<%BANNER%>

Liquidity Risk Measurement and Management

Permanent Link: http://ufdc.ufl.edu/UFE0044137/00001

Material Information

Title: Liquidity Risk Measurement and Management
Physical Description: 1 online resource (71 p.)
Language: english
Creator: Sun, Pengyi
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: allocation -- liquidity -- measure -- optimization -- risk -- strategy -- trading
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this paper, we introduced the liquidity risk measures. The coherent risk measure introduced by Artzner1 does not include the liquidity effects. That is the trading amount is so large that it will affect the market price. In this situation, the positive homogeneityaxiom will no longer hold. So we set up a new axioms system to deal with liquidity effects. We call it the liquidity risk measure. In addition, we found the acceptance sets and established the relationship between the liquidity risk measures and the acceptance sets. We also developed appropriate liquidity risk measures for multi-asset portfolios and explored the corresponding optimal trading strategies. We further studied variations of liquidity risk measures and checked that the liquidity costs mentioned by Cetin et al.7 is in fact a liquidity risk measure, but not coherent. We also gave an algorithm to calculate the optimal trading strategy under liquidity risk measures. Finally, we discussed some applications in risk allocations and tested liquidity risk measures in the real markets.
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.
Statement of Responsibility: by Pengyi Sun.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Yan, Liqing.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044137:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044137/00001

Material Information

Title: Liquidity Risk Measurement and Management
Physical Description: 1 online resource (71 p.)
Language: english
Creator: Sun, Pengyi
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: allocation -- liquidity -- measure -- optimization -- risk -- strategy -- trading
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this paper, we introduced the liquidity risk measures. The coherent risk measure introduced by Artzner1 does not include the liquidity effects. That is the trading amount is so large that it will affect the market price. In this situation, the positive homogeneityaxiom will no longer hold. So we set up a new axioms system to deal with liquidity effects. We call it the liquidity risk measure. In addition, we found the acceptance sets and established the relationship between the liquidity risk measures and the acceptance sets. We also developed appropriate liquidity risk measures for multi-asset portfolios and explored the corresponding optimal trading strategies. We further studied variations of liquidity risk measures and checked that the liquidity costs mentioned by Cetin et al.7 is in fact a liquidity risk measure, but not coherent. We also gave an algorithm to calculate the optimal trading strategy under liquidity risk measures. Finally, we discussed some applications in risk allocations and tested liquidity risk measures in the real markets.
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.
Statement of Responsibility: by Pengyi Sun.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Yan, Liqing.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044137:00001


This item has the following downloads:


Full Text

PAGE 1

LIQUIDITYRISKMEASUREMENTANDMANAGEMENTByPENGYISUNADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

PAGE 2

c2012PengyiSun 2

PAGE 3

Idedicatethisdissertationtomyfamilythathelpedandsupportedmethroughmygraduatestudy. 3

PAGE 4

ACKNOWLEDGMENTS SpecialthankstoProfessorStanislavUryasev,ProfessorWilliamHager,ProfessorLeiZhangandProfessorMuraliRaoforalltheassistantsandsuggestionsaboutmydissertation.AndgreatthankstomyadvisorProfessorLiqingYanforhissupportandguidanceduringmyPh.D.study. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1INTRODUCTION ................................... 9 1.1FinancialRiskEnvironment .......................... 9 1.2ScopeofThisPaper .............................. 11 2LIQUIDITYRISKMEASURES ........................... 13 2.1ReviewonCoherentRiskMeasures ..................... 13 2.2Notations .................................... 17 2.3AxiomsonAcceptanceSets .......................... 17 2.4AxiomsonLiquidityRiskMeasures ...................... 17 3CORRESPONDENCEBETWEENACCEPTANCESETSANDLIQUIDITYRISKMEASURES .................................. 19 3.1DualDenitionsofRiskMeasuresandAcceptanceSets .......... 19 3.2CorrespondenceTheorems .......................... 20 4LIQUIDITYRISKMEASURESWITHDISCRETETHRESHOLDS ........ 22 4.1SingleRiskyAssetPortfolio .......................... 22 4.2Bid-AskSpread ................................. 27 4.3ExtensiontoMultipleStages ......................... 29 4.4ExtensiontoTwo-AssetPortfolios ....................... 31 4.5ExtensiontoMultipleTradingPeriods ..................... 34 5OTHERVARIATIONSOFLIQUIDITYRISKMEASURES ............. 37 5.1ConditionalDiversication ........................... 37 5.2PowerLiquidityRiskMeasures ........................ 38 5.3SomeExamples ................................ 40 6OPTIMALBALANCEDTRADINGSTRATEGY .................. 44 6.1SeparateTradingStrategy ........................... 45 6.2SolutiontoASimpleCase ........................... 47 6.3OptimalSolutionsSet ............................. 49 6.4GeneralSolutions ............................... 51 6.5NumericalExamples .............................. 54 5

PAGE 6

7LIQUIDITYRISKALLOCATIONS .......................... 57 7.1BasicConcept ................................. 57 7.2LowerandUpperBound ............................ 59 7.3RiskAllocationsUsingLiquidityRiskMeasures ............... 60 7.4PropertiesofRiskAllocations ......................... 62 8EMPIRICALSTUDIESONLIQUIDITYRISKMEASURES ............ 64 9CONCLUSION .................................... 68 REFERENCES ....................................... 69 BIOGRAPHICALSKETCH ................................ 71 6

PAGE 7

LISTOFFIGURES Figure page 1-1LiquidityRiskComparison .............................. 11 5-1Non-LinearRiskMeasure .............................. 43 6-1AssetPriceProcess ................................. 48 6-2DescendingPriceSequencewithTradingVolume ................. 52 6-3ConvergenceoftheVariance ............................ 55 6-4ConvergenceoftheMean .............................. 55 6-5OptimalTradingStrategy .............................. 56 8-1GCNTradeRecap .................................. 65 8-2T-TestforGCN .................................... 66 8-3T-TestforIBM ..................................... 66 8-4CorrelationBetweenPriceandVolume ...................... 67 7

PAGE 8

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyLIQUIDITYRISKMEASUREMENTANDMANAGEMENTByPengyiSunMay2012Chair:LiqingYanMajor:Mathematics Inthispaper,weintroducedtheliquidityriskmeasures.ThecoherentriskmeasureintroducedbyArtzner[ 1 ]doesnotincludetheliquidityeffects.Thatisthetradingamountissolargethatitwillaffectthemarketprice.Inthissituation,thepositivehomogeneityaxiomwillnolongerhold.Sowesetupanewaxiomssystemtodealwithliquidityeffects.Wecallittheliquidityriskmeasure.Inaddition,wefoundtheacceptancesetsandestablishedtherelationshipbetweentheliquidityriskmeasuresandtheacceptancesets.Wealsodevelopedappropriateliquidityriskmeasuresformulti-assetportfoliosandexploredthecorrespondingoptimaltradingstrategies.WefurtherstudiedvariationsofliquidityriskmeasuresandcheckedthattheliquiditycostsmentionedbyCetinetal.[ 7 ]isinfactaliquidityriskmeasure,butnotcoherent.Wealsogaveanalgorithmtocalculatetheoptimaltradingstrategyunderliquidityriskmeasures.Finally,wediscussedsomeapplicationsinriskallocationsandtestedliquidityriskmeasuresintherealmarkets. 8

PAGE 9

CHAPTER1INTRODUCTION 1.1FinancialRiskEnvironment Overthepastdecade,thenancialmarkethasbeensovolatilethateveryonerealizedtheactualriskwasmuchhigherthantheythought.Mostinvestmentmanagersnowtakeriskintoaccountfortheirperformance.Insteadofusingexcessreturns,fundmanagersareusingriskadjustedreturnstomanagetheirportfolios.TherearethreetypesofrisksconsideredbyBaselII:marketrisk,creditrisk,andoperationalrisk[ 23 ].Marketriskistheriskbasedonpricevolatilityofanassetoveratimeperiodbecauseofeconomicandmarketconditionchanges.Creditriskistheriskthataborrowermaydefaultbyfailingtorepaypartoralloftheprincipalandinterestatapredetermineddate.Operationalriskistheriskcreatedbyinefcientorfailedprocess,people,systemsorexternalevents.Itiscloselyrelatedtoeverydayoperationofabusiness[ 24 ]. Wecandivideinvestmentassetsintofourbroadcategories:moneymarket,stockmarket,bondmarketandderivatives[ 4 ].Eachcategoryhasdifferentfocusondifferenttypesofrisk,buttheyhaveoneincommon:liquidityrisk.Moneymarketistypicallyshortterminvestment,usuallylessthanoneyear.ThemostcommonformisT-bills.ThematuritiesforT-billsare4weeks,3months,6monthsand1year.Anditisusuallyconsideredasrisk-freeinstrument.Stockmarketisthemostimportantandcomplexmarketintheworldandcontrollingriskiscriticalinthemarket.Thereareallkindsofrisksinvolvedinthestockmarket.Themostobviousoneismarketrisk.Stockpriceraisesandfallseverydayandourinvestmenthighlydependsonit.Creditriskisalsocommoninthestockmarket.Therearecompaniesgoingtobankruptcyeverydayandyoumayloseallormostofyourinvestmentfromthatcompany.Operationalriskisalsoinvolved.Ifacompanyhasahugeoperationalrisk,themarketmayconsideritasapotentiallossanditsstockpricewillfall.Thebondmarketisprimarilyforlongterminvestment.Itislessvolatilethanthestockmarketanditsmajorconcerniscredit 9

PAGE 10

riskoftheissuer.Itspriceisnotaffectedbytheperformanceofthecompany,butbytheinterestrate.Thelongerthematurityofthebond,thehigheritsliquidityrisk.Thederivativesmarketishighlyvolatileandhardtovalue.Itsmajorconcernisthemarketriskoftheunderlyingassets.Thelongerthematurityofthederivatives,thehighertheliquidityrisk.Forthepurposeofresearch,weusuallyassumeonemonthT-billshavenoliquidityrisk. Therearethreestagesdealingwithrisk:riskmeasurement,riskallocationandriskmanagement.Themostintuitivemeasureisthestandarddeviation.Itisthemostcommonmeasureofvolatilityinstatistics.Asamatteroffact,alotoffundmanagersarestillusingittoestimatetheriskoftheirportfolios.However,itcanbeaffectedbyalotofotherfactorslikeoutliers.Italsocannotgiveyouthedirectionofthedispersion.Inordertosystematicallystudytherisk,coherentriskmeasurewasestablishedbyArtzneretal.[ 1 ].Itwasthersttimethatriskwasstudiedinarigoroustheoreticalmanner.Theirworkissoinspiringthatmanyotherriskmeasureshavebeencreatedbasedontheirtheory.Thecoherentriskmeasureisbasedonfouraxiomsthatmentionedinthepaper[ 1 ],anditsfollow-upworksareconductedbymodifyingtheseaxioms.Thegreatestadvantageofcoherentriskmeasureisthatithasperfecttheoreticalresults.However,itisnotveryusefulinpractice.Therehavebeenmanyresearchersdedicatedtothisapproachandproposedvariousmodicationsbasedonit.Forexample,theriskmeasuredenedbyRockafellar[ 27 ]completelyremovedthepositivehomogeneityaxiom.Therearesomanyotherfactorsweneedtoconsidertomakeitapplicableinthenancialmarkets.Onefactorthatisnotconsideredistheliquidityeffect,whichisveryimportantforbigtraders.Anotherfactoristhevolatilityofinterestrate,whichisverydramaticduringnancialcrisis.Figure1-1showsthedemandcurvefortwoassetswithdifferentliquiditypropertiesinthesameperiodoftime.Asweknow,short-termT-billisquiteliquidanditspricedoesnotchangemuchfordifferenttradingvolume.Ontheother 10

PAGE 11

hand,thestockUEICisnotasliquidasT-billandshowsupslopingdemandcurve.Thatis,priceincreaseswithlargedemandtobuy. Figure1-1. LiquidityRiskComparison 1.2ScopeofThisPaper Inthispaper,wearegoingtointroduceanewriskmeasurethattsbetterinanilliquidmarketwithinterestratevolatility.Liquidityeffecthasbeenstudiedforalong 11

PAGE 12

time.BaumandBank[ 2 ]describedanilliquidnancialmarketmodelwheremarketpricescanbedecidedbysomelargetradingparties.UndergeometricBrownianmotion,DufeandZiegler[ 14 ]studiedthecorrelationbetweenbid-askspreadandmid-priceandmodeledthemasstochasticprocesses.Cetinetal.[ 7 ]usedstochasticsupplycurvetoapproximateliquidityeffectanddevelopedcorrespondingpricingmethod. Interestrateisalsoanimportantfactorinriskmeasures.Thecoherentriskmeasureassumesriskypositionsarediscountedbeforeapplyingtheriskmeasureandthediscountingprocessdoesnotinvolveanyadditionalvolatility.However,wheninterestrateisvolatile,thisproceduredoesnotseparatetheriskofthenancialpositionandtheriskassociatedwiththediscountingprocess[ 20 ].Riskoptimizationhasbeenthoroughlystudiedbymanyresearchers.UryasevestablishedoptimizationtechniquesforC-Var[ 29 ],acoherentriskmeasure.Longstaff[ 22 ]studiedtheoptimaltradingstrategiesinanilliquidmarketasboundedvariations.Byconsideringliquidityeffect,multipleresearchersstudiedtheoptimalliquidationprocess[ 3 16 ].Pricingtechniqueshavealsobeenstudiesforcoherentriskmeasures[ 5 8 ]. Theorganizationofthispaperisasfollowing:Chapter2reviewscoherentriskmeasuresandstatesthebasicaxiomsforliquidityriskmeasuresandacceptancesets.Chapter3studiestherelationshipbetweentheaxiomsonacceptancesetsandtheaxiomsonliquidityriskmeasures.Chapter4denesspecicliquidityriskmeasuresfordifferentscenarios.Chapter5introducesconditionaldiversicationandsomevariationsofliquidityriskmeasures.Chapter6proposestheoptimaltradingstrategyproblemandprovidescertainsolutionsincludingcomputationalalgorithmandnumericalexamples.Chapter7explorestheapplicationofliquidityriskmeasuresinriskallocationproblems.Finally,Chapter8providessomeempiricalstudytosupportliquidityriskmeasureasabetteralternativeforpracticaluses. 12

PAGE 13

CHAPTER2LIQUIDITYRISKMEASURES 2.1ReviewonCoherentRiskMeasures Innancialmarkets,riskisusuallydividedintothreecategories:marketrisk,creditriskandoperationalrisk[ 23 ].Marketriskistheriskofanancialpositionduetotheuncertaintyofthemarketpriceorvaluationofthecontainingassets.Forexample,aportfolioconsistsofseveralstocksorbondsisgreatlyexposedtopricechangesinstockandbondmarkets.Creditriskisalsocalleddefaultrisk.Itistheriskthattheborrowerswillnotfullypaybacktheirloans.Anynancialloansaresubjecttocreditrisk.Forexample,peoplemaynotbeabletopaytheirautoloansandthuslenderssufferhugecreditrisk.Thelessstudiedoneistheoperationalrisk.Itisthepotentiallossesduetoinefciencyorfailureofinternalprocess,systemsorthemanagement.Forexample,company'spolicychangescanresultgreatuncertaintyinoperationandunstablecashows.Thesethreecategoriesarenotmutuallyexclusive.Theboundariesarenotclearandoneriskcanaffectanother.Anaxiomaticapproachtomodelmarketriskisthroughcoherentriskmeasures. CoherentriskmeasurewasrstintroducedbyArtzneretal.[ 1 ]anddevelopedandextendedtobroadersituations[ 13 17 26 ].Itwasestablishedthroughfourbasicaxiomsdescribedbelow.Beforeexploringthedetailsoftheaxioms,weneedtogetageneralunderstandingofwhatcoherentmeans.ByMerriam-Webstercdictionary,coherentmeansrelatingtoorcomposedofwaveshavingaconstantdifferenceinphase.Thisisjustthekeyofthisriskmeasure:nomatterhowthepositionchanges,theriskwillalwayschangeinproportion.Itisaneatconditionandwecangetbeautifulresultsunderthiscondition,liketherepresentationtheorem.However,itisalsoaverystrongassumptionandinmostcasesitdoesnothold.Thusitcallsforaweakeningofthecoherentassumptionandthatisthegoalofthispaper.Werstcarryoutabriefstudyofthecoherentriskmeasuresinaxiomaticsettings. 13

PAGE 14

Oneimportantconceptusedtointroducecoherentriskmeasureisacceptanceset,whichisasetofacceptablefuturenetworth[ 1 ].Theacceptanceorrejectionofanancialpositioncanbebasedonwhetheritisintheacceptanceset.Intuitively,allpositivefuturenetworthsareacceptable,sotheyshouldbeincludedintheacceptanceset.Andallstrictlynegativepositionscannotbeincludedintheacceptanceset.Italsoassumesriskaversionwhichleadstotheconvexityoftheset.Finally,toreectthecoherentproperty,theacceptancesetisrequiredtobepositivelyhomogeneous.Thefollowingisthelistofalltheaxiomsforacceptanceset[ 1 ]: Positive:TheacceptancesetAcontainsL+,theconeofnonnegativeelements.Nonnegative:ThesetAdoesnotintersectthesetL\000=fXjX(!)<0g.Convexity:TheacceptancesetAisconvex.Coherence:TheacceptancesetAisapositivelyhomogeneouscone. Afterdeningacceptanceset,wecanroughlymeasuretheriskbysimplydeterminingwhetheritisacceptableornot.Strictlyspeaking,wecaninduceariskmeasurebasedontheacceptanceset.Andluckilyenough,theriskmeasureinducedbytheacceptancesetdenedaboveiscoherent.Wewilldiscussthisrelationshipfurtherinlatersections.Whatwewanttodonextistointroducecoherentriskmeasureasanindependentconceptandstudyitsproperties. Ifapositionisdescribedbytheresultingdiscountednetworthattheendofagivenperiod,denedasareal-valuedfunctionXonsomesetofpossiblescenarios,thenaquantitativemeasureofriskisgivenbyamappingfromacertainspaceGoffunctionsontotherealline.Formallyspeaking,wehavethefollowingdenitions[ 1 ]: Denition2.1AmeasureofriskisamappingfromGintoR. Denition2.2Areferenceinstrumentisacommonlyacceptedinstrumentthatisriskfree. Nowassumeisniteandristhetotalreturnofareferenceinstrument. 14

PAGE 15

Denition2.3Thecoherentmeasureofrisk:G!Rsatisesthefollowingaxioms: Monotonicity:IfXY,then(X)(Y) (2) TranslationInvariance:Ifm2R,then(X+mr)=(X))]TJ /F6 11.955 Tf 11.95 0 Td[(m (2) PositiveHomogeneity:Forany0,(X)=(X) (2) Subadditivity:(X+Y)(X)+(Y) (2) MonotonicityisjustsayingwepreferXtobepositivelargenumbers.SinceXisdenedasthenalnetworth,wearebetteroffwhenourpositivenetworthisgreat.Thatalsomeanswehavelittleriskoflosingmoney.Thatiswhytherelationshipisthereversedmonotonicity.Translationinvariancemeansanyconstantcanbetakenoutoftheriskmeasure.IfyouhaveariskypositionXandarisk-freepositionmtoday,theriskofyourwholeportfolioisjusttheriskofyourriskypositionminustherisk-freeposition.Therisk-freepositionservesasaninsuranceinthiscase.Thatiswhyitlowersyourrisk.Theonlyissuewiththisaxiomisthatinterestrateisabsolutelynotconstantandtheequationdoesnotholdundervolatileinterestrate.PositiveHomogeneityisjustalinearassumptionneededfortherepresentationtheorem.ForsmallandX,itisapproximatelytrue.Butforlargepositionsandmultipliers,thisequationdoesnotholdeither.Subadditivityconveysriskdiversications.Theriskofholdingtwopositionsseparatelywillbegreaterthanholdingthemtogethersincesometimetheymaycanceloutsomerisk.However,riskdiversicationisnotalwaystrue.Wewillseesomeexampleslater. Themostimportantresultofcoherentriskmeasureistherepresentationtheorem.Typically,acoherentmeasureofriskarisesfromsomefamilyPofprobabilitymeasuresonbycomputingtheexpectedlossunderP2Pandthentakingthe 15

PAGE 16

worstresultasPvariesoverP: (X)=supP2PEP[)]TJ /F6 11.955 Tf 10.49 8.09 Td[(X r] (2) forthecasewhereisniteandristhetotalreturnofareferenceinstrument.ThisistherepresentationtheoremofcoherentmeasureofriskgivenbyArtzner[ 1 ].Itisaverystrongtheoremandthemodernriskmeasuresystemisestablishedonthisground.Theproblemofthetheoremisthatithasverystrongassumptionswhicharenotgoodestimatesoftherealmarketconditions.WewilltestthemarketconditionsfortheassumptionsinChapter8. Whenwethinkabouttheassumptionsintherealmarketconditions,thePositiveHomogeneityandSubadditivityareoftennotvalid.Becauseinanilliquidmarket,whichistrueinmostcases,tradinglargeamountofassetscanaffectthepriceoftheseassetsandthenaffectthebenetsyouwillget.Forexample,ifyouwanttotradenX,whereXisverylargeandn>1,thentheriskencounteredshouldbebiggerthann(X)becauseyoualsoshouldconsidertheliquiditycosts.HencethePositiveHomogeneityaxiomisviolated.Similarly,ifX=Yisverylarge,then(X+Y)=(2X)shouldbebiggerthan2(X),whichviolatestheSubadditivityaxiom.Therefore,inthenewsystemofmeasuresofrisk,weshouldreplacethesetwoaxiomsbyothers.OneistheConvexityaxiomandtheotheristheLiquidityaxiom: Convexity:Forany2[0,1],(X+(1)]TJ /F9 11.955 Tf 11.96 0 Td[()Y)(X)+(1)]TJ /F9 11.955 Tf 11.95 0 Td[()(Y) (2) Liquidity:If>1,then(X)(X);If0<1,then(X)(X) (2) Convexitymeansthatdiversicationdoesnotincreasetherisk,i.e.,theriskofadiversiedpositionX+(1)]TJ /F9 11.955 Tf 12.85 0 Td[()Yislessthanorequaltotheweightedaverageoftheindividualrisks.Liquiditymeansforalargeposition,theriskoftradingthewholepositionisbiggerthantheriskoftradingdividedpartsofthepositionseparately. 16

PAGE 17

2.2Notations First,weclarifythenotationswewillusethroughoutthispaper.Wedenetobethesetofstatesofnatureandassumeitisnite.Thenalnetworthofapositionforeachelementofisarandomvariable,denotedbyX.LetGbethesetofallrisks,thatisthesetofallreal-valuedfunctionson.Sinceisnite,GcanbeidentiedwithRn,wheren=card().DeneL\000=fXjX<0g.LetAbethesetsofnalnetworthwhichareacceptableandwecallittheacceptanceset. 2.3AxiomsonAcceptanceSets Theacceptablesetsunderliquidityriskmeasureshavethesamemeaningoftheacceptancesetsundercoherentriskmeasuresbutwithdifferentconditions.Sincewetakeoutthecoherentcondition,theacceptancesetisnolongerapositivelyhomogeneouscone,butwestillneedittobeconvex.Weadoptsomeaxiomsfromcoherentriskmeasuresandaddsomenewones.Wesummarizetheminthefollowing: Axiom1TheacceptancesetAdoesnotintersectthesetL\000. Axiom2TheacceptancesetAisconvex. Axiom3IfX2AandY2G,thenf2[0,1]jX+(1)]TJ /F9 11.955 Tf 11.88 0 Td[()Y2Agisclosedin[0,1]. Axiom4For1,ifX=2A,thenX=2A.For0<1,ifX2A,thenX2A. Remark:Thatmeanstheacceptancesetisaconvexset,whichreectsriskaversiononthepartoftheregulators.TheAxiom3isnotobvious,andwewillseeitisnecessaryforlateruse. 2.4AxiomsonLiquidityRiskMeasures Weputtheoldandournewaxiomstogethertogetthefollowingaxiomaticsystemforliquidityriskmeasure: AxiomSSeparation.IfX<0,then(X)>0. AxiomTTranslationInvariance.Ifm2R,then(X+mr)=(X))]TJ /F6 11.955 Tf 11.95 0 Td[(m AxiomCConvexity.Forany2[0,1],(X+(1)]TJ /F9 11.955 Tf 11.95 0 Td[()Y)(X)+(1)]TJ /F9 11.955 Tf 11.96 0 Td[()(Y) AxiomLLiquidity.If>1,then(X)(X);If2(0,1],(X)(X) 17

PAGE 18

Nowwecometotheformaldenitionofaliquidityriskmeasure: Denition2.1AliquidityriskmeasureisameasureofrisksatisesAxiomsS,T,andL.AconvexliquidityriskmeasureisariskmeasuresatisesAxiomsS,T,C,andL. WehaveexplainedTranslationInvariancebefore.WedonothaveMonotonicityhere.Becausesomeassetsmayhavelargenegativeliquidityeffecttomakethemlessvaluablethanotherassets.WewillcarefullyexaminethisconditioninChapter4.TounderstandConvexity,wecanthinkaboutthetermstructuremodelforinterestrisk.Theyieldcurveisusuallyconcave,forexample,in30years.Youcanalwaysobservethepatternthatthelongtermratesareusuallygreaterthantheshorttermrates.Theincreaseintherateisquickinthebeginningandslowwhenmaturitypasses5years.Soitisaconcaveincreasingcurve.Theriskhereisdenedasthenegativeoftheinterestrateandhenceitshouldbeaconvexcurve.TheLiquidityaxiomistrueforlargetransactionsinmostmarkets.Thecommoditiesmarketsaretheleastliquidmarketsandwecanoftenseepriceincreaseduetolargebuyingpowers.TheLiquidityaxiomisareasonableassumptionanditcontainsthePositiveHomogeneitycondition.Forsmalltransactions,wecanassumenoliquidityeffectsandtheequalityholds. Forgeneralriskmeasure,itcanbeinterpretedasthecashamountweneedtomakeourpositionriskfree.Whenpositive,thenumber(X)willbeinterpretedastheminimumextracashneededtoaddtotheriskypositionXandinvestinthereferenceinstrumenttomakeitacceptable.Ifitisnegative,thecashamount)]TJ /F9 11.955 Tf 9.3 0 Td[((X)canbewithdrawnfromtheposition. 18

PAGE 19

CHAPTER3CORRESPONDENCEBETWEENACCEPTANCESETSANDLIQUIDITYRISKMEASURES 3.1DualDenitionsofRiskMeasuresandAcceptanceSets AswementionedinChapter2,theacceptancesetsandriskmeasuresarecloselyrelatedtoeachother.Infact,wecanderiveonefromtheother.Thefollowingsaregeneraldenitionsofriskmeasuresandacceptancesets[ 1 ].AndwewillseethedenitionsproposedinChapter2areconsistentwiththem. Denition3.1Giventhetotalreturnronareferenceinstrument,theriskmeasureassociatedwiththeacceptancesetAisthemappingfromGtoRdenotedbyAanddenedbyA(X)=inffmjmr+X2Ag. Correspondingly,wecandenetheacceptancesetAassociatedwithariskmeasure: Denition3.2TheacceptancesetassociatedwithariskmeasureisthesetdenotedbyAanddenedbyA=fX2Gj(X)0g. Theriskmeasurecanbeconsideredastheminimumcashweneedtoaddtothethepositiontomakeitacceptablebyinvestors.Andtheacceptancesetisthecollectionofpositionsthatwithnon-positiverisks.Theyareperfectlycorrelatedwitheachotherbasedonthefactthatapositionisacceptablebyinvestorsifandonlyiftheinvestor'sriskmeasureofthispositionisnon-positive.Thesedenitionsaretrueforanykindsofacceptancesetsandanykindsofriskmeasures.Theproblemisthatthesedenitionsholdwhenconsideredseparatelybutmayconictwitheachotherwhenputtogetherduetotheinconsistenceoftheacceptancesetsandriskmeasuresdenedbyinvestors.ConsistenceheremeansA=andA=A. Forexample,inonedimensionalcase,wedeneA=fX>0gand(X)=5)]TJ /F6 11.955 Tf 12.19 0 Td[(X.SoA=fXj(X)0g=fX5g.Clearly,A6=A. However,wewillseeinthecaseofconvexliquidityriskmeasures,theyareperfectlyconsistentwitheachother. 19

PAGE 20

3.2CorrespondenceTheorems Inthissection,wewilldiscusstherelationshipbetweenconvexliquidityriskmeasuresandthecorrespondingacceptancesets.Weapplytheabovetwodenitionstoconvexliquidityriskmeasuresandchecktheconsistencyinsidethesystem.Givenaconvexliquidityriskmeasure,thenaturallyinducedsetAisanacceptanceset. Theorem3.1Ifisaconvexliquidityriskmeasure,thenAsatisesAxioms1,2,3,and4.Inaddition,wehaveA=. Proof: (1)ForanyX2L\000wehaveX<0.ByAxiomS,weget(X)>0.Bythedenitionofacceptanceset,X=2A.SoAxiom1issatised. (2)AssatisesAxiomC,isaconvexfunction,thenitiscontinuous.Hence,A=fXj(X)0gisclosedandconvex. (3)DeneF:7)166(!(X+(1)]TJ /F9 11.955 Tf 12.75 0 Td[()Y).ThenFiscontinuous,asitisconvex.Moreover,f2[0,1]jX+(1)]TJ /F9 11.955 Tf 11.91 0 Td[()Y2Ag=f2[0,1]j(X+(1)]TJ /F9 11.955 Tf 11.91 0 Td[()Y)0gisclosed. (4)IfX=2Aand1,thenbydenition,(X)>0.ByAxiomL,wehave(X)(X)>0.SoX=2A. IfX2Aand0<1,then(X)0andbyAxiomL,(X)(X)0.SoX2A.SoAxiom4issatised. (5)ForanyX,A(X)=inffmjmr+X2Ag=inffmj(mr+X)0g=inffmj(X))]TJ /F6 11.955 Tf 11.95 0 Td[(m0g=inffmj(X)mg=(X),byAxiomT.2 Correspondingly,wehavethefollowingtheoremfortheconvexliquidityriskmeasureinducedbyanacceptanceset. Theorem3.2IfthesetAsatisesAxioms1,2,3,and4,thentheriskmeasureAisaconvexliquidityriskmeasure.Moreover,wehaveA=AA. Proof: (1)WehavetheequalityinffpjX+(+p)r2Ag=inffqjX+qr2Ag)]TJ /F9 11.955 Tf 42.94 0 Td[(.SoitimpliesA(X+r)=A(X))]TJ /F9 11.955 Tf 11.96 0 Td[(.SoAxiomTissatised. 20

PAGE 21

(2)IfX<0,thenX2L\000.ByAxiom1,L\000\A=,thenX=2A.Bydenitionofacceptanceset,(X)>0.SoAxiomSissatised. (3)LetX1,X22Xandm1,m22R,suchthatXi+mir2A.AsAisconvex,forany2[0,1],(X1+m1r)+(1)]TJ /F9 11.955 Tf 12.31 0 Td[()(X2+m2r)2A.ThenbyAxiomTprovedabove,0A((X1+m1r)+(1)]TJ /F9 11.955 Tf 12.1 0 Td[()(X2+m2r))=A(X1+(1)]TJ /F9 11.955 Tf 12.1 0 Td[()X2))]TJ /F8 11.955 Tf 12.1 0 Td[((m1+(1)]TJ /F9 11.955 Tf 12.1 0 Td[()m2).ThatisA(X1+(1)]TJ /F9 11.955 Tf 12.06 0 Td[()X2)m1+(1)]TJ /F9 11.955 Tf 12.06 0 Td[()m2.Asthisistrueforanym1,m2,suchthatXi+mir2A,itisalsotruethatA(X1+(1)]TJ /F9 11.955 Tf 12.35 0 Td[()X2)A(X1)+(1)]TJ /F9 11.955 Tf 12.35 0 Td[()A(X2).SoAxiomCholds. (4)IfX=2A,thenA(X)>0.Foranym0.ThatmeansX+mr=2A.ByAxiom4,forany1,(X+mr)=2A.SoA((X+mr))>0.ThatisA(X)m.Asitistrueforanym=2fnjX+nr2Ag,wecantakesupremumofmwhichistheinmumoffnjX+nr2Ag,andtheinequalitystillholds.Hence,wehaveA(X)A(X).IfX2A,thenA(X)0.ForanymA(X),wehaveA(X+mr)0.Forany0<1,byAxiom4,(X+mr)2A.SoA(X)m.Similarasabove,aftertakinginmumofm,westillgettheinequalityandhaveA(X)A(X). SoeitherX2Aornot,wehaveatleastoneinequality.Andbasedonthisinequalitywecanderivetheotherinequalitybysubstituting1 .SoAxiomLholds. (5)ForanyX2A,A(X)0,henceX2AA.SoAAA.NowassumeX=2A.Foranym>0,m2A.ByAxiom3,exists2(0,1)suchthatm+(1)]TJ /F9 11.955 Tf 11.86 0 Td[()X=2A.Thus,mA((1)]TJ /F9 11.955 Tf 12.52 0 Td[()X)(1)]TJ /F9 11.955 Tf 12.52 0 Td[()A(X).ThatisA(X) 1)]TJ /F12 7.97 Tf 6.58 0 Td[(m>0.Hence,X=2AA.Therefore,A=AA.2 Remark:WecanseeDenition3.1andDenition3.2aredualtoeachother.Andtheconvexliquidityriskmeasurewedenedisconsistentwithinthissystem. 21

PAGE 22

CHAPTER4LIQUIDITYRISKMEASURESWITHDISCRETETHRESHOLDS Theriskmeasuredenedaboveisquitegeneralandhardtoapplyintherealmarket.Inthischapter,wewilldenesomespecicliquidityriskmeasuresthatarereadytouse.Thebasicassumptionforsuchriskmeasuresisthethresholdstructureoftheassetprices.Wewillbeginwithsomedenitionsandnotationsrst. 4.1SingleRiskyAssetPortfolio Tobettermodelliquidityrisk,weadopttheconceptacceptableportfoliobyKu[ 21 ].Aportfolioisconsideredtobeacceptableifitcanbeturnedintoanacceptablecashonlypositionwithpositivefuturecashowatsomexeddate.Assumewehaveunlimitedaccesstoriskfreeassetsinthemarketaswellasriskyassets.WedenotetheseriskyassetsasS1,S2,...,SN.Weassumetheseassets'pricesS1,S2,...,SNareadaptedstochasticprocessesonaprobabilityspace(,F,P)withaltration. Denition4.1Atradingstrategyt=(0t,1t,...,Nt)isa(N+1)-dimensionalfFtgadaptedprocess.0tisthenumberofunitsoftherisk-freeassetandntisthenumberofsharesofassetsSntheldattimet. Thereisliquidityriskinthemarketbecauseoftwofacts.Therstiswhenyousellanasset,itishardtondacounterpartytobuy.Andthesecond,evenifyoundsomecounterpartytobuy,theymayonlybeabletopayapricemuchlowerthanyouraskingprice.Soevenifyouareholdingassetswithgreatvalue,youmaynotbeabletosellthematafairpriceinashortperiodoftime.Inthebankruptcycaseyoumustpaybackyourdebt.Thereisnotmuchtimeallowedforyoutosellatafairprice.Soifyouonlykeepjusttheassetswithfairvalueequaltoyourdebt,youmaynotbeabletocoveralltheliabilitiesandmakethecompanyinsolvent.Thatiswhyliquidityriskissoimportantinriskregulation.Inthewordsofriskmeasure,liquidityriskmeasuresshouldalwaysbegreaterthancoherentriskmeasures.Wecallthedifferencebetweenthesetworiskmeasurestheliquidityspread. 22

PAGE 23

Denition4.2Aliquidityspreadisafunctions(X)=L(X))]TJ /F9 11.955 Tf 12.19 0 Td[(C(X),whereLisaliquidityriskmeasureandCisacoherentriskmeasure. Proposition4.1AnyliquidityspreadsatisestheLiquidityaxiom.Thatisforany1,s(X)s(X). Proof:Bydenition,forany1s(X)=L(X))]TJ /F9 11.955 Tf 12 0 Td[(C(X)L(X))]TJ /F9 11.955 Tf 11.99 0 Td[(C(X)=(L(X))]TJ /F9 11.955 Tf 11.95 0 Td[(C(X))=s(X).2 WebeginwithacoherentriskmeasureCwithasetofscenarioprobabilitiesfPi,i2Ig.BythedenitioninChapter3,arandomvariableXiscalledacceptableifC(X)0.ThatisequivalenttoEPi[X]0foralli2Iusingtherepresentationtheoremforcoherentriskmeasures.AccordingtoKu[ 21 ],aportfolioissaidtobepositiveifitentailsonlynon-negativecashowsinthefuture. Denition4.3AportfolioXisacceptableifthereisanadmissibletradingstrategytandadateTsuchthatXcanbedecomposedbytradingintoacashonlypositionCandapositiveportfoliobydateT.Thatis, (i)n,CT=0forall1nN,wheren,CTdenotesthenumberofshares(correspondingtocashonlypartC)ofassetSnheldattimeT, (ii)therandomvariablee)]TJ /F11 7.97 Tf 6.59 0 Td[(rT0,CTsatisesEPi[e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT0,CT]0foralli2I. WedenotethesetofallacceptableportfoliostobeAp.ItisnotthesameastheacceptancesetAofthecorrespondingcoherentriskmeasure.BecauseApmaynotbepositivelyhomogeneous.ForanacceptableportfolioX,itispossiblethatyoumaynothaveatradingstrategytoliquidatelargerpositionslikeX,where>1.Thisisthedifferencebetweenliquidityriskmeasureandcoherentriskmeasure. Forsimplicity,werstonlyconsideraportfoliowithonerisk-freeassetandoneriskyassetS.WeassumethepriceoftheriskyassetSfollowsanfFgadaptedgeometricBrownianmotiondSt St=dt+dBt.Andtheinterestratefortherisk-freeassetisr.Themarkethasdifferentpricesforbuyersandsellers.Weassumethepriceforthebuyersis 23

PAGE 24

P+t=St+ 2StandthepriceforthesellersisP)]TJ /F11 7.97 Tf -.98 -7.58 Td[(t=St)]TJ /F12 7.97 Tf 13.39 4.71 Td[( 2St.Thehereisthebid-askspread. Foragivenshortperiodoftime,itisusuallyimpossibletotradealargenumberofsharescompletely.Thereforeweassumethenumberofsharesthatcanbetradedismultiplyingthelengthoftimeinterval,i.e.t.Anytradingstrategysatisfyingthisconditioniscalledanadmissibletradingstrategy[ 21 ]. Denition4.4Atradingstrategytisadmissibleifj1t1)]TJ /F9 11.955 Tf 12.67 0 Td[(1t2jjt1)]TJ /F6 11.955 Tf 12.68 0 Td[(t2jforallt1,t20. Bytheabovedenition,anyadmissibletradingstrategyisLipschitzcontinuousint.So1tisaprocessofboundedvariation. ThetotalwealthattimetisdenotedbyXt=0t+1tSt.Wealsoneedtoassumeadmissibletradingstrategiesareself-nancing,i.e.besidestradingnoadditionalcashowwillbegenerated. TheinitialwealthisX0=00+10S0with100.Andforsimplicity,weassumeX0=0.Thatmeans00=)]TJ /F9 11.955 Tf 9.3 0 Td[(10S0.Inordertostudyliquidityeffect,wedistinguishthewealthbeforeandafterliquidatingallriskypositions.TheendingwealthbeforeliquidatingallsharesofriskyassetsisX)]TJ /F11 7.97 Tf -1.59 -8.28 Td[(T=0T+1TST,andtheendingwealthafterliquidatingallsharesofriskyassetsisX+T=0,CT.ThedifferencebetweencoherentriskmeasureandliquidityriskmeasureisthatcoherentriskmeasurecalculatesthechangeX)]TJ /F11 7.97 Tf -1.59 -8.27 Td[(T)]TJ /F6 11.955 Tf 12.67 0 Td[(X0whileliquidityriskmeasurecalculatesthechangeX+T)]TJ /F6 11.955 Tf 12.67 0 Td[(X0.SotheimpliedliquidityeffectisX+T)]TJ /F6 11.955 Tf 12.41 0 Td[(X)]TJ /F11 7.97 Tf -1.59 -8.28 Td[(T.AccordingtoKu[ 21 ],thediscountedwealthprocessbeforeliquidationX)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(isasupermartingalesunderanyriskneutralprobabilitymeasureQ. Inordertoderivealiquidityriskmeasurefromtherestrictionontrading,werewritetheLipschitzconditiontodeneatradinglimitforacertainperiodoftime.Wedenet=ttobethetradinglimitforthetimeperiodt.Thatmeansifthetradingamountislessthan,thenwecanfreelytradethemwithoutaffectingtheprice.However,ifthetradingamountisgreaterthan,thenwemustpayapricepremiumifwebuyor 24

PAGE 25

sellatdiscount.Thatmeansthepriceisaffectedbyaliquidityfactor.Ifj10j,thenX+T=erT00+10ST.Ifj10j>,thenX+T=erT00+ST+(j10j)]TJ /F9 11.955 Tf 17.93 0 Td[()S0T. Denition4.5Foranylongposition1,wedene(X+T)=supi2IEPi[)]TJ /F9 11.955 Tf 9.3 0 Td[(00)]TJ /F6 11.955 Tf -419.26 -23.91 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTminf,10gST)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTmaxf10)]TJ /F9 11.955 Tf 11.96 0 Td[(,0gS0T],withS0TST. Weshallprovethattheriskmeasuredenedaboveisaconvexliquidityriskmeasure.Itismorerealisticbecausesmalltradingrarelyaffectsthepricebutlargetradingusuallyhasbigimpactontheprice.Thefollowingtheoremveriesthatthismeasureisaconvexliquidityriskmeasure. Theorem4.1Theriskmeasuredenedby4.5isaconvexliquidityriskmeasure. Proof: TranslationInvariance.(X+merT)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+m+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTminf,j10jgST+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTmaxfj10j)]TJ /F9 11.955 Tf 19.59 0 Td[(,0gS0T]=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTminf,j10jgST+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTmaxfj10j)]TJ /F9 11.955 Tf -422.74 -23.91 Td[(,0gS0T])]TJ /F6 11.955 Tf 11.95 0 Td[(m=(X))]TJ /F6 11.955 Tf 11.95 0 Td[(m. Liquidity.For1,wehave(X)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTminf,10gST+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTmaxf10)]TJ /F9 11.955 Tf 11.96 0 Td[(,0gS0T]. If10and10,then(X)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT10ST]=()]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT10ST])=(X). If10and10,then(X)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 9.36 0 Td[()S0T]and(X)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT10ST].So(X))]TJ /F9 11.955 Tf 11.49 0 Td[((X)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 10.78 0 Td[()S0T)]TJ /F9 11.955 Tf 10.78 0 Td[(00)]TJ /F6 11.955 Tf 10.78 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT10ST]=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(ST)]TJ /F6 11.955 Tf 10.78 0 Td[(S0T)+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT10(S0T)]TJ /F6 11.955 Tf 10.78 0 Td[(ST)]=e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTsupi2IEPi[(ST)]TJ /F6 11.955 Tf 11.95 0 Td[(S0T)(10)]TJ /F9 11.955 Tf 11.96 0 Td[()]0.Therefore,(X)(X). If10,then10.So(X)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 12.12 0 Td[()S0T]and(X)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 12.53 0 Td[()S0T].So(X))]TJ /F9 11.955 Tf 12.53 0 Td[((X)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 12.03 0 Td[()S0T]+infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf -456.21 -23.9 Td[()S0T]=infi2IEPi[()]TJ /F8 11.955 Tf 10.55 0 Td[(1)e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+(1)]TJ /F9 11.955 Tf 10.55 0 Td[()e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTS0T]=()]TJ /F8 11.955 Tf 10.55 0 Td[(1)e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTinfi2IEPi[ST)]TJ /F6 11.955 Tf 10.55 0 Td[(S0T]0.Hence,(X)(X). 25

PAGE 26

Therefore,foranyandanyX,wehave(X)(X).Andwecanseefromtheabovethattheinequalityisnottrivial.Forexample,ifST>S0T,wehavethestrictinequality(X)>(X)inthelastcase. Convexity.For01,withX(00,10)andY(00,10),wehave(X+(1)]TJ /F9 11.955 Tf 11.34 0 Td[()Y)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+(1)]TJ /F9 11.955 Tf 12.39 0 Td[()00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTminf,10+(1)]TJ /F9 11.955 Tf 12.4 0 Td[()10gST+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTmaxf10+(1)]TJ /F9 11.955 Tf -443.23 -23.9 Td[()10)]TJ /F9 11.955 Tf 11.96 0 Td[(,0gS0T] If10+(1)]TJ /F9 11.955 Tf 9.63 0 Td[()10,then10and(1)]TJ /F9 11.955 Tf 9.64 0 Td[()10.Sowehave(X+(1)]TJ /F9 11.955 Tf 9.64 0 Td[()Y)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+(1)]TJ /F9 11.955 Tf 11.63 0 Td[()00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10+(1)]TJ /F9 11.955 Tf 11.63 0 Td[()10)ST]=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT10ST])]TJ /F8 11.955 Tf -458.7 -23.91 Td[(infi2IEPi[(1)]TJ /F9 11.955 Tf 11.95 0 Td[()00+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(1)]TJ /F9 11.955 Tf 11.96 0 Td[()10ST]=(X)+(1)]TJ /F9 11.955 Tf 11.96 0 Td[()(Y). If10and(1)]TJ /F9 11.955 Tf 12.14 0 Td[()10,then10+(1)]TJ /F9 11.955 Tf 12.14 0 Td[()10.Then(X+(1)]TJ /F9 11.955 Tf 12.13 0 Td[()Y)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+(1)]TJ /F9 11.955 Tf 11.33 0 Td[()00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10+(1)]TJ /F9 11.955 Tf 11.33 0 Td[()10)]TJ /F9 11.955 Tf 11.33 0 Td[()S0T]=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 9.32 0 Td[()S0T])]TJ /F8 11.955 Tf 9.32 0 Td[(infi2IEPi[(1)]TJ /F9 11.955 Tf 9.32 0 Td[()00+(1)]TJ /F9 11.955 Tf 9.32 0 Td[()e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(1)]TJ /F9 11.955 Tf 9.33 0 Td[()(10)]TJ /F9 11.955 Tf 9.32 0 Td[()S0T]=(X)+(1)]TJ /F9 11.955 Tf 11.95 0 Td[()(Y). WOLOG,weassume10(1)]TJ /F9 11.955 Tf 13.05 0 Td[()10,then10+(1)]TJ /F9 11.955 Tf 13.05 0 Td[()10.So(X+(1)]TJ /F9 11.955 Tf 10.48 0 Td[()Y)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+(1)]TJ /F9 11.955 Tf 10.48 0 Td[()00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10+(1)]TJ /F9 11.955 Tf 10.49 0 Td[()10)]TJ /F9 11.955 Tf 10.49 0 Td[()S0T]=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.77 0 Td[()S0T])]TJ /F8 11.955 Tf 11.77 0 Td[(infi2IEPi[(1)]TJ /F9 11.955 Tf 11.77 0 Td[()00+(1)]TJ /F9 11.955 Tf 11.76 0 Td[()e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(1)]TJ /F9 11.955 Tf 11.85 0 Td[()(10)]TJ /F9 11.955 Tf 11.85 0 Td[()S0T]=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.86 0 Td[()S0T]+(1)]TJ /F9 11.955 Tf 11.86 0 Td[()(Y).Therearetwocasesforthissituation.If10,then)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 12.68 0 Td[()S0T]=(X).Soweget(X+(1)]TJ /F9 11.955 Tf 12.69 0 Td[()Y)=(X)+(1)]TJ /F9 11.955 Tf 12.69 0 Td[()(Y).Theothercaseis10.Then)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 12.77 0 Td[()S0T])]TJ /F9 11.955 Tf -429.95 -23.91 Td[((X)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 12.94 0 Td[()S0T)]TJ /F9 11.955 Tf 12.94 0 Td[(00)]TJ /F9 11.955 Tf 12.94 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT10ST]=e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTsupi2IEPi[)]TJ /F9 11.955 Tf 9.3 0 Td[(ST)]TJ /F9 11.955 Tf 12.07 0 Td[(10S0T+S0T+10ST]=e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTsupi2IEPi[(S0T)]TJ /F6 11.955 Tf 12.08 0 Td[(ST)()]TJ /F9 11.955 Tf 12.07 0 Td[(10)]0.Thatmeans(X+(1)]TJ /F9 11.955 Tf 11.96 0 Td[()Y)(X)+(1)]TJ /F9 11.955 Tf 11.96 0 Td[()(Y). Thelastsituationis10and(1)]TJ /F9 11.955 Tf 12.77 0 Td[()10,but10+(1)]TJ /F9 11.955 Tf 12.76 0 Td[()10.So(X+(1)]TJ /F9 11.955 Tf 10.48 0 Td[()Y)=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+(1)]TJ /F9 11.955 Tf 10.48 0 Td[()00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10+(1)]TJ /F9 11.955 Tf 10.49 0 Td[()10)]TJ /F9 11.955 Tf 10.49 0 Td[()S0T]=)]TJ /F8 11.955 Tf 11.29 0 Td[(infi2IEPi[00+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.77 0 Td[()S0T])]TJ /F8 11.955 Tf 11.77 0 Td[(infi2IEPi[(1)]TJ /F9 11.955 Tf 11.77 0 Td[()00+(1)]TJ /F9 11.955 Tf 11.76 0 Td[()e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(1)]TJ /F9 11.955 Tf 13.17 0 Td[()(10)]TJ /F9 11.955 Tf 13.17 0 Td[()S0T].Foreachpart,therearetwocases.Oneis10and 26

PAGE 27

theotheris10.Sothisisthesamesituationasabove.SoWecanconclude(X+(1)]TJ /F9 11.955 Tf 11.96 0 Td[()Y)(X)+(1)]TJ /F9 11.955 Tf 11.95 0 Td[()(Y)andtheinequalityisnottrivial. Insummary,nomatterwhichsituation,wealwayshave(X+(1)]TJ /F9 11.955 Tf 13.01 0 Td[()Y)(X)+(1)]TJ /F9 11.955 Tf 11.95 0 Td[()(Y). Therefore,theriskmeasuredenedin4.5isaconvexliquidityriskmeasure.2 Remark:AliquidityriskmeasuremaynothaveMonotonicity.Becausewithtwopositions,XYwith1010,wemusthaveSTerTS0and10(ST)]TJ /F6 11.955 Tf 12.41 0 Td[(erTS0)10(ST)]TJ /F6 11.955 Tf 12.25 0 Td[(erTS0).So(X))]TJ /F9 11.955 Tf 12.26 0 Td[((Y)=supi2IEPi[(10)]TJ /F9 11.955 Tf 12.26 0 Td[(10)(S0)]TJ /F6 11.955 Tf 12.26 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTS0T)].Ife)]TJ /F11 7.97 Tf 6.59 0 Td[(rTS0TS0,then(X)(Y),whichcontradictstheMonotonicityaxiom. Generaldiversicationdoesnotholdforliquidityriskmeasureeither.Generaldiversicationsays(X+Y)(X)+(Y).However,thereisnocertainrelationbetween(X+Y)and(X)+(Y)forliquidityriskmeasures.Suppose10and10,then(X+Y))]TJ /F9 11.955 Tf 12.22 0 Td[((X))]TJ /F9 11.955 Tf 12.23 0 Td[((Y)=supi2IEPi[(10+10)S0)]TJ /F6 11.955 Tf 12.22 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST)]TJ /F6 11.955 Tf 12.22 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10+10)]TJ /F9 11.955 Tf 12.18 0 Td[()S0T)]TJ /F9 11.955 Tf 12.18 0 Td[(10S0+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 12.18 0 Td[()S0T)]TJ /F9 11.955 Tf 12.18 0 Td[(10S0+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 12.18 0 Td[()S0T]=supi2IEPi[e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTS0T]0.So(X+Y)(X)+(Y). Ontheotherhand,if1010,wehave10+10.So(X+Y))]TJ /F9 11.955 Tf 10.25 0 Td[((X))]TJ /F9 11.955 Tf 10.25 0 Td[((Y)=supi2IEPi[(10+10)S0)]TJ /F6 11.955 Tf 9.94 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST)]TJ /F6 11.955 Tf 9.94 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10+10)]TJ /F9 11.955 Tf 9.94 0 Td[()S0T)]TJ /F9 11.955 Tf 9.94 0 Td[(10S0+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT10ST)]TJ /F9 11.955 Tf 9.94 0 Td[(10S0+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTST+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 12.4 0 Td[()S0T]=supi2IEPi[e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT10(S0T)]TJ /F6 11.955 Tf 12.4 0 Td[(ST)]0.So(X+Y)(X)+(Y).Inaddition,theinequalitiesarenottrivial,sothereisnocertainrelationshipbetween(X+Y)and(X)+(Y). 4.2Bid-AskSpread Allnancialtransactionsareconductedbetweenatleasttwoparties:thebuyerandtheseller.Eventually,everytransactionisanegotiationprocess.Thesellersaskapricethattheyarewillingtosellandthebuyersbidapricethattheyarewillingtobuy.Ifthetwopricesarethesame,thenthetransactioniscompletedandeachpartygetexactlywhattheywant.Butusually,themarketisnotefcientenoughtomatchbuyersandsellerswiththesamebidandaskprice.Thegapbetweenthehighestbidpriceand 27

PAGE 28

thelowestaskpriceiscalledbid-askspread.Itisonewaytomeasuretheliquidityofthemarket.Forperfectlyliquidmarket,thebid-askspreadiszero.Themoreilliquid,thegreaterthebid-askspread.Tobemorespecic,wecansimplymodelliquidityeffectbybid-askspread.ThatmeansS0T=(1)]TJ /F12 7.97 Tf 13.67 4.71 Td[( 2)STandtheliquidityriskmeasurebecomesexplicitlydened. Denition4.6Foranylongposition(0,1),wedenethebid-askspreadriskmeasureasfollowing: (X+T)=supi2IEPi[10S0)]TJ /F8 11.955 Tf 11.95 0 Td[((minf,10g+(1)]TJ /F12 7.97 Tf 13.15 4.71 Td[( 2)maxf10)]TJ /F9 11.955 Tf 11.95 0 Td[(,0g)e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST]. Theorem4.2IfStisageometricBrownianmotionthenthebid-askspreadriskmeasurehasalowerboundthatdoesnotdependsonST. Proof:SinceStisageometricBrownianmotion,E[St]=ertS0.So(X+T)=supi2IEPi[10S0)]TJ /F8 11.955 Tf 11.01 0 Td[((minf,10g+(1)]TJ /F12 7.97 Tf 12.21 4.71 Td[( 2)maxf10)]TJ /F9 11.955 Tf 11.01 0 Td[(,0g)e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTST]=10S0)]TJ /F8 11.955 Tf 11.01 0 Td[((minf,10g+(1)]TJ /F12 7.97 Tf -457.5 -19.2 Td[( 2)maxf10)]TJ /F9 11.955 Tf 12.01 0 Td[(,0g)e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTinfi2IEPi[ST]10S0)]TJ /F8 11.955 Tf 12.01 0 Td[((minf,10g+(1)]TJ /F12 7.97 Tf 13.21 4.71 Td[( 2)maxf10)]TJ /F9 11.955 Tf 12.01 0 Td[(,0g)S0=(10)]TJ /F8 11.955 Tf 11.95 0 Td[(minf,10g)]TJ /F8 11.955 Tf 20.59 0 Td[((1)]TJ /F12 7.97 Tf 13.15 4.71 Td[( 2)maxf10)]TJ /F9 11.955 Tf 11.96 0 Td[(,0g)S02 Weuseanexampletoillustratethatbid-askspreadriskmeasureismoreappropriatethancoherentriskmeasure.Supposeinamarket,weonlyhavetwoparties,sellerAandbuyerB.Andthereareonlytwoassetsinthemarket:risk-freeassetandstockS.Supposeatt=0,thestockistradingatS0=10.ThereisadecreasingdemandforthestockSandthefairpriceofthestockattheendoftheholdingperiodt=Tispredictedtobe$8withprobability0.7and$5withprobability0.3.ButBisonlywillingtobuyatmost5sharesatprice$8andallothersharesatprice$5.Theestimatedbid-askspreadcoefcientinthismarketis.4.Wedonotconsiderinterestrateinthiscase.ThesellerAisnowholding10sharesofthestock.Hiscoherentriskisc=0.720+0.350=29.Butthebid-askspreadriskisl=1010)]TJ /F8 11.955 Tf 10.87 0 Td[((5+(1)]TJ /F8 11.955 Tf 10.87 0 Td[(0.2)5)E[ST]=100)]TJ /F8 11.955 Tf 10.87 0 Td[(97.1=36.1.Nowwecancalculatetherealrisk.ThegaintoAis58+55=65.SothelossofAis$100)]TJ /F8 11.955 Tf 12.29 0 Td[($65=$35.Sothecoherentriskmeasurehighlyunderestimatestheriskandbid-askspreadriskmeasuredoesabetterjobinthisilliquidmarket. 28

PAGE 29

Onekeyelementinthebid-askspreadriskmeasureisthecoefcient.Itisusuallyapercentageofthestockpriceanditchangesovertime.Therearedifferentwaystoestimatethebidaskspreadcoefcient.HereweusetheapproachfromShaneCorwinandPaulSchultz[ 11 ].Thisapproachsimplyusesdailyhighandlowprices.Thereasonisthatdailyhighpricesarebuyerinitiatedanddailylowpricesaresellerinitiated.Theratioofhigh-to-lowreectsboththevolatilityandbid-askspreadofthestock.LetHAt,LAtdenotetheactualhighandlowpricesondaytandHOt,LOtdenotetheobservedhighandlowpricesondayt.Wehavethefollowingrelationship:[ln(HOt LOt)]2=[ln(HAt(1+=2) LAt(1)]TJ /F12 7.97 Tf 6.59 0 Td[(=2))]2.Thisisequivalentto[ln(HOt LOt)]2=[ln(HAt LAt)]2+2[ln(HAt LAt)][ln(2+ 2)]TJ /F12 7.97 Tf 6.59 0 Td[()]+[ln(2+ 2)]TJ /F12 7.97 Tf 6.59 0 Td[()]2.Thebid-askspreadestimatorgiveninthatpaperis=2(e)]TJ /F7 7.97 Tf 6.59 0 Td[(1) 1+e,where=p 2a)]TJ 6.59 5.91 Td[(p a 3)]TJ /F7 7.97 Tf 6.58 0 Td[(2p 2)]TJ /F18 11.955 Tf 12.9 12.92 Td[(q b 3)]TJ /F7 7.97 Tf 6.59 0 Td[(2p 2.a=E[1Pj=0[ln(HOt+j LOt+j)]2]andb=[ln(HOt,t+1 LOt,t+1)]2.Hence,wecancalculatethecoefcientbyusingdailyhighandlowprices. 4.3ExtensiontoMultipleStages Uptonow,wehavestudiedthesimplestcaseofliquidityriskmeasures,thatisonlyonethresholdforpricechanges.Butinreality,thereareusuallymultiplethresholdsforpricechangesandthegreaterthetradingvolumethegreaterthepricechanges.Supposethethresholdsare1,2,...,nandthepricesafterpassingeachthresholdareS(1)T,S(2)T,...,S(n)T.Wehave0=0<1<2<
PAGE 30

Basedonthisrelationship,wecanhavebid-askspreadriskmeasureintheform(X)=supi2IEPi[10S0)]TJ /F6 11.955 Tf 12.01 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTkPj=1(j)]TJ /F9 11.955 Tf 12.01 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)S(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T)]TJ /F6 11.955 Tf 12.01 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 12.01 0 Td[(k)S(k)T]=supi2IEPi[10S0)]TJ /F6 11.955 Tf -456.6 -32.13 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTkPj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)(1)]TJ /F12 7.97 Tf 13.15 4.71 Td[( 2)j)]TJ /F7 7.97 Tf 6.58 0 Td[(1ST)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.95 0 Td[(k)(1)]TJ /F12 7.97 Tf 13.15 4.71 Td[( 2)kST]. Proposition4.2Forlongpositions,then-stagebid-askspreadriskmeasurehasalowerboundthatdoesnotdependonSTifSTfollowsgeometricBrownianmotion. Proof:Wehave(X)=supi2IEPi[10S0)]TJ /F6 11.955 Tf 11.73 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTkPj=1(j)]TJ /F9 11.955 Tf 11.74 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)(1)]TJ /F12 7.97 Tf 12.93 4.71 Td[( 2)j)]TJ /F7 7.97 Tf 6.58 0 Td[(1ST)]TJ /F6 11.955 Tf 11.74 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf -458.7 -32.12 Td[(k)(1)]TJ /F12 7.97 Tf 12.93 4.7 Td[( 2)kST]=10S0)]TJ /F6 11.955 Tf 11.73 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT[kPj=1(j)]TJ /F9 11.955 Tf 11.73 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)(1)]TJ /F12 7.97 Tf 12.93 4.7 Td[( 2)j)]TJ /F7 7.97 Tf 6.59 0 Td[(1+(10)]TJ /F9 11.955 Tf 11.73 0 Td[(k)(1)]TJ /F12 7.97 Tf 12.93 4.7 Td[( 2)k]supi2IEPi[ST]10S0)]TJ /F8 11.955 Tf 12.11 0 Td[([kPj=1(j)]TJ /F9 11.955 Tf 12.11 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)(1)]TJ /F12 7.97 Tf 13.3 4.71 Td[( 2)j)]TJ /F7 7.97 Tf 6.59 0 Td[(1+(10)]TJ /F9 11.955 Tf 12.1 0 Td[(k)(1)]TJ /F12 7.97 Tf 13.3 4.71 Td[( 2)k]S0=[10)]TJ /F11 7.97 Tf 16.66 11.36 Td[(kPj=1(j)]TJ /F9 11.955 Tf 12.1 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)(1)]TJ /F12 7.97 Tf 13.3 4.71 Td[( 2)j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F8 11.955 Tf -453.45 -23.91 Td[((10)]TJ /F9 11.955 Tf 11.95 0 Td[(k)(1)]TJ /F12 7.97 Tf 13.15 4.71 Td[( 2)k]S0.2 Mostofthecase,itisconvenienttoassumeinnitestagesforthepricesoftheasset.Nowweassumeaninnitesetofthresholds=f0,1,...gwith0<1<.Inaddition,wewantthethresholdsettoreectsomepropertyoftheliquidityconditioninthemarket.Wedenetheclosestthresholdfrombelowtothetradingvolumetobethegreatestlowerbound(GLB)ofin.OnepropertyofanilliquidmarketisthattheGLBthresholdgoesupmorethanproportionaltothegrowthoftradingsize.Forexample,iftheGLBthresholdforis100,thentheGLBthresholdfor2shouldbegreaterthan200.Thisisthediminishingincreasepropertyofilliquidmarket.WealsowanttheorderoftheGLBthresholdtobebounded.Continuefromtheaboveexample,ifk=100andm=200,thenwerequirem2k.Thatisbecausewewantthethresholdsettoberelativelyevenlydistributed.Wesummarizeourkeyassumptionsaboutthethresholdsetinthefollowing: ThresholdAssumption:Thethresholdset=f0,1,...gissaidtoevenlyreectthemarketliquidityconditioniffortradingvolumeand1,wehavethefollowing 30

PAGE 31

results: (1)TheGLBthresholdforiskandtheGLBthresholdforism,(2)mk,and(3)mk. Remark:Withoutspecicannouncement,wewillassumetheThresholdAssumptionthroughoutthispaper. Theorem4.3TheriskmeasureinDenition4.7isaliquidityriskmeasure. Proof:TranslationInvariance. (X+merT)=supi2IEPi[)]TJ /F6 11.955 Tf 9.3 0 Td[(m+10S0)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTkXj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)S(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.95 0 Td[(k)S(k)T]=supi2IEPi[10S0)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTkXj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)S(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.96 0 Td[(k)S(k)T])]TJ /F6 11.955 Tf 11.95 0 Td[(m=(X))]TJ /F6 11.955 Tf 11.95 0 Td[(m Liquidity.Suppose1andtheGLBthresholdforXism.Wehave (X)=supi2IEPi[10S0)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTmXj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)S(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.95 0 Td[(m)S(m)T]supi2IEPi[10S0)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTmXj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)S(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.95 0 Td[(k)S(k)T]supi2IEPi[10S0)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTkXj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)S(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.95 0 Td[(k)S(k)T]=supi2IEPi[10S0)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTkXj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)S(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.96 0 Td[(k)S(k)T]=(X).2 4.4ExtensiontoTwo-AssetPortfolios Therearethousandsofinvestmentassetsintherealworldandtypicalportfoliosusuallycontainmorethanjustoneriskyasset.Webeginthediscussionwithtwoassetscase.Theriskmeasureoftwoassetsisnotjustthesummationoftheriskmeasuresofthetwoassetsseparately.Inotherwords,thereisnolinearityindifferentassets. 31

PAGE 32

Therearetwofactorsweneedtoconsider.Oneisthecorrelationbetweenthepricesofdifferentassets.Forexample,ifthepricesoftwodifferentassetsUandVarepositivelycorrelatedandtheportfoliocontainslongpositionsinbothUandV.ThensellinglargeportionofUwilldrawdownthepricepuofU.SincethepriceofVpvispositivelycorrelatedwithpu,pvwillfallatthesametime.SothelossoftheportfolioismorethanthelossofjustUposition.Therefore,itmakessensethecorrespondingriskmeasureshouldalsobegreater.Theotherfactoristheimpactontheliquidityoftheotherassets.Tradingoneassetcoulddirectlyaffectthesupplyanddemandoftheotherassetsandhenceinuencetheliquidityspreadandnalpriceoftheotherassets.Thesetwofactorsaredifferent.Correlationisanexogenousfactorwhichisdeterminedbythemarketconditionswhileliquidityimpactiscausedbytransactionsandtradingvolumes.Soitisbettertomodelthemseparatelyinriskmeasures. Nowsupposeamarketcontainsonerisk-freeassetandtwodifferentriskyassetsUandV.Aportfolioconsistsofthethreeassetsanditsnetvalueiszeroatt=0.Thesharesoftherisk-freeasset,assetUandassetVare0,1,2respectively.Soweshouldhavetherelationship00=10U0+20V0.ThethresholdsforUandVare1,...,mand1,...,n.Inadditionassumek10
PAGE 33

j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 10.98 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(20)]TJ /F9 11.955 Tf 10.98 0 Td[(l)V(l)T+CorrPi(U,V)(10+20)+U,V10+V,U20].Thenotationsandassumptionsarethesameasabove. Theorem4.4TheriskmeasureinDenition4.8isaliquidityriskmeasure. Proof:Translationinvariance:(UT,VT,merT)=supi2IEPi[)]TJ /F9 11.955 Tf 9.3 0 Td[(00)]TJ /F6 11.955 Tf 12.07 0 Td[(m)]TJ /F6 11.955 Tf 12.07 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTkPj=1(j)]TJ /F9 11.955 Tf -454.67 -32.12 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 12.97 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 12.97 0 Td[(k)U(k)T)]TJ /F6 11.955 Tf 12.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTlPj=1(j)]TJ /F9 11.955 Tf 12.97 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 12.96 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(20)]TJ /F9 11.955 Tf 12.97 0 Td[(l)T(l)T+CovPi(U,V)(10+20)+U,V10+V,U20]=supi2IEPi[10U0+20V0)]TJ /F6 11.955 Tf 12.65 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTkPj=1(j)]TJ /F9 11.955 Tf -434.36 -32.12 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 12.97 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 12.97 0 Td[(k)U(k)T)]TJ /F6 11.955 Tf 12.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTlPj=1(j)]TJ /F9 11.955 Tf 12.97 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 12.96 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(20)]TJ /F9 11.955 Tf 12.97 0 Td[(l)T(l)T+CovPi(U,V)(10+20)+U,V10+V,U20])]TJ /F6 11.955 Tf 11.95 0 Td[(m=(U,V))]TJ /F6 11.955 Tf 11.95 0 Td[(m. Liquidity:Forsimplicity,denotethefunctioninsidetheexpectationofthesingleassetriskmeasuretobef(X).Then(X)=supi2IEPi[f(X)].For1,(U,V)=supi2IEPi[f(U)+f(V)+CovPi(U,V)(10+20)+U,V10+V,U20]supi2IEPi[f(U)+f(V)+CovPi(U,V)(10+20)+U,V10+V,U20]=supi2IEPi[10U0+20V0)]TJ /F6 11.955 Tf 12.33 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTkPj=1(j)]TJ /F9 11.955 Tf 12.34 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 12.33 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 12.33 0 Td[(k)U(k)T)]TJ /F6 11.955 Tf 12.33 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTlPj=1(j)]TJ /F9 11.955 Tf -445.3 -25.9 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 11.98 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(20)]TJ /F9 11.955 Tf 11.98 0 Td[(l)T(l)T+CorrPi(U,V)(10+20)+U,V10+V,U20]=(U,V).2 Remark:Theliquidityimpactiscausedbyinequilibriumofthemarket,i.e.instantdemandorsupplyisnotmatched.Thedemandorsupplysurpluscausesthepricetoincreaseordecreaserespectively.Thisisalsothesourceforliquiditycost.Theliquidityimpactbetweentwodifferentriskyassetsisformedinasimilarsituation.Thedemandsurplusofoneassetmaycauseademandorsupplysurplusoftheotherasset.Theinteractionbetweenthetwoassetsisanothersourceofliquiditycost.Ifthedemandandsupplycurveofoneassetisnotaffectedbytheotherasset,wemusthaveU,V=V,U=0.Moregenerally,wecandenethesituationwithzeronetimpactbetweentwoassets. Denition4.9Thetradingstrategy(10,20)oftwoassetsUandViscalledabalancedtradingstrategyifU,V10+V,U20=0. 33

PAGE 34

Thebalancedtradingstrategytakestheimpactbetweentwoassetsoutandleavesonlythecorrelationeffectbetweenthetwoassetsintheriskmeasure.Also,weshouldnoticethatbalancedtradingstrategymaynotalwaysexist.Forthecasessuchstrategydoesexist,weintroduceanewriskmeasureintheChapter5,theconditionaldiversicationriskmeasure.Thebalancedtradingstrategysimplysayszeronettradingeffectonbothassets.Itisawaytopairthetradingvolumesofthetwoassetstoeliminateanycrosseffectcausedbyeachother.Hereweassumeweholdlongpositionsofbothassets.So10>0and20>0.Inordertoreachbalancedtradingstrategy,wemustneedU,VV,U<0,orwecanjustassumeU,V>0andV,U<0.ThatmeanssellingUwillcausethepriceofVtodropandsellingVwillcausethepriceofUtorise. 4.5ExtensiontoMultipleTradingPeriods Thenalextensionweneedisconsideringmultipletimeperiods.Alltheabovediscussionsareconductedinonesingletimeperiod.However,weoftenneedtodividebigtransactionintosmallermultipletransactionsinconsiderationoftradinglimitorliquiditycost.Weconstructourmodelunderthetworiskyassetsassumption.Forsimplicity,weconsidertwoperiodsatrst.Sothetimelineisfromt=0tot=T1forthersttransactionandtot=T2forthesecondandnaltransaction.Sinceweneedtoliquidateallriskyassetsintheend,wemusthave12=22=0.Sothetradingvolumeintherstperiodis10)]TJ /F9 11.955 Tf 12.8 0 Td[(11and20)]TJ /F9 11.955 Tf 12.8 0 Td[(21respectively.Andthetradingvolumeinthesecondandnalperiodis11and21respectively.Topreciselycalculatetheriskofeachposition,weneedtondthecorrespondingthresholdforeachtransaction.Weassumethatk110)]TJ /F9 11.955 Tf 12.06 0 Td[(11
PAGE 35

e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT1(10)]TJ /F9 11.955 Tf 11.98 0 Td[(11)]TJ /F9 11.955 Tf 11.99 0 Td[(k1)U(k1)T1)]TJ /F6 11.955 Tf 11.98 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT2k2Pj=1(j)]TJ /F9 11.955 Tf 11.99 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T2)]TJ /F6 11.955 Tf 11.98 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT2(11)]TJ /F9 11.955 Tf 11.99 0 Td[(k2)U(k2)T2)]TJ /F6 11.955 Tf 11.98 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT1l1Pj=1(j)]TJ /F9 11.955 Tf -457.6 -32.24 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T1)]TJ /F6 11.955 Tf 11.98 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1(20)]TJ /F9 11.955 Tf 11.98 0 Td[(21)]TJ /F9 11.955 Tf 11.98 0 Td[(l1)V(l1)T1)]TJ /F6 11.955 Tf 11.98 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT2l2Pj=1(j)]TJ /F9 11.955 Tf 11.98 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T2)]TJ /F6 11.955 Tf 11.99 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT2(21)]TJ /F9 11.955 Tf 11.98 0 Td[(l2)V(l2)T2+CorrPi(U,V)(10+20)+U,V10+V,U20]. Thisisareallylongdenition.Butifwearefamiliarwithpreviousdenitions,wecanclearlycatchthestructurehere.Wearejustaddingthesecondtimeperiodtermstothetworiskyassetsdenition.Thereisnochangeinthecorrelationtermandliquidityimpactterm.Thatisbecausethesetwotermsassumelinearrelationintradingvolumes.Thatis,thesumofthetwotradingvolumeswillbethesameastheinitialholdingvolumes.Sothereisnoneedtochangetheseterms.However,thesetwoassetsmayhavedifferentcorrelationindifferenttimeperiodsandthecoefcientcouldbedifferentindifferenttimeperiods.SothecorrelationtermwillchangetoT2Pt=1tCorrPi(Ut,Vt)(1t+2t).Thesamecouldoccurtotheliquidityimpactterm.Thatistheimpactcoefcientsaredifferentfromtimetotime.SotheliquidityimpacttermwillchangetoT2Pt=1(Ut,Vt1t+Vt,Ut2t).Wecalldifferenttimeperiodsconsistentifthecorrelationsandallcoefcientsarethesameineverytimeperiod.Sopreciselyspeaking,Denition4.10isunderconsistenttimeperiodsassumption. Nowwecanfurtherextendtheaboveriskmeasuretomulti-periodsituations.SupposethetotalnumberoftimeperiodsisN.Sothetimelineisfromt=0tot=T1,,t=TNforthenaltransaction.Afterliquidatingallassets,wehave1T=2T=0.SothetradingvolumeforeachtransactionisiTj)]TJ /F16 5.978 Tf 5.76 0 Td[(1)]TJ /F9 11.955 Tf 12.56 0 Td[(iTjfordifferentassets.Wealsoassumethatkj1Tj)]TJ /F16 5.978 Tf 5.75 0 Td[(1)]TJ /F9 11.955 Tf 11.51 0 Td[(1Tj
PAGE 36

Remark:WeaddalltheNtimeperiodsinsidethesupreme.Thatisbecauseweassumethesamemarketconditionthroughoutthewholetradingperiods.Morespecically,weassumethewholetransactionwillbecarriedoutinthesamescenariounderprobabilityexpectation.Thatiswhywecancalculatealltheriskandtakethesupremeofallscenarios. 36

PAGE 37

CHAPTER5OTHERVARIATIONSOFLIQUIDITYRISKMEASURES 5.1ConditionalDiversication Whenwetalkaboutriskdiversication,wealwaysassumethatthemorethebetter.ThereasoningbehindthisisthatifwehavetwopositionsXandY,thenwemaybebetteroffiftheymovetooppositedirections.Assumeisariskmeasure.Thenwewouldarguethat(X+Y)(X)+(Y),whichmeansriskdiversication.IfthepricesofXandYgotothesamedirection,thenwehavenodifferenceholdingthemtogetherorseparately.Actually,theequalitywillholdinthiscase.IfthepricesofXandYgotooppositedirection,wearebetteroffholdingthemtogether.Inthiscase,wehave(X+Y)<(X)+(Y).ThebasicassumptionofthisreasoningisthatchangingthepositionofXwillnotaffectthepriceofY,orviceversa.However,itisnotthecasewhenthemarketisilliquid[ 6 ]. IfthemarketisilliquidandXispositivelycorrelatedwithY,thenwemayhave(X+Y)>(X)+(Y).IfwewanttosellXandYseparatelyinanilliquidmarket,thepriceofXwilldropby$1andthepriceofYwilldropby$2.Sowehave(X)=1and(Y)=2.However,ifwesellXandYtogether,thepriceofXwilldropby$1andthepriceofYwilldropby$2.Butatthesametime,thedownwardpressureofXwilldrawthepriceofYfurtherandviceversa.SotheactualpricedropofXcouldbe$1.1andtheactualpricedropofYcouldbe$2.2.Sosellingthemtogetherwillendupwithrisk(X+Y)=3.3whichisbiggerthan(X)+(Y). IfthemarketisilliquidandXisnegativelycorrelatedwithY,thenwecouldhavediversication(X+Y)<(X)+(Y).IfwewanttosellXandYseparatelyintheilliquidmarket,thepriceofXwilldropby$1andthepriceofYwilldropby$2.Sowehave(X)=1and(Y)=2.However,ifwesellXandYtogether,thepriceofXwilldropby$1andthepriceofYwilldropby$2.Butsincetheyarenegativelycorrelatedwitheachother,thedownwardpressureofXwillpushupthepriceofYalittlebitand 37

PAGE 38

viceversa.SotheactualdropofXcouldbe$0.9andtheactualdropofYcouldbe$1.8.Thatmeans(X+Y)<(X)+(Y). Nowweseetwodifferentresultsmayhappenwithriskdiversicationinanilliquidmarket.Soitisnecessarytodeneariskmeasuretospecifythissituation.Wedenethisclassofriskmeasuresasconditionaldiversicationriskmeasures.Theformaldenitionisasfollows: Denition5.1Ariskmeasureiscalledaconditionaldiversicationriskmeasureifitsatisesthefollowingaxioms: AxiomMMonotonicity.IfXY,then(X)Y. AxiomTTranslationInvariance.Ifm2R,then(X+mr)=(X))]TJ /F6 11.955 Tf 11.96 0 Td[(m. AxiomLLiquidity.If>1,(X)(X);if2[0,1],(X)(X). AxiomCConditionalDiversication.Ifcorr(X,Y)<0,(X+Y)(X)+(Y). Thedifferencebetweenconditionaldiversicationriskmeasureandconvexliquidityriskmeasureisthatconditionaldiversicationriskmeasureisnotconvex.Intheory,wepreferaconvexriskmeasurebecausetheconvexitycanyieldnicerepresentationtheorem.However,theevidencesfromnancialmarketsdonotalwayssupportconvexity.Forexample,theyieldcurveforMortgageBackedSecuritiesandcallablebondsusuallyhavenegativeconvexity.Thatmeanstheyareconcave.Togetabetterunderstandingofwhatisliquidityriskmeasureandconditionaldiversicationriskmeasure,weexaminesomespecicexampleslater. 5.2PowerLiquidityRiskMeasures Inaddition,theTranslationInvarianceaxiomdoesnottakeinterestrateriskintoconsideration.TheTranslationInvariancesayswhenm rdollarsaresubtractedfromthecurrentposition,thecapitalrequirementissubtractedbytheamountmdollars.Itassumesaconstantinterestrater.However,intherealworld,asmentionedbyKarouiandRavanelli[ 20 ],anyformofuncertaintyininterestrateswillviolatetheTranslationInvariance.Inanextremecase,whenthemoneymarketisilliquid,theinterestrates 38

PAGE 39

mayalsodependonm.Inordertobereal-lifeapplicable,wedenesomespecicriskmeasureswithoutTranslationInvariance.Becauseweonlycareaboutthecaseswhenwelosemoney,wedenetheriskmeasuresonlyforthecaseX0.WecanmakethemeasureconstantforanypositiveX. Denition5.2AliquidityfactorisafunctionL:R+!R+,suchthatL()when1. Liquidityfactorcanmorepreciselydescribehowliquidityissuemayaffectouroverallrisk.Sincedifferentmarketshavedifferentliquiditysituations,itmakessensetousedifferentliquidityfactorsfordifferentmarkets.Also,thesamemarketmayhavedifferentliquidityduringdifferentperiodsoftime.Soweneedtospecifyacertainliquidityfactortoaspecicperiodoftime.Insummary,liquidityfactormaychangeovertimeandmarkets.Thenextquestionishowcanwechooseappropriateliquidityfactors.Wewilldenesomesimpleliquidityfactorsandapproximategeneralsituationsbythesimplefactors. Denition5.3PowerfactorisaliquidityfactorsuchthatL(X)=kL(X),fork2Z+. Denition5.4AliquidityriskmeasurewithoutTranslationInvarianceiscalledapowerliquidityriskmeasureifitsatisesthefollowingPoweraxiom: Poweraxiom:(X)=k(X)forsomek2Z+. Powerliquidityriskmeasureisusefulbecauseitiseasytocalculateandgoodtoapproximateotherriskmeasures.Thefollowingtheoremdenesariskmeasurebasedonthepowerliquidityriskmeasures. Theorem5.1Theriskmeasurep(X)=E[Pni=1ai()]TJ /F6 11.955 Tf 9.3 0 Td[(X)bi]forX0isaconvexliquidityriskmeasurewithoutTranslationInvarianceifai>0andbiarepositiveintegers. Proof:ForanyX<0,p(X)=E[Pni=1ai()]TJ /F6 11.955 Tf 9.3 0 Td[(X)bi]>E[Pni=1ai(0)bi]=0. Sinceallai()]TJ /F6 11.955 Tf 9.3 0 Td[(X)biareconvexfunctionsgivenai>0,theirsummationisalsoaconvexfunction.SoitsatisestheConvexityaxiom. 39

PAGE 40

Forany>1,p(X)=E[Pni=1ai()]TJ /F9 11.955 Tf 9.3 0 Td[(X)bi]=E[Pni=1aibi()]TJ /F6 11.955 Tf 9.3 0 Td[(X)bi]=E[Pni=1aibi)]TJ /F7 7.97 Tf 6.59 0 Td[(1()]TJ /F6 11.955 Tf 9.3 0 Td[(X)bi]E[Pni=1ai()]TJ /F6 11.955 Tf 9.3 0 Td[(X)bi]=p(X)sincebi)]TJ /F8 11.955 Tf 11.95 0 Td[(10. ItdoesnotsatisfyTranslationInvariancebecause(X)]TJ /F6 11.955 Tf 11.96 0 Td[(m)26=X2)]TJ /F6 11.955 Tf 11.95 0 Td[(m.2 5.3SomeExamples Inordertoillustratetheliquidityriskmeasureismoreappropriatethancoherentriskmeasure,weneedtondaliquidityriskmeasurethatisnotcoherent.Itisnotatrivialexampleandweneedsomepreparations. WeconsidertheliquiditycostproposedbyCetinetal[ 7 ]asacandidate.Forsimplicity,weonlyhaveoneperiod,[0,T].Weconsiderastockmarkethere,althoughthesubsequentmodelappliesequallywelltobonds,commodities,foreigncurrencies,etc.Andweassumethatthespotrateofinterestiszero.LetthesupplycurveSt(x)representthestockpricepershareattimetwithxsharestraded.Apositiveorder(x>0)representsabuyandanegativeorder(x<0)representsasale.Normallyweassumethesamepriceforanyordersize.ThatisahorizontalsupplycurveSt(0).Butinanilliquidmarket,thesupplycurveshouldbeSt(x),anincreasecurve.Sincethemorewebuy,thehigherthepriceofthestockwillbe.Wewilladopttheassumptionin[ 7 ],thatthesupplycurvehastheformSt(x)=exSt(0)with>0.Andthroughoutthispaper,wewillassume=S0(0). Nowweassumeatthebeginning,t=0,wehavexshares,sotheinitialwealthisxS0(0).Ifthemarketisliquid,attimet=T,thenetworthX=x(ST(0))]TJ /F6 11.955 Tf 12.11 0 Td[(S0(0)).Butinanilliquidmarket,thenalnetworthisXl=x(ST()]TJ /F6 11.955 Tf 9.3 0 Td[(x))]TJ /F6 11.955 Tf 11.96 0 Td[(S0(0)).Nowwedene l(X)=supP2PfEP[)]TJ /F6 11.955 Tf 9.3 0 Td[(Xl]g=supP2PfEP[xS0(0))]TJ /F6 11.955 Tf 11.95 0 Td[(xST()]TJ /F6 11.955 Tf 9.3 0 Td[(x)]g. (5) Theorem5.2Theriskmeasureldenedin(5-1)isaconvexliquidityriskmeasure. Proof:WeneedtochecklsatisesAxiomsS,T,CandL: Ifwehaveanotherposition,sayY,itmeanswehavethesameamountofmoneyatthebeginning,butinvestedinanotherstock. 40

PAGE 41

SoyU0(0)=xS0(0),whereUisthesupplycurveofanotherstockandxisthesharesheld.SoYl=y(UT()]TJ /F6 11.955 Tf 9.29 0 Td[(y))]TJ /F6 11.955 Tf 11.95 0 Td[(U0(0)). AxiomS:Thisoneistrivial.IfX<0,thenl(X)1,asST(x)isstrictlyincreasinginx,ST()]TJ /F6 11.955 Tf 9.3 0 Td[(kx)ksupP2PfEP[)]TJ /F6 11.955 Tf 9.3 0 Td[(x(ST(x))]TJ /F6 11.955 Tf 11.95 0 Td[(S0(0))]g=kl(X) Thecase0k1issimilar.2 Remark:AswehavestrictinequalityinAxiomL,thePositiveHomogeneitydoesnothold.Soitisnotacoherentriskmeasure.Andthesituationconstructedaboveismorerealistic,sincepricesdogetaffectedbylargeorders.Hence,liquidityriskmeasureismoreappropriatedinthiscase. Anexampleofconditionaldiversicationriskmeasureisthefollowinglog-exponentialriskmeasure,given>0: (X)=log(E[e)]TJ /F15 5.978 Tf 8.24 3.26 Td[(x ]) (5) 41

PAGE 42

Wewillprovethisriskmeasuresatisestheaxiomsofconditionaldiversicationriskmeasure. Theorem5.3Theriskmeasuredenedin(5-2)isaconditionaldiversicationriskmeasurebutitisnotcoherent. Proof:Clearly,thefunctionisdecreasing,sotheMonotonicityaxiomissatised. Forsimplicity,weconsiderherer=1.Wecalculate(X+m)=log(E[e)]TJ /F16 5.978 Tf 5.76 0 Td[((x+m) ])=log(E[e)]TJ /F15 5.978 Tf 5.76 0 Td[(x e)]TJ /F15 5.978 Tf 5.75 0 Td[(m ])=log(e)]TJ /F15 5.978 Tf 5.75 0 Td[(m E[e)]TJ /F15 5.978 Tf 5.76 0 Td[(x ])=()]TJ /F11 7.97 Tf 6.59 .01 Td[(m +log(E[e)]TJ /F15 5.978 Tf 5.75 0 Td[(x ]))=(X))]TJ /F6 11.955 Tf 12.02 0 Td[(m.SoTranslationInvarianceissatised. TotestLiquidity,welet>1,so(X)=log(E[e)]TJ /F17 5.978 Tf 5.76 0 Td[(x ])=log(E[(e)]TJ /F15 5.978 Tf 5.76 0 Td[(x )])log(E[e)]TJ /F15 5.978 Tf 5.76 0 Td[(x ])=log(E[e)]TJ /F15 5.978 Tf 5.76 0 Td[(x ])=(X).WecangetthisrelationshipbyapplyingJensen'sinequality.If>1,thenXisconvex. ForConditionalDiversication,weassumecorr(X,Y)0.SincewehaveE[XY]=E[X]E[Y]+cov(X,Y),E[XY]E[X]E[Y].So(X+Y)=log(E[e)]TJ /F15 5.978 Tf 7.78 3.53 Td[(x+y ])=log(E[e)]TJ /F15 5.978 Tf 8.24 3.25 Td[(x e)]TJ /F15 5.978 Tf 8.18 3.52 Td[(y ])log(E[e)]TJ /F15 5.978 Tf 8.24 3.25 Td[(x ]E[e)]TJ /F15 5.978 Tf 8.18 3.52 Td[(y ])=log(E[e)]TJ /F15 5.978 Tf 8.23 3.25 Td[(x ])+log(E[e)]TJ /F15 5.978 Tf 8.18 3.52 Td[(y ])=(X)+(Y).2 Remark:Ifwetaketheweightedaverageofaseriesofconditionaldiversicationriskmeasures,theresultisalsoaconditionaldiversicationriskmeasure. Foranexampleofpowerliquidityriskmeasure,wewantaforeignstockintheUSmarket,whichisnotveryliquid.ThestockwepickedhereisChinaMassMediaCorp.(CMM).Becauseitisverydifculttondhistoricaltradingdataandvolumeofacertainparty,weconsiderthemarkethasonlytwoparticipants,thebuyerandseller.Belowisagraphofhowtheadjustedpricechangesaccordingtothetradingvolumes.Theadjustedpriceistheclosepriceadjustedtodividends. Forthisstock,asshowninFigure5-1,itsriskmeasurecannotbelinear,andpowerliquidityriskmeasurecandoabetterjob.Eitherinsmalltradingvolumerange(below20,000shares)orlargetradingvolumerange(above80,000shares),thepriceisdrivenupbybuyingpowers.Inthemiddlerange(between20,000and80,000 42

PAGE 43

Figure5-1. Non-LinearRiskMeasure shares),thepriceisdrivenbysellingpowers.Andineachrange,itisnotasimplelinearrelationship.Anotheradvantageofpowerliquidityriskmeasureisthatitcanbeappliedtoapproximateotherriskmeasuresincomputationwithoutexplicitexpressions. 43

PAGE 44

CHAPTER6OPTIMALBALANCEDTRADINGSTRATEGY Riskoptimizationproblemshavebeenwidelystudiedbynumerousresearchers[ 12 29 ].Tradingconstraintsarehighlyimportantinoptimizingriskmeasures[ 15 ].Liquidityriskcanalsobeaffectedbytradingstrategies.Atradingstrategyhasthreemajorelements:thetradingvolume,thelengthoftradingandthedirectionoftrading.Thereisdenitelyalimitoftheamountyoucantradeinashortperiodoftime.Tradingtoomuchinashorttimecouldcauseeitherfailureorhugeliquiditycost.Thetotaldurationoftradingalsomatters.Wecanmodelthedurationbythenumberofstepsofthetrading.Forexample,thereisonly5buyorsellcanbemadeintheperiodoftime.Thetradingdirectionsimplymeanseitherbuyingorsellinginatransaction.Wepreferabalancedtradingstrategyifitexists.Thatisbecauseiteliminatestheinteractionbetweenthetradingassetsduringthetransaction.Inthissection,wearegoingtondthebalancedtradingstrategythatwillminimizetheoverallriskofthetransactionandwecallsuchstrategytheoptimalbalancedtradingstrategy,orOBTS.Wewillonlyconsidertwoassetscasehere.WewilladoptDenition4.11andassumeconsistentN-periodtransactions.SotheriskmeasureisN(U,V)=supi2IEPi[10U0+20V0)]TJ /F11 7.97 Tf 14.86 11.36 Td[(NPh=1(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rThkhPj=1(j)]TJ /F9 11.955 Tf -458.7 -32.47 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)Th)]TJ /F6 11.955 Tf 12.13 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTh(1h)]TJ /F7 7.97 Tf 6.58 0 Td[(1)]TJ /F9 11.955 Tf 12.13 0 Td[(1h)]TJ /F9 11.955 Tf 12.13 0 Td[(kh)U(kh)Th)]TJ /F6 11.955 Tf 12.13 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rThlhPj=1(j)]TJ /F9 11.955 Tf 12.14 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)Th)]TJ /F6 11.955 Tf 12.13 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTh(2h)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 12.13 0 Td[(2h)]TJ /F9 11.955 Tf -452.51 -25.9 Td[(lh)V(lh)Th)+CorrPi(U,V)(10+20)+U,V10+V,U20]. Undertheseassumptions,onlyacertainkindoftransactionswillhavetheOBTS.Bydenition,nomatterhowmanystepsoftrading,thereisonlyonerequirementforexistenceofbalancedtradingstrategyforeachstep.Thatisthetotalamountoftradingofasset110=)]TJ /F12 7.97 Tf 10.49 5.78 Td[(V,U U,V20,where20isthetotalamountoftradingofasset2.Wewilldiscussthesimplestversionoftheproblemrst,i.e.N=2.Sotheriskmeasurebecomes2(U,V)=supi2IEPi[10U0+20V0)]TJ /F6 11.955 Tf 12 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1k1Pj=1(j)]TJ /F9 11.955 Tf 12 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T1)]TJ /F6 11.955 Tf 12 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1(10)]TJ /F9 11.955 Tf 12 0 Td[(11)]TJ /F9 11.955 Tf -457.13 -32.23 Td[(k1)U(k1)T1)]TJ /F6 11.955 Tf 12.24 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT2k2Pj=1(j)]TJ /F9 11.955 Tf 12.25 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T2)]TJ /F6 11.955 Tf 12.25 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT2(11)]TJ /F9 11.955 Tf 12.25 0 Td[(k2)U(k2)T2)]TJ /F6 11.955 Tf 12.24 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT1l1Pj=1(j)]TJ /F9 11.955 Tf 12.24 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T1)]TJ ET BT /F1 11.955 Tf 227.35 -687.85 Td[(44

PAGE 45

e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT1(20)]TJ /F9 11.955 Tf 9.86 0 Td[(21)]TJ /F9 11.955 Tf 9.86 0 Td[(l1)V(l1)T1)]TJ /F6 11.955 Tf 9.86 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT2l2Pj=1(j)]TJ /F9 11.955 Tf 9.87 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T2)]TJ /F6 11.955 Tf 9.86 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT2(21)]TJ /F9 11.955 Tf 9.86 0 Td[(l2)V(l2)T2+CorrPi(U,V)(10+20)+U,V10+V,U20].Andtheoptimizationproblemturnstobeminimize11,21supi2IfEPi[)]TJ /F6 11.955 Tf 9.3 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1k1Xj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1(10)]TJ /F9 11.955 Tf 11.96 0 Td[(11)]TJ /F9 11.955 Tf 11.95 0 Td[(k1)U(k1)T1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT2k2Xj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T2)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT2(11)]TJ /F9 11.955 Tf 11.95 0 Td[(k2)U(k2)T2)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1l1Xj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1(20)]TJ /F9 11.955 Tf 11.96 0 Td[(21)]TJ /F9 11.955 Tf 11.96 0 Td[(l1)V(l1)T1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT2l2Xj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T2)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT2(21)]TJ /F9 11.955 Tf 11.95 0 Td[(l2)V(l2)T2]+CorrPi(U,V)(10+20)gsubjecttoU,V11+V,U21=0,01110,02120. 6.1SeparateTradingStrategy Tosolvethisoptimizationproblem,wecanrstsolvefor21usingtherstconstraintandsubstituteitintotheobjectivefunctiontoeliminatethevariable21.Fromtheconstraint,21=)]TJ /F12 7.97 Tf 10.5 5.77 Td[(U,V V,U11.Sotheoptimizationproblembecomesminimize11supi2IfEPi[)]TJ /F6 11.955 Tf 9.3 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1k1Xj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1(10)]TJ /F9 11.955 Tf 11.96 0 Td[(11)]TJ /F9 11.955 Tf 11.95 0 Td[(k1)U(k1)T1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT2k2Xj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T2)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT2(11)]TJ /F9 11.955 Tf 11.95 0 Td[(k2)U(k2)T2)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1l1Xj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1(20+U,V V,U11)]TJ /F9 11.955 Tf 11.95 0 Td[(l1)V(l1)T1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT2l2Xj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T2+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT2(U,V V,U11+l2)V(l2)T2]+CorrPi(U,V)(10+20)gsubjectto01110. Thisisnotasimpleoptimizationproblembecausetheobjectivefunctionwillchangedependingonthevalueof11.When11changes,k1willchangedependingon10)]TJ /F9 11.955 Tf 12.37 0 Td[(11andk2willchangedependingon11.Thesechangesalsodependonthestructureof 45

PAGE 46

thethresholdsofthestochasticpriceprocesses.Sothereisnogeneralexplicitsolution.Toexplorethepossibletrivialsolutions,weprovebelowthattradingseparatelyisbetterthantradingtogetherandhenceeliminatestrivialsolutions. Theorem6.1Itmakessensetotradethewholeportfolioseparatelyunderconsistentmulti-periodassumption.Moreprecisely,2(U,V)(U,V). Proof:Wedenotetheterminsidetheexpectationtobef2(U,V)andf(U,V)respectively.Underconsistentmulti-periodassumption,wehave f(U,V))]TJ /F6 11.955 Tf 11.95 0 Td[(f2(U,V)=10U0+20V0)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTkXj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.95 0 Td[(k)U(k)T)]TJ /F6 11.955 Tf 9.3 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTlXj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(20)]TJ /F9 11.955 Tf 11.96 0 Td[(l)V(l)T+CorrPi(U,V)(10+20)+U,V10+V,U20)]TJ /F8 11.955 Tf 11.95 0 Td[((10U0+20V0)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT1k1Xj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT1(10)]TJ /F9 11.955 Tf 11.95 0 Td[(11)]TJ /F9 11.955 Tf 11.95 0 Td[(k1)U(k1)T1)]TJ /F6 11.955 Tf 9.3 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT2k2Xj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T2)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT2(11)]TJ /F9 11.955 Tf 11.95 0 Td[(k2)U(k2)T2)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1l1Xj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T1)]TJ /F6 11.955 Tf 9.3 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT1(20)]TJ /F9 11.955 Tf 11.95 0 Td[(21)]TJ /F9 11.955 Tf 11.96 0 Td[(l1)V(l1)T1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT2l2Xj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T2)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT2(21)]TJ /F9 11.955 Tf 11.96 0 Td[(l2)V(l2)T2+CorrPi(U,V)(10+20)+U,V10+V,U20)=e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1k1Xj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T1+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT1(10)]TJ /F9 11.955 Tf 11.95 0 Td[(11)]TJ /F9 11.955 Tf 11.96 0 Td[(k1)U(k1)T1+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT2k2Xj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T2+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT2(11)]TJ /F9 11.955 Tf 11.95 0 Td[(k2)U(k2)T2+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1l1Xj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T1+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT1(20)]TJ /F9 11.955 Tf 11.96 0 Td[(21)]TJ /F9 11.955 Tf 11.95 0 Td[(l1)V(l1)T1+e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT2l2Xj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T2+e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT2(21)]TJ /F9 11.955 Tf 11.96 0 Td[(l2)V(l2)T2)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTkXj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T)]TJ /F6 11.955 Tf 9.3 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.95 0 Td[(k)U(k)T)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTlXj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(20)]TJ /F9 11.955 Tf 11.96 0 Td[(l)V(l)T 46

PAGE 47

Sincek110)]TJ /F9 11.955 Tf 12.85 0 Td[(11
PAGE 48

Figure6-1. AssetPriceProcess optimaltradingstrategy11tomaximizeourgain.Thatisequivalenttominimizetheriskmeasure. If1110,thenourtotalgainisG(11)=1010+9(20)]TJ /F9 11.955 Tf 10.74 0 Td[(11)]TJ /F8 11.955 Tf 10.75 0 Td[(10)+1111=190+211.Itisasimplelinearoptimization.Itsoptimalsolutionis11=10andG=210. If1110,thenG(11)=10(20)]TJ /F9 11.955 Tf 11.41 0 Td[(11)+1110+8(11)]TJ /F8 11.955 Tf 11.41 0 Td[(10)=230)]TJ /F8 11.955 Tf 11.41 0 Td[(211.Theoptimalsolutiontothislinearproblemis11=10andG=210. Inbothcases,wegetthesameoptimalsolutions.Thatmeans11=10istheoptimalsolutiontothewholeproblem.Itisanontrivialtradingstrategyanditdemonstratesseparatetradingisbetterthanwholesales.Forgeneralsituations,we 48

PAGE 49

havemorecomplexscenarios,andthesolutionsbecomemoreandmoredifculttocalculate.Sincethereisnogeneralsolution,itmakesmoresensetosolveitusingsimulations. 6.3OptimalSolutionsSet Generallyspeaking,thereisnouniquesolutiontothisoptimizationproblem.Instead,thereareinnitelymanysolutions.Infact,thisisatrivialproblemforriskmeasureswithoutliquidityeffects.Thatis,nomatterhowmanyyouaretradingineachstep,youwillgetabsolutelythesamepriceandhencethesametotalreturn.Sotheriskmeasureisthesame.Everynumberbetween10and0isavalidsolutionandalsoanoptimalsolution.Sotheoptimalsolutionsetis[0,10].Forliquidityriskmeasures,thedifferenceiswehaveadecreasingsupplyfunctionfortheasset.Butsincethisfunctionisdiscrete,itdoesnotmatterhowmuchitpassesthethreshold.Theonlythingmattersiswhetherthesupplypassesthethreshold.Therefore,theoptimalsolutionsetshouldbeasubsetof[0,10]. Toexplorethesolutionsets,webeginwiththesimplecase:assumingnointerestrateandconsistentmulti-period.Sotheoptimizationproblemchangestothefollowing:minimize11supi2IfEPi[k1Xj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1))]TJ /F8 11.955 Tf 11.96 0 Td[((10)]TJ /F9 11.955 Tf 11.95 0 Td[(11)]TJ /F9 11.955 Tf 11.95 0 Td[(k1)U(k1))]TJ /F11 7.97 Tf 16.61 15.06 Td[(k2Xj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1))]TJ /F8 11.955 Tf 11.95 0 Td[((11)]TJ /F9 11.955 Tf 11.96 0 Td[(k2)U(k2))]TJ /F11 7.97 Tf 17.73 15.06 Td[(l1Xj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1))]TJ /F8 11.955 Tf 11.95 0 Td[((20+U,V V,U11)]TJ /F9 11.955 Tf 11.95 0 Td[(l1)V(l1))]TJ /F11 7.97 Tf 17.72 15.06 Td[(l2Xj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)+(U,V V,U11+l2)V(l2)]+CorrPi(U,V)(10+20)gsubjectto01110. Wewillshowbelowthatthereisanexplicitoptimalsolutionsetforthisoptimizationproblem.Beforethat,wewillexaminetheassumptionsrst.Weareconsideringtwoassetsintwotimeperiods.Werstassumethereexistsoptimalbalancedtrading 49

PAGE 50

strategy,i.e.U,V10+V,U20=0andU,V11+V,U21=0.SowemusthaveU,VV,U<0.Herewearestudyingliquidationinashortperiodoftime.Therefore,thetimevalueofmoneycouldbeignored.Strictlyspeaking,thetradinginpreviousperiodmayhavepriceimpactonthenextperiod.Butherewejustassumethepricedistributionsarethesameinthetwotimeperiods.Wewillseehowresultschangelaterfordifferentpricedistributions.Undertheseassumptions,wehavethefollowingresult: Theorem6.2Underoptimalbalancedtradingstrategy,nointerestrate,andconsistentmulti-periodassumptions,theaboveoptimizationproblemhasanoptimalsolutionsetOSS=(k,k+1]\[10)]TJ /F9 11.955 Tf 12.29 0 Td[(k+1,10)]TJ /F9 11.955 Tf 12.29 0 Td[(k)\()]TJ /F12 7.97 Tf 10.5 5.78 Td[(V,U U,Vl,)]TJ /F12 7.97 Tf 10.49 5.78 Td[(V,U U,Vl+1]\[V,U U,V(l+1)]TJ /F9 11.955 Tf -446.91 -23.91 Td[(20),V,U U,V(l)]TJ /F9 11.955 Tf 11.96 0 Td[(20)),wherek<10 2k+1andl<20 2l+1. Proof:Sincethepricedistributionofthesameassetisthesameinthetwotimeperiod,theequallydistributedtradingamount11=10 2shouldbeanoptimalsolution.Ifnot,thereshouldbeanotherpricestrategyachievingasmallervalue.Nowassumethethresholdsfor10 2arekandk+1.Thenk<10 2k+1.Toachievesmallervalue,wewantsmallerkvalue.Tondadifferentstrategy,wemusthave11k.Thisisnotoptimalbecause11doesn'ttakethefulladvantageofthepricestagekand10)]TJ /F9 11.955 Tf 10.91 0 Td[(11mayinthethresholdthatevenfurtherthank+1.Sotheresultofthissolutionmustbegreaterthan11=10 2.Inaddition,anytradingvolumebetweenthesetwothresholdsshouldbeanoptimalsolution.Therefore,wemusthavek<11k+1andk<10)]TJ /F9 11.955 Tf 11.95 0 Td[(11k+1.Sothesolutionistheset(k,k+1]\[10)]TJ /F9 11.955 Tf 11.96 0 Td[(k+1,10)]TJ /F9 11.955 Tf 11.96 0 Td[(k). Wecanconductthesameargumentforthesecondasset.Wendthethresholdsl<20 2l+1.Thenthetradingvolumemustsatisfy l<)]TJ /F9 11.955 Tf 10.5 8.09 Td[(U,V V,U11l+1l<20+U,V V,U11l+1 Solvingtheinequalities,wehavetheset()]TJ /F12 7.97 Tf 10.5 5.78 Td[(V,U U,Vl,)]TJ /F12 7.97 Tf 10.49 5.78 Td[(V,U U,Vl+1]\[V,U U,V(l+1)]TJ /F9 11.955 Tf 12.11 0 Td[(20),V,U U,V(l)]TJ /F9 11.955 Tf -453.12 -23.9 Td[(20). 50

PAGE 51

Since11shouldsatisfybothconditions,theoptimalsolutionsetisOSS=(k,k+1]\[10)]TJ /F9 11.955 Tf 11.95 0 Td[(k+1,10)]TJ /F9 11.955 Tf 11.95 0 Td[(k)\()]TJ /F12 7.97 Tf 10.49 5.77 Td[(V,U U,Vl,)]TJ /F12 7.97 Tf 10.49 5.77 Td[(V,U U,Vl+1]\[V,U U,V(l+1)]TJ /F9 11.955 Tf 11.95 0 Td[(20),V,U U,V(l)]TJ /F9 11.955 Tf 11.96 0 Td[(20).2 Remark:Thisoptimalsolutionsetdoesnotdependontheprobabilityscenarios.Thatisbecauseunderanyscenario,wehavethesameoptimalsolution.Sothesupremumonallscenariosdoesnotaffecttheoptimalsolutionset.Thisisnotageneralconclusion.Wecanseelaterthatdifferentscenariosdohavedifferentsolutions. 6.4GeneralSolutions Nowwediscussthesolutionstogeneralcases.Bygeneralcase,wemeantwodifferentriskyassetswithdifferentpricedistributionsforNtimeperiodswithnonzerointerestrates.Theonlyassumptionwemakehereisthattheyhavethesamepriceimpactfactorsovertime,i.e.dosenotdependsontime.Asmentionedbefore,thereisnoexplicitsolutionsforthegeneralcases.However,wecandesignanalgorithmtocalculatetheoptimalsolutionsnumerically.Beforethat,wewillexaminetheconditionsfortheoptimizationproblemtohaveanuniquesolution.Wecanwritethegeneralcaseoptimizationproblemaftertakingoutnon-inuentialtermsasbelow:minimize11,21supi2IfEPi[)]TJ /F11 7.97 Tf 16.63 14.95 Td[(NXh=1(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rThkhXj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)Th)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTh(1h)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 11.95 0 Td[(1h)]TJ /F9 11.955 Tf 11.95 0 Td[(kh)U(kh)Th)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rThlhXj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)Th)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTh(2h)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 11.96 0 Td[(2h)]TJ /F9 11.955 Tf 11.95 0 Td[(lh)V(lh)Th)]+CorrPi(U,V)(10+20)gsubjectto01h10,02h20forh=1N. Fortheconvenienceonnotations,wedenethesetofallexpectationsofdiscountedassetspricesinthefollowing: Denition6.1TheexpectationpricesetforassetUwithNtradingperiodsUNisasetofallexpectationsofdiscountedassetprices.Specically,UiN=fe)]TJ /F11 7.97 Tf 6.59 0 Td[(rTiEPiUhjjj=1,N,h2Z+. 51

PAGE 52

Figure6-2. DescendingPriceSequencewithTradingVolume Theorem6.3TheoptimizationproblemhasanuniqueoptimalsolutioniftherearenotwoelementsinUiNandViNarethesame. Proof:SincetherearenotwoelementsinUiNandViNarethesame,wecansortthesetwosetsinstrictlydescendingorders.Andwecandrawagraphwithy-axistobetheelementsinthesetandx-axistobethecorrespondingmaximaltradingvolume(Figure6-2).Then10mustfallsinanintervalsays(Thm)]TJ /F7 7.97 Tf 6.59 0 Td[(1,Thm].ThenwegetasetMofintervalsthatontheleftof(Thm)]TJ /F7 7.97 Tf 6.58 0 Td[(1,Thm].ThentheoptimalsolutionforassetUis10)]TJ /F9 11.955 Tf 11.95 0 Td[(11=PM(T1j)]TJ /F9 11.955 Tf 11.95 0 Td[(T1j)]TJ /F7 7.97 Tf 6.59 0 Td[(1),,1h)]TJ /F7 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 11.95 0 Td[(1h=PM(T1j)]TJ /F9 11.955 Tf 11.96 0 Td[(T1j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)+(10)]TJ /F9 11.955 Tf 11.96 0 Td[(Thm)]TJ /F7 7.97 Tf 6.59 0 Td[(1),. Theabovesolutionisuniqueandoptimal.Theareaunderthepiecewisefunctionisthenegativeoftheriskmeasure.Tominimizetherisk,weneedtomaximizethisarea.ItisoptimalbecauseifweexchangeanyintervalinMwithsomearbitraryintervalsoutsideM,theareaunderthefunctionwilldecrease.Thatisbecausethefunctionisdecreasing.Thissolutionisuniquebecauseanyothersolutionwilldecreasetheareaunderthefunction,henceleadingtosuboptimalsolutions.2 Remark:Intheaboveproof,wefoundtheoptimalsolutionandwriteitintheformofsummationsofcorrespondingintervals.ThisresultissoimportantthatwewillrestateitseparatelyinTheorem6.4. 52

PAGE 53

Theorem6.4Underthesameassumptions,iftherearenotwoelementsinUiNarethesamethentheoptimalsolutionforassetUis10)]TJ /F9 11.955 Tf 12.75 0 Td[(11=PM(T1j)]TJ /F9 11.955 Tf 12.76 0 Td[(T1j)]TJ /F7 7.97 Tf 6.58 0 Td[(1),,1h)]TJ /F7 7.97 Tf 6.58 0 Td[(1)]TJ /F9 11.955 Tf 11.96 0 Td[(1h=PM(T1j)]TJ /F9 11.955 Tf 11.96 0 Td[(T1j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)+(10)]TJ /F9 11.955 Tf 11.95 0 Td[(Thm)]TJ /F7 7.97 Tf 6.58 0 Td[(1),. Remark:TheproofisthesameasTheorem6.3.Theresultactuallyprovidesacomputeralgorithmtondtheoptimalsolutions.Thecompletealgorithmisinthefollowing. Algorithm (1)GeneraterandomvariablesU(i)ThandV(j)Thaccordingtothejointdistributionf(U,V)forh=1toN,i=0tokhandj=0tojh. (2)Sorte)]TJ /F11 7.97 Tf 6.58 0 Td[(rThU(i)Thande)]TJ /F11 7.97 Tf 6.58 0 Td[(rThV(j)Thaccordingtodescendingorderseparately. (3)FindthecorrespondingtradingvolumessequencesMforUandQforV. (4)Sets=0andt=0. (5)s=s+(Thi+1)]TJ /F9 11.955 Tf 11.95 0 Td[(Thi)intheorderofthesequencein(2). (6)t=t+(Thi+1)]TJ /F9 11.955 Tf 11.95 0 Td[(Thi)intheorderofthesequencein(2). (7)Repeat(5)untils10andrecordtheinterval(Tam)]TJ /F7 7.97 Tf 6.58 0 Td[(1,Tam]. (8)Repeat(6)untilt20andrecordtheinterval(Tbq)]TJ /F7 7.97 Tf 6.59 0 Td[(1,Tbq]. (9)1N)]TJ /F7 7.97 Tf 6.58 0 Td[(1=PM(TN)]TJ /F16 5.978 Tf 5.76 0 Td[(1j)]TJ /F9 11.955 Tf 13.1 0 Td[(TN)]TJ /F16 5.978 Tf 5.76 0 Td[(1j)]TJ /F7 7.97 Tf 6.59 0 Td[(1),1N)]TJ /F7 7.97 Tf 6.59 0 Td[(2=1N)]TJ /F7 7.97 Tf 6.59 0 Td[(1+PM(TN)]TJ /F16 5.978 Tf 5.76 0 Td[(2j)]TJ /F9 11.955 Tf 13.1 0 Td[(TN)]TJ /F16 5.978 Tf 5.76 0 Td[(2j)]TJ /F7 7.97 Tf 6.58 0 Td[(1),,1a=1a+1+PM(TN)]TJ /F16 5.978 Tf 5.76 0 Td[(2j)]TJ /F9 11.955 Tf 11.95 0 Td[(TN)]TJ /F16 5.978 Tf 5.75 0 Td[(2j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)+(10)]TJ /F9 11.955 Tf 11.95 0 Td[(Tam)]TJ /F7 7.97 Tf 6.58 0 Td[(1),. (10)2N)]TJ /F7 7.97 Tf 6.58 0 Td[(1=PQ(TN)]TJ /F16 5.978 Tf 5.76 0 Td[(1i)]TJ /F9 11.955 Tf 12.94 0 Td[(TN)]TJ /F16 5.978 Tf 5.76 0 Td[(1i)]TJ /F7 7.97 Tf 6.58 0 Td[(1),2N)]TJ /F7 7.97 Tf 6.59 0 Td[(2=2N)]TJ /F7 7.97 Tf 6.58 0 Td[(1+PQ(TN)]TJ /F16 5.978 Tf 5.76 0 Td[(2i)]TJ /F9 11.955 Tf 12.94 0 Td[(TN)]TJ /F16 5.978 Tf 5.75 0 Td[(2i)]TJ /F7 7.97 Tf 6.59 0 Td[(1),,2b=2b+1+PQ(TN)]TJ /F16 5.978 Tf 5.75 0 Td[(2i)]TJ /F9 11.955 Tf 11.96 0 Td[(TN)]TJ /F16 5.978 Tf 5.76 0 Td[(2i)]TJ /F7 7.97 Tf 6.59 0 Td[(1)+(20)]TJ /F9 11.955 Tf 11.95 0 Td[(Tbq)]TJ /F7 7.97 Tf 6.58 0 Td[(1),. (11)CalculatethevalueinsidesupremumaccordingtotheformulaunderscenarioianddenotethevalueasGi. (12)Repeat(1)-(11)untilgetallGifori2I. (13)=maxGi. (14)Repeat(1)-(13)forRtimes. (15)PickthemaximalintheRtimesandthecorrespondingtradingstrategyistheoptimaltradingstrategy. 53

PAGE 54

6.5NumericalExamples Inthissection,wewillapplytheAlgorithmtosolvearealproblemandcomparetheresultwithordinarytradingstrategies.Forsimplicity,weconsiderthecasewithonlyoneriskyassetSinthreetradingperiods.WealsoassumethatSt,S(1)t,S(2)tfollowgeometricBrownianmotion.RecallthatastochasticprocessStisageometricBrownianmotionifSt=S0e()]TJ /F17 5.978 Tf 7.78 3.26 Td[(2 2)t+Bt,whereBtisanordinaryBrownianmotion.TheinitialpriceisknownS0=10.Assumetheyhavethesamedeviation=0.2.Theirvaluesareasfollow:=0.2,(1)=0.15,(3)=0.1.Wehavethefollowingthresholds(1)1=10,(1)2=30,(3)3=100,(2)1=20,(2)2=70,(2)3=100,(3)1=10,(3)2=20,(3)3=100.Wealsoassumeequaltimeperiods,i.e.T=0,1,2,3.Wealsoassumethreedifferentscenarios:log-normaldistribution,exponentialdistributionanduniformdistributioncorrespondingtoi=1,2,3. WeknowthemeanofgeometricBrownianmotionunderlog-normaldistributionisE1[St]=S0et.Butweneedtosimulatetheexpectationsundertheothertwodistributions.Fornumericalcomputation,wecanassumetheperiodinterestrateisr=3%. WeruntheprograminMatlabRandgetconvergentresultsshowninFigure6-3.Theaverageoptimaltradingstrategiesare8.88,56.28,34.84andthecorrespondingriskmeasureis=25.41showninFigure6-4andFigure6-5.Wealsocompareouroptimaltradingstrategywithgeneraltradingstrategies.Thegeneraltradingstrategyistotradeequalamountineachtimeperiod.Inourcase,thegeneraltradingstrategyistrading33.33inperiod1,2,and3andthecorrespondingriskmeasureis=29.16.Ourresultshowsthattheoptimaltradingstrategyisalwaysbetterthanthegeneraltradingstrategy.Therefore,ouroptimaltradingstrategyisabetterwaytotradelargeamountofassetsintherealmarket. 54

PAGE 55

Figure6-3. ConvergenceoftheVariance Figure6-4. ConvergenceoftheMean 55

PAGE 56

Figure6-5. OptimalTradingStrategy 56

PAGE 57

CHAPTER7LIQUIDITYRISKALLOCATIONS 7.1BasicConcept Onedirectapplicationofriskmeasuresistostudytheriskallocationproblems[ 9 ].Thebasicprobleminriskmanagementishowtheriskofeachcomponentinaportfoliocontributestotheriskofthewholeportfolio.Riskallocationisgenerallyusedtodescribesuchcontributions.Morespecically,letXbeaportfoliothatconsistsofsubportfoliosX1,...,Xn.SoX=X1++Xn.Inordertoevaluatetherisk,weneedtoassignariskmeasuretotheportfoliosystem.Thenthetotalriskoftheportfolioisdenotedby(X).WewanttostudytheallocationoftherisktothesubportfoliosX1,...,Xn.Clearlythechoiceofriskmeasuredirectlyaffectsthepropertiesoftheriskallocation.Ascoherentriskmeasurereectsproportionalchangeofassets,wecangetlinearriskallocationsunderthisriskmeasure[ 10 ].However,aswediscussedearlier,liquidityriskmeasuredoesnothavelinearity.Sotheriskallocationinducedbyliquidityriskmeasurewillnolongersupportlinearityeither.Conversely,whenweallocaterisktoeachcomponent,wealsoassignheavierweightsonilliquidassetsinsteadofallocatingproportionally.Therefore,nonlinearriskallocationsmakesenseforliquidityriskmeasures. Thewidelyaccepteddenitionisthefollowing[ 18 ]: Denition7.1Let:X!Rbeanyriskmeasure.Ariskallocationwithrespecttoisafunction:XX!RsuchthatforanyX2X,(X,X)=(X). Inaddition,theriskallocationiscalledaliquidityriskallocationifforany1,(X,Y)(X,Y).Foranyriskmeasure,wecandeneanewfunction (X,Y)=lim!0(Y+X))]TJ /F9 11.955 Tf 11.96 0 Td[((Y) (7) Ifwetakeintheabovetobeacoherentriskmeasure,thenwecangetalinearriskcontribution.ThelinearitypartiallycomesfromthePositiveHomogeneityofthecoherentriskmeasure.Thedenitionforlinearriskcontributionisasfollows[ 10 ]: 57

PAGE 58

Denition7.2Alinearriskcontributionisafunctionl(X,Y)denedonL1L1satisfyingtheaxioms: (1)(Linearity)l(a1X1+a2X2,Y)=a1l(X1,Y)+a2l(X2,Y)fora1,a22R(2)(Diversication)l(X,Y)(X)(3)(Consistency)l(X,X)=(X)(4)(LawInvariance)l(X,Y)dependsonlyonthejointlawof(X,Y)(5)(Continuity)IfjXnj1andXn!Xinprobability,thenl(Xn,Y)!l(X,Y) Fortheliquidityriskmeasure,thelinearitydoesnotholdanymoreanditbecomesmorecomplex.ThereasonisthatliquidatingX1+X2couldbemoredifcultthanliquidatingX1andX2separately.Soyoumayendupliquidatingatdiscount.ButitdoeshavetheConsistencypropertyandotherspecialproperties. Noticethatthelimitdenedin(7-1)maynotalwaysexist.Soweneedonemoreconditiontomakeitavaliddenition.Thatis[ 18 ] lim!0(X,Y+X)=(X,Y) (7) Proposition7.1Ifisaliquidityriskmeasure,thenthelimitin(7-1)existsifandonlyifthecondition(7-2)issatised[ 18 ]. Utilizingtheaboveproposition,wecanshowthattheriskcontributiondenedin(7-1)usingcoherentriskmeasuresatisesthegeneraldenitionofriskcontribution. Theorem7.1(X,Y)inducedbycoherentriskmeasureisavalidriskallocationfunction. Proof:Bydenition,weonlyneedtoverifythat(X,X)=(X)foranyX2X.LetF(X)=(X,X))]TJ /F9 11.955 Tf 12.21 0 Td[((X).WeneedtoshowthatF(X)0.LetF(X)=((1+)X))]TJ /F12 7.97 Tf 6.58 0 Td[((X) )]TJ /F9 11.955 Tf -449.7 -23.9 Td[((X).When>0,1+>1.ByLiquidityaxiom,((1+)X)(1+)(X).SoF(X)=((1+)X))]TJ /F12 7.97 Tf 6.59 0 Td[((X) )]TJ /F9 11.955 Tf 11.96 0 Td[((X)(1+)(X))]TJ /F12 7.97 Tf 6.59 0 Td[((X) )]TJ /F9 11.955 Tf 11.95 0 Td[((X)=(X))]TJ /F9 11.955 Tf 11.95 0 Td[((X)=0. 58

PAGE 59

When<0,1+<1.ByLiquidityaxiom,((1+)X)(1+)(X).ThenF(X)=((1+)X))]TJ /F12 7.97 Tf 6.59 0 Td[((X) )]TJ /F9 11.955 Tf 11.96 0 Td[((X)(1+)(X))]TJ /F12 7.97 Tf 6.59 0 Td[((X) )]TJ /F9 11.955 Tf 11.95 0 Td[((X)=(X))]TJ /F9 11.955 Tf 11.95 0 Td[((X)=0. SincethelimitexistsbyProposition7.1,thenforanyX, F(X)=lim!0F(X)=lim#0F(X)=lim"0F(X)=0.2 7.2LowerandUpperBound Riskcontributionsaremostusefulwhentheyarebounded.Anyriskallocationinducedbyaconvexriskmeasurehasalowerandupperbound.Inthissection,wederivealowerboundofifiscontinuousandanupperboundof.Theyalsogiveawaytoestimatetheriskallocation. Theorem7.2Ifisdenedby(7-1)usingaconvexriskmeasure,then(X,Y)(X+Y))]TJ /F9 11.955 Tf 11.95 0 Td[((Y)foranyX,Y2X. Proof:Scienceisconvex, (Y+X))]TJ /F9 11.955 Tf 11.96 -.01 Td[((Y) =((1)]TJ /F9 11.955 Tf 11.95 0 Td[()Y+(X+Y)))]TJ /F9 11.955 Tf 11.95 0 Td[((Y) (1)]TJ /F9 11.955 Tf 11.95 0 Td[()(Y)+(X+Y))]TJ /F9 11.955 Tf 11.95 0 Td[((Y) =(X+Y))]TJ /F9 11.955 Tf 11.95 0 Td[((Y) =(X+Y))]TJ /F9 11.955 Tf 11.96 0 Td[((Y) So (X,Y)=lim!0(Y+X))]TJ /F9 11.955 Tf 11.96 0 Td[((Y) (X+Y))]TJ /F9 11.955 Tf 11.96 0 Td[((Y).2 Theorem7.3Ifisdenedby(7-1)usingacontinuousconvexriskmeasure,then(X,Y)(Y))]TJ /F9 11.955 Tf 11.95 0 Td[((Y)]TJ /F6 11.955 Tf 11.95 0 Td[(X)foranyX,Y2X. Proof:Scienceisconvex,letZ=Y+Xwehave (Y+X))]TJ /F9 11.955 Tf 11.95 0 Td[((Y) =(Z))]TJ /F9 11.955 Tf 11.96 0 Td[((Z)]TJ /F9 11.955 Tf 11.96 0 Td[(X) =(Z))]TJ /F9 11.955 Tf 11.95 0 Td[(((1)]TJ /F9 11.955 Tf 11.95 0 Td[()Z+(Z)]TJ /F6 11.955 Tf 11.96 0 Td[(X)) (Z))]TJ /F8 11.955 Tf 11.96 0 Td[((1)]TJ /F9 11.955 Tf 11.96 0 Td[()(Z))]TJ /F9 11.955 Tf 11.96 0 Td[((Z)]TJ /F6 11.955 Tf 11.95 0 Td[(X) =(Z))]TJ /F9 11.955 Tf 11.96 0 Td[((Z)]TJ /F6 11.955 Tf 11.95 0 Td[(X) =(Y+X))]TJ /F9 11.955 Tf 11.95 0 Td[((Y+()]TJ /F8 11.955 Tf 11.95 0 Td[(1)X) 59

PAGE 60

Sinceiscontinuous,wehave (X,Y)=lim!0(Y+X))]TJ /F9 11.955 Tf 11.95 0 Td[((Y) lim!0[(Y+X))]TJ /F9 11.955 Tf 11.96 0 Td[((Y+()]TJ /F8 11.955 Tf 11.95 0 Td[(1)X)]=(Y))]TJ /F9 11.955 Tf 11.96 0 Td[((Y)]TJ /F6 11.955 Tf 11.95 0 Td[(X)2 7.3RiskAllocationsUsingLiquidityRiskMeasures Inthissection,wewillderivetheriskallocationfunctionsforliquidityriskmeasures.Weusethesamebasicdenitionforriskallocationfunction,butsubstitutetheriskmeasurebyaliquidityriskmeasure.RecallfromChapter4,theliquidityriskmeasureofU+Vis(U+V)=supi2IEPi[10U0+20V0)]TJ /F6 11.955 Tf 9.78 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTkPj=1(j)]TJ /F9 11.955 Tf 9.78 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 9.79 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 9.79 0 Td[(k)U(k)T)]TJ /F6 11.955 Tf -458.7 -32.13 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTlPj=1(j)]TJ /F9 11.955 Tf 10.64 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 10.64 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(20)]TJ /F9 11.955 Tf 10.64 0 Td[(l)V(l)T+CorrPi(U,V)(10+20)+U,V10+V,U20].Ifweassumeexistenceofbalancedtradingstrategyandthereisnocorrelationeffect,wewillget(U+V)=supi2IEPi[10U0+20V0)]TJ /F6 11.955 Tf 12.23 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTkPj=1(j)]TJ /F9 11.955 Tf 12.23 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T)]TJ /F6 11.955 Tf 12.23 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(10)]TJ /F9 11.955 Tf -448.89 -32.13 Td[(k)U(k)T)]TJ /F6 11.955 Tf 12.66 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTlPj=1(j)]TJ /F9 11.955 Tf 12.65 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T)]TJ /F6 11.955 Tf 12.65 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT(20)]TJ /F9 11.955 Tf 12.66 0 Td[(l)V(l)T].Substitutethisintotheriskallocationfunction,wewillgetanexplicitexpressionfortheriskallocationfunction.AnotherthingweneedtodealwithistheprobabilitydistributionofeachscenarioPi.Sincewewillfocusonshorttermtradinginthesamemarketcondition,allriskmeasuresshouldbecalculatedatthesameprobabilitydistributionandtakesupremumoverthecalculatedresults.Thatmeanswecantaketheoutsideoperationsintothesupremumandstillhavethesameresults.Formallyspeaking,wehavethefollowingassumption: SameMarketAssumption:Whenweevaluateriskallocation,weassumeallassetsaretradedinthesamemarketcondition.Inotherwords,theyhavethesamepricedistributions. Theorem7.4UnderSameMarketAssumption,fortwoassetsUandV,ifthereisnocorrelationeffectandthereexistsbalancedtradingstrategy,thentheriskallocationfunctioninducedbytheliquidityriskmeasureinDenition4.8isacoherentriskmeasureofV,i.e.(V,U)=supi2IEPi[20V0)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT20VT]=c(V). 60

PAGE 61

Proof:Werstevaluatethenumeratorinthedenitionofriskallocation. (U+V))]TJ /F9 11.955 Tf 11.95 0 Td[((U)=supi2IEPi[10U0+20V0)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTkXj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.95 0 Td[(k)U(k)T)]TJ /F6 11.955 Tf 9.3 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTlXj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(20)]TJ /F9 11.955 Tf 11.95 0 Td[(l)V(l)T])]TJ /F8 11.955 Tf 11.95 0 Td[(supi2IEPi[10U0)]TJ /F6 11.955 Tf 9.3 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTkXj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.95 0 Td[(k)U(k)T]=supi2IEPi[20V0)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rTlXj=1(j)]TJ /F9 11.955 Tf 11.95 0 Td[(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.58 0 Td[(1)T)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(20)]TJ /F9 11.955 Tf 11.95 0 Td[(l)V(l)T] Asgoesto0,20alsoapproachesto0.SotheGLBthresholdfor20alsogoesto0.Forsmallenough,wemusthavel=0=0.Sothemiddleterme)]TJ /F11 7.97 Tf 6.58 0 Td[(rTlPj=1(j)]TJ /F9 11.955 Tf -430.77 -25.9 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T=0.Therefore,(U+V))]TJ /F9 11.955 Tf 11.96 0 Td[((U)=supi2IEPi[20V0)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT20VT]. (V,U)=lim!0(U+V))]TJ /F9 11.955 Tf 11.96 0 Td[((U) =lim!0supi2IEPi[20V0)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT20VT] =supi2IEPi[20V0)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT20VT]=c(V).2 Theaboveresultistheriskallocationfunctioninwhichoneassetisindependentoftheotherasset.However,whenweallocaterisk,alotoftheassetswillbecorrelated.Therefore,weneedtointroduceariskallocationfunctioninamuchrelaxedenvironment.Whenoneassetispositivelycorrelatedwiththewholeportfolio,alossinthisassetwillalsobringdownthevalueoftheotherassetsthatarepositivelycorrelatedwithit.Sothetotallossofthewholeportfolioshouldbegreaterthanthelossofthesingleasset.Sincetheextralossiscreatedbythissingleasset,weshouldallocateallorpartoftheextralosstothistroubleasset.Therefore,theriskallocationfunctionshouldbegreaterthanthatwithoutcorrelationeffect. Theorem7.5UnderthebalancedtradingstrategyandSameMarketAssumption,theriskallocationfunctionoftwoassetsUandVinducedbyaliquidityriskmeasure 61

PAGE 62

isalsoaliquidityriskmeasure,i.e.(V,U)=supi2IEPi[20V0)]TJ /F6 11.955 Tf 13.58 0 Td[(e)]TJ /F11 7.97 Tf 6.58 0 Td[(rT20VT+Corr(U,V)20]=(V,0). Proof:Sincethisisinthetwo-assetmarket,weshouldusethetwo-assetdenitionofliquidityriskmeasure. (U,V))]TJ /F9 11.955 Tf 11.95 0 Td[((U,0)=supi2IEPi[10U0+20V0)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTkXj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.96 0 Td[(k)U(k)T)]TJ /F6 11.955 Tf 9.3 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTlXj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(20)]TJ /F9 11.955 Tf 11.96 0 Td[(l)V(l)T+Corr(U,V)(10+20)])]TJ /F8 11.955 Tf 11.96 0 Td[(supi2IEPi[10U0)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTkXj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)U(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 9.3 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(10)]TJ /F9 11.955 Tf 11.95 0 Td[(k)U(k)T+Corr(U,V)10]=supi2IEPi[20V0)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rTlXj=1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)V(j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)T)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT(20)]TJ /F9 11.955 Tf 11.96 0 Td[(l)V(l)T+Corr(U,V)20] Sameargumentasabove,asgoesto0,wecanhave (U,V))]TJ /F9 11.955 Tf 11.95 0 Td[((U,0)=supi2IEPi[20V0)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT20VT+Corr(U,V)20]. Therefore, (V,U)=lim!0(U+V))]TJ /F9 11.955 Tf 11.96 0 Td[((U) =lim!0supi2IEPi[20V0)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT20VT+Corr(U,V)20] =supi2IEPi[20V0)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F11 7.97 Tf 6.59 0 Td[(rT20VT+Corr(U,V)20].2 Remark:ThecorrelationtermistheproportionoftheextrariskcreatedbyVtothewholeportfolioUallocatedbacktoV.Thebackallocationwilldenitelyincreasetheriskallocationfunction.However,wecanseetheriskallocatedfromVhasnothingtodowiththevolume10ofU.ThisisbecauseitdoesnotallocationalltheextrariskbacktoV. 7.4PropertiesofRiskAllocations Inthissection,weinvestigatethespecialpropertiesoftheriskallocationfunctionsinducedfromliquidityriskmeasuresduetothespecialLiquidityaxiom.Recallthata 62

PAGE 63

riskallocationiscalledaliquidityriskallocationif(X,Y)(X,Y)if1and(X,Y)(X,Y)if0<1. Theorem7.6Theriskallocationinducedbyaliquidityriskmeasurein(7-1)isaliquidityriskallocation. Proof:Wewanttoshowthatforany0<1,(X,Y)(X,Y). (X,Y))]TJ /F9 11.955 Tf 11.95 0 Td[((X,Y)=lim!0(Y+X))]TJ /F9 11.955 Tf 11.96 0 Td[((Y))]TJ /F9 11.955 Tf 11.96 0 Td[((Y+X)+(Y) =lim!0(Y+X))]TJ /F9 11.955 Tf 11.96 0 Td[((Y+X)+()]TJ /F8 11.955 Tf 11.96 0 Td[(1)(Y) =lim!0(Y+X))]TJ /F8 11.955 Tf 11.96 0 Td[([(Y+X)+(1)]TJ /F9 11.955 Tf 11.96 0 Td[()(Y)] lim!0(Y+X))]TJ /F9 11.955 Tf 11.96 0 Td[(((Y+X)+(1)]TJ /F9 11.955 Tf 11.96 0 Td[()Y) =lim!0(Y+X))]TJ /F9 11.955 Tf 11.96 0 Td[((Y+X) =02 Fromthis,wecanderivethecasefor1. CorollaryFor1,wehave(X,Y)(X,Y). Proof:(X,Y)=(1 X,Y)1 (X,Y)2 Theriskallocationdoesnotnecessarilydiversifytheriskoftheportfolio.SotheDiversicationpropertyoflinearcontributiondoesnotholdforliquidityriskcontribution.Insomecases,itcanevenincreasetherisk.Forexample,wehavetwostocksXandY,andCorr(X,Y)=1.IfwesellX,thenthepriceofXfallsbyconsideringtheliquidityeffect.SinceXisperfectlycorrelatedwithY,thepriceofYalsofalls.ThismeanstheriskXcontributedtotheportfolioX+YisgreaterthantheriskofXalone.So(X,X+Y)>(X,X)=(X). 63

PAGE 64

CHAPTER8EMPIRICALSTUDIESONLIQUIDITYRISKMEASURES Uptonow,wehavealreadyestablishedalotofaxioms,assumptionsandmodelsregardingliquidityrisk.Therestwewanttodoistotesthowthisriskmeasureworksintherealnancialmarketandwhetherourassumptionsarereasonable.Duetolimitationofourresources,weareunabletotesteverythinginthepaper.OurmajorconcerniswhetherthestrictinequalityinLiquidityaxiomwillholdinrealtransactions.Thisisthemajordistinctionbetweentheriskmeasureestablishedinthispaperandothers.Inordertoapplytheriskmeasureintherealmarket,wearealsoconcernedabouttheestimationofcoefcientsintheformula.Thereareseveralcomputationalstudiesconcerningriskmeasures[ 28 30 ].Thereareseveraldifferentaxiomsproposedbyresearcherstobuildriskmeasurementsystems.ThemostpopularoneisthePositiveHomogeneityproposedbyArtzner[ 1 ]incoherentriskmeasures.Onewaytoseewhichaxiomisbettersuitedtotherealworldistotesttheaxiomsintherealmarket.However,itisusuallydifculttoperformsuchtest.HerewecarryoutastatisticaltesttostudytheLiquidityaxiom.TheLiquidityaxiomstatesthatwhen1,wehave(X)(X).Todistinctouraxiomfromcoherentriskmeasure,wetestthestrictversionoftheLiquidityaxiom.Thatmeanswewanttotestgiven>1,wehave(X)>(X).Sowehavethefollowingnullhypothesis: H0:(X))]TJ /F9 11.955 Tf 11.95 0 Td[((X)0 (8) Inordertoconductstatisticaltest,weneedtogatherthedatainspecictransactionsandcomparethemwithsimilartransactionswithdifferentsizes.Ourtestcomparestwotransactions,oneisthetradingXandtheotheristhetradingX.Thesetwotransactionsarecarriedoutintheexactlysamemarketconditionatthesametime.Inacertainmarket,(X)=P1Xand(X)=P2X.Soifwewanttocomparethesetwovalues,weonlyneedtocompareP1andP2,whereP1isthepricechangewith 64

PAGE 65

Figure8-1. GCNTradeRecap largetradingvolumeandP2isthepricechangewithsmalltradingvolume.Duetolimitedinformationabouttrading,wehaveonlytwosecuritiestostudy.OneisGCNandtheotherisIBM.ThedataisextractedfromBloombergTerminalTM.Tostudythepriceimpact,weonlyextractedthedatawithtradingvolumegreaterthan100million.OnesnapshotofthetradingdataisdisplayedinFigure8-1. Therstthingwewanttotestisthatthepricechangeduetolargetradingisstatisticallysignicant.Thatistotestthemeanofthepricechangeissignicantlydifferentfrom0.TheSASRtestresultisshowninFigure8-2. Thet-testrejectedthenullhypothesis,meaningthepricechangeisstatisticallysignicant.Thisiswhatweexpect:LiquidityaxiomisamorerealisticapproachthanPositiveHomogeneityaxiom.Inaddition,wegetsimilarresultsfromcorporatebondofIBMinFigure8-3. Thenexttestwecanconductisthecorrelationbetweenpricechangeandtradingtype,eitherbuyingorselling.Fortestpurpose,wedenotesellingby0andbuyingby1.ThetestresultshowsasignicantpositivecorrelationinFigure8-4. 65

PAGE 66

Figure8-2. T-TestforGCN Figure8-3. T-TestforIBM 66

PAGE 67

Figure8-4. CorrelationBetweenPriceandVolume Thispositivecorrelationshowsthatlargeamountofbuyingwillincreasethepriceandlargeamountofsellingwilldecreasetheprice.ThisisalsoconsistentwithLiquidityaxiom. Insummary,themarketsaremorelikelytoacceptliquidityriskmeasuresbecauseanylargeenoughtradingwillcausethemarkettobecomeilliquidatleastforashortperiodoftime.Soitisbetterformarketparticipantstoevaluatetheriskincludingliquidityeffectbeforemakinginvestmentdecisions.Accuratelymeasuringtheriskisthekeytochoseoptimaltradingstrategies. 67

PAGE 68

CHAPTER9CONCLUSION Inthispaper,wehavedevelopedanewriskmeasurementsystemforilliquidmarketsandlargetradings.Itcanhelpinvestmentmanagerstomoreaccuratelyassesstheirafter-tradingriskandactualreturns.Itisimportanttounderstandthechangeinsupplyanddemandandtodetermineoptimaltradingstrategiesbasedonpriceimpact.Wehavedenedriskmeasuresformultiple-assetportfoliosandcorrespondingtradingstrategiestominimizethetotalrisk.Althoughthereisnogeneralsolutionforcomplexoptimaltradingstrategies,itispossibletocalculatesuchstrategiesbycomputationalmethodinpractice.Therearestillnumerousfactorstobeconsideredwhendeningriskmeasures.Duetolimitedtimeandresources,wecannotexploreeverythinginthispaper.Butwebelievemoresophisticatedrisksystemswillbedevelopedbyusorotherresearchersinthenearfuture. 68

PAGE 69

REFERENCES [1] Artzner,P.,Delbaen,F.,Eber,J.-M.,Heath,D.:Coherentmeasuresofrisk.MathematicalFinance9(3),203-228(1999) [2] Bank,P.andBaum,D.:Hedgingandportfoliooptimizationinfmancialmarketswithalargetrader.MathematicalFinance14,1-18(2004) [3] Bayraktar,E.,Ludkovski,M.:Optimaltradeexecutioninilliquidmarkets.MathematicalFinance21,681-701(2011) [4] Brigham,E.,Ehrhardt,M.:Financialmanagement,ThomsonSouth-Western,2008 [5] Cassese,G.:Assetpricingwithnoexogenousprobabilitymeasure.MathematicalFinance18,23-54(2008) [6] Cerreia-Vioglio,S.,Maccheroni,F.,Marinacci,M.,Montrucchio,L.:Riskmeasures:rationalityanddiversication.MathematicalFinance21,743-774(2011) [7] Cetin,U.,Jarrow,R.,Protter,P.:Liquidityriskandarbitragepricingtheory.FinanceandStochastics8,311-341(2004) [8] Cherny,A.:Pricingwithcoherentrisk.TheoryProbab.Appl.52,389-415(2008) [9] Cherny,A.:Capitalallocationandriskcontributionwithdiscrete-timecoherentrisk.MathematicalFinance19,13-40(2009) [10] ChernyA.,OrlovD.:Ontwoapproachestocoherentriskcontribution.MathematicalFinance21557-571(2011) [11] Corwin,ShaneandSchultz,Paul:Asimplewaytoestimatebid-askspreadsfromdailyhighandlowprices,workingpaper [12] Cvitanic,J.,Karatzas,I.:Ondynamicmeasuresofrisk.FinanceandStochastics3,451-482(1999) [13] Detlefsen,K.,Scandolo,G.:Conditionalanddynamicconvexriskmeasures.FinanceandStochastics9,539-561(2005) [14] Dufe,D.andZiegler,A.:Liquidationrisk.FinancialAnalystsJournal59,42-51(2003) [15] Follmer,H.,Schied,A.:Convexmeasuresofriskandtradingconstraints.FinanceandStochastics6,429-447(2002) [16] Henderson,V.,Hobson,D.:Optimalliquidationofderivativeportfolios.MathematicalFinance21,365-382(2011) 69

PAGE 70

[17] Jobert,A.,Rogers,L.:Valuationsanddynamicconvexriskmeasures.MathematicalFinance18,1-22(2008) [18] Kalkbrener,M.:Anaxiomaticapproachtocapitalallocation.MathematicalFinance15,425-437(2005) [19] Karatzas,I.andShreve,S.:Brownianmotionandstochasticcalculus,2nd,NewYork:Springer-Verlag,(1988) [20] Karoui,N.andRavanelli,C.:Cashsubadditiveriskmeasuresandinterestrateambiguity,MathematicalFinance11,325-343(2009) [21] Ku,H.:Liquidityriskwithcoherentriskmeasures.AppliedMathematicalFinance13,131-141(2006) [22] Longstaff,F.:Optimalportfoliochoiceandthevaluationofilliquidsecurities.ReviewofFinancialStudies14,403-431(2001) [23] McNeil,A.J.,Frey,R.andEmbrechts,P.:Quantitativeriskmanagement:concepts,techniques,tools.PrincetonUniversityPress,2005 [24] Ong,M.:Riskmanagement.AcademicPress,2006 [25] Potter,P.:Stochasticintegrationanddifferentialequations,2nd,Springer-Verlag,(2004) [26] Riedel,F.:Dynamiccoherentriskmeasures.StochasticProcessesandtheirApplications112,185-200(2004) [27] Rockafellar,R.:Coherentapproachestoriskinoptimizationunderuncertainty.Informs,38-61(2007) [28] Tsukahara,H.:One-parameterfamiliesofdistortionriskmeasures.MathematicalFinance19,691-705(2009) [29] Uryasev,S.:ConditionalValue-at-Risk:optimizationalgorithmsandapplications.FinancialEngineeringNews,14,1-5(2000) [30] Zarepour,M.,Bedard,T.,Dabrowski,A.:ReturnandvalueatriskusingtheDirichletprocess.AppliedMathematicalFinance15,205-218(2008) 70

PAGE 71

BIOGRAPHICALSKETCH PengyiSunwasborninTangshan,China.HeattendedTsinghuaUniversityinChinafrom2002to2006.Aftergraduation,hewasadmittedtothePh.D.programinDepartmentofMathematicsatUniversityofFloridaandwasawardedtheAlumniFellowship.Hisconcentrationwasmathematicalnance.DuringhisstudyatUniversityofFlorida,healsoearnedseveralmaster'sdegrees,includingMasterofStatisticsandMasterofBusinessAdministration.HereceivedhisPh.Dinthespringof2012. 71