Optimization with Generalized Deviation Measures in Risk Management

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Permanent Link:: https://ufdc.ufl.edu/UFE0044163/00001

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Title:: Optimization with Generalized Deviation Measures in Risk Management
Creator:: Kalinchenko, Konstantin P
Place of Publication:: [Gainesville, Fla.]
Florida
Publisher:: University of Florida
Publication Date:: 2012
Language:: english
Physical Description:: 1 online resource (85 p.)

Thesis/Dissertation Information

Degree:: Doctorate ( Ph.D.)
Degree Grantor:: University of Florida
Degree Disciplines:: Industrial and Systems Engineering
Committee Chair:: Uryasev, Stanislav
Committee Members:: Pardalos, Panagote M
Boginski, Vladimir L.
Yan, Liqing
Graduation Date:: 5/5/2012

Subjects

Subjects / Keywords:: Connectivity ( jstor )
Financial portfolios ( jstor )
Investment risks ( jstor )
Investors ( jstor )
Linear regression ( jstor )
Market prices ( jstor )
Risk preferences ( jstor )
Scheduling ( jstor )
Sensors ( jstor )
Standard deviation ( jstor )
Industrial and Systems Engineering -- Dissertations, Academic -- UF
capm -- cvar -- deviation -- risk
Genre:: bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Industrial and Systems Engineering thesis, Ph.D.

Notes

Abstract:: Our work provides an overview of the so-called generalized deviation measures and generalized risk measures, and develops stochastic optimization approaches utilizing them. These measures are designed to quantify risk when implied distributions are known. We provide useful examples of deviation and risk measures, which can be efficiently applied in situations, when the classical measures either do not properly account for risk, or do not satisfy properties desired for efficient application in stochastic optimization. We discuss the importance of considering alternative risk and deviation measures in the classical models, such as the capital asset pricing model and quantile regression. We apply stochastic optimization and risk management techniques based on the conditional value-at-risk (CVaR) to solve a dynamic sensor scheduling problem with robustness constraints on a wireless connectivity network. We also develop an efficient application of the generalized Capital Asset Pricing Model based on mixed CVaR deviation to estimating risk preferences of investors using S&P500 stock index option prices. In the first part we provide an overview of the main classes of generalized deviation measures and corresponding risk measures, and compare them to the classical risk and deviation measures, such as maximum risk, value-at-risk and standard deviation. In addition, we provide a relation between deviation measures and measures of error, which are used in regression models. In some applications, such as simulation, a distribution of the residual term has to be specified. We apply the entropy maximization principle to identify the appropriate distribution for the quantile regression (factor) model. In the second part we consider several classes of problems that deal with optimizing the performance of dynamic sensor networks used for area surveillance, in particular, in the presence of uncertainty. The overall efficiency of a sensor network is addressed from the aspects of minimizing the overall information losses, as well as ensuring that all nodes in a network form a robust connectivity pattern at every time moment, which would enable the sensors to communicate and exchange information in uncertain and adverse environments. The considered problems are solved using mathematical programming techniques that incorporate CVaR, which allows one to minimize or bound the losses associated with potential risks. The issue of robust connectivity is addressed by imposing explicit restrictions on the shortest path length between all pairs of sensors and on the number of connections for each sensor (i.e., node degrees) in a network. Specific formulations of linear 0-1 optimization problems and the corresponding computational results are presented. In the third part we apply the generalized Capital Asset Pricing Model based on mixed CVaR deviation to calibrate risk preferences of investors. We introduce the new generalized beta to capture tail performance of S&P500 returns. Calibration is done by extracting information about risk preferences from option prices on S&P500. Actual market option prices are matched with the estimated prices from the pricing equation based on the generalized beta. These results can be used for various purposes. In particular, the structure of the estimated deviation measure conveys information about the level of fear among investors. High level of fear reflects a tendency of market participants to hedge their investments and signals investors' anticipation of poor market trend. This information can be used in risk management and for optimal capital allocation. ( en )
General Note:: In the series University of Florida Digital Collections.
General Note:: Includes vita.
Bibliography:: Includes bibliographical references.
Source of Description:: Description based on online resource; title from PDF title page.
Source of Description:: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:: Thesis (Ph.D.)--University of Florida, 2012.
Local:: Adviser: Uryasev, Stanislav.
Statement of Responsibility:: by Konstantin P Kalinchenko.

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Source Institution:: UFRGP
Rights Management:: Copyright Kalinchenko, Konstantin P. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Classification:: LD1780 2012 ( lcc )

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Aggregations:: Institutional Repository at the University of Florida (IR@UF)
University of Florida Theses & Dissertations
INTERNAL UF RETRO ETDS

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OPTIMIZATIONWITHGENERALIZEDDEVIATIONMEASURESINRISKMANAGEMENTByKONSTANTINP.KALINCHENKOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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2012KonstantinP.Kalinchenko 2

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IdedicatethisthesistomyparentsPavelandOlga,andmybrotherAlexander,whosupportedmeinallmyendeavours. 3

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ACKNOWLEDGMENTS IamverythankfultomyadvisorProf.StanUryasevandtoProf.R.TyrrellRockafellarfortheirsupportduringmydoctoratestudyingattheUniversityofFlorida.WiththeirresearchguidanceandinvaluablehelpIwasabletogrowonbothprofessionalandpersonallevels.Iwouldliketoexpressmygratitudetoothermembersofmydoctoratecommittee,Prof.PanosPardalos,Prof.VladimirBoginskiandProf.LiqingYanfortheircontributiontomyresearch.Also,IamparticularlythankfultoProf.MichaelZabarankin(StevensInstituteofTechnology),Prof.MarkJ.Flannery(UniversityofFlorida,WarringtonCollegeofBusinessAdministration)andProf.OlegBondarenko(UniversityofIllinois)forvaluablefeedbacks.IwouldalsoliketoexpressmygreatestappreciationtomycolleaguesfromtheRiskManagementandFinancialEngineeringlabandCenterforAppliedOptimization.Intensivediscussions,exchangeofideasandjointresearchwithmyfellowgraduatestudentsfromthesetwolabshelpedmesignicantlytoachievemygoals.Also,Iwouldliketothankmyfamilyandfriends,whosupportedandencouragedmeinallofmybeginnings. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2GENERALIZEDMEASURESOFDEVIATION,RISKANDERROR ....... 14 2.1ClassicalRiskandDeviationMeasures ................... 14 2.2GeneralizedRiskandDeviationMeasures .................. 19 2.3ConditionalValue-at-Risk ........................... 22 2.4ApplicationtoGeneralizedLinearRegressions ............... 23 2.4.1MeasuresofError ........................... 23 2.4.2GeneralizedLinearRegressions ................... 25 2.4.3DistributionofResidual ......................... 27 3ROBUSTCONNECTIVITYISSUESINDYNAMICSENSORNETWORKSFORAREASURVEILLANCEUNDERUNCERTAINTY .............. 33 3.1Multi-SensorSchedulingProblems:GeneralDeterministicSetup ..... 36 3.1.1FormulationwithBinaryVariables ................... 36 3.1.2CardinalityFormulation ......................... 38 3.2QuantitativeRiskMeasuresinUncertainEnvironments:ConditionalValue-at-Risk ........................ 39 3.3OptimizingtheConnectivityofDynamicSensorNetworksUnderUncertainty ............................ 41 3.3.1EnsuringShortTransmissionPathsvia2-clubFormulation ..... 44 3.3.2EnsuringBackupConnectionsviak-plexFormulation ........ 46 3.4ComputationalExperiments .......................... 47 4CALIBRATINGRISKPREFERENCESWITHGENERALIZEDCAPMBASEDONMIXEDCVARDEVIATION ....................... 52 4.1DescriptionoftheApproach .......................... 55 4.1.1GeneralizedCAPMBackground .................... 55 4.1.2PricingFormulasinGCAPM ...................... 59 4.1.3MixedCVaRDeviationandBetas ................... 60 4.1.4RiskPreferencesofaRepresentativeInvestor ............ 63 4.2CaseStudyDataandAlgorithm ........................ 66 5

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4.3CaseStudyComputationalResults ...................... 70 5CONCLUSIONS ................................... 75 5.1DissertationContribution ........................... 75 5.2FutureWork ................................... 76 APPENDIX:PROOFS ................................... 77 REFERENCES ....................................... 81 BIOGRAPHICALSKETCH ................................ 85 6

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LISTOFTABLES Table page 3-1CPLEXResults:Problemwith2-clubConstraints ................. 48 3-2CPLEXResults:Problemwithk-plexConstraints ................. 49 3-3CPLEXandPSGResults:StochasticSetup .................... 50 4-1CaseStudyDataforSelectedDates ........................ 68 4-2CaseStudyCommonData ............................. 68 4-3DeviationMeasureCalibrationResults ....................... 69 7

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LISTOFFIGURES Figure page 2-1RelationsBetweenMeasuresofError,DeviationMeasures,RiskMeasuresandStatistics ............................ 24 2-2ProbabilityDensityFunctionsforDExp(,1) .................... 31 2-3ProbabilityDensityFunctionsforDExp(,2) .................... 32 3-1GraphicalrepresentationofVaRandCVaR. .................... 40 4-1CVaR-typeRiskIdentierforaGivenOutcomeVariableX ............ 62 4-2CalculatedPricesandMarketPricesintheScaleofImpliedVolatilitiesPart1outof3 .................................... 71 4-3CalculatedPricesandMarketPricesintheScaleofImpliedVolatilitiesPart2outof3 .................................... 72 4-4CalculatedPricesandMarketPricesintheScaleofImpliedVolatilitiesPart3outof3 .................................... 73 4-5S&P500ValueandRiskAversityDynamics .................... 74 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyOPTIMIZATIONWITHGENERALIZEDDEVIATIONMEASURESINRISKMANAGEMENTByKonstantinP.KalinchenkoMay2012Chair:StanUryasevMajor:IndustrialandSystemsEngineering Ourworkprovidesanoverviewoftheso-calledgeneralizeddeviationmeasuresandgeneralizedriskmeasures,anddevelopsstochasticoptimizationapproachesutilizingthem.Thesemeasuresaredesignedtoquantifyriskwhenimplieddistributionsareknown.Weprovideusefulexamplesofdeviationandriskmeasures,whichcanbeefcientlyappliedinsituations,whentheclassicalmeasureseitherdonotproperlyaccountforrisk,ordonotsatisfypropertiesdesiredforefcientapplicationinstochasticoptimization.Wediscusstheimportanceofconsideringalternativeriskanddeviationmeasuresintheclassicalmodels,suchasthecapitalassetpricingmodelandquantileregression.Weapplystochasticoptimizationandriskmanagementtechniquesbasedontheconditionalvalue-at-risk(CVaR)tosolveadynamicsensorschedulingproblemwithrobustnessconstraintsonawirelessconnectivitynetwork.WealsodevelopanefcientapplicationofthegeneralizedCapitalAssetPricingModelbasedonmixedCVaRdeviationtoestimatingriskpreferencesofinvestorsusingS&P500stockindexoptionprices. Intherstpartweprovideanoverviewofthemainclassesofgeneralizeddeviationmeasuresandcorrespondingriskmeasures,andcomparethemtotheclassicalriskanddeviationmeasures,suchasmaximumrisk,value-at-riskandstandarddeviation.Inaddition,weprovidearelationbetweendeviationmeasuresandmeasuresoferror,whichareusedinregressionmodels.Insomeapplications,suchassimulation,a 9

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distributionoftheresidualtermhastobespecied.Weapplytheentropymaximizationprincipletoidentifytheappropriatedistributionforthequantileregression(factor)model. Inthesecondpartweconsiderseveralclassesofproblemsthatdealwithoptimizingtheperformanceofdynamicsensornetworksusedforareasurveillance,inparticular,inthepresenceofuncertainty.Theoverallefciencyofasensornetworkisaddressedfromtheaspectsofminimizingtheoverallinformationlosses,aswellasensuringthatallnodesinanetworkformarobustconnectivitypatternateverytimemoment,whichwouldenablethesensorstocommunicateandexchangeinformationinuncertainandadverseenvironments.TheconsideredproblemsaresolvedusingmathematicalprogrammingtechniquesthatincorporateCVaR,whichallowsonetominimizeorboundthelossesassociatedwithpotentialrisks.Theissueofrobustconnectivityisaddressedbyimposingexplicitrestrictionsontheshortestpathlengthbetweenallpairsofsensorsandonthenumberofconnectionsforeachsensor(i.e.,nodedegrees)inanetwork.Specicformulationsoflinear0-1optimizationproblemsandthecorrespondingcomputationalresultsarepresented. InthethirdpartweapplythegeneralizedCapitalAssetPricingModelbasedonmixedCVaRdeviationtocalibrateriskpreferencesofinvestors.WeintroducethenewgeneralizedbetatocapturetailperformanceofS&P500returns.CalibrationisdonebyextractinginformationaboutriskpreferencesfromoptionpricesonS&P500.Actualmarketoptionpricesarematchedwiththeestimatedpricesfromthepricingequationbasedonthegeneralizedbeta.Theseresultscanbeusedforvariouspurposes.Inparticular,thestructureoftheestimateddeviationmeasureconveysinformationabouttheleveloffearamonginvestors.Highleveloffearreectsatendencyofmarketparticipantstohedgetheirinvestmentsandsignalsinvestors'anticipationofpoormarkettrend.Thisinformationcanbeusedinriskmanagementandforoptimalcapitalallocation. 10

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CHAPTER1INTRODUCTION Themainobjectiveofourstudyistodevelopmodelsutilizinggeneralizedmeasuresofdeviation,riskanderrorinstochasticoptimizationandriskmanagementapplications.Uncertaintyisgenerallymodeledusingrandomvariables,anddifferentmodelsutilizevariousfunctionals(ormeasures)onthespaceofrandomvariablestoproperlyaccountforrisk.Dependingonaparticularapplication,thefunctionalmusthavecertainpropertiestoquantifyacertainaspectofuncertainty.Althoughmanydifferentfunctionalsmaysatisfytheseproperties,mostmodelsutilizejustseveralclassicfunctionals,suchasstandarddeviationorquantile. Optimalityofanyparticularchoiceofmeasureaccountingforuncertaintycanoftenbeargued.Thisledtomultiplestudiesintroducingnewmeasuresanddevelopingalternativemodelsutilizingthem.Theconceptsofgeneralizedmeasuresofdeviation,riskanderrorweredevelopedtowraptheseandothermeasuresinseveralclasses,whereeachclasssatisescertainproperties(axioms)requiredinaparticularapplication.Ifacertainmodelbasedonsomeriskordeviationfunctionalisadjustedtocertainassumptions,itsapplicationcanoftenbegeneralizedbysubstitutingthefunctionalwithageneralizedmeasureofriskordeviation.Differentinstancesofthegeneralizedmodelcanthusbecompared.Moreover,ifthefunctionalhasaparameter,itcanalsobeoptimized. Inourwork,weutilizethreeclassesoffunctionals:generalizedmeasuresofrisk,generalizeddeviationmeasures,andgeneralizedmeasuresoferror.Generalizedmeasuresofriskweredesignedtoquantifypotentiallosses.Generalizeddeviationmeasuresaccountonlyforvariabilityoflosses.Generalizedmeasuresoferrorcanbeviewedastoolstoestimatesignicanceofaresidualterminapproximation,oritsdeviationfrom0. 11

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Itisimportanttomentionthatthereisaone-to-onecorrespondencebetweenmeasuresofriskandmeasuresofdeviation,andeverymeasureoferrorhasaparticularmeasureofdeviationcorrespondingtoit.Thisfeaturelinkstogethercertainverydifferentmodels,andprovidesasolutiontoproperlychoosingfunctionalinonemodeldependingonthefunctionalusedinanothermodel,whenthetwomodelsareappliedtothesameproblem. Althoughthetheoreticalfoundationsofthedeviationmeasureshavealreadybeendeveloped,theirpracticalapplicationshavenotyetbecomepopular.Inourwork,wedemonstrateseveralapplicationsofthegeneralizedmeasuresofdeviation,riskanderrorinstochasticoptimization.Inparticular,weconsiderinstancesbasedonconditionalvalue-at-risk(CVaR)andmixedCVaR.CVaRisariskmeasure,whichgainedsubstantialattentioninacademicpublicationsduetoseveralreasons.First,CVaRhasanintuitivedenitionasexpectedlossescorrespondingtothe1)]TJ /F4 11.955 Tf 12.56 0 Td[(tailofdistribution.Second,CVaRisacoherentmeasureofrisk,andisthereforeapplicableinoptimization.Third,theproblemofoptimizingCVaRhasalinearprogrammingformulation.MixedCVaRisaconvexcombinationofseveralCVaRtermswithdifferentvalues.Byvaryingthenumberoftermsandthevaluesofcoefcientsand,onecanpreciselyspecifysignicanceofdifferentpartsofadistributionaccordingtohisperceptionofrisk. InChapter2weappliedentropymaximizationmethodologytospecifyingdistributionoftheresidualtermingeneralizedlinearregression.Generalizedlinearregressionisdenedasastochasticoptimizationproblemofminimizingageneralizedmeasureoferroroftheresidual.Itisimportanttomentionthatthegeneralizedlinearregressionhasanalternativeformulationutilizingdeviationmeasureandso-calledstatistic,bothcorrespondingtothesamemeasureoferror.Twoinstancesofthegeneralizedlinearregressionarewellknown:classicallinearregression,basedonmeansquarederror,andquantileregression,basedonKoenker-Bassetterror.Incertainapplications, 12

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suchassimulation,thedistributionoftheresidualterminthelinearregressionhastobespecied.AcommonwaytospecifythedistributionunderlimitedinformationisbymaximizingShannonentropy.Thisapproachisjustiedbythecommonviewthatentropyisameasureofuncertainty.Inthecaseoftheclassicallinearregression,whenexpectationandvarianceoftheresidualtermareknown,thedistributionwithmaximumentropyisnormal.Followingthesameintuition,inquantileregressionweestimatedthedistributionbymaximizingentropysubjecttoconstraintsonquantileandCVaRdeviation. InChapter3weappliedCVaR-basedoptimizationtoasensorschedulingproblem.Suchproblemsarecommoninapplicationswhereinformationlossesoccurduetoinabilitytocollectinformationfromallsourcessimultaneously.Informationlossesassociatedwithnotobservingacertainsiteatsomemomentintimearemodeledasapenalty,whichconsistsoftwocomponents:axedpenaltyandapenaltyproportionaltothetimethesitewasnotobserved.InthissetupCVaRisappliedtominimizetheaverageofthe1)]TJ /F4 11.955 Tf 12.32 0 Td[(greatestpenalties.Themodelalsoincludestwotypesofwirelessconnectivityrobustnessconstraints:2-clubandk-plex. WediscussanexampleoftheproblemrequiringoptimizationofadeviationmeasureinChapter4.WeconsideredthegeneralizedCapitalAssetPricingModelbasedonmixedCVaRdeviationtoestimateriskpreferencesofinvestors.Theproblemofriskpreferencesestimationwasactivelydiscussedinmanystudies.OnemotivationforthesediscussionsisthecriticismoftheclassicalCAPM,whichisbasedontheassumptionthatinvestors'perceptionofriskcanberepresentedbystandarddeviation.Thiscriticismalignswithourmotivationforapplyingtheconceptofgeneralizedmeasures.ContrarytotheclassicalCAPMandsomerecentmodications,generalizedCAPMconsidersaclassofmixedCVaRdeviationswithcoefcientsspecifyingparameterization. 13

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CHAPTER2GENERALIZEDMEASURESOFDEVIATION,RISKANDERROR Inthischapterweprovideanoverviewofgeneralizeddeviationmeasures1andrelatedquantitativemeasures,e.g.riskmeasures,measuresoferror,statisticsandentropy.MostofthesemeasureshavebeenintroducedinarecentlineofresearchbyR.T.Rockafellar,S.Uryasev,M.ZabarankinandS.Sarykalin.Wedemonstratethatsomesubclassesofthesemeasureshaveproperties,whichallowthemtobemoreefcientlyappliedinriskmanagementapplications. Anapplicationofthedeviationmeasuresinregressionmodelsisproposedinthischapter.Inparticular,weidentiedthedistributionwhichisthemostapplicabletomodeltheresidualinthequantileregression. Thefollowingsectionprovidesanoverviewofthepopularmeasuresusedinriskmanagement.Section3introducesso-calledgeneralizeddeviationmeasuresandgeneralizedriskmeasures,andprovidesmathematicalrelationsbetweenthem.SomeimportantsubclassesofthesemeasuresarealsointroducedinSection3.Anoverviewofconditionalvalue-at-riskdeviationandrelatedmeasuresisprovidedinsection4.Section5containsanoverviewofso-calledmeasuresoferrorandtheirrelationtodeviationmeasures.Thesamesectionintroducesapplicationsofthedeviationmeasuresandthemeasuresoferrorinregressionmodels. 2.1ClassicalRiskandDeviationMeasures Sinceriskmanagementbecameastandardpracticeforalmostallinstitutionsandcommercialcompanies,anumberofquantitativemeasureshavebeendevelopedto 1Weusethewordgeneralizedtoaccentuatethatthesemeasuresbelongtoaclassgeneralizingthestandarddeviation.Inthecontextofthisandthefollowingchaptersthetermsdeviationmeasureandgeneralizeddeviationmeasurehavethesamemeaning. 14

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provideanaturalwaytoestimaterisk.Themostpopularandcommonlyusedmeasureshavebeenmaximumrisk,value-at-riskandstandarddeviation. Maximumrisk(maxrisk)providesaquantitativeevaluationoflossesintheworstpossiblescenario.Theexpressionbelowprovidesaformaldenitionofmaxrisk:maxrisk(X)=)]TJ /F2 11.955 Tf 11.29 0 Td[(inf(X) Itistheleastconvenientmeasureduetoitsover-conservatism:formostcommondistributionsitprovidesameaninglessvalueofinnity.Evenwhenthismeasureisapplicable(e.g.whenasetofpossibleoutcomesisnite),thismeasurelacksrobustness.Forexample,ifthesetofpossibleoutcomesisextendedbyaddingonemoreoutcomecorrespondingtolosses,whicharesubstantiallygreaterthanthepreviousvalueofmaxrisk,thenthevalueofmaxriskchangesbythesameamountregardlessoftheprobabilityofthisoutcome.Itisimportanttomention,however,thatthismeasurecanbeefcientlyusedinsomeoptimizationapplications.Inparticular,anyconstraintonmaxriskisequivalenttoasetofsimilarconstraintsforeachpossibleoutcome.Thiswasdemonstrated,forexample,in Boykoetal. ( 2011 ). Value-at-riskhasbeenapopularmeasureinthelast20yearsduetoitsnaturalinterpretationasanamountofreservesrequiredtopreventdefaultwithagivenprobability.Belowistheformaldenitionfrom Artzneretal. ( 1999 ): Denition.Given2(0,1),andareferenceinstrumentr,thevalue-at-riskVaRatlevelofthenalnetworthXwithdistributionP,isdenedbythefollowingrelation:VaR(X)=)]TJ /F2 11.955 Tf 11.29 0 Td[(inffxjP[Xxr]>gBaselCommitteeonBankingSupervision(BCBS)issuesso-calledBaselAccords,whicharerecommendationsonbankinglawsandregulations.Accordingtotheserecommendations,value-at-riskisthepreferredapproachtomarketriskmeasurement(forexample, Bas ( 2004 )).Inparticular,theserecommendationsspecifyminimum 15

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capitalrequirements,estimatedwithvalue-at-risk.Inmanycountries(includingtheUSA)regulatorsenforcenancialcompaniestocomplywithsomeoralloftheserecommendations.Thisledtovalue-at-riskbecomingoneofthemostcommonlyusedriskmeasures. Despiteitspopularity,value-at-riskhasaseriousdrawback.TheproblemisthatthefunctionalVaRdoesnothaveaconvexityproperty.Inriskmanagementconvexityisoftenanecessaryrequirement.Forexample,inportfoliomanagement,convexityofariskmeasurejustiesdiversicationofinvestments.Also,thelackofconvexitymakesthevalue-at-riskmeasureinefcientinoptimization,whereconvexityoftheoptimizedfunctionorconstraintsisalwaysadesiredproperty. Standarddeviation()isafunctiondenedonthespaceofrandomvariablesasasquarerootofvariance.Insomeapplications,atermvolatilityisusedinsteadofstandarddeviation.Contrarytovalue-at-risk,standarddeviationsatisestheconvexityproperty.Thismeasureisveryuseful,because,inparticular,volatilityinmanymodelsisaparameter.Forexample,inportfoliomanagementastockpricerandomprocessStisdescribedbythefollowingstochasticdifferentialequation:dSt=tStdt+tStdWt withtdenotingstockvolatility. Volatilityandvariancecanbeviewedasthemostpopularmeasuresinportfoliooptimizationandriskmanagement.Forexample,theCapitalAssetPricingModel(CAPM, Sharpe ( 1964 ), Lintner ( 1965 ), Mossin ( 1966 ), Treynor ( 1961 ), Treynor ( 1999 ))andtheArbitragePricingTheory(APT, Ross ( 1976 ))arefactormodelsfocusingonexplainingvariabilityinstockreturns.CAPMassumptionsimply,inparticular,thatallinvestorsinthemarketareoptimizingtheirinvestmentportfoliosconsideringvarianceasameasureofrisk.Basedonvariance,CAPMintroducestheso-calledBeta,aquantity 16

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determiningexposureofstock(orportfolio)returnstofuturemarkettrend:i=iM 2M whereiMdenotescovariancebetweenstockireturnsandmarketreturns,andMdenotesthemarketvolatility.IfallCAPMassumptionshold,thetotalvarianceofastockreturncanbeseparatedintosystematicandnonsystematic(idiosyncratic)components,wheresystematicpartofthevariancecorrespondstomarketreturns:2i=2i2M+2i,n IntheexpressionaboveidenotestheBetaofthestocki,andi,nisthenonsystematic(stock-specic)partofvolatility.Thisexpressionprovidesatoolforriskmanagement.Forexample,ifamanagerislookingforaportfoliowithweights(wi)withnoexposuretomarkettrend,hehastoconsideronlyportfolioswithtotalBetaequal0:NXi=1wii=0 Theconvexitypropertyofvolatilityguaranteesthatthenonsystematiccomponentoftheportfoliovariancecanbereducedbydiversication. ItisalsoimportanttomentionthatforanormalrandomvariableX,apair(E[X],(X))providescompleteinformationaboutthedistributionofX.Moreover,assumethatthedistributionofYhastobespecied,andtheonlyavailableinformationaboutarandomvariableYisthevaluesy=E[Y]andy=(Y).Followingthemaximumentropyprinciple,whichwasrstintroducedin Jaynes ( 1957 1968 ),itisnaturaltoassumetheleast-informativedistributionofYwithgivenmeanandvariance.Specically,considerthefollowingoptimizationproblem: maxEntr(f)s.t.Z1tf(t)dt= 17

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Z1t2f(t)dt)]TJ /F4 11.955 Tf 11.95 0 Td[(2=2fisaPDF whereEntr(f)denotestheShannonentropy( Shannon ( 1948 )):Entr(f)=)]TJ /F12 11.955 Tf 11.29 16.27 Td[(Z1f(t)lnf(t)dt whichisacommonmeasureofuncertainty.Theoptimalsolutionf(t)=1 p 2e)]TJ /F15 5.978 Tf 7.78 3.86 Td[((t)]TJ /F18 5.978 Tf 5.76 0 Td[()2 22istheprobabilitydensityfunctionofanormaldistributionwithmeanandvariance2(theproofcanbefoundin CozzolinoandZahner ( 1973 )).Therefore,iftheonlyavailableinformationaboutadistributionismeanandvariance,incanbenaturaltoassumeanormaldistribution.Itisveryconvenient,becauseinmanymodels,suchasfactormodels,uncertaintyismodeledbynormaldistribution. Thestandarddeviation,however,hasdisadvantages.Inparticular,itdoesn'tsatisfythemonotonicitypropertyX
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canbefoundin BradleyandTaqqu ( 2003 )).Theworst-casep%outcomesparticularlyinterestinvestors,becausetheyaretheonesmostlikelytocauseadefault. Theinefcienciesofthestandardmeasures,mentionedabove,canbesummarizedasfollows: 1. Theclassicalvalue-at-riskmeasurelackstheconvexityproperty,thereforeitissometimesdifculttoimplementVaRinstochasticoptimization. 2. Themaximumriskmeasurecanbeeasilyimplementedinstochasticoptimization,butitsover-conservatismoftenmakesitmeaningless. 3. Thestandarddeviationandthevariancedonotsatisfythemonotonicityproperty,whichwouldbenaturalforariskmeasure. 4. Asadeviationmeasure,thestandarddeviationdoesnotdistinguishbetweenthepositiveoutcomes(gains)andnegativeoutcomes(losses),andmeasuresoverallvariability,whileriskmanagerisprimarilyconcernedaboutthepartofvariabilityassociatedwiththemostundesirablescenarios. 2.2GeneralizedRiskandDeviationMeasures AnewsystematizationofmeasuresevaluatingprobabilitydistributionswasintroducedbyRockafellar,Sarykalin,UryasevandZabarankin( Rockafellaretal. ( 2006a ), Sarykalin ( 2008 )).Theyintroduceanumberofaxiomsdeningtwoseparateclasses:deviationmeasuresandriskmeasures.Theyalsoprovideaone-to-onecorrespondencebetweentheseclassesandspecialsubclasses. Considerthefollowingsetofaxioms: (D1)D(X+C)=D(X)forallXandconstantsC, (D2)D(0)=0andD(X)=D(X)forallXandall>0, (D3)D(X+Y)D(X)+D(Y)forallXandY, (D4)D(X)0forallXwithD>0fornonconstantX, (D5)fXjD(X)CgisclosedforeveryconstantC, (D6)D(X)EX)]TJ /F2 11.955 Tf 11.96 0 Td[(infXforallX. Theaxiom(D2)denespositivehomogeneity,axioms(D2)and(D3)togetherdeneconvexity.Accordingtothedenitionin Rockafellaretal. ( 2006a ),afunctional 19

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D:L2![0,1]isadeviationmeasureifitsatisesaxioms(D1)-(D4).Theaxiom(D5)denesclosedness.Inthisdocumentwewillonlyconsidercloseddeviationmeasures.Theproperty(D6)deneslowerrangedominance. Undertheseaxioms,D(X)dependsonlyonX)]TJ /F3 11.955 Tf 12.27 0 Td[(EX(fromthecaseof(D1)whereC=)]TJ /F3 11.955 Tf 9.29 0 Td[(EX),anditvanishesonlyifX)]TJ /F3 11.955 Tf 10.59 0 Td[(EX=0(asseenfrom(D4)withX)]TJ /F3 11.955 Tf 10.59 0 Td[(EXinplaceofX.ThiscapturestheideathatDmeasuresthedegreeofuncertaintyinX.Proposition4in Rockafellaretal. ( 2006a )provestheconvexitypropertyoftheclassofdeviationmeasuresandthesubclassoflowerrangedominateddeviationmeasures.Itcanbeseenthatthestandarddeviationtsintotheclassofdeviationmeasures,butitdoesn'tsatisfythelowerrangedominanceaxiom(D6). AlthoughthedeviationmeasuresasmeasuresofuncertaintyprovidesomeinformationabouttheriskinessassociatedwiththeoutcomeofX,theyarenotriskmeasuresinthesenseproposedin Artzneretal. ( 1999 ).Consider,forexample,asituationinthenancialmarketwithanarbitrageopportunitywithanetpayoffX.Bydenitionofthearbitrage,X0almostsurelyandP(X>0)>0.Arbitrageisgenerallyviewedasaprotablerisklessopportunity,therefore,forariskmeasureRthevalueR(X)shouldnotbegreaterthan0.IfXisrandom,adeviationmeasurewillalwaysbegreaterthan0. Rockafellaretal. ( 2006a )introducestheclassofcoherentriskmeasures,whichextendstheriskmeasuresdenedin Artzneretal. ( 1999 ).Considerthefollowingaxioms: (R1)R(X+C)=R(X))]TJ /F3 11.955 Tf 11.96 0 Td[(CforallXandconstantsC, (R2)R(0)=0,andR(X)=R(X)forallXandconstants>0, (R3)R(X+Y)R(X)+R(Y)forallXandY, (R4)R(X)R(Y)forallXY, (R5)R(X)>E[)]TJ /F3 11.955 Tf 9.29 0 Td[(X]forallnon-constantX. 20

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Theaxiom(R2)denespositivehomogeneity,(R3)denessubadditivity.Theaxioms(R2)and(R3)combinedimplythatRisaconvexfunctional.(R4)iscalledmonotonicity. Accordingtothedenitionsin Rockafellaretal. ( 2006a ),afunctionalR:L2!(,1]is a) acoherentriskmeasureifitsatisesaxioms(R1)-(R4), b) strictlyexpectationboundedriskmeasureifitsatises(R1)-(R3)and(R5),and c) coherent,strictlyexpectationboundedriskmeasureifitsatisesallaxioms(R1)-(R5). ItcanbenoticedthatVaRdoesn'ttinanyofthesecategoriesbecauseitdoesn'thavethesubadditivityproperty(R3). Thesamepaperdenesrelationsbetweendeviationmeasuresandriskmeasures: D(X)=R(X)]TJ /F3 11.955 Tf 11.96 0 Td[(EX) (2) R(X)=E[)]TJ /F3 11.955 Tf 9.29 0 Td[(X]+D(X) (2) Inparticular,equations( 2 )and( 2 )provideone-to-onecorrespondencebetweenstrictlyexpectationboundedriskmeasures(satisfying(R1)-(R3)and(R5))anddeviationmeasures(satisfyingaxioms(D1)-(D4)),andone-to-onecorrespondencebetweencoherent,strictlyexpectationboundedriskmeasures(satisfying(R1)-(R5))andlowerrangedominateddeviationmeasures(satisfying(D1)-(D4)and(D6)).Itcanbeshowndirectlyorusingtheserelationsthatthesetsofcoherentriskmeasures,strictlyexpectationboundedriskmeasures,andcoherent,strictlyexpectationboundedriskmeasuresareallconvex. Accordingtotheserelations,thestandarddeviationcorrespondstothestrictlyexpectationboundedriskmeasureR(x)=E[)]TJ /F3 11.955 Tf 9.3 0 Td[(X]+(X),whichisnotacoherentriskmeasure. 21

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2.3ConditionalValue-at-Risk Theclassicalriskmeasuresdiscussedinthebeginningofthechapterhaveanumberofimperfections.Thisledtothedevelopmentofnewkindsofriskmeasures.Oneofthemostnoticeableriskmeasuresistheconditionalvalue-at-risk(CVaR),alsoknownasexpectedshortfall,ortail-VaR.Wedenethismeasureaccordingto Pug ( 2000 ):CVaR(X)=minCC+(1)]TJ /F4 11.955 Tf 11.96 0 Td[())]TJ /F11 7.97 Tf 6.59 0 Td[(1E[X+C])]TJ /F12 11.955 Tf 7.09 11.48 Td[( where[y])]TJ /F1 11.955 Tf 10.41 1.79 Td[(equals)]TJ /F3 11.955 Tf 9.3 0 Td[(yfory<0and0otherwise.Theparametermusthaveavalueintheinterval(0,1).ItisimportanttonoticethattheoptimalCequalsvalue-at-risk:argminCC+(1)]TJ /F4 11.955 Tf 11.95 0 Td[())]TJ /F11 7.97 Tf 6.58 0 Td[(1E[X+C])]TJ /F12 11.955 Tf 7.09 11.48 Td[(=VaR(X) whereVaRdenotesthevalue-at-risk:VaR(X)=)]TJ /F2 11.955 Tf 11.29 0 Td[(supfzjFX(z)<1)]TJ /F4 11.955 Tf 11.95 0 Td[(g Anequivalentdenitioncanbefoundin Acerbi ( 2002 ):CVaR(X)=(1)]TJ /F4 11.955 Tf 11.96 0 Td[())]TJ /F11 7.97 Tf 6.59 0 Td[(1Z1VaR(X)d IfCVaRiscontinuousat)]TJ /F2 11.955 Tf 9.3 0 Td[(VaR,thenitcanbeexpressedviathefollowingrelation: CVaR(X)=)]TJ /F3 11.955 Tf 9.3 0 Td[(E[XjX)]TJ /F2 11.955 Tf 21.91 0 Td[(VaR(X)](2) Equation( 2 )showsthatconditionalvalue-at-riskequalstheweightedaverageoflossesexceedingvalue-at-risk.Therefore,conditionalvalue-at-riskestimateshowseverecanbepotentiallossesassociatedwithatailofdistribution.Thiscanbeviewedasanadvantageovertheclassicalvalue-at-risk,whichisnotsensitivetochangesinthetailofdistribution.Also,conditionalvalue-at-riskisacoherent,strictlyexpectationboundedriskmeasure.ThisfeatureallowsconsideringCVaRinavarietyofstochasticoptimizationapplications.Inparticular,asitwillbediscussedinChapter3,conditional 22

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value-at-riskcanbeusedinlinearprogramming.Thelowerrangedominateddeviationmeasure,denedvia( 2 )forR=CVaRiscalledaconditionalvalue-at-riskdeviation(CVaRdeviation),denotedasCVaR.Duetoconvexityofcoherent,strictlyexpectationboundedriskmeasures,aconvexcombinationofseveralCVaRswithdifferentcondencelevelsisalsoacoherentstrictlyexpectationboundedriskmeasure.ConvexcombinationofseveralCVaRsiscalledamixedconditionalvalue-at-risk(mixedCVaR).Thelowerrangedominateddeviationmeasure,denedvia( 2 )forR=mixedCVaR,iscalledamixedconditionalvalue-at-riskdeviation(mixedCVaRdeviation).InChapter4wedemonstrateapplicationofmixedCVaRdeviationintheframeworkofthegeneralizedcapitalassetpricingmodelforestimatingriskpreferencesofinvestors. 2.4ApplicationtoGeneralizedLinearRegressions Inthissectionweprovideanoverviewoftheso-calledmeasuresoferror,includingtheirrelationtogeneralizeddeviationmeasuresandapplicationtogeneralizedlinearregressions. 2.4.1MeasuresofError ConsiderafunctionalE:L2![0,1]andthesetofaxioms: (E1)E(0)=0butE(X)>0whenX6=0;also,E(C)<1forconstantsC, (E2)E(X)=E(X)forconstants>0, (E3)E(X+Y)E(X)+E(Y)forallX,Y, (E4)fX2L2()jEcgisclosedforallC<1, (E5)infX:EX=CE(X)>0forconstantsC6=0. Accordingtothedenitionin Rockafellaretal. ( 2008 ),thefunctionalEisameasureoferrorifitsatisesaxioms(E1)-(E4).Theproperty(E5)iscallednondegeneracy. ConsiderafunctionalD:L2![0,1]denedaccordingtothefollowingrelation: D(X)=infCE(X)]TJ /F3 11.955 Tf 11.96 0 Td[(C)(2) 23

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Figure2-1. RelationsBetweenMeasuresofError,DeviationMeasures,RiskMeasuresandStatistics whereEisanondegeneratemeasureoferror.Then,accordingtoTheorem2.1in Rockafellaretal. ( 2008 ),Disadeviationmeasure.Wewillfurtherusethetermprojecteddeviationmeasuretospecifythatthisdeviationmeasureisobtainedaccordingtoequation( 2 ).ThesetS(X)=argminCE(X)]TJ /F3 11.955 Tf 11.95 0 Td[(C)iscalledstatistic.InmanycasesS(X)isreducedtoasinglevalue. Considerameansquareerror:EMS(X)=E[X2].Itisawell-knownfactthatSMS(X)=EX.Accordingto( 2 ),thecorrespondingdeviationmeasureisvariance:DMS=2. Figure 2-1 illustratesrelationsbetweenmeasuresoferror,statistics,deviationmeasuresandriskmeasures. AnotherimportantexampleistheKoenker-Bassetterror,introducedin KoenkerandBassett ( 1978 ):EKB=EX++ 1)]TJ /F4 11.955 Tf 11.95 0 Td[(X)]TJ /F12 11.955 Tf 7.09 18.65 Td[( whereX+=maxf0,XgandX)]TJ /F2 11.955 Tf 10.78 1.8 Td[(=maxf0,)]TJ /F3 11.955 Tf 9.3 0 Td[(Xg.Itwasshownin Rockafellaretal. ( 2008 )thatEKBisanondegeneratemeasureoferror,whichcorrespondstotheconditional 24

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value-at-riskdeviation:DKB(X)=infCEKB(X)]TJ /F3 11.955 Tf 11.96 0 Td[(C)=CVaR(X) Moreover,thecorrespondingSKB(X)equalsvalue-at-risk:SKB(X)=argminCEKB(X)]TJ /F3 11.955 Tf 11.96 0 Td[(C)=VaR(X) 2.4.2GeneralizedLinearRegressions Generalizedlinearregressionwasdenedin Rockafellaretal. ( 2008 )asthefollowingproblem: minE(Z(c0,c1,...,cn)) (2) s.t.Z(c0,c1,...,cn)=Y)]TJ /F2 11.955 Tf 11.95 0 Td[((c0+c1X1+...+cnXn) (2) IntheaboveformulationtherandomvariablesX1,...,Xnarefactorsandc0,c1,...,cnareregressioncoefcients.IfE(Y)<1,thenthesetofoptimalregressioncoefcients(c0,c1,...,cn)alwaysexists. Theorem3.2inthesamepaperprovesequivalenceoftheproblem( 2 )( 2 )tothefollowingproblemofminimizingtheprojecteddeviationmeasure: minD(Z(c0,c1,...,cn)) (2) s.t.02S(Z(c0,c1,...,cn)) (2) Z(c0,c1,...,cn)=Y)]TJ /F2 11.955 Tf 11.95 0 Td[((c0+c1X1+...+cnXn) (2) Fortheoptimalregressioncoefcientsc0,c1,...,cnwecanwrite: Y=c0+c1X1+...+cnXn+"(2) where"istheerrorterm,equaltotheresidualfortheoptimalregressioncoefcients:"=Z(c0,c1,...,cn) 25

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TheTheorem3.2andequation( 2 )implythatfortheoptimalsetofcoefcientsstatisticofYequalsstatisticoftheoptimalcombinationoffactors:S(Y)=S(c0+c1X1+...+cnXn) TheyalsoimplythattheoptimalsetofcoefcientsminimizesthedeviationmeasureoftheresidualZ(c0,c1,...,cn): D(")=minc0,c1,...,cnD(Z(c0,c1,...,cn))(2) Itisimportanttonoticethatthevalueofc0hasnoeffectontherighthandsideofequation( 2 ). Belowweconsidertwoimportantexamples,whichwillbeusedinfurtheranalysis. ConsiderthemeansquarederrorEMS.Forthismeasureoferrortheproblem( 2 )( 2 )isequivalenttotheclassicallinearregression.SMS()=E[]impliesthattheexpectationoftheoptimalcombinationoffactorsequalsexpectationofY:EY=E[c0+c1X1+...+cnXn] Equation( 2 )impliesthattheerror"canbeinterpretedasarandomvariable,minimizingresidualvariance:2(")=minc1,...,cn2(Y)]TJ /F3 11.955 Tf 11.95 0 Td[(c1X1)]TJ /F2 11.955 Tf 11.96 0 Td[(...)]TJ /F3 11.955 Tf 11.96 0 Td[(cnXn) Anotherexampleisthequantileregression(forexample, Koenker ( 2005 )).Itcanbeformulatedaccordingto( 2 )( 2 )withE=EKB,or,equivalently,accordingto( 2 )( 2 )withageneralizeddeviationmeasureD=CVaRandstatisticS=VaR.Followingthesamelogicthatweusedfortheclassicallinearregression,SKB()=VaR()impliesthefollowing:VaR(Y)=VaR(c0+c1X1+...+cnXn) 26

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Equation( 2 )forquantileregressionimpliesthefollowinginterpretationoftheoptimalresidual":CVaR(")=minc1,...,cnCVaR(Y)]TJ /F3 11.955 Tf 11.96 0 Td[(c1X1)]TJ /F2 11.955 Tf 11.95 0 Td[(...)]TJ /F3 11.955 Tf 11.95 0 Td[(cnXn) 2.4.3DistributionofResidual Insomeapplicationsitisconvenienttospecifythedistributionoftheresidualerror".Thechoiceofthedistributionshouldbeonlybasedontheavailableinformation.Forthegeneralizedlinearregression,onlythestatisticanddeviationmeasureoftheerrortermareknown.Giventhisinformation,itisnaturaltopickthedistributionwhichhasthegreatestuncertainty. WeconsidertheShannonentropyEntr(f),whichiscommonlyusedasameasureofuncertainty( Shannon ( 1948 )):Entr(f)=)]TJ /F12 11.955 Tf 11.29 16.28 Td[(Z1f(t)lnf(t)dt wherefistheprobabilitydensityfunction. Forconvenience,considerthefollowingnotation.ForafunctionalF:L2![,1],deneF:ffjfisaPDFg![,1] accordingtothefollowing:F(f)=F(X)forXsuchthatfisaPDFofX WeapplythisnotationforF()=E[],F()=(),F()=CVaR(),F()=VaR(). Forageneralizedlinearregression,denedin( 2 )( 2 ),wechoosethedistributionf"bysolvingthefollowingentropymaximizationproblem: maxEntr(f) (2) s.t.S(f)=0 (2) 27

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D(f)=D(") (2) fisaPDF (2) whereS(f)andD(f)arethestatisticandthedeviationmeasureofarandomvariablewiththeprobabilitydensityfunctionf. Fortheclassicallinearregressiontheoptimizationproblem( 2 )( 2 )istheentropymaximizationproblemwithconstraintsonexpectationandexpectedvalue.Thesolutiontothisproblemisthenormaldistribution. Forthequantileregressiontheoptimizationproblem( 2 )( 2 )isequivalenttothefollowing: maxEntr(f) (2) s.t.VaR(f)=0 (2) CVaR(f)=CVaR(") (2) fisaPDF (2) Wederivethesolutiontothisproblemfromthesolutiontoasimilarproblem: maxEntr(g) (2) s.t.E(g)=0 (2) CVaR(g)=v (2) gisaPDF (2) Thesolutionto( 2 )( 2 )forv=1canbefoundin Grechuketal. ( 2009 ): g",1(t)=8>><>>:(1)]TJ /F4 11.955 Tf 11.96 0 Td[()exp)]TJ /F11 7.97 Tf 6.68 -4.98 Td[(1)]TJ /F9 7.97 Tf 6.59 0 Td[( )]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t)]TJ /F11 7.97 Tf 13.15 4.7 Td[(2)]TJ /F11 7.97 Tf 6.58 0 Td[(1 1)]TJ /F9 7.97 Tf 6.59 0 Td[(t2)]TJ /F11 7.97 Tf 6.58 0 Td[(1 1)]TJ /F9 7.97 Tf 6.58 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[()exp)]TJ /F5 11.955 Tf 5.48 -9.68 Td[()]TJ /F12 11.955 Tf 11.29 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t)]TJ /F11 7.97 Tf 13.15 4.7 Td[(2)]TJ /F11 7.97 Tf 6.59 0 Td[(1 1)]TJ /F9 7.97 Tf 6.59 0 Td[(t2)]TJ /F11 7.97 Tf 6.58 0 Td[(1 1)]TJ /F9 7.97 Tf 6.58 0 Td[( 28

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Derivationofthesolutionforothervaluesofvisidenticaltothecasev=1.Thefollowingfunctionistheoptimalprobabilitydensityfunction: g",v(t)=8>><>>:1)]TJ /F9 7.97 Tf 6.59 0 Td[( vexp)]TJ /F11 7.97 Tf 6.68 -4.97 Td[(1)]TJ /F9 7.97 Tf 6.59 0 Td[( v)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t)]TJ /F11 7.97 Tf 13.15 4.71 Td[(2)]TJ /F11 7.97 Tf 6.59 0 Td[(1 1)]TJ /F9 7.97 Tf 6.58 0 Td[(t2)]TJ /F11 7.97 Tf 6.59 0 Td[(1 1)]TJ /F9 7.97 Tf 6.58 0 Td[(1)]TJ /F9 7.97 Tf 6.59 0 Td[( vexp)]TJ /F5 11.955 Tf 5.48 -9.68 Td[()]TJ /F11 7.97 Tf 10.68 4.71 Td[(1 v)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t)]TJ /F11 7.97 Tf 13.15 4.71 Td[(2)]TJ /F11 7.97 Tf 6.59 0 Td[(1 1)]TJ /F9 7.97 Tf 6.59 0 Td[(t2)]TJ /F11 7.97 Tf 6.59 0 Td[(1 1)]TJ /F9 7.97 Tf 6.58 0 Td[((2) Note,that VaR(g",v)=)]TJ /F2 11.955 Tf 10.5 8.09 Td[(2)]TJ /F2 11.955 Tf 11.96 0 Td[(1 1)]TJ /F4 11.955 Tf 11.96 0 Td[((2) whatfollowsfrom( 2 )andthefollowingequality:Z2)]TJ /F15 5.978 Tf 5.76 0 Td[(1 1)]TJ /F18 5.978 Tf 5.76 0 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[( vexp1)]TJ /F4 11.955 Tf 11.95 0 Td[( vt)]TJ /F2 11.955 Tf 13.15 8.08 Td[(2)]TJ /F2 11.955 Tf 11.95 0 Td[(1 1)]TJ /F4 11.955 Tf 11.95 0 Td[(dt=1)]TJ /F4 11.955 Tf 11.96 0 Td[( Theorem.Theoptimalprobabilitydensityfunctionfintheproblem( 2 )( 2 )isthefollowingfunctionf",v: f",v(t)=8>><>>:1)]TJ /F9 7.97 Tf 6.59 0 Td[( vexp)]TJ /F11 7.97 Tf 6.68 -4.98 Td[(1)]TJ /F9 7.97 Tf 6.59 0 Td[( vtt01)]TJ /F9 7.97 Tf 6.59 0 Td[( vexp)]TJ /F5 11.955 Tf 5.48 -9.68 Td[()]TJ /F11 7.97 Tf 10.68 4.7 Td[(1 vtt0(2) wherev=CVaR("). ProofofTheorem.First,considerthefollowingentropyproperty:Entr(f)=Entr(g)forg(t)=f(t)]TJ /F3 11.955 Tf 11.96 0 Td[(c) wherecisanyconstant.Thispropertyguaranteesthattheproblem( 2 )( 2 )isequivalenttotheproblem: maxEntr(g) (2) s.t.g(t)=f(t+E(f))8t (2) VaR(f)=0 (2) CVaR(f)=v (2) f,garePDFs (2) 29

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Theaxiom(D1)allowsustosubstituteconstraint( 2 )withconstraintCVaR(g)=d.Alsonote:( 2 )guaranteesthatE(g)=0fortheoptimalg,and( 2 )togetherwith( 2 )guaranteethatf(t)=g(t)]TJ /F2 11.955 Tf 12.11 0 Td[(VaR(g)).Therefore,thisproblemisalsoequivalenttothefollowing: maxEntr(g) (2) s.t.g(t)=f(t+E(f))8t (2) f(t)=g(t)]TJ /F2 11.955 Tf 11.95 0 Td[(VaR(g))8t (2) VaR(f)=0 (2) CVaR(g)=v (2) E(g)=0 (2) f,garePDFs (2) Notethatinthisproblemtheconstraints( 2 ),( 2 )and( 2 )areredundant:foranyginthefeasibleset,iffisdenedaccordingto( 2 ),thensuchfalreadysatises( 2 ),and( 2 )isenforcedby( 2 ).Therefore,thisproblemisequivalenttotheproblem( 2 )( 2 ).g",din( 2 )istheoptimalPDFg,andtheoptimalfisobtaineddirectlyfrom( 2 )and( 2 ). Thisdistributionhasparametersandv.Wewillfurthersaythatarandomvariable"isdistributedaccordingtoDExp(,v),ifitsprobabilitydensityfunctionf"isexpressedby( 2 ). Figures 2-2 and 2-3 depicttheprobabilitydensityfunctionsofdistributionsDExp(,v)fordifferentvaluesandv.Eachdistributioncanbeviewedasatwo-sidedexponentialdistribution. 30

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Figure2-2. ProbabilityDensityFunctionsforDExp(,1) 31

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Figure2-3. ProbabilityDensityFunctionsforDExp(,2) 32

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CHAPTER3ROBUSTCONNECTIVITYISSUESINDYNAMICSENSORNETWORKSFORAREASURVEILLANCEUNDERUNCERTAINTY Inthischapter1,weaddressseveralproblemsandchallengesarisinginthetaskofutilizingdynamicsensornetworksforareasurveillance.Thistaskneedstobeefcientlyperformedindifferentapplications,wherevarioustypesofinformationneedtobecollectedfrommultiplelocations.Inadditiontoobtainingpotentiallyvaluableinformation(thatcanoftenbetime-sensitive),onealsoneedstoensurethattheinformationcanbeefcientlytransmittedbetweenthenodesinawirelesscommunication/sensornetwork.Inthesimpleststaticcase,thelocationofsensors(i.e.,nodesinasensornetwork)isxed,andthelinks(edgesinasensornetwork)aredeterminedbythedistancebetweensensornodes,thatis,twonodeswouldbeconnectediftheyarelocatedwithintheirwirelesstransmissionrange.However,inmanypracticalsituations,thesensorsareinstalledonmovingvehicles(forinstance,unmannedairvehicles(UAVs))thatcandynamicallymovewithinaspeciedareaofsurveillance.Clearly,inthiscasethelocationofnodesandedgesinanetworkandtheoverallnetworktopologycanchangesignicantlyovertime.Thetaskofcrucialimportanceinthesesettingsistodevelopoptimalstrategiesforthesedynamicsensornetworkstooperateefcientlyintermsofbothcollectingvaluableinformationandensuringrobustwirelessconnectivitybetweensensornodes. Intermsofcollectinginformationfromdifferentlocations(sites),oneneedstodealwiththechallengethatthenumberofsitesthatneedtobevisitedtogatherpotentiallyvaluableinformationisusuallymuchlargerthanthenumberofsensors.Undertheseconditions,oneneedstodevelopefcientschedulesforallthemovingsensorssuchthattheamountofvaluableinformationcollectedbythesensorsismaximized.Arelevant 1ThischapterisbasedonthejointpublicationwithA.Veremyev,V.Boginski,D.E.JeffcoatandS.Uryasev( Kalinchenkoetal. ( 2011 )) 33

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approachthatwaspreviouslyusedbytheco-authorstoaddressthischallengedealtwithformulatingthisproblemintermsofminimizingtheinformationlossesduetothefactthatsomelocationsarenotundersurveillanceatcertaintimemoments.Inthesesettings,theinformationlossescanbequantiedasbothxedandvariablelosses,wherexedlosseswouldoccurwhenagivensiteissimplynotundersurveillanceatsometimemoment,whilevariablelosseswouldincreasewithtimedependingonhowlongasitehasnotbeenvisitedbyasensor.Takingintoaccountvariablelossesofinformationisoftencriticalinthecasesofdealingwithstrategicallyimportantsitesthatneedtobemonitoredascloselyaspossible.Inaddition,theparametersthatquantifyxedandvariableinformationlossesareusuallyuncertain,therefore,theuncertaintyandriskshouldbeexplicitlyaddressedinthecorrespondingoptimizationproblems. Theotherimportantchallengethatwillbeaddressedinthischapterisensuringrobustconnectivitypatternsindynamicsensornetworks.Theserobustnesspropertiesareespeciallyimportantinuncertainandadverseenvironmentsinmilitarysettings,whereuncertainfailuresofnetworkcomponents(nodesand/oredges)canoccur. Theconsideredrobustconnectivitycharacteristicswilldealwithdifferentparametersofthenetwork.First,thenodeswithinanetworkshouldbeconnectedbypathsthatarenotexcessivelylong,thatis,thenumberofintermediarynodesandedgesintheinformationtransmissionpathshouldbesmallenough.Second,eachnodeshouldbeconnectedtoasignicantnumberofothernodesinanetwork,whichwouldprovidethepossibilityofmultiple(backup)transmissionpathsinthenetwork,sinceotherwisethenetworktopologywouldbevulnerabletopossiblenetworkcomponentfailures. Clearly,theaforementionedrobustconnectivitypropertiesaresatisediftherearedirectlinksbetweenallpairsofnodes,thatis,ifthenetworkformsaclique.Cliquesareveryrobustnetworkstructures,duetothefactthattheycansustainmultiplenetworkcomponentfailures.Notethatanysubgraphofacliqueisalsoaclique,whichimpliesthatthisstructurewouldmaintainrobustconnectivitypatternsevenifmultiplenodesin 34

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thenetworkaredisabled.However,thepracticaldrawbacksofcliquesincludethefactthatthesestructuresareoftenoverlyrestrictiveandexpensivetoconstruct. Toprovideatradeoffbetweenrobustnessandpracticalfeasibility,certainothernetworkstructuresthatrelaxthedenitionofacliquecanbeutilized.Thefollowingdenitionsaddresstheserelaxationsfromdifferentperspectives.GivenagraphG(V,E)withasetofvertices(nodes)VandasetofedgesE,ak-cliqueCisasetofverticesinwhichanytwoverticesaredistanceatmostkfromeachotherinG Luce ( 1950 ).LetdG(i,j)bethelengthofashortestpathbetweenverticesiandjinGandd(G)=maxi,j2VdG(i,j)bethediameterofG. Thus,iftwoverticesu,v2Vbelongtoak-cliqueC,thendG(u,v)k,howeverthisdoesnotimplythatdG(C)(u,v)k(thatis,othernodesintheshortestpathbetweenuandvarenotrequiredtobelongtothek-clique).ThismotivatedMokken( Mokken ( 1979 ))tointroducetheconceptofak-club.Ak-clubisasubsetofverticesDVsuchthatthediameterofinducedsubgraphG(D)isatmostk(thatis,thereexistsapathoflengthatmostkconnectinganypairofnodeswithinak-club,whereallthenodesinthispathalsobelongtothisk-club).Also,~VVissaidtobeak-plexifthedegreeofeveryvertexintheinducedsubgraphG(~V)isatleastj~Vj)]TJ /F3 11.955 Tf 18.02 0 Td[(k( SeidmanandFoster ( 1978 )).Acomprehensivestudyofthemaximumk-plexproblemispresentedinarecentworkby Balasundarametal. ( 2010 ). Inthischapter,weutilizetheseconceptstodeveloprigorousmathematicalprogrammingformulationstomodelrobustconnectivitystructuresindynamicsensornetworks.Moreover,theseformulationswillalsotakeintoaccountvariousuncertainparametersbyintroducingquantitativeriskmeasuresthatminimizeorrestrictinformationlosses.Overall,wewilldevelopoptimalschedulesforsensormovementsthatwilltakeintoaccountboththeuncertainlossesofinformationandtherobustconnectivitybetweenthenodesthatwouldallowonetoefcientlyexchangethecollectedinformation. 35

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3.1Multi-SensorSchedulingProblems:GeneralDeterministicSetup Thissectionintroducesapreliminarymathematicalframeworkfordynamicmulti-sensorschedulingproblems.Thesimplestdeterministicone-sensorversionofthisproblemwasintroducedin YavuzandJeffcoat ( 2007 ).Theone-sensorschedulingproblemwasthenextendedandgeneralizedtomorerealisticcasesofmulti-sensorschedulingproblems,includingthesetupsinuncertainenvironmentsin Boykoetal. ( 2011 ).Inthesubsequentsectionsofthischapter,thissetupwillbefurtherextendedtoincorporaterobustconnectivityissuesintotheconsidereddynamicsensornetworkmodels. Tofacilitatefurtherdiscussion,werstintroducethefollowingmathematicalnotationsthatwillbeusedthroughoutthischapter.Assumethattherearemsensorsthatcanmovewithinaspeciedareaofsurveillance,andtherearensitesthatneedtobeobservedateverydiscretetimemomentt=1,...,T.Onecaninitiallyassumethatasensorcanobserveonlyonesiteatonepointoftimeandcanimmediatelyswitchtoanothersiteatthenexttimemoment.Sincemisusuallysignicantlysmallerthann,therewillbebreachesinsurveillancethatcancauselossesofpotentiallyvaluableinformation. Apossibleobjectivethatarisesinpracticalsituationsistobuildastrategythatoptimizesapotentiallossthatisassociatedwithnotobservingcertainsitesatsometimemoments. 3.1.1FormulationwithBinaryVariables Onecanintroducebinarydecisionvariables xi,t=8><>:1,ifi-thsiteisobservedattimet0,otherwise(3) andintegervariablesyi,tthatdenotethelasttimesiteiwasvisitedasoftheendoftimet,i=1,...,n,t=1,...,T,m
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Onecanthenassociateaxedpenaltyaiwitheachsiteiandavariablepenaltybiofinformationloss.Ifasensorisawayfromsiteiattimepointt,thexedpenaltyaiisincurred.Moreover,thevariablepenaltybiisproportionaltothetimeintervalwhenthesiteisnotobserved.Weassumethatthevariablepenaltyratecanbedynamic;therefore,thevaluesofbimaybedifferentateachtimeinterval.Thusthelossattimetassociatedwithsiteiis ai(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi,t)+bi,t(t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)(3) Intheconsideredsetup,wewanttominimizethemaximumpenaltyoveralltimepointstandsitesi maxi,tfai(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t)g(3) Furthermore,xi,tandyi,tarerelatedviathefollowingsetofconstraints.Nomorethanmsensorsareusedateachtimepoint;therefore nXi=1xi,tm,8t=1,...,T(3) Timeyi,tisequaltothetimewhenthesiteiwaslastvisitedbyasensorbytimet.Thisconditionissetbythefollowingconstraints: 0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F11 7.97 Tf 6.59 0 Td[(1txi,t,8i=1,...,n,8t=1,...,T (3) txi,tyi,tt,8i=1,...,n,8t=1,...,T (3) Further,usinganextravariableCandstandardlinearizationtechniques,wecanformulatethemulti-sensorschedulingoptimizationprobleminthedeterministicsetupasthefollowingmixedintegerlinearprogram: minC (3) s.t.Cai(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bi,t(t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t),8i=1,...,n,8t=1,...,T (3) nXi=1xi,tm,8t=1,...,T (3) 37

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0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F11 7.97 Tf 6.58 0 Td[(1txi,t,8i=1,...,n,8t=1,...,T (3) txi,tyi,tt,8i=1,...,n,8t=1,...,T (3) yi,0=0,8i=1,...,n (3) xi,t2f0,1g,8i=1,...,n,8t=1,...,T (3) yi,t2IR,8i=1,...,n,8t=0,...,T (3) Weallowedrelaxation( 3 )ofvariablesyi,ttothespaceofrealnumbers,becausetheconstraints( 3 )and( 3 )enforcethefeasiblevaluesofvariablesyi,ttobeinteger. 3.1.2CardinalityFormulation Lemma.Constraint( 3 )isequivalenttothefollowingcombinationoftwoconstraints: 0xi,t18i=1,...,n,8t=1,...,T (3) card(xt)nXi=1xi,t8t=1,...,T (3) wherext=(x1,t,...,xn,t)T,andcard(xt)denotesthecardinalityfunctionforthevectorxt.Bydenition,card(xt)equalsthenumberofnon-zeroelementsintheinputvectorxt. Proof.Assumethematrix(xi,t)satisesconstraint( 3 ).Obviously,itthensatises( 3 ).Atthesametime,foreveryt,sumofallelementsisequaltothenumberofvalues1init.Andthesearetheonlynon-zeroelementsinit.Therefore,constraint( 3 )isalsosatised. Nowassumethematrix(xi,t)doesnotsatisfyconstraint( 3 ).Thusthereisapair(i,t),forwhichxi,t=and6=0and6=1.If<0or>1,thenconstraint( 3 )isviolated.Thus,forallpairs(i,t),0xi,t1,and0<<1.Therefore,forallpairs(i,t),card(xi,t)xi,t,andcard()>.Takingintoaccountthatcard(xt)=Picard(xi,t)weconcludethat( 3 )isviolated. 38

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Nowwecanwritealternative,cardinalityformulationforthegeneraldeterministicsensor-schedulingproblem. minC (3) s.t.Cai(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bi,t(t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t),8i=1,...,n,8t=1,...,T (3) nXi=1xi,tm,8t=1,...,T (3) 0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F11 7.97 Tf 6.58 0 Td[(1txi,t,8i=1,...,n,8t=1,...,T (3) txi,tyi,tt,8i=1,...,n,8t=1,...,T (3) yi,0=0,8i=1,...,n (3) 0xi,t1,8i=1,...,n,8t=1,...,T (3) card(xt)nXi=1xi,t,8t=1,...,T (3) yi,t2IR,8i=1,...,n,8t=0,...,T (3) Althoughthetwoformulationsareequivalent,someoptimizationsolvers,suchasPortfolioSafeguard(thatwillbementionedlaterinthischapter),canprovideanear-optimalsolutionfasteriftheformulationwithcardinalityconstraintsisusedinsteadoftheonewithbooleanvariables,whichmaybeimportantintime-criticalsystemsinmilitarysettings. 3.2QuantitativeRiskMeasuresinUncertainEnvironments: ConditionalValue-at-Risk Tofacilitatefurtherdiscussionontheformulationsoftheaforementionedproblemsunderuncertainty,inthissectionwebrieyreviewbasicdenitionsandfactsrelatedtotheConditionalValue-at-Riskconcept. ConditionalValue-at-Risk(CVaR) RockafellarandUryasev ( 2000 2002 ); Sarykalinetal. ( 2008 )isaquantitativeriskmeasurethatwillbeusedinthemodelsdevelopedinthenextsection,whichwilltakeintoaccountthepresenceofuncertainparameters.CVaRiscloselyrelatedtoawell-knownquantitativeriskmeasurereferredtoas 39

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Value-at-Risk(VaR).Bydenition,withrespecttoaspeciedprobabilitylevel(inmanyapplicationsthevalueofissetratherhigh,e.g.95%),the-VaRisthelowestamountsuchthatwithprobability,thelosswillnotexceed,whereasforcontinuousdistributionsthe-CVaRistheconditionalexpectationoflossesabovethatamount.Asitcanbeseen,CVaRisamoreconservativeriskmeasurethanVaR,whichmeansthatminimizingorrestrictingCVaRinoptimizationproblemsprovidesmorerobustsolutionswithrespecttotheriskofhighlosses(Figure 3-1 ). Figure3-1. GraphicalrepresentationofVaRandCVaR. Formally,-CVaRforcontinuousdistributionscanbeexpressedas CVaR(x)=(1)]TJ /F4 11.955 Tf 11.95 0 Td[())]TJ /F11 7.97 Tf 6.58 0 Td[(1ZL(x,w)(x)L(x,w)p(w)dw(3) whereL(x,w)istherandomloss(penalty)variabledrivenbydecisionvectorxandhavingadistributioninIRinducedbythatofthevectorofuncertainparametersw. 40

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CVaRisdenedinasimilarwayfordiscreteormixeddistributions.ThereadercanndtheformaldenitionofCVaRforgeneralcasein RockafellarandUryasev ( 2002 ); Sarykalinetal. ( 2008 ). Ithasbeenshownin RockafellarandUryasev ( 2000 )thatminimizing( 3 )isequivalenttominimizingthefunction F(x,)=+(1)]TJ /F4 11.955 Tf 11.96 0 Td[())]TJ /F11 7.97 Tf 6.59 0 Td[(1Zw2IRd[L(x,w))]TJ /F4 11.955 Tf 11.95 0 Td[(]+p(w)dw(3) overwand,where[t]+=twhent>0but[t]+=0whent0,andoptimalvalueofthevariablecorrespondstotheVaRvalue,introducedabove. 3.3OptimizingtheConnectivityofDynamicSensorNetworks UnderUncertainty Thissectionextendstheprevioussensorsschedulingproblemtoastochasticenvironment.WeuseCVaRmeasuretomodelandoptimizevariousobjectivesassociatedwiththeriskoflossofinformation. Inthestochasticformulation,thepenaltiesaiandbi,tarerandom.WegenerateSdiscretescenarios,whichapproximateimpliedjointdistribution.Thus,everyscenarioconsistsoftwoarrays:one-dimensionalfaigsandtwo-dimensionalfbi,tgs. Now,considerthetermofthelossfunctioncorrespondingtothesitei,timet,andscenarios:Ls(x,y;i,t)=asi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t) Underuncertainty,itisoftenmoreimportanttomitigatethebiggestpossiblelosses,ratherthantheaveragedamage.Followingthisidea,wetake(1)]TJ /F4 11.955 Tf 12.17 0 Td[()biggestpenalties,andminimizeaveragepenaltyoveralli,tands.Thisobjectivefunctionisexactlytheconditionalvalue-at-risk. Wenowhavethefollowingclassofoptimizationproblems: minx,yCVaRfL(x,y;i,t)g(3) 41

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Thisclasshasoneextremecase:=1,whentheproblembecomesequivalenttominimizingmaximumpossiblepenaltyoverallscenarios,locationsandtimepoints: minx,ymaxi,t,s(asi(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t))(3) ThisproblemhasanequivalentLPformulation: minC (3) s.t.Casi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t) (3) 8i=1,...,n,8t=1,...,T,8s=1,...,S InordertoformulateageneralCVaRoptimizationprobleminLPtermswehavetointroduceadditionalvariablessi,t,s=1,...,S,i=1,...,n,t=1,...,T,and.WiththesevariablestheproblemofminimizingCVaRwillbereducedtothefollowing: minC (3) s.t.C+1 (1)]TJ /F4 11.955 Tf 11.96 0 Td[()nSTXs=1,...,Si=1,...,nt=1,...,Tsi,t (3) si,tasi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t))]TJ /F4 11.955 Tf 11.95 0 Td[( (3) 8i=1,...,n,t=1,...,T,s=1,...,Ssi,t0,8i=1,...,n,t=1,...,T,s=1,...,S (3) Wehavediscussedvariousobjectivefunctionswithobjective-specicconstraintsforsensorschedulingproblemsinthestochasticenvironment.Inadditiontothat,everysensorschedulingproblem,includingthoseinstochasticenvironment,musthaveconstraintslimitingnumberofsensors( 3 )anddeningvariablesofthelasttimeofobservation( 3 )( 3 ).Theseconstraintsarereferredtoasmandatoryconstraintsforeverysensor-schedulingproblem. FurtherwedeneawirelessconnectivitynetworkG(V,E)onthesetoflocationsV.Weinterpretitintermsofthe0-1adjacencymatrixE=feijgi,j=1,...,n,whereeacheijisa 42

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0-1indicatorofwirelesssignalreachabilitybetweennodesiandj,thatis,iflocationsiandjarewithindirecttransmissiondistancefromeachother,thentheyareconnectedbyanedge,andeij=1(eij=0otherwise).Wealsodeneasubnetwork~GofG(V,E)containingonlythosemnodes(locations)thataredirectlyobservedbysensorsataparticulartimemoment. Schedulingofobservationoftenrequiressensorstomaintainacertainlevelofwirelessconnectivityrobustness.Ifanenemysendsajammingsignalthatbreaksconnectivitybetweenapairofnodes,thenthesubnetwork~Geithershouldstayconnected,oratleastshouldmaintainunitywithprobabilitycloseto1.Further,wewillutilizeseveraltypesofnetworkstructuresthatcanbeappliedtoensurethatthenetworksatisescertainrobustnessconstraints. Themostrobustnetworkstructureisaclique,whichimpliesthateachpairofnodesisdirectlyconnectedbyanedge.Obviously,maintainingacliquestructureofthesubnetwork~Gateverymomentintimeisveryexpensiveintermsofpenalty,andcanbeevenimpossible,iftheoverallwirelessconnectivitynetworkisnotdenseenough.Hence,itisreasonabletoutilizeappropriatetypesofcliquerelaxationstoensurerobustnetworkconnectivityateverytimemoment. Oneoftheconsideredconceptsisak-plex.Bydenition,asmentionedabove,ak-plexisasubgraphinwhicheverynodeisconnectedtoatleastm)]TJ /F3 11.955 Tf 12.05 0 Td[(kothernodesinit(wheremisthenumberofnodesinthissubgraph).Thisnetworkcongurationensuresthateachnodeisconnectedtomultipleneighbors,whichmakesitmorechallengingforanadversarytodisconnectthenetworkandisolatethenodesbydestroying(jamming)theedges. Anotherconsideredclassofnetworkcongurationsisak-club.Recallthateverypairofnodesink-clubisconnectedinitthroughachainofnomorethankarcs(edges).Themotivationforstudyingthistypeofconstraintsisbasedonthefactthatiftwosensorsareconnectedthroughashorterpath,itlowerstheprobabilityoferrorsin 43

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informationtransmissionthroughintermediaries,sincethenumberofintermediariesissmaller.Laterinthepaper,wewillspecicallyuseastrongerrequirementonthelengthofthesepaths.Werequirethatanytwonodesareconnectedeitherdirectlybyanedge,orthroughatmostoneintermediarynode,whichisoftenadesiredrobustnessrequirementundertheconditionswhenthenumberofintermediaryinformationtransmissionsneedstobeminimizedduetoadversarialconditions.Clearly,a2-clubisastructurethatsatisesthisrequirement.Inthenextsubsection,weshowthatthisconditioncanbeincorporatedintheconsideredoptimizationmodels. 3.3.1EnsuringShortTransmissionPathsvia2-clubFormulation Thegeneralrequirementforasubnetwork~Gtorepresentak-clubcanbeformulatedasthefollowingsetofconstraints: eij+...+nXq=1eiqeqjxq,t+...+nXq=1nXl=1eiqeqleljxq,txl,t+...+nXq=1nXl=1nXp=1eiqeqlelpepjxq,txl,txp,t+...+nXi1=1nXi2=1nXik)]TJ /F15 5.978 Tf 5.76 0 Td[(2=1nXik)]TJ /F15 5.978 Tf 5.75 0 Td[(1=1eii1ei1i2...eik)]TJ /F15 5.978 Tf 5.76 0 Td[(2ik)]TJ /F15 5.978 Tf 5.76 0 Td[(1eik)]TJ /F15 5.978 Tf 5.76 0 Td[(1jxi1,t...xik)]TJ /F15 5.978 Tf 5.75 0 Td[(1,txi,t+xj,t)]TJ /F2 11.955 Tf 11.96 0 Td[(1(3) wherei=1,...,n)]TJ /F2 11.955 Tf 12.26 0 Td[(1,j=i+1,...,n,t=1,...,T.Foreveryktheseconstraintscanbelinearized,however,thesizeoftheproblemmaysubstantiallyincrease.Inthispaper,welimitourdiscussiononlyto2-clubconstraintsduetothepracticalreasonsmentionedearlierinthissectionandduetothefactthattheformulationforthecaseofk=2willnotaddtoomanynewentities(nomorethanO(n2))totheproblemformulation.Theyrequireeverypairofnodes(i,j)tobeconnecteddirectly,orthroughsomeothernodep. 44

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Suchtypeofcommunicationbetweensensors(i,j)hasaconciseformulation:eij+nXp=1eipepjxp,txi,t+xj,t)]TJ /F2 11.955 Tf 11.96 0 Td[(18i=1,...,n)]TJ /F2 11.955 Tf 11.96 0 Td[(1,8j=i+1,...,n8t=1,...,T Here,theleft-handsideisalwaysnonnegative.Theright-handsidebecomespositiveonlyifbothlocationsiandjareobservedbysensors,andthenitequals1.Accordingtothe2-clubdenition,thesesensorshavetobeconnected(andexchangeinformation)eitherdirectly,orthroughoneotherintermediarysensornode.Intherstcaseeijequals1.Inthesecondcase,thesumPnp=1eipepjxp,twillalsobepositive. Itisalsoimportanttonotethatthoseconstraints,forwhicheij=1,canbeomitted.Thus,a2-clubwirelessnetworkcongurationcanbeensuredbythefollowingsetofconstraints:Xp2(i)\(j)xp,txi,t+xj,t)]TJ /F2 11.955 Tf 11.95 0 Td[(18i=1,...,n)]TJ /F2 11.955 Tf 11.95 0 Td[(1,8j=i+1,...,n,j=2(i),8t=1,...,T where(i)and(j)arethesetsofneighborsofnodesiandj,respectively. Belowwepresentthecompletegeneralformulationforthedynamicsensorschedulingoptimizationprobleminastochasticenvironmentwith2-clubwirelessconnectivityconstraints. minCs.t.C+1 (1)]TJ /F4 11.955 Tf 11.96 0 Td[()nSTXs=1,...,Si=1,...,nt=1,...,Tsi,tsi,tasi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t))]TJ /F4 11.955 Tf 11.95 0 Td[(8i=1,...,n,t=1,...,T,s=1,...,Ssi,t0,8i=1,...,n,t=1,...,T,s=1,...,SnXi=1xi,tm,8t=1,...,T 45

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0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F11 7.97 Tf 6.58 0 Td[(1txi,t,8i=1,...,n,8t=1,...,Ttxi,tyi,tt,8i=1,...,n,8t=1,...,Tyi,0=0,8i=1,...,nXp2(i)\(j)xp,txi,t+xj,t)]TJ /F2 11.955 Tf 11.95 0 Td[(18i=1,...,n)]TJ /F2 11.955 Tf 11.95 0 Td[(1,8j=i+1,...,n,j=2(i),8t=1,...,Txi,t2f0,1g,8i=1,...,n,8t=1,...,Tyi,t2IR,8i=1,...,n,8t=0,...,T 3.3.2EnsuringBackupConnectionsviak-plexFormulation Constraintsthatrequireawirelessnetworktohavethek-plexstructure,canbedenedusingasymmetricadjacencymatrixE=feijgi,j=1,...,n,asdenedabove.Recallthatxt=(x1,t,...,xn,t)T.Considerthevectorzt=(z1,t,...,zn,t)=Ext.Theelementzi,tcanbeinterpretedasthenumberofsensorswhichhaveawirelessconnectionwithnodeiattimet.Thus,theconstraintExtxtor(E)]TJ /F3 11.955 Tf 11.96 0 Td[(I)xt0ensuresthateachsensornodehasatleastoneneighbor,i.e.,itisnotisolated.Ifwewanteachsensortohaveatleast(m)]TJ /F3 11.955 Tf 11.99 0 Td[(k)wirelessconnections(edges)withothersensors,thenweshouldmaketheconstraintsmorerestrictive:Ext(m)]TJ /F3 11.955 Tf 11.95 0 Td[(k)xt,or(E)]TJ /F2 11.955 Tf 11.95 -.16 Td[((m)]TJ /F3 11.955 Tf 11.95 0 Td[(k)I)xt08t=1,...,T Theserestrictionsbydenitionensurethatasubnetwork~Gisak-plex. Belowwepresentthecompletegeneralformulationforthedynamicsensorschedulingoptimizationprobleminastochasticenvironmentwithk-plexwirelessconnectivityconstraints. minCs.t.C+1 (1)]TJ /F4 11.955 Tf 11.96 0 Td[()nSTXs=1,...,Si=1,...,nt=1,...,Tsi,t 46

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si,tasi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t))]TJ /F4 11.955 Tf 11.95 0 Td[(8i=1,...,n,t=1,...,T,s=1,...,Ssi,t0,8i=1,...,n,t=1,...,T,s=1,...,SnXi=1xi,tm,8t=1,...,T0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F11 7.97 Tf 6.58 0 Td[(1txi,t,8i=1,...,n,8t=1,...,Ttxi,tyi,tt,8i=1,...,n,8t=1,...,Tyi,0=0,8i=1,...,n(E)]TJ /F2 11.955 Tf 11.95 -.17 Td[((m)]TJ /F3 11.955 Tf 11.95 0 Td[(k)I)xt0,8t=1,...,Txi,t2f0,1g,8i=1,...,n,8t=1,...,Tyi,t2IR,8i=1,...,n,8t=0,...,T 3.4ComputationalExperiments ComputationalexperimentsonsampleprobleminstanceshavebeenperformedonIntelXeonX53552.66GHzCPUwith16GBRAM,usingtwocommercialoptimizationsolvers:ILOGCPLEX11.2andAORDAPSG64bit(MATLAB64bitenvironment).Itshouldbenotedthatduetothenatureoftheconsideredclassofproblems,theyarecomputationallychallengingevenonrelativelysmallnetworks.Therefore,inmanypracticalsituations,ndingnear-optimalsolutionsinareasonabletimewouldbesufcient.ThePSGpackagewasusedinadditiontoCPLEXbecauseithasattractivefeaturesintermsofcodingtheoptimizationproblems,andthereforeitmaybemorepreferabletouseinpracticaltime-criticalsettings.Inparticular,inadditiontolinearandpolynomialfunctions,PSGsupportsanumberofdifferentclassesoffunctions,suchasCVaRandcardinalityfunctions.ForthepurposesofthecurrentcasestudywedenedinPSGtheobjectiveusingtheCVaRfunction,andwealsousedcardinalityfunctionforthecardinalityconstraintonxi,tinsteadofbooleanconstraint. 47

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Table3-1. CPLEXResults:Problemwith2-clubConstraints Case PSGCARPSGTANKCPLEX26secCPLEX1minnm valuetimevaluetimevaluetimevaluetime 104 97.9622.3100.6121.684.6226.184.6260.1105 79.2922.883.6925.674.3026.071.5760.0106 74.7924.271.1523.663.3226.163.3260.1107 64.8225.661.6826.557.5426.157.5460.0108 52.2726.751.8625.250.7626.150.2160.0114 106.1923.2105.1624.292.2126.192.2160.1115 86.6123.085.2122.475.5426.175.5460.1116 75.0823.976.2122.570.7926.166.3260.1117 68.4325.469.9324.058.3726.058.3760.1118 60.4424.461.0523.857.0126.157.0160.1124 122.3224.1124.0022.8105.4526.1105.4560.1125 91.2522.898.0222.682.0826.281.9660.1126 84.6922.881.7623.072.6526.072.6560.1127 73.4423.676.9522.164.1126.164.1160.1128 64.7825.661.9524.056.1026.156.1060.1134 126.3724.5119.7628.998.4626.098.4660.1135 94.4824.0104.7826.586.3126.088.2060.1136 82.4624.283.2927.276.6126.176.6160.1137 73.9725.574.5930.870.5326.167.9360.0138 71.5726.469.7933.959.9226.159.9260.1144 135.7525.6139.0627.3118.4126.0112.7460.1145 109.7527.1114.2727.295.0126.194.8760.1146 89.5824.393.8227.279.5426.179.5460.1147 80.7025.980.8923.370.5326.270.3760.1148 75.3126.076.8826.665.2626.161.6760.1154 155.6727.9145.0026.7127.0026.2126.8260.1155 113.1825.8115.6528.8104.0626.1102.3460.1156 95.5124.899.1128.190.8026.182.7460.1157 85.9625.086.4927.874.2226.174.2260.1158 77.8126.176.8334.468.0926.168.2260.0 avgavgavgavg 24.826.026.160.1 Forcomparisonpurposes,multipleexperimentshavebeenperformed.Allexperimentsweredividedintotwogroups:with2-clubconnectivityconstraintsonsubnetwork~G,andk-plexconstraintswithk=m 2.Ineachofthesegroups,numberoflocationsn=10,11,12,13,14,15andnumberofsensorsm=4,5,6,7,8.AllproblemshaveCVaR-typeobjectivewith=0.9,deterministicsetup(1scenario),20 48

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Table3-2. CPLEXResults:Problemwithk-plexConstraints Case PSGCARPSGTANKCPLEX27secCPLEX1minnm valuetimevaluetimevaluetimevaluetime 104 106.5225.1102.9422.785.3427.185.3460.1105 79.9823.783.4822.072.4027.172.4060.1106 74.2225.078.2526.063.2227.063.2260.1107 65.7127.062.2726.956.7327.056.7360.0108 55.7025.850.9124.350.2627.050.2660.0114 117.6123.7101.2826.887.5727.087.5760.1115 89.1624.792.7722.975.9727.175.9760.0116 77.2724.384.0826.268.2527.167.0360.0117 70.426.067.2423.558.6127.158.6160.0118 67.9825.964.8423.453.8027.153.1560.0124 134.6231.1128.6026.1109.8927.1102.5260.1125 100.8024.4103.7023.080.1127.180.1160.1126 83.2124.387.5625.570.5427.170.5460.1127 69.9125.874.9926.263.2627.163.2660.1128 70.0327.969.2822.756.7527.156.7360.1134 134.3932.2121.7228.5103.8727.1103.8760.1135 97.2425.4103.8923.784.6327.184.5760.1136 90.5125.789.4029.577.9427.177.9460.1137 78.1526.277.3428.768.5227.168.5260.1138 77.4926.772.6124.363.3627.159.6960.1144 134.0130.3140.1232.1119.1727.1112.1860.1145 114.7226.9113.6127.690.0027.189.3460.1146 97.8327.496.7129.178.3727.178.3760.1147 86.0926.487.0031.470.4627.170.2460.1148 77.7827.275.7526.062.4827.062.4860.1154 153.1834.4189.4132.3136.8927.1120.8160.1155 123.6128.6123.8427.0110.9427.198.1560.2156 97.5327.2101.7231.483.1127.082.5960.1157 93.2127.386.4830.174.4327.174.4360.0158 80.2928.975.7025.767.0827.167.3760.1 avgavgavgavg 26.926.527.160.1 timeintervals.Theedgedensityoftheconsideredoverallwirelessconnectivitynetworkwas=0.8(80%pairsofnodesareconnected). WehaverunPSGusingtwobuilt-insolvers:CARandTANK.Thesesolverstookonaverage26secondstodeliversolutionoverallcaseswith2-clubconstraints,and27secondsforthecaseswithk-plexconstraints.AfterthatwerunCPLEXoncases 49

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Table3-3. CPLEXandPSGResults:StochasticSetup CPLEXPSGCARPSGTANKtypeS valuetimegapvaluetimevaluetime k-plex10 85.14300.127.0%96.7825.396.8129.4k-plex20 89.36300.234.6%95.1035.396.1037.1k-plex50 92.27300.544.4%110.06154.797.40273.1k-plex100 93.81301.749.7%104.57300.6115.69300.62-club10 86.92300.130.5%100.13229.9100.13300.12-club20 84.92300.132.0%97.5579.497.55300.12-club50 89.75300.544.3%104.30300.1103.42300.12-club100 95.87301.850.7%116.63300.2116.23300.2 with2-clubconstraintswithtimelimit26seconds,oncaseswithk-plexconstraintswithtimelimit27seconds.Then,weadditionallyrunCPLEXonallcaseswithtimelimit1minute.ComputationalresultsarepresentedinTable 3-1 andTable 3-2 ,forthecaseswith2-clubconstraintsandk-plexconstraints,respectively. TheresultsshowthatonaveragethebestsolutionisproducedbyCPLEX1minuterun.Values,obtainedbyCPLEXrunswith26and27secondslimitsareby1.2%and2.2%greaterfor2-clubandfork-plexrespectively.Inmostcasessolutionsobtainedbytworunswereequal.Therefore,CPLEXobtainssolutionclosetooptimalinaboutlessthan30seconds.PSGTANKsolutionvalueisgreaterthanCPLEX1minutesolutionvalueby15.8%and22.4%for2-clubandfork-plexrespectively.PSGCARperformsslightlybetterthananothersolver,providingthesolutionvaluesgreaterthanCPLEX1minutesolutionvaluesby15.0%and22.0%. Inadditiontodeterministicsetup,wehaveruntheaforementionedoptimizationproblemsunderuncertaintyonseveralstochasticprobleminstanceswiththenumberofsensorsm=6,thenumberoflocationsn=12,theCVaR-typeobjectivewith=0.9,T=10timeintervals,fordifferentnumbersofscenarios:S=10,20,50,100.Asbefore,thewirelessconnectivitynetworkedgedensitywas=0.8.Thetimelimitwassetto5minutes.PSGsolversinmostcasesprovidedsolutionbeforethetimelimitwasreached 50

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(Table 3-3 ).However,thequalityofsolutionwasworsethenprovidedbyCPLEXby15%onaverage. 51

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CHAPTER4CALIBRATINGRISKPREFERENCESWITHGENERALIZEDCAPMBASEDONMIXEDCVARDEVIATION ThischapterisbasedonthejointpublicationwithS.UryasevandR.T.Rockafellar( Kalinchenkoetal. ( 2012 )) TheCapitalAssetPricingModel(CAPM, Sharpe ( 1964 ), Lintner ( 1965 ), Mossin ( 1966 ), Treynor ( 1961 ), Treynor ( 1999 ))afteritsfoundationinthe1960'sbecameoneofthemostpopularmethodologiesforestimationofreturnsofsecuritiesandexplanationoftheircombinedbehavior.Thismodelassumesthatallinvestorswanttominimizeriskoftheirinvestments,andallinvestorsmeasureriskbythestandarddeviationofreturn.ThemodelimpliesthatalloptimalportfoliosaremixturesoftheMarketFundandriskfreeinstrument.TheMarketFundiscommonlyapproximatedbysomestockmarketindex,suchasS&P500. AnimportantpracticalapplicationoftheCAPMmodelisthepossibilitytocalculatehedgedportfoliosuncorrelatedwiththemarket.Toreducetheriskofaportfolio,aninvestorcanincludeadditionalsecuritiesandhedgemarketrisk.TheriskoftheportfoliointermsofCAPMmodelismeasuredbybeta.Thevalueofbetaforeverysecurityorportfolioisproportionaltothecorrelationbetweenitsreturnandmarketreturn.Thisfollowsfromtheassumptionthatinvestorshaveriskattitudesexpressedwiththestandarddeviation(volatility).Thehedgingisdesignedtoreduceportfoliobetawiththeideatoprotecttheportfolioincaseofamarketdownturn.However,betaisjustascaledcorrelationwiththemarketandthereisnoguaranteethathedgeswillcoverlossesduringsharpdownturns,becausetheprotectionworksonlyonaverageforthefrequentlyobservedmarketmovements.Recentcreditcriseshaveshownthathedgeshavetendenciestoperformverypoorlywhentheyaremostneededinextrememarketconditions.Theclassicalhedgingproceduresbasedonstandardbetasetupadefencearoundthemeanofthelossdistribution,butfailinthetails.ThisdeciencyhasledtomultipleattemptstoimprovetheCAPM. 52

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OneapproachtoCAPMimprovementistoincludeadditionalfactorsinthemodel.Forexample, KrausandLitzenberger ( 1976 ), FriendandWestereld ( 1980 ),and Lim ( 1989 )providetestsforthethree-momentCAPM,includingco-skewnessterm.Thismodelaccountsfornon-symmetricaldistributionofreturns. FamaandFrench ( 1996 )addedtotheassetreturnlinearregressionmodeltwoadditionalterms:thedifferencebetweenthereturnonaportfolioofsmallstocksandthereturnonaportfoliooflargestocks,andthedifferencebetweenthereturnonaportfolioofhigh-book-to-marketstocksandthereturnonaportfoliooflow-book-to-marketstocks.Recently, BarberisandHuang ( 2008 )presentedCAPMextensionbasedonprospecttheory,whichallowstopricesecurity'sownskewness. Thesecondapproachistondalternativeriskmeasures,whichmaymorepreciselyrepresentriskpreferencesofinvestors.Forinstance, KonnoandYamazaki ( 1991 )appliedanL1riskmodel(basedonmeanabsolutedeviation)totheportfoliooptimizationproblemwithNIKKEI225stocks.TheirapproachledtolinearprogramminginsteadofquadraticprogrammingintheclassicalMarkowitz'smodel,butcomputationalresultsweren'tsignicantlybetter.Furtherresearchhasbeenfocusedonriskmeasuresmorecorrectlyaccountingforlosses.Forexample, Estrada ( 2004 )applieddownsidesemideviation-basedCAPMforestimatingreturnsofInternetcompanystocksduringtheInternetbubblecrisis.Downsidesemideviationcalculatesonlyforthelossesunderperformingthemeanofreturns.Nevertheless,semideviation,similarlytostandarddeviation,doesn'tpayspecialattentiontoextremelosses,associatedwithheavytails. SortinoandForsey ( 1996 )alsopointoutthatdownsidedeviationdoesnotprovidecompleteinformationneededtomanagerisk. AmuchmoreadvancedlineofresearchisconsideredinpapersofRockafellar,UryasevandZabarankin( Rockafellaretal. ( 2006a ), Rockafellaretal. ( 2006b ), Rockafellaretal. ( 2007 )).Theassumptionhereisthattherearedifferentgroupsofinvestorshavingdifferentriskpreferences.ThegeneralizedCapitalAssetPricing 53

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Model(GCAPM)proposesthatthereisacollectionofdeviationmeasures,representingriskpreferencesofthecorrespondinggroupsofinvestors.Thesedeviationmeasuressubstituteforthestandarddeviationoftheclassicaltheory.WiththegeneralizedpricingformulafollowingfromGCAPMonecanestimatethedeviationmeasureforaspecicgroupofinvestorsfrommarketprices.Thisisdonebyconsideringparametricclassesofdeviationmeasuresandcalibratingparametersofthesemeasure.TheGCAPMprovidesanalternativetotheclassicalCAPMmeasureofsystematicrisk,so-calledgeneralizedbeta.Similarlytoclassicalbeta,thegeneralizedbetacanbeusedinportfoliooptimizationforhedgingpurposes. Weconsidertheclassofso-calledmixedCVaRdeviations,havingseveralattractiveproperties.First,differenttermsinthemixedCVaRdeviationgivecredittodifferentpartsofthedistribution.Therefore,byvaryingparameters(coefcients),onecanapproximatevariousstructuresofriskpreferences.Inparticular,so-calledtail-betacanbebuiltwhichaccountsforheavytaillosses(e.g.,lossesinthetop5%ofthetaildistribution).Second,mixedCVaRdeviationisacoherentdeviationmeasure,anditthereforesatisesanumberofdesiredmathematicalproperties.Third,optimizationofproblemswithmixedCVaRdeviationcanbedoneveryefciently.Forinstance,fordiscretedistributions,theoptimizationproblemscanbereducedtolinearprogramming. Weconsiderasetupwithonegroupofinvestors(representativeinvestor).WeassumethattheseinvestorsestimateriskswiththemixedCVaRdeviationshavingxedquantilelevels:50,75,85,95and99percentofthelossdistribution.Bydenition,thismixedCVaRdeviationisaweightedcombinationofaveragelossesexceedingthesequantilelevels.TheweightsforCVaRswiththedifferentquantilelevelsdetermineaspecicinstanceoftheriskmeasure.Thegeneralizedpricingformulaandgeneralizedbetaforthisclassofdeviationmeasuresareusedinthisapproach.Withmarketoptionpricestheparametersofthedeviationmeasurearecalibrated,thusestimatingriskpreferencesofinvestors. 54

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Severalnumericalexperimentscalibratingriskpreferencesofinvestorsatdifferenttimemomentswereconducted.Wehavefoundthatthedeviationmeasure,representinginvestors'riskpreferences,hasthebiggestweightontheCVaR50%term,whichequalstheaveragelossbelowmedianreturn.Onaverage,about11%oftheweightisassignedtoCVaR85%,CVaR95%andCVaR99%evaluatingheavy-lossscenarios.Experimentsalsoshowedthatriskpreferencestendtochangeovertimereectinginvestors'opinionsaboutthestateofmarket. Wearenottherstwhoattemptedtoextractriskpreferencesfromoptionprices.Itisacommonknowledgethatoptionpricesconveyriskneutralprobabilitydistribution.Somestudies,suchas Ait-SahaliaandLo ( 2000 ), Jackwerth ( 2000 ), BlissandPanigirtzoglou ( 2002 ),containvariousapproachestoextractingriskpreferencesintheformofutilityfunctionbycomparingobjective(orstatistical)probabilitydensityfunctionwithriskneutralprobabilitydensityfunction,estimatedfromoptionprices.Inourworkriskpreferencesareexpressedintheformofdeviationmeasure,thusmakingitimpossibletocompareresultswithpreviousstudies.Webelieve,however,thatawiderangeofapplicabilityofgeneralizedCAPMframeworkmakeourresultsbeingusefulinagreatervarietyofapplicationsinpracticalnance. Theremainingpartofthischapterisstructuredasfollows.Section2recallsthenecessarybackground,describestheassumptionsofthemodel,providesthemaindenitionsandstatements,andpresentsthederivationofthegeneralizedpricingformula.Section3containsdescriptionofthecasestudy.Section4presentstheresultsofthecasestudy.Theconclusionsectionprovidesseveralideasforfurtherresearchthatcanbeperformedinthisarea. 4.1DescriptionoftheApproach 4.1.1GeneralizedCAPMBackground IntheclassicalMarkovitzportfoliotheory( Markowitz ( 1952 ))allinvestorsaremean-varianceoptimizers.Contrarytotheclassicalapproach,considernowagroupof 55

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investorswhoformtheirportfoliosbysolvingoptimizationproblemsofthefollowingtype:P()minx0r0+xTErr0+x0+xTe=1D)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(x0r0+xTr whereDissomemeasureofdeviation(notnecessarystandarddeviation),r0denotestherisk-freerateofreturn,risacolumnvectorof(uncertain)ratesofreturnonavailablesecurities,andeisacolumn-vectorofones.ProblemP()minimizesdeviationoftheportfolioreturnsubjecttoaconstraintonitsexpectedreturnandthebudgetconstraint.Differentinvestorswithintheconsideredgroupmaydemanddifferentexcessreturn.Unlikeclassicaltheory,insteadofvarianceorstandarddeviation,investorsmeasureriskwiththeirgeneralizeddeviationmeasureD.Accordingtothedenitionin Rockafellaretal. ( 2006a ),afunctionalD:L2![0,1]isadeviationmeasureifitsatisesthefollowingaxioms: (D1)D(X+C)=D(X)forallXandconstantsC. (D2)D(0)=0andD(X)=D(X)forallXandall>0. (D3)D(X+Y)D(X)+D(Y)forallXandY. (D4)D(X)0forallXwithD>0fornonconstantX. Similarlyto Rockafellaretal. ( 2006b ),wecaneliminatex0,whichisequalto1)]TJ /F3 11.955 Tf 11.95 0 Td[(xTe:P0()minxT(Er)]TJ /F10 7.97 Tf 6.58 0 Td[(r0e)D)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xTr Apair(x0,x)isanoptimalsolutiontoP()ifandonlyifxisanoptimalsolutiontoP0()andx0=1)]TJ /F3 11.955 Tf 11.95 0 Td[(xTe.Theorem1in Rockafellaretal. ( 2006c )showsthatanoptimalsolutiontoP0()exists,ifdeviationmeasureDsatisestheproperty(D5): (D5)fXjD(X)CgisclosedforeveryconstantC. AdeviationmeasureDsatisfyingthispropertyiscalledlowersemicontinuous.Forfurtherresultswewillalsorequireanadditionalproperty,calledlowerrangedominance: (D6)D(X)EX)]TJ /F2 11.955 Tf 11.96 0 Td[(infXforallX. 56

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Inthispaperweconsideronlylowersemicontinuous,lowerrange-dominateddeviationmeasures. Rockafellaretal. ( 2006b )showthatifagroupofinvestorssolvesproblemsP(),theoptimalinvestmentpolicyischaracterizedbytheGeneralizedOne-FundTheorem(Theorem2inthatpaper).Accordingtotheresult,theoptimalportfolioshavethefollowinggeneralstructure:x=x1,x0=1)]TJ /F2 11.955 Tf 11.95 0 Td[((x1)Te wherex0istheinvestmentinriskfreeinstrument,xisavectorofpositionsinriskyinstruments,and(x10,x1)isanoptimalsolutiontoP(),with=1.Portfolio(x10,x1)iscalledabasicfund.Itisimportanttonotethat,infullgenerality,(x1)TErcouldbepositive,negative,orequal0(thresholdcase),althoughformostsituationsthepositivecaseshouldprevail. Accordingtothesamepaper,aportfolioxDiscalledamasterfundofpositive(negative)typeif(xD)Te=1((xD)Te=)]TJ /F2 11.955 Tf 9.3 0 Td[(1),andxDisasolutiontoP0()forsome>0.Fromthedenitionfollowsthatmasterfundcontainsonlyriskysecurities,withnoinvestmentinriskfreesecurity.Withthisdenition,thegeneralizedOne-FundTheoremcanbereformulatedintermsofthemasterfund.Belowwepresentitsformulationasitwasgivenin Rockafellaretal. ( 2006b ). Theorem1(One-FundTheoreminMasterFundForm).Supposeamasterfundofpositive(negative)typeexists,furnishedbyanxD-portfoliothatyieldsanexpectedreturnr0+forsome>0.Then,forany>0,thesolutionfortheportfolioproblemP()isobtainedbyinvestingthepositiveamount=(negativeamount)]TJ /F2 11.955 Tf 9.3 0 Td[(=)inthemasterfund,andtheamount1)]TJ /F2 11.955 Tf 12.43 -.16 Td[((=)(amount1+(=)>1)intheriskfreeinstrument. 57

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FromTheorem1followsthatforeveryinvestorintheconsideredgroup,theoptimalportfoliocanbeexpressedasacombinationofinvestmentinthemasterfund,andinvestmentintheriskfreesecurity. Rockafellaretal. ( 2007 )extendstheframeworktothecasewithmultiplegroupsofinvestors.Everygroupofinvestorsi,wherei=1,...,I,solvestheproblemP()withtheirowndeviationmeasureDi.Itwasshownthatthereexistsamarketequilibrium,andoptimalpolicyforeverygroupofinvestorsisdenedbytheGeneralizedOne-FundTheorem.Inthisframeworkinvestorsfromdifferentgroupsmayhavedifferentmasterfunds.Fromnowonweassumethatageneralizeddeviationmeasurerepresentsriskpreferencesofagivengroupofinvestors. ConsideraparticulargroupofinvestorswithriskpreferencesdenedbyageneralizeddeviationD.Iftheirmasterfundisknown,thecorrespondingGeneralizedCAPMrelationscanbeformulated.Theexactrelationdependsonthetypeofthemasterfund.LetrMdenotetherateofreturnofthemasterfund.ThenrM=(xD)Tr=nXj=1xDjrj wheretherandomvariablesrjstandforratesofreturnonthesecuritiesintheconsideredeconomy,xDjarethecorrespondingweightsofthesesecuritiesinthemasterfund,andPnj=1xDj=1. Thegeneralizedbetaofasecurityj,replacingtheclassicalbeta,isdenedasfollows: j=cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDM) D(rM)(4) InthisformulaQDMdenotestheriskidentiercorrespondingtothemasterfund,takenfromtheriskenvelopecorrespondingtothedeviationmeasureD.Examplesofriskidentiersforspecicdeviationmeasureswillbepresentedinthenextsubsection. Rockafellaretal. ( 2006c )derivesoptimalityconditionsforproblemsofminimizingageneralizeddeviationofthereturnonaportfolio.Theoptimalityconditionsareapplied 58

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tocharacterizethreetypesofmasterfunds.Theorem5inthatpaper,presentedbelow,formulatestheoptimalityconditionsintheformofCAPM-likerelations. Theorem2.LetthedeviationDbeniteandcontinuous. Case1.AnxD-portfoliowithxD1+...+xDn=1isamasterfundofpositivetype,ifandonlyifErM>r0andErj)]TJ /F3 11.955 Tf 11.95 0 Td[(r0=j(ErM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0)forallj. Case2.AnxD-portfoliowithxD1+...+xDn=)]TJ /F2 11.955 Tf 9.3 0 Td[(1isamasterfundofnegativetype,ifandonlyifErM>)]TJ /F3 11.955 Tf 9.29 0 Td[(r0andErj)]TJ /F3 11.955 Tf 11.96 0 Td[(r0=j(ErM+r0)forallj. Case3.AnxD-portfoliowithxD1+...+xDn=0isamasterfundofthresholdtype,ifandonlyifErM>0andErj)]TJ /F3 11.955 Tf 11.95 0 Td[(r0=jErMforallj. FromnowonwecalltheconditionsspeciedintheTheorem2theGeneralizedCAPM(GCAPM)relations. 4.1.2PricingFormulasinGCAPM Letrj=j=j)]TJ /F2 11.955 Tf 12.03 0 Td[(1,wherejisthepayofforthefuturepriceofsecurityj,andjisthepriceofthissecuritytoday. Similarlytoclassicaltheory,pricingformulascanbederivedfromtheGeneralizedCAPMrelations,asitwasdonein Sarykalin ( 2008 ).ThefollowingLemmapresentsthesepricingformulasbothincertaintyequivalentform,andriskadjustedform. Lemma1.Case1.Ifthemasterfundisofpositivetype,thenj=Ej 1+r0+j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0=1 1+r0Ej+cov(j,QDM) D(rDM))]TJ /F3 11.955 Tf 5.48 -9.69 Td[(ErDM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0Case2.Ifthemasterfundisofnegativetype,thenj=Ej 1+r0+j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM+r0=1 1+r0Ej+cov(j,QDM) D(rDM))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM+r0Case3.Ifthemasterfundisofthresholdtype,thenj=Ej 1+r0+jErDM=1 1+r0Ej+cov(j,QDM) D(rDM)ErDM 59

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SeeproofinAppendix. Rockafellaretal. ( 2007 )provedtheexistenceofequilibriumformultiplegroupsofinvestorsoptimizingtheirportfoliosaccordingtotheirindividualriskpreferences,andthereforethepricingformulasinLemma1holdtrueforallgroupsofinvestors. 4.1.3MixedCVaRDeviationandBetas ConditionalValue-at-Riskhasbeenstudiedbyvariousresearchers,sometimesunderdifferentnames(expectedshortfall,Tail-VaR).Wewillusenotationsfrom RockafellarandUryasev ( 2002 ).FormoredetailsonstochasticoptimizationwithCVaR-typefunctionssee Uryasev ( 2000 ), RockafellarandUryasev ( 2000 ), RockafellarandUryasev ( 2002 ), Krokhmaletal. ( 2002 ), Krokhmaletal. ( 2006 ), Sarykalinetal. ( 2008 ). SupposerandomvariableXdeterminessomenancialoutcome,futurewealthorreturnoninvestment.Bydenition,Value-at-Riskatlevelisthe-quantileofthedistributionof()]TJ /F3 11.955 Tf 9.3 0 Td[(X):VaR(X)=q()]TJ /F3 11.955 Tf 9.3 0 Td[(X)=)]TJ /F3 11.955 Tf 9.29 0 Td[(q1)]TJ /F9 7.97 Tf 6.59 0 Td[((X)=)]TJ /F2 11.955 Tf 11.29 0 Td[(inffzjFX(z)>1)]TJ /F4 11.955 Tf 11.95 0 Td[(g whereFXdenotestheprobabilitydistributionfunctionofrandomvariableX. ConditionalValue-at-RiskforcontinuousdistributionsequalstheexpectedlossexceedingVaR:CVaR(X)=)]TJ /F3 11.955 Tf 9.3 0 Td[(E[XjX)]TJ /F2 11.955 Tf 21.91 0 Td[(VaR(X)] ThisformulaunderliesthenameofCVaRasconditionalexpectation.Forthegeneralcasethedenitionismorecomplicated,andcanbefound,forexample,in RockafellarandUryasev ( 2000 ).ConditionalValue-at-Riskdeviationisdenedasfollows:CVaR(X)=CVaR(X)]TJ /F3 11.955 Tf 11.95 0 Td[(EX) AsfollowsfromTheorem1in Rockafellaretal. ( 2006a ),thereexistsaone-to-onecorrespondencebetweenlower-semicontinuous,lowerrange-dominateddeviation 60

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measuresDandconvexpositiveriskenvelopesQ:Q=nQQ0,EQ=1,EXQEX)-222(D(X)forallXo D(X)=EX)]TJ /F2 11.955 Tf 15.05 0 Td[(infQ2QEXQ(4) TherandomvariableQX2Q,forwhichD(X)=EX)]TJ /F3 11.955 Tf 12.06 0 Td[(EXQX,iscalledtheriskidentier,associatedbyDwithX. ForagivenXandCVaRdeviation,theriskidentiercanbeviewedasastepfunction,withajumpatthequantilepoint: QX(!)=1 1)]TJ /F4 11.955 Tf 11.95 0 Td[(1InX(!)q1)]TJ /F9 7.97 Tf 6.59 0 Td[((X)o(4) where!denotesanelementaryeventontheprobabilityspace,and1Inconditionoisanindicatorfunction,denedonthesameprobabilityspace,whichequals1ifconditionistrue,and0otherwise.Figure 4-1 illustratesthestructureoftheCVaRriskidentier,correspondingtosomerandomoutcomeX.Forsimplicity,theprobabilityspace,assumedintheFigure 4-1 ,isthespaceofvaluesoftherandomvariableX. IfthegroupofinvestorsconstructsitsmasterfundbyminimizingCVaRdeviation,andallrjarecontinuouslydistributed,betaforsecurityjhasthefollowingexpression,derivedinRockafellaretal.(2006c): j=cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QM) CVaR(rM)=E[Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rjjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaR(rM)] E[ErM)]TJ /F3 11.955 Tf 11.96 0 Td[(rMjrM)]TJ /F2 11.955 Tf 21.91 0 Td[(VaR(rM)](4) Classicalbetaisascaledcovariancebetweenthesecurityandthemarket.Thenewbetafocusesoneventscorrespondingtobiglossesinthemasterfund.Forbig(>0.8),thisexpressioncanbecalledtail-beta. ThefollowingtwotheoremsleadtothedenitionofmixedCVaRdeviation,whichisusedforthepurposeofthispaper. 61

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Figure4-1. CVaR-typeRiskIdentierforaGivenOutcomeVariableX Theorem3.LetdeviationmeasureDlcorrespondtoriskenvelopeQlforl=1,...,L.IfdeviationmeasureDisaconvexcombinationofthedeviationmeasuresDl:D=LXl=1lDl,withl0,LXl=1l=1 thenDcorrespondstoriskenvelopeQ=PLl=1lQl. SeeproofinAppendix. Thefollowingtheorempresentsaformulaforthebetacorrespondingtoadeviationmeasurethatisaconvexcombinationofanitenumberofdeviationmeasures. Theorem4.IfthemasterfundM,correspondingtothedeviationmeasureD,isknown,andDisaconvexcombinationofanitenumberofdeviationmeasuresDl,l=1,...,L:D=1D1+...+LDL 62

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thenj=1cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QD1M)+...+Lcov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDLM) 1D1(rM)+...+LDL(rM) whereQDlMisariskidentierofmasterfundreturncorrespondingtodeviationmeasureDl. SeeproofinAppendix. Foragivensetofcondencelevels=(1,...,L)andcoefcients=(1,...,L)suchthatl0foralll=1,...,L,andPLl=1l=1,mixedCVaRdeviationCVaR;isdenedinthefollowingway: CVaR,(X)=1CVaR1(X)+...+LCVaRL(X)(4)Corollary1.IfD=CVaR,,where=(1,...,L)and=(1,...,L),anddistributionofrMiscontinuous,then j=1E[Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rjjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaR1(rM)]+...+LE[Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rjjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRL(rM)] 1CVaR1(rM)+...+LCVaRL(rM)(4)SeeproofinAppendix. 4.1.4RiskPreferencesofaRepresentativeInvestor Howcanriskpreferencesofinvestorsbeextractedfrommarketprices? AccordingtoGCAPM,riskpreferencesofagroupofinvestorsarerepresentedbyadeviationmeasure.Thisdeviationmeasuredeterminesthestructureofamasterfund.Foraknowndeviationmeasureandamasterfund,ariskidentierforthemasterfundcanbespecied.Ifajointdistributionofpayoffsforsecuritiesisalsoknown,thenonecancalculatethebetasforsecurities,andthencalculateGCAPMpricesforthesesecurities.Therefore,accordingtoGCAPM,thedeviationmeasureandthedistributionofpayoffdeterminethepriceforeachsecurity.Toestimatethedeviationmeasure,havingexpectedreturnsonsecuritiesandmarketprices,onecanndacandidatedeviationmeasureDforwhichtheGCAPMpricesareequaltothemarketprices. 63

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Inthisandfollowingsectionsthepaperconsidersasetupwithonegroupofinvestors.Inotherwords,allinvestorsevaluaterisksoftheirinvestmentsaccordingtothesamedeviationmeasure.Therefore,allfurtherresultscanbereferredtoasdescribingaso-calledrepresentativeinvestor.Frommarketequilibriumfollowsthatthemasterfundforarepresentativeinvestorisknown,and,therefore,canbeapproximatedwithamarketindex,suchasS&P500. Alternativelytostandarddeviation,whichmeasuresthemagnitudeofpossiblepricechangesinbothdirections,ConditionalValue-at-Riskdeviationmeasurestheaveragelossfortheworst-casescenarios.WeassumethatriskpreferencescanbeexpressedwithamixedCVaRdeviation,denedbyformula( 4 ),whichisaweightedcombinationofseveralCVaRdeviationswithappropriateweights,tocapturedifferentpartsofthetailofthedistribution. Amongthewholevarietyofsecuritiestradedinthemarket,inadditiontotheIndexfunditself,weconsiderS&P500putoptionswithonemonthtomaturity.Byconstruction,putoptions'pricesprovidemonetaryevaluationofthetailsofdistribution,sotheyareexpectedtobeperfectcandidatetocalibratecoefcientsinthemixedCVaRdeviation. Toestimatethecoefcients1,...,LwewilluseGCAPMformulas,presentedinTheorem2.LetPKdenotethemarketpriceofaputoptionwithstrikepriceKand1monthtomaturity,Kdenoteits(random)monthlyreturn,andrK=K PK)]TJ /F2 11.955 Tf 12.51 0 Td[(1denoteits(random)returninonemonth.LetrMbe(random)returnonthemasterfund,withitsdistributionatthismomentassumedtobeknown;r0isthereturnonariskfreesecurity.IfmarketpricesareexactlyequaltoGCAPMprices,andthedeviationmeasureisamixedCVaRdeviationwithxedcondencelevels1,...,L,thenthesetofcoefcients1,...,Lisasolutiontothefollowingsystemofequations: ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(r0=K()(ErM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0),K=K1,K2...,KJ)]TJ /F11 7.97 Tf 6.59 0 Td[(1,KJ(4) 64

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where K()=PLl=1lE[ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(rKjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRl(rM)] PLl=1lCVaRl(rM)(4) LXl=1l=1(4) and l0,l=1,...,L(4) Equations( 4 )areGCAPMformulasfromTheorem2,appliedtomarketpricesPKofputoptionswithstrikepricesK=K1,...,KJ,andrandompayoffsK.Systematicriskmeasure()isexpressedthroughthecoefcientslaccordingtoCorollary1. Bymultiplyingbothsidesofequation( 4 )byPLl=1lCVaRl(rM)andtakingintoaccount( 4 ),weget (ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)LXl=1lCVaRl(rM)=(ErM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)LXl=1lE[ErK)]TJ /F3 11.955 Tf 11.96 0 Td[(rKjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRl(rM)],K=K1,...,KJ or,equivalently, LXl=1)]TJ /F2 11.955 Tf 5.48 -9.68 Td[((ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)CVaRl(rM))]TJ /F2 11.955 Tf 11.95 0 Td[((ErM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)E[ErK)]TJ /F3 11.955 Tf 11.96 0 Td[(rKjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRl(rM)]l=0,K=K1,...,KJ(4) Ifthenumberofequations(optionswithdifferentstrikepricesK)isgreaterthanthenumberofvariables,thensystemofequations( 4 )maynothaveasolution.Forthisreasonwereplacetheequations( 4 )withalternativeexpressionswitherrortermseK: LXl=1)]TJ /F2 11.955 Tf 5.48 -9.68 Td[((ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)CVaRl(rM))]TJ /F2 11.955 Tf 11.95 0 Td[((ErM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)E[ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(rKjrM)]TJ /F2 11.955 Tf 21.91 0 Td[(VaRl(rM)]l=eK,K=K1,...,KJ(4) 65

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Weestimatethecoefcients1,...,Lastheoptimalpointtothefollowingoptimizationproblemminimizinganormofvector(eK1,...,eKJ): min1,...,Lk(eK1,...,eKJ)k(4) subjectto LXl=1)]TJ /F2 11.955 Tf 5.48 -9.69 Td[((ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)CVaRl(rM))]TJ /F2 11.955 Tf 11.95 0 Td[((ErM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)E[ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(rKjrM)]TJ /F2 11.955 Tf 21.91 0 Td[(VaRl(rM)]l=eK,K=K1,...,KJ(4) l0,l=1,...,L,LXl=1l=1(4) Intheaboveformulationkkissomenorm.Weconsidertwonorms: L1-norm: k(eK1,...,eKJ)k1=1 JJXj=1jeKjj(4) andL2-norm:k(eK1,...,eKJ)k2=vuut 1 JJXj=1e2Kj 4.2CaseStudyDataandAlgorithm Wedid153experimentsofestimatingriskpreferences,eachforaseparatedate(henceforth:dateofexperiment)startingwith1/22/1998.Dateswerechosenwithintervalsapproximately1monthinsuchawaythateachdateis1monthpriortoanextmonthoptionexpirationdate.However,wepresentdetailedanalysisfor12dateswithintervalsapproximately1=2yearstartingwith12/23/2004.ForeveryexperimentweusedasetofS&P500putoptionswithstrikepricesK1,...,KJ,whereKJisastrikepriceoftheat-the-moneyoption(optionwithstrikepriceclosesttotheIndexvalue).WedeneoptionmarketpricePKasanaverageofBIDandASKprices:PK=1 2Pask,K+Pbid,K 66

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WechoseK1asaminimumstrikeprice,forwhichthefollowingtwoconditionsaresatised:1)StartingwiththeoptionK1,pricesPKjarestrictlyincreasing,i.e.PKj+1>PKj;2)Openinterestforalloptionsintherangeisgreaterthan0. ForeveryexperimentwedesignedasetofscenariosofmonthlyIndexratesofreturninthefollowingway.ObservinghistoricalvaluesofS&P500overtheperiodfrom1/1/1994to10/1/2010,foreverytradingdaysfromhistoricalobservationswerecordedthevaluefr(s)I=Is+21 Is)]TJ /F2 11.955 Tf 11.95 0 Td[(1,whereIsistheIndexvalueondays. Wefurthercalculateimpliedvolatilityoftheat-the-moneyoption(theoptionwithstrikepriceKJ),andthevalueb=standarddeviation(fr(s)I) Next,everyscenarioreturnwasmodiedasfollows: r(s)I= bfr(s)I)]TJ /F3 11.955 Tf 11.96 0 Td[(EerI+r0+(4) wherethevalueforthemonthlyrisk-freerateofreturnr0wasselectedequalto0.01%,and>0issomeparameter.Thenewscenarioswillhavevolatilityequaltothevolatilityoftheat-the-moneyoptions,andexpectedreturnr0+.Informula( 4 )thevalueofwaschosensuchthatexpectedreturnsonoptionsarenegative.Weselected=1 3.Numericalexperimentsshowedthatresultsarenotverysensitivetotheselectionoftheparameter. Suppose,formodelingpurposes,thattheinvestors'preferencesaredescribedbyamixedCVaRdeviationwithcondencelevels50%,75%,85%,95%and99%: D()=LXl=1lCVaRl(4) whereL=5,1=99%,2=95%,3=85%,4=75%,5=50% 67

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Table4-1. CaseStudyDataforSelectedDates DateofexperimentIndexvalueLoweststrikepriceHigheststrikepriceNotationI0KminKmax 12/23/041210.13112012106/16/051210.961080121012/22/051268.12110012706/22/061245.601150124512/21/061418.30131014206/21/071522.191375152012/20/071460.12125514606/19/081342.831110134512/18/08885.286308856/18/09918.3773592012/17/091096.0890010956/17/101116.049401115 Table4-2. CaseStudyCommonData DecriptionNotationValue Risk-freemonthlyinterestrater00.4125%NumberoftermsinmixedCVaRdeviationL5Condencelevel1199%Condencelevel2295%Condencelevel3385%Condencelevel4475%Condencelevel5550%Numberofscenarios(days)S5443 and l0,5Xl=1l=1(4) TheinputdataforthecasestudyarelistedinTable 4-1 andTable 4-2 Multipletestsdemonstratedthattheresultsdonotdependsignicantlyonthechoiceofnormintheoptimizationproblem( 4 ).FurtherinthispaperwepresentresultsobtainedusingL1-norm. Thefollowingstepsdescribethealgorithm,whichwasusedtoestimateriskpreferencesfromtheoptionprices. 68

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Step1.Calculatescenariosindexedbys=1,...,Sforpayoffsandnetreturnsofputoptionsaccordingtotheformula:(s)K=max(0,K)]TJ /F3 11.955 Tf 11.96 0 Td[(I0(1+r(s)I)),r(s)K=(s)K PK)]TJ /F2 11.955 Tf 11.96 0 Td[(1 whereK=K1,...,KJarethestrikeprices,andI0istheIndexvalueattimeoftheexperiment. Step2.Calculatethefollowingvalues:E[ErK)]TJ /F3 11.955 Tf 11.96 0 Td[(rKjrI)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRl(rI)]forallK=K1,...,KJandl=1,...,L andCVaRl(rI)foralll=1,...,L Step3.Buildthedesignmatrixfortheconstrainedregression( 4 )-( 4 )withrM=rI. Table4-3. DeviationMeasureCalibrationResults DateofExperiment99%95%85%75%50% 12/23/20040.0000.0200.2350.0000.7456/16/20050.0360.0160.0000.0000.94812/22/20050.0580.0000.0000.0000.9426/22/20060.0710.0330.0000.0000.89512/21/20060.0810.0000.0000.0000.9196/21/20070.0550.0400.0000.0000.90512/20/20070.0000.0410.2750.0000.6846/19/20080.0000.0550.1810.0000.76512/18/20080.0000.0000.1150.0000.8856/18/20090.0150.0140.1680.0000.80312/17/20090.0490.0010.0830.0000.8686/17/20100.0410.0480.0230.0000.889mean(12dates)0.0340.0220.0900.0000.854standarddeviation(12dates)0.0300.0200.1020.0000.085mean(153dates)0.0290.0290.0520.0070.883standarddeviation(153dates)0.0280.0330.0770.0470.078 69

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Step4.Findasetofcoefcientslbysolvingconstrainedregression( 4 )-( 4 )withL1norm,givenbyequation( 4 ).VectorgivescoefcientsinmixedCVaRdeviation. 4.3CaseStudyComputationalResults ComputationswereperformedonthelaptopPCwithIntelCore2DuoCPUP88002.66GHz,4GBRAMandWindows7,64-bit.Algorithm,describedinprevioussection,wasprogrammedinMATLAB.Bothoptimizationproblems,theconstrainedregressionandCVaRportfoliooptimization,oneachiterationofthealgorithmweresolvedwithAORDAPortfolioSafeguarddecisionsupporttool( PSG ( 2009 )).Foronedatethecomputationaltimeisaround15seconds. ThesetofcoefcientsinthemixedCVaRdeviationforeverydateispresentedinTable 4-3 .ThistableshowsthatinallexperimentstheobtaineddeviationmeasurehasthebiggestweightonCVaR50%,andsmallerweightsoneitherCVaR85%,CVaR95%,orCVaR99%.Thiscanbeinterpretedasthatinvestorsareconcernedbothwiththemiddlepartofthelossdistribution,expressedwithCVaR50%,andextremelossesexpressedwithCVaR85%,CVaR95%,orCVaR99%. LetusdenotebyKtheGCAPMoptionprices,calculatedwithpricingformulasinLemma1,usingcalculatedmixedCVaRdeviationmeasureandthemasterfund.WemappedtheobtainedoptionpricesKandthemarketpricesPKintotheimpliedvolatilityscale.ThismappingisdenedbytheBlack-Scholesformulainimplicitform.ThegraphsofKandPKfor12datesinthescaleofmonthlyimpliedvolatilitiesarepresentedinFigures 4-2 4-3 and 4-4 .AllgraphsshowthattheGCAPMpricesareclosetomarketprices,exceptforthegraphsfor6/16/2005andfor12/22/2005. Figure 4-5 comparesdynamicsofthevalue=1)]TJ /F4 11.955 Tf 10.81 0 Td[(50%on153datesofexperimentwithS&P500dynamics.Highvaluesofindicategreaterinvestors'apprehensionaboutpotentialtaillossesandgreaterinclinationtohedgetheirinvestmentsinS&P500.Itcanbeseenthatriskpreferenceswererelativelystableuntil2008,whenthedistressed 70

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Figure4-2. CalculatedPricesandMarketPricesintheScaleofImpliedVolatilitiesPart1outof3 71

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Figure4-3. CalculatedPricesandMarketPricesintheScaleofImpliedVolatilitiesPart2outof3 72

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Figure4-4. CalculatedPricesandMarketPricesintheScaleofImpliedVolatilitiesPart3outof3 73

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Figure4-5. S&P500ValueandRiskAversityDynamics periodbegan.Itcanalsobeseenthatmarketparticipantsdidn'talwaysproperlyanticipatefuturemarkettrends.Inparticular,inDecember2008thevalueofwaslow(0.115),whichindicatedthatmarketwronglyanticipatedthatIndexreacheditsbottomandwillgoup.Nevertheless,2009startedwithfurtherdeclineintheIndex. 74

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CHAPTER5CONCLUSIONS 5.1DissertationContribution Inthisdissertationweprovidedanoverviewofthefollowingclasses:generalizeddeviationmeasures,riskmeasures,measuresoferror,andsomeoftheirsubclasses.Webrieydiscussedthemotivationforapplyingthesemeasuresinstochasticoptimizationapplications. Wereviewedgeneralizedlinearregressionmodels.Forthequantileregressionwedeterminedthedistributionfortheresidual.Thisresultcanbeusedinapplications,whichrequirethedistributionoftheerrorterminaquantilefactormodeltobespecied,suchassimulationprocedures.Interpretationofthisdistributionasatwo-sidedexponentialdistributionmakesitpossibletoestimatevariouspropertiesofthedistributionwithoutapplyingnumericalintegrationmethods.Therefore,implementationsofcomputationalmodelsbasedonthisdistributionareexpectedtobehighlyefcient. Wehavedenedandextendedaclassofdynamicsensorschedulingproblems,basedonconditionalvalue-at-risk,byintroducingexplicitrobustconnectivityrequirements,specically,k-clubandk-plexconstraints,takingintoaccountwirelessconnectivityrequirementsforsensorsateverytimemoment.Wehavealsopresentedcomputationalresultsformoderate-sizeinstancesinbothdeterministicandstochasticproblemsetups.SincethesizeofthestochasticversionoftheproblemisStimeslargerthanforthedeterministicversion(whereSisthenumberofimpliedpenaltyscenarios),solvingthesestochasticproblemsisclearlychallengingfromthecomputationalperspective. Theclassesofproblemsconsideredinthisresearchareprimarilymotivatedbymilitaryapplications;however,thedevelopedformulationsaregeneralenoughsothattheycanbeappliedinavarietyofsettings. Wehavedescribedanewtechniqueofexpressingriskpreferenceswithgeneralizeddeviationmeasures.Wehavepresentedamethodforextractingriskpreferences 75

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frommarketoptionpricesusingtheseformulas.Wehaveconductedacasestudyforextractingriskpreferencesofarepresentativeinvestorfromputoptionprices. Weextractedriskpreferencesfor153dateswith1monthintervals,andexpressedthemwithmixedCVaRdeviation.Resultsdemonstratethatinvestorsareconcernedbothwiththemiddlepartofthelossdistribution,expressedwithCVaR50%,andextremelossesexpressedwithCVaR85%,CVaR95%,orCVaR99%.Exactproportionsvary,reectinginvestorsanticipationofhighorlowreturns. Animportantapplicationofthetheoryisthatitprovidesanalternative,morebroadviewonsystematicrisk,comparedtotheclassicalCAPMbasedonstandarddeviation.SimilarlytotheclassicalCAPM,wecalculatednewbetasforsecurities,whichmeasuresystematicriskinadifferentway,capturingtailbehaviorofamasterfundreturn.Thesebetascanbeusedforhedgingagainsttaillosses,whichoccurindownmarket. Potentialapplicationsgobeyondidentifyingriskpreferencesofconsideredinvestors.Aninvestorcanexpressriskattitudesintheformofadeviationmeasure,andthenrecalculatebetasforsecuritiesusingthisdeviationmeasure.Withthesebetastheinvestorcanbuildaportfoliohedgedaccordingtohisriskpreferences. 5.2FutureWork Sensorsschedulingproblemformulationscanbefurtherextendedbyaddingmovementnetworkandcorrespondingconstraintsasintroducedin Boykoetal. ( 2011 ),thusmodelingthemapofpossiblesensormovements. Itwouldbeinterestingtoexamineinvestors'perceptionofriskfordifferenttimehorizonsbyestimatingriskpreferencesusingmarketpricesofputoptionswith2,3,andmoremonthstomaturity. 76

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APPENDIX:PROOFS Belowwepresentproofsofstatementsformulatedinthearticle.Forthereader'sconvenience,werepeatformulationsbeforeeveryproof.Lemma1.Case1.Ifthemasterfundisofpositivetype,thenj=Ej 1+r0+j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(ErDM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0=1 1+r0Ej+cov(j,QDM) D(rDM))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0Case2.Ifthemasterfundisofnegativetype,thenj=Ej 1+r0+j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM+r0=1 1+r0Ej+cov(j,QDM) D(rDM))]TJ /F3 11.955 Tf 5.48 -9.69 Td[(ErDM+r0Case3.Ifthemasterfundisofthresholdtype,thenj=Ej 1+r0+jErDM=1 1+r0Ej+cov(j,QDM) D(rDM)ErDMProofofLemma1.Proofsforallthreecasesaresimilar,sowepresenttheproofonlyforamasterfundofpositivetype.AccordingtotheGCAPMrelationspeciedinCase1,Erj)]TJ /F3 11.955 Tf 11.95 0 Td[(r0=j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0Sincerj=j=j)]TJ /F2 11.955 Tf 11.95 0 Td[(1,thenErj=Ej=j)]TJ /F2 11.955 Tf 11.95 0 Td[(1,fromwhichweget Ej j)]TJ /F2 11.955 Tf 11.96 0 Td[((1+r0)=j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0(A)ThisyieldstheGeneralizedCapitalAssetPricingFormulainthecertaintyequivalentform: j=Ej 1+r0+j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0(A)Usingtheexpressionforbeta( 4 )wecanalsowrite j=Ej 1+r0+cov()]TJ /F10 7.97 Tf 6.59 0 Td[(rj,QDM) D(rDM))]TJ /F3 11.955 Tf 5.48 -9.69 Td[(ErDM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0(A) 77

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Bymultiplyingbothsidesoftheequality( A )byj,weget Ej)]TJ /F4 11.955 Tf 11.95 0 Td[(j(r0+1)=jj)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(ErDM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0(A)Withexpressionforbeta( 4 )weget jj=jcov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDM) D(rDM)=cov()]TJ /F4 11.955 Tf 9.29 0 Td[(jrj,QDM) D(rDM)=cov()]TJ /F4 11.955 Tf 9.3 0 Td[(j(1+rj)+j,QDM) D(rDM)=cov()]TJ /F4 11.955 Tf 9.3 0 Td[(j(1+rj),QDM) D(rDM)+cov(j,QDM) D(rDM)(A)Herejisaconstant,consequentlythesecondterminthelastsumequals0.Therefore,jj=cov()]TJ /F4 11.955 Tf 9.3 0 Td[(j(1+rj),QDM) D(rDM)Sincej(1+rj)=j,thenjj=)]TJ /F2 11.955 Tf 10.49 8.09 Td[(cov(j,QDM) D(rDM)Substitutingexpressionforjjinto( A )gives:Ej)]TJ /F4 11.955 Tf 11.96 0 Td[(j(r0+1)=)]TJ /F2 11.955 Tf 10.49 8.08 Td[(cov(j,QDM) D(rDM))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0Thelastequationimpliestherisk-adjustedformofthepricingformula: j=1 1+r0Ej+cov(j,QDM) D(rDM))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0(A)Theorem3.LetdeviationmeasureDlcorrespondtoriskenvelopeQlforl=1,...,L.IfdeviationmeasureDisaconvexcombinationofthedeviationmeasuresDl:D=LXl=1lDl,withl0,LXl=1l=1thenDcorrespondstoriskenvelopeQ=PLl=1lQl. 78

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ProofofTheorem3.Withformula( 4 )weget: D(X)=LXl=1lDl(X)=EX)]TJ /F10 7.97 Tf 18.18 14.94 Td[(LXl=1linfQ2QlEXQ==EX)]TJ /F2 11.955 Tf 48.68 0 Td[(inf(Q1,...,QL)2(Q1,...,QL)EX LXl=1lQl!=EX)]TJ /F2 11.955 Tf 30.58 0 Td[(infQ2PLl=1lQlEXQ(A)Theorem4.IfthemasterfundM,correspondingtothedeviationmeasureD,isknown,andDisaconvexcombinationofanitenumberofdeviationmeasuresDl,l=1,...,L:D=1D1+...+LDLthenj=1cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QD1M)+...+Lcov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDLM) 1D1(rM)+...+LDL(rM)whereQDlMisariskidentierofmasterfundreturn,correspondingtodeviationmeasureDl.ProofofTheorem4.FromTheorem3follows:j=cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDM) D=cov()]TJ /F3 11.955 Tf 9.29 0 Td[(rj,1QD1M+...+LQDLM) 1D1(rM)+...+LDL(rM)= =1cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QD1M)+...+Lcov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDLM) 1D1(rM)+...+LDL(rM)(A)Next, cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDlM)=E(Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rj)(QDlM)]TJ /F3 11.955 Tf 11.96 0 Td[(EQDlM)(A)Accordingtothedenitionofriskenvelope,EQDlM=1.Therefore,from( A )wehave:cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDlM)=E(Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rj)(QDlM)]TJ /F2 11.955 Tf 11.96 0 Td[(1)=E(Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rj)QDlM)]TJ /F3 11.955 Tf 11.96 0 Td[(E(Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rj)=E(Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rj)QDlM 79

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Corollary1.IfD=CVaR,,where=(1,...,L)and=(1,...,L),anddistributionofrMiscontinuous,then j=1E[Erj)]TJ /F3 11.955 Tf 11.95 0 Td[(rjjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaR1(rM)]+...+LE[Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rjjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRL(rM)] 1CVaR1(rM)+...+LCVaRL(rM)(A)ProofofCorollary1.ForDl=CVaRl,accordingto( 4 ):QDlM(!)=1 1)]TJ /F4 11.955 Tf 11.96 0 Td[(l1InrM(!))]TJ /F2 11.955 Tf 21.92 0 Td[(VaRl(rM)oThen, cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDlM)=E(Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rj)1 1)]TJ /F4 11.955 Tf 11.95 0 Td[(l1InrM(!))]TJ /F2 11.955 Tf 21.91 0 Td[(VaRl(rM)o==E[Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rjjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRl(rM)](A)Substitutingexpressionforcov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDlM)andexpressionformixedCVaRdeviation( 4 )into( A )gives:j=1E[Erj)]TJ /F3 11.955 Tf 11.95 0 Td[(rjjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaR1(rM)]+...+LE[Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rjjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRL(rM)] 1CVaR1(rM)+...+LCVaRL(rM) 80

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Treynor,JackL.1961.Marketvalue,time,andrisk.Unpublishedmanuscript. Treynor,JackL.1999.Towardatheoryofmarketvalueofriskyassets.AssetPricingandPortfolioPerformance:Models,StrategyandPerformanceMetrics15. Uryasev,Stan.2000.Conditionalvalue-at-risk:optimizationalgorithmsandapplications.FinancialEngineeringNews14. Yavuz,Mesut,andDavidE.Jeffcoat.2007.Singlesensorschedulingformulti-sitesurveillanceTechnicalreport,AirForceResearchLaboratory. 84

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BIOGRAPHICALSKETCH KonstantinKalinchenkowasbornin1985inProtvino,Russia.Hereceivedhis5-yearSpecialistdegreeinmathematicsfromMoscowStateUniversityUniversityin2007.KonstantinKalinchenkoworkedasaneconomistandsenioreconomistin2005-2008foraleadingRussiancommercialbank:SberabankRossii,wherehehadanopportunitytotakepartindevelopingcreditriskmeasurementmethodologyandcomputermethodsincontrolofoperationswithprivatesecurities.In2008,KonstantinKalinchenkojoinedthegraduateprograminindustrialandsystemsengineeringdepartmentattheUniversityofFlorida.HereceivedhisMasterofSciencedegreeinindustrialandsystemsengineeringfromtheUniversityofFloridainthespringof2011,andhereceivedhisPh.D.fromtheUniversityofFloridainthespringof2012.KonstantinKalinchenkoalsoworkedasaninternin2011foranassetmanagementcompanyStateStreetGlobalAdvisors(Boston,MA),whereheworkedondevelopingquantitativefactorsbasedonnewsparsingsolutionsforstockranking.KonstantinKalinchenkoistheauthorofseveralscienticpapers.HewasalsoaTAinEngineeringEconomyandFinancialOptimizationCaseStudiesclasses. 85