observationsforeachdatasetisconsidered.Estimationofparametersofthesemodels areobtainedusingExpectation-Maximization(EM)algorithm.TheEMalgorithmisa powerfultechniqueformaximumlikelihoodestimationfordatacontainingmissingvalues or datathatcanbeconsideredascontainingmissingvalues.Themixingoperationcanbe consideredresponsibleforproducingmissingvalues. TwoimportantriskmeasuresinliteratureareValueatRisk(VaR)andExpectedShortfall (ES). InthisworkwehaveobtainedVaRandESfortheNWIGdistributions.Backtesting of thismeasuresisalsoperformed. Wehavealsoconsidereddependencemodellingoffinancialreturnsusingcopulas.The marginal distributionsarebasedonNormalWeightedInverseGaussiandistributions. Wehighlightthefollowingcontributionstothiswork 1. WehaveconstructedanewclassofWeightedInverseGaussiandistributions. 2. WehaveusedthisclassasamixingdistributiontotheNormalVarianceMeanMixture to obtainaclassofNormalWeightedInverseGaussiandistributions. 3. All worksonparameterestimationofEMalgorithmatthemaximizationstepisbased on explicitsolutiontonormalequations.Oftenthisinvolvesnumericaltechniques which aredifficulttoimplement.Inthiswork,weshowthattheiterativeschemes arenotnecessarilybasedonexplicitsolutions.Theycanalsobedesignedusing a representationbasedonthenormalequations.Thissubtleapproachiseasily programmableandpreservesthemonotonicconvergencepropertyoftheEMalgorithm with eachiterationincreasingthelikelihood. vi 4. Fromthedatasetsused,thisclassofNWIGdistributionsisshowntobeagood alternativetotheNormalInverseGaussiandistribution.However,onespecialcaseof the GIGdistributionwhenusedasamixingdistributiontoNVMMoutperformsthe NIG andonefinitecasewhenusedasamixingdistributionoutperformsallmodels. 5. Using backtestingproceduresitcanbeshownthatthisclassofdistributions,NWIG, which haveheavytailed,isanalternativecandidateforfinancialriskmanagement. 6. The modelshavealsobeenusedasmarginalsindependencemodellingusingcopulas approach.