1IntroductionThequanticationofmarketriskforderivativepricing,forportf - PDF document

1IntroductionThequanticationofmarketriskforderivativepricing,forportfoliooptimizationandpricingriskforinsurancepurposeshasgeneratedalargeamountoftheoreticalandpracticalwork,withavarietyofinterconnections.Twolinesofresearchintheseareasarebasedupon,anduseaspointofdepartureaxiomsthatboth,themarketriskmeasureandtheriskpricingmeasure,havetosatisfy.ValueatRisk(VaR)isoneofthemostpopularriskmeasures,duetoitssimplic-ity.VaRindicatestheminimallossincurredintheworseoutcomesaportfolio.Butthisriskmeasureisnotalwayssub-additive,norconvex.Toovercomethis,Artzner,Delbaen,EbnerandHeath(1999)proposedthemainpropertiesthatariskmeasuresmustsatisfy,thusestablishingthenotionofcoherentriskmeasure.Aftercoherentriskmeasuresandtheirpropertieswereestablished,otherclassesofmeasureshavebeenproposed,eachwithdistinctiveproperties:convex(FollmerandShied,2004),spectral(Acerbi,2002)ordeviationmeasures(Rockafellaretal.2006).ThecoherentriskmeasureshavebeenusedforcapitalallocationandportfoliooptimizationasinRockafellar,UryasevandZabarankin(2002),aswellastopriceoptionsinincompletemarkets,asinCherny(2006).Spectralriskmeasuresarecoherentriskmeasuresthatsatisfytwoadditionalcon-ditions.Thesemeasureshavebeenappliedtofuturesclearinghousemarginre-quirementsinCotterandDowd(2006).AcerbiandSimonetti(2002)extendtheresultsofP ugandRockafellar-Uryasevmethodologytospectralriskmeasures.AdescriptionoftheaxiomsofriskpricingmeasureswithmanyapplicationstoinsurancecanbefoundinWang,YoungandPanjer(1997),andinthemonographbyKass,Goovaerts,DhaeneandDenuit(2001).Theconceptofdistortedriskmeasuresevolvedfromthislineofworkandtiesinwiththenotionofcapacity.Capacitiesarenon-additive,monotonesetfunctionswhichextendthenotionofintegralinapeculiarway.Theevolutionofthisconcept,fromChoquet'sworkinthe1950'suntilthe1990'scanbetracedbackfromthereviewbyDenneberg(1997).Interestinglyenough,therehavebeensomenaturalpointsofcontactbetweenactuarialandnancialrisktheory.Ontheonehand,conceptsinactuarialrisk2 2.Positivehomogeneity:ForanyX2L2and0;wehave(X)=(X):3.Monotonicity:ForanyXandY2L2,suchthatXYthen(X)(Y):4.Subadditivity:ForanyXandY2L2,(X+Y)(X)+(Y):Thesepropertiesinsurethatdiversicationreducestheriskoftheportfolioandifpositionsizedirectlyincreasesrisk(consequencesoflackofliquidity)thenitisaccountedinthefuturenetworthoftheposition.OneexampleofcoherentriskmeasuresistheConditionalValueatRisk(CVaR).Thismeasureindicatestheexpectedlossincurredintheworstcasesoftheposition.ItisthemostpopularalternativetotheValueatRiskmeasures.CVaR(X)=�1 Z0VaRt(X)dt;whereVaRt(X)=supfx:P[X�x]�tgSpectralriskmeasuresaredenedbyageneralconvexcombinationofConditionalValueatRisk.Denition2.2Anelement2L1([0;1])iscalledanadmissibleriskspectrumif1.02.isdecreasing3.kk=R10j(t)jdt=1:Denition2.3Foranadmissibleriskspectrum2L1([0;1])theriskmeasure(X)=�Z10qX(u)(u)duiscalledthespectralriskmeasuregeneratedby.iscalledtheRiskAversionFunctionandassigns,infact,dierentweightstodierentp-condencelevelsofthelefttail.Anyrationalinvestorcanexpresshersubjectiveriskaversionbydrawingadierentprolefortheweightfunction.Spectralriskmeasuresareasubsetofcoherentriskmeasuresasmovedby4 2.gisnon-decreasingfunction.Forapplicationstoinsuranceriskpricingitisconvenienttothinkoftheliabilitiesaspositivevariables,werestrictourselvestoX2L+2(P),i.e.,topositiverandomvariableswithnitevariance,whichwethinkaboutaslossesorliabilities.Ifweweretorelatethistothepreviousinterpretation,wewouldsaythatourpositionis�X:Thecompaniontheoremcharacterizingthedistortedriskmeasureinducedbygisthefollowing.Theorem2.2DenethedistortedriskmeasureDg(X)inducedbygontheclassL2(P)byDg(X)=Z10g(S(x))dx+Z0�1[g(S(x))�1]dx;(3)whereS(x)=1�FX(x).ThenDg(X)hasthefollowingproperties:1.XYimpliesDg(X)Dg(Y):2.Dg(X)=Dg(X)forallpositive:Dg(c)=cwhenevercisaconstantrisk.3.IftherisksXandYarecomonotone,thenDg(X+Y)=Dg(X)+Dg(Y):4.IfgisconcavethenDg(X+Y)Dg(X)+Dg(Y):5.IfgisconvexthenDg(X+Y)Dg(X)+Dg(Y):HardyandWirch(2001)haveshownthatariskmeasurebasedonadistortionfunctioniscoherentifandonlyifthedistortionfunctionisconcave.Itcanbeshownthatifgisconcavethegeneratedriskmeasureisspectral.Adistortionriskmeasureistheexpectationofanewvariable,withchangedprobabilities,re-weightingtheinitialdistribution.Example2.3TheVaRcanbedenedbythedistortionfunction:g(x)=(0ifx1ifx;(4)6 Thefollowingcharacterizationsareimportant:foru2(0;1)andx2R.Wehave:q+X(u)x,P(Xx)uqX(u)x,P(Xx)u:(6)Also,foru2(0;1);thefactthatQX(u)=[qX(u);q+X(u)];canbeusedtoestablishthatQ�X(u)=�QX(1�u);andinparticularthatq�X(u)=�qX(1�u):Fortheproofofthersttheoremofsection3,weshallneedthefollowingversionofthetransferencetheorem(seesection6.5inKingmanandTaylor(1966)).SetG(x)=P(Xx)=F(x�):ThenclearlyG(x)isincreasingandleftcontinuous.WehaveTheorem2.4(Transferencetheorem)(a)Foreverypositive,measurableh:(0;1)!RwehaveZ10h(u)dq+(u)=ZRh(G(x))dx;whereq+denotestherightquantileofF(x)=P(Xx):(b)Foreverypositive,measurableh:R!R;andeverycontinuousincreasingg:[0;1]![0;1],wehaveZ10h(qX(u))dg(u)=Z1�1h(x)d(gFX)(x):Proof:Toprove(a)itsucestoprovetheresultforh(u)=I(a;b](u)with0ab1:Inthiscase,involvingthecharacterization(6),wehavethatZ10I(a;b](u)dq+(u)=q+(b)�q+(a)=ZRI(q+(a);q+(b)](x)dx=ZRI(a;b](G(x))dx;whichconcludestheproofof(a):Toprove(b),weconsiderqX:((0;1);dg)!R;andtoidentifythetransferedmeasureitsucestoconsiderh(x)=I(a;b](x):Denotingby~gthetransferedmeasure,wehaveZ10h(qX(u))dg(u)=Z1�1h(x)d~g(x);therefore~g(b)�~g(a)=R10I(a;b](x)(qX(u))dg(u)=R10I(FX(a);FX(b)](u)dg(u)=(gFX)(b)�(gFX)(a);forwhichweinvoke(6)oncemore.28 Example3.1TheriskmeasureCVaRisaspectralriskmeasure(seeExample2.1).Ifweapplytheprevioustheoremto(2)wehavethattheConditionalValueatRiskisadistortionriskmeasuredenedby:g(u)=Zu0(s)ds=Zu01 1f0sg=(u ifu1ifu:WethusreobtaintheresultofExample2.4Theprevioustheoremadmitsthefollowingreciprocal,theproofofwhichfollowsreversingthestepsoftheproofoftheprevioustheorem.Theorem3.2Letgaconcavedistortionfunction,andletDgbetheassociateddistortedriskmeasure.Then(u)=g0(u)denesaspectralmeasuresuchthat(X)=Dg(�X):Example3.2WenowcalculatetheRiskAversionFunctionforthedistortionriskfunctionslistedinExample2.5.1.Dual-powermeasure:(u)=(1�u)�1with1.2.Proportionalhazardmeasure:(u)=1 u1 �1,with 1.3.Wang'smeasure:(u)=e[��1(u)�2 2].ObservethatforProportionalHazardandWang'smeasures,theRiskAversionfunctionisnotboundedatzero.Moreover,theRiskAversionfunctiontheWang'smeasuredecreasesmorequicklythanthatofProportionalHazard's.Therefore,theinvestorusingWang'sriskmeasureismoreriskaversethananinvestorthatmeasurestheriskbytheProportionalHazarddistortionbecausetherstinvestorgivesmoreimportancetohigherlosesthanthelatter.Wehavethusestablishedthatbothmethodstoconstructriskmeasures,eitherbymeansofdistortionriskfunctionsorbyadmissiblespectralfunctions,areequivalent.Inboth,theriskmeasurecanbethoughtofasare-weightingoftheinitialdistribution.Moreover,thederivativeofthedistortionriskfunctionindicatesthewayofthisre-weighting,asBalbaset.al.(2006)haveindicated.Comment3.2Thesecorrespondencesalsoprovideanindirectproofofthefactthatforaconcavedistortionfunctiong,theriskmeasuredenedby3isacoherentriskmeasure.10 Proof:Deneg(u)=Ru0 (q(s))ds,wherefor0u1.Wedenotebyq(u)theleftcontinuousinverseofF:Clearly,gisincreasing,continuous,withg(0)=0andg(1)=1:Letusnowverifythatg(F(x))=F(x):Anapplicationofthetransferencethe-orem(oravariationonthechangeofvariablestheme,seeKingmanandTaylor,(1970))yieldsthatF(x)=Rx�1 (t)dF(t)=RRI(�1;x])(t) (t)dF(t)=R10I(�1;x])(F�1(u)) (F�1(u))du=RF(x)0 (F�1(u))du=g(F(x));sinceuF(x),F�1(u)x.2Comment4.1Thistheoremassertsthatanytwopossibledistributionsassignedtoagivenrandomvariable,canberelatedbymeansofadistortionfunction.Theorem4.2Letgbeapiecewisecontinuouslydierentiabledistortionfunctionasabove,andFX(x)beacontinuous,andstrictlyincreasingdistributionfunctiononitssupport.Thenthemeasuredm=dFXon(R;B(R)),inducedbythedis-torteddistributionfunctionFX,isabsolutelycontinuouswithrespecttodm=dFXhavingdensity (x)=g0(FX(x)):Proof:ItboilsdowntonoticingthatFX(x)=g(FX(x))impliesthatdFX(x)=g0(FX(x))dFX(x).2Comment4.2Inthiscasetherelationshipbetweendistortedmeasuresandspec-tralriskmeasuresiseasytoestablish.NotethatE[X]=�E[�X]=�R1�1xdF�X(x)=�R1�1xg0(F�X(x))dF�X(x)=�R10q�X(u)g0(u)du=(�X)ifweidentify(u)withg0(u):Thefollowingresultshowstherelationshipbetweenspectralriskmeasures,andthosebasedondistortionfunctionsandrelativeentropy.Theorem4.3LetXbearisksuchthatFXiscontinuousandstrictlyincreas-ingonitssupport.Thenthespectral,thedistortedandtherelativeentropyrisk12 distortionfunctionsarestrictlyincreasingandcontinuous,thedierenceoftheagent'sriskaversionisgivenbyh=g1g�12.5ConclusionsWehaveestablishedthatspectralriskmeasuresarerelatedtodistortedriskpric-ingmeasures.Thuswehavetworepresentationsathandforagivenriskmeasure,andmaychoosewhicheverrepresentationismoreconvenientfortheapplicationathand.Also,distortedriskpricingmeasuresturnouttobeabsolutelycontinuouswithrespecttothemeasurethattheydistort.Thisallowsus,forexample,tointerpretdierentphysicalprobabilities(ordierentgeneralizedscenarios)asdis-tortedviewsofreality,onewithrespecttotheother.Moreover,welinkspectralriskmeasures,andthosebasedondistortionfunctionsandrelativeentropy.AcknowledgmentWewanttothanktherefereesandtheeditorfortheircom-ments,whichimprovedthepaper.References[1][A]Acerbi,C.(2002)Spectralmeasuresofrisk:acoherentrepresentationofsubjectiveriskaversion,JournalofBankingandFinance,7,1505{1518.[2][AS]Acerbi,C.,Simonetti,P.(2002)Portfoliooptimizationwithspectralmeasuresofrisk,WorkingPaper,Abaxbank,www.gloriamundi.com.[3][ADEH]Artzner,P.,Delbaen,F.,Eber,J.M.andHeath,D.(1999)Coherentmeasuresofrisk,MathematicalFinance,9203-228.[4][B]Balbas,A.,Garrido,J.,Mayoral,S.(2006)Propertiesofdistortionriskmeasures,WorkingPaper,UniversityofNavarra.[5][Ch]Cherny,A.S.(2006)Pricingwithcoherentrisk,www.citebase.org.[6][C]Cotter,J.,Dowd,K.(2006)Extremespectralriskmeasures:anapplica-tiontofuturesclearinghousemarginrequirements.JournalofBankingandFinance,30,3469{3485.[7][De]DennebergD.(1997).Non{AdditiveMeasureandIntegral.Dordrecht-Boston-London:KluwerAcademicPublishers.14 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