Optimal Contract Under Asymmetric Information - the Role of Options on Futures

7/31/2019 Optimal Contract Under Asymmetric Information - the Role of Options on Futures

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Optimal contract under asymmetric information:

the role of options on futures

Andrea Beccarini

19/2011

Department of Economics, University of Mnster, Germany


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1

Optimalcontractunderasymmetricinformation:

theroleofoptionsonfutures.

AndreaBeccarini1,WestflischeWilhelmsUniversittMnster(DE),

Abstract

Theaimofthispaperistoshowthatanoptiononanappropriatefuturemaysolvesomemarket

failurescausedbyasymmetricinformation.Somemodelsrelatedtotheadverseselection,moral

hazard andverificationcostsareanalyzedandtheperformanceoftheseoptionsonfuturesis

evaluated.Thetypicalsituationregardsaconsumer(oraninvestor)whowishestodiscount

his/herfutureincomeinordertofinancehis/herpresentconsumption(investment);under

asymmetricinformationthisagentmayincurinliquidityconstraints(creditrationing),whichisnot

thecasewhenbuyingtheoptiononafuturescontract.Thiscontractisconstructedsothatthe

(future)agentsincomeiscorrelatedwithsomefuturescontract(butthisisprivateinformation)onwhichtheoptionisissued.Someexamplesshowthatthisisnotaverystringentassumption.

JELclassification:D82,G14

Keywords:Asymmetricinformation,creditrationing,optionsonfutures.

1

Addressforcorrespondence:InstitutfrkonometrieundWirtschaftsstatistik,AmStadtgraben9,48143,Mnster(DE).Email:05anbc.wiwi.unimuenster.de


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Introduction

Ineconomics,informationasymmetrydealswiththestudyofdecisionsintransactionswhereone

agenthasmoreorbetterinformationthantheother.Thiscreatesanimbalanceofpowerin

transactionswhichcansometimescausesomemarketfailure.Examplesofthisproblemare

adverseselection,moralhazardandcostlymonitoring.Atpresent,informationasymmetryisonly

apartiallysolvedproblemineconomictheoryalthoughavastliteraturehasshownmicroand

macroeconomiceffectsofthisphenomenon. Infact,asymmetricinformationmaycompromise

thegoodfunctioningofthemarketinfixingthepriceofgoodsorservicesandcausesserious

consequencesatthemacroeconomiclevel.Forexample,ifconsumersorfirmsexpectahigh

futureincome,theyfinditoptimaltosellthestreamoftheirfutureincometothebankingsystem

inordertoobtainpresentcashflow(forconsumptionorinvestment,respectively);however,

underasymmetricinformation,themarketmayfail:peoplewhoarewillingtopayapricehigher

thanthemarketpricearerationedandconsequentlyconsumptionorinvestmentsufferfrom

liquidityconstraint.Putinanotherway,thetraditionalneoclassicaleconomicsliteratureassumes

thatmarketsareefficientexceptforsomelimitedandwelldefinedmarketfailures.However,

someauthorsarguethatitisonlyunderexceptionalcircumstancesthatmarketsareefficient.In

2001,theNobelPrizeinEconomicswasawardedtoGeorgeAkerlof,MichaelSpence,andJoseph

E.Stiglitz"fortheiranalysesofmarketswithasymmetricinformation".

Thisworkproposesamicrofoundedsolutiontotheprobleminquestion.Theproposedsolution

reliesonthemarketofoptionsissuedonfutures(seeHull,chapter162).Inthispaper,asa

benchmark,oneconsidersageneralmodelembeddingeitheradverseselectionormoralhazardor

verificationcosts(whichall implycreditrationing).Onethenshowsthattheoptiononfutures

providesanalternativewaytoaccesscreditand(undercertainconditions)thisalternativedoes

notsufferfrommarketfailureorcreditrationing.Theconditionswherebytheoptiononafutures

contract(OF) ischeaperthanatraditionaldebtcontract (TD)willbealsoshown.Thisisan

importantfeaturetoshowwhenborrowersarenotrationed.Onethenshowsthatborrowers

withgoodexpectationsontheirincomealwayspreferanOFcontractinsteadofaTDcontract.

2Inthesecontracts,theexerciseoftheoptiongivestheholderapositioninafuturescontract.


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AclassicpaperonadverseselectionisGeorgeAkerlof's"TheMarketforLemons"of1970.George

Akerlofnotes thattheaveragevalueofthecommoditytendstogodown,evenforproductsof

perfectlygoodquality.Becauseofinformationasymmetry,somesellerscanhidethetruequality

of

their

items

and

defraud

the

buyer.

As

a

result,

buyers

prove

to

be

unwilling

to

purchase

this

commoditysincethisentailsacertainamountofrisk.Consequently,itisevenpossibleforthe

markettodecaytothepointofnonexistence.JaffeandRussell(1976)showthatinthecredit

market,underasymmetricinformationbetweenborrowersandlender,adverseselectionmay

causecreditrationing.Inapoolingequilibrium,lendersofferthesameloancontracttoall

borrowers(whoareeitherhonestordishonest)sincetheyareunabletodistinguishbetweenthe

twotypes.Theinterestrateiscalculatedundertheassumptionthatborrowersaredrawn

randomlyfromthepopulation.Butatthisrate,thepoolofborrowersisnolongerthesameasin

thepopulation:adverseselectionoccurs.Infact,thelenderismorelikelytoattractthedishonest

borrowerandhencethelendersexpectedprofittendstofallandthemarketmayfail.However,

loansarenotcharacterizedonlybyinterestratesbutalsobytheamountofsomecollateral.Given

thecollateral,equilibriummayimplycreditrationing.Inotherwords,creditrationingisthemarket

responsetoadverseselection.Townsend(1979)providedthefirstanalysisofoptimalcontracts

whenitiscostlytoverifythestate.Heshowsthatunderthepresenceof asymmetricinformation

betweenlenderandborrower,adebtcontractwithcostlystateverificationcanbeconstructedso

astobeParetoefficient.GaleandHellwig(1985)followthisstrandandsettheimplicationofa

firmsinvestmentlevel.StiglitzandWeiss(1981)shedfurtherlightonthecreditrationingdueto

thepresenceofadverseselectionandmoralhazard,showingthattheprofitfunctionishump

shapedwithrespecttotheinterestrate.ForasurveyofdefinitionsofcreditrationingseeWalsh

(2003,chapter7).However,asWalshargues:Thecriticalaspectofthisdefinitionisthat,atthe

marketequilibriuminterestrate,thereisanunsatisfieddemandforloansthatcannotbe

eliminatedthroughhigherinterestrates. Williamson(1986,1987a,1987b)hasillustratedhow

debtcontractsandcreditrationingcanariseevenintheabsenceofadverseselectionormoral

hazard,iflendersmustincurcoststomonitorborrowers.Gertler(1988)andBernankeandGertler

(1989) showhowagencycostsdriveawedgebetweenthecostsofinternalandexternalsources

offinancing;theyfurthershowhowinvestmentdecisionsalsodependontheinternalfunds.

Anotherhugestrandofliteratureshowstherelationshipwithcreditrationingandthecredit

channel

through

which

monetary

factors

and

policy

may

affect

the

real

economy.

See

Walsh


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2003,chapter8forasurvey.MoschiniandLapan(1995)outlineahedgingrole ofOptionson

futuresbuttheyaimedtoshowthatacombinationofoptionsonfuturesandfuturescontracts

hedgeaproducerwhofacesproductionandpriceuncertainty.Instead,inthispaper,oneshows

that

(a

portfolio

of)

options

on

futures

may

prevent

the

credit

rationing

and

liquidity

constraint.

Therestofthispaperisorganizedasfollows.Thesecondsectionformallysetsupnotationsand

theassumptionsoftheeconomy;thethirdsectionshowshowthisOFcontractmayworkfrom

theborrowersside.Thefourthsectionshows, inacontextofsomemodelswithmarketfailures,

theremedyprovidedbytheOFcontract.Thefifthsectionshowshowalender(bank)setsthe

optimalpriceoftheTDcontractwithverificationcostsandthepriceoftheOFcontract.Thesixth

sectionshowstheconditionwherebyarepresentativeagentwithnonlinearobjectivefunction,

giventhepriceofbothcontracts,chooseseithercontract.Conclusionsthenfollow.

Notationsandtheassumptionsintheeconomy

ThefollowingassumptionsmostlyrecalltheassumptionsofStiglitzandWeiss(1981)andof

Williamson(1986,1987a,1987b).

a) Theeconomyiscomposedofthefutures,TD,OF,andriskfreebondsmarkets.Theeconomylasts1,2,..,Nperiods,N 3.

b) Thefuturesmarketiscompleteandefficient: participantsarerational,riskneutral,competitiveanddetainnoprivateinformation.Inthismarket,acontinuumoffuturesare

issuedwhoseunderlyingvalueisdenotedbyX:X .ThefuturesfixthepriceofeachXNinperiod2;henceitspriceistheexpecteddiscountvalueofXN.

c) IntheTD andintheOFmarkets,thereareidenticallenders(banks)whoarerational,riskneutral,haveconstantreturntoscaleandoperatecompetitively.

d) Onthedemandsideofthesemarkets,thereisacontinuumofrationalborrowers(consumersorinvestors)indexedby: , whoevaluateanincomestreamYinperiodN. maymeasuretheaprioridegreeofhonestyoftheborrowers(seeJaffeandRussell,1976)ordifferentattitudesoftheirexpostbehaviorinaffectingtheprojectyieldingYN.

e) Yislognormallydistributedsuchthat: ElnYand VarlnY.FYandfYare thedistributionfunctionandprobabilitydensityfunction(respectively)ofYinperiodiconditionaloninformationj,withji.


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f) Atperiod2,borrowersdetainsomeprivate(butnotperfect)informationonYN,banksdonot.

g) EachincomeYiscorrelatedwithone(combinationof)Xbutthiscorrelationisalsoprivateinformation

of

each

borrower.

Without

loss

of

generality,

it

is

assumed

that

this

correlation

isperfect.Thisimpliesthattheborrowercannotexploithisprivateinformationaboutthe

correlationbetweenYandsomeXtoobtainsomefunding(andthebankcannotinferthe

type(riskiness)ofborrowerthroughtheknowledgeofX).Assumealsothateachactionof

theborroweronYissuchthatYremainscorrelatedwithsomeotherX,X .h) Iftheborrowerisaconsumer,hewishestodiscountthepotentialvalueYNinorderto

smoothhisconsumption(heisriskaverter)inperiod2;ifheisaninvestor,hewishesto

discountYNinordertofinancehisinvestmentprojectyielding YN.HealsodetainsY andYinperiod1and2respectively.i) Theparameter (thetypeofborrowers)andtheactionofeachborrowerontheproject

yieldingYarenotknownbythelenders(orbyotheragentsoftheothermarkets)butthey

knowthedistributionfunction .Thisfunctiondescribesthefrequencyofparticipationinthemarketofeachtypeofborrower. However,thisisaffectedbythepriceofeach

marketsincethoseborrowerswhofinditunprofitableexitthemarket,therebyalteringthe

shapeof.Assumethatthisprocessoccurswithinperiod2.j) IntheTDmarket,debtcontractsaresetinperiod2andredeemedinperiodN;the

(competitive)marketinterestrateisRNB andeachbankbidsitsoptimalquantityofdebtD(inperiod2).AssumethatthisrateisneitherarandomvariablenorD.

k) AsregardstheTDcontract,some(notmutuallyexclusive)alternativesarepossible: inperiodN,borrowersknowYNforcertainandwithoutacost,hencelendershaveto

payaverificationcostC;

lendersrequireacollateralcinordertograntthefunds.l) IntheriskfreebondsmarketgrossinterestratesaredenotedasRN 1 r,NNas

theperiod2isthereferenceperiodinthiseconomy.Assumethatthetermstructureof

interestratesisnotarandomvariable.

m) IntheOFmarket,(call)optioncontractsareissuedinperiod1andexpireinperiod2.Denote thepriceas,thequantityasQandthemultipleoftheoptionasm.Eachoption

gives

the

right

to

buy

a

futures

contract

at

the

strike

price

K.

Since

the

value

of

each


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futureisthediscountedexpectedvalueofsomeX,andifYNis(perfectly)correlatedwithXN,thenthepayoffoftheoptionis,inperiod2, maxEXN/RN K, 0

maxEYN/RN K,0TheOptiononFuturescontractfromtheborrowersside

ConsideranoptimizingnonriskneutralconsumerwhoseekstosmoothconsumptionoveranN

timeshorizonperiod.Considertwocasesintime2. Thatis,heorshemayormaynotface

liquidityconstraintsand,inthelattercase,theconstraintisbinding.Supposethatincomeintime

Nissufficientlyhighsuchthattheconsumerdesirestodiscountit(tosmoothconsumption).Call

this(desired)levelofconsumption

C

YN.Thisimpliesthatiftheconsumerwereconstrained,

consumptionintime2is:C Y RY C CYN. Hence,intime2,itholds:C minCYN, C (1)Iftheconsumerhasquadraticutilityfunction,itcanbeshownthatthisoutcomeforChasthesamepayoffofaputoptionwrittenonYN.Infact,supposenowthatCYN CYN(C 0andthatC 1, theneq.(1)canberepresentedingraph1.Thismeansthattheconsumptionfunctionintime2replicatesaputoptionwithstrikepriceK.

Intuitively,intime2,thefollowingcasesmayoccur:

theconsumerfacesnobudgetconstraint:theconsumptionfunctionisthelinewithslopeC andinterceptQ;

theconsumerfacesabudgetconstraintbutitisnotbinding:EYN/RN K;whereKisthesmallestvalueofYNsuchthattheconsumerdesirestodiscounthisorherfutureincome;

theconsumerfacesabindingbudgetconstraintEYN/RN Kbutitmustbethat:C C


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Graph1:

EYN/RNThestructureoutlinedaboveshowsthepossibilityofrecoveringasmoothedconsumptionintime

2bybuying(intime1)acalloptionwhosepriceisCO,withstrikepriceK,writtenonasecuritycorrelatedwithYN, EYN/RN; thissecuritycanbeafuturescontractwhoseunderlyingis(correlatedwith)YN.Theeffectofbuyingthiskindofoptionistoobtaina(parallel)linewithslopeC andinterceptQCO.Seegraph2.

Graph2:

SimplemathematicsshowsthatQ isafunctionoftheinterestratesandYandY. Thesameexamplecanbeusedtodescribethesituationofanentrepreneurwithaprofitableinvestment

projectyieldingYNattimeNwheretheinternalfundsofhisfirmarenotenoughtofinancehisproject.Theentrepreneurwantstodiscountthepotentialcashflow(derivedfromYN) inorderto

C

Q

K

C

Y

Budgetconstraintnotbinding

Budgetconstraintbinding

CO

C

Q

K

C

EYN/RN

CallOption onFutures

Hedgedconsumption

OriginalconsumptionY


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implementthedesiredinvestment.Ifhecannotaccesstosometraditionaldebt(thefirmiscredit

rationed),hecanbuytheOFcontractatperiod1inordertohavehisfundingintime2.Evenif

thefirmisnotrationed,theentrepreneurwantstominimizethecostoffundingandhence

compares

the

cost

of

the

T

D

contract

with

the

O

F

contract.

Asapracticalexample,consideraneconomicagentwhoisconfidentaboutinvestinginabakery

thatpromises(inhis/heropinion)afuturehighprofit.Theprofitofthisinvestmentdependson

thepriceofthebread(theoutput)andthepriceofsomerawstuffssuchasflourasinput.Evenif

itisunrealistictostatethatthereisafuturesmarketforthebakery,itismorerealistictoassume

theexistenceofafuturesmarketforbread,flourandotherrawstuffs.Theeconomicagentshould

onlyseekaportfolioofoptionsissuedonfuturesofthesestuffs.Inparticular,he/sheshould

detainacalloptiononfuturesissuedonbreadandaputoptiononfuturesissuedonflour.

Nowsupposethatabankfacesanagent(consumerorfirm)whohasprivateinformationabout

his/itsfutureincome.Consider,forexample,traditionallendingcontractsthatareaffectedby

adverseselection.FollowingJaffeandRussel(1976),supposethatagentsareoftwotypes:honest

anddishonest.Thefirstalwaysstateshis/itsinformationaboutfutureincomewhereasthesecond

istruthfulonlyifitisprofitabletohim/it. Supposingthebankknowsforeachtypeofborrower

theprobabilitythatheisabletorepaythedebt,thenthebankisabletosetaninterestratethat

incorporatesthisinformation3.So,thepresenceofasymmetricinformationfirstlyimplies

inefficiencyofthemarketbecauseagentswhoseprobabilitynottorepaythedebtislow,borrow

atahigherinterestrate;secondlyandconsequently,thelatteragentmayfinditnotprofitableto

borrowanymore,henceadverseselectionoccurs.

NowconsiderthefactthatthebankhasthepossibilitytoofferanOFcontract insteadofaTD

contract.Thequestionisnowwhethertheagentmayexploithis/itsprivateinformation.Suppose

thedishonestagentconsidersbuyingtheOFcontract;anyonewhobuysanOFcontractmustbe

longandhencecannotexploithisprivateinformationthathisincomemaybelow.HencetheOF

contractyieldsafavorableselectioninsteadofanadverseselection.Notethat, inallofthe

3

Supposenowthatthebankcannotdistinguishbetweenthem,hencethelendinginterestrateisanaverageofalowinterestrateandahigherinterestrate.


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examplesabove,onehasimplicitlyassumedthatagentswerelongabouttheirfuturediscounted

income.

ThecomparisonofOptionsonFutureswiththeAdverseSelection,

MoralHazardandVerificationcostsmodels.

ThissectionbuildsacomprehensivemodelincludingtheStiglitzandWeissmodel(1981)of

adverseselection andmoralhazardandWilliamsonsmodel(1986,1987a,1987b)of

verificationcosts4.Oneshowshowthemarketfailurecanoccurinthisgeneralmodelbutthisis

notthecasewherethelenderoffersanOFcontract.

Withtheassumptionthatborrowersareacontinuumlabeledby

,theexpectedprofitofabank

whenbiddingaTDcontractis:

ENB EminYN c,DRNBCE minYN cDRNB , 0

YN cDRNB : dH RND 2

Ifc=0(c=collateral)thenonehastheWilliamsonmodel;ifC=0(C=verificationcosts)thenone

obtainstheStiglitzandWeissmodels.

Notethat,giventheinformationstructurebetweenborrowersandlender,Townsend(1979)

showsthattheoptimalcontractthatthebankiswillingtooffertotheborrower isadebtcontract

EminYN, DRNB.HealsoshowsthatverificationcostsCarepaidonlyinthecaseofbankruptcy(thatisYN c RNB .Notethatthiseventmaybedescribedbytheoperator: , .Forsimplicity,assume(fornow)thatborrowersareariskneutralinvestor,then,theexpected

profitofeachtypeofborroweris5

EN EYNEminYN, DRNBcE minYN DRNB , 0

YN DRNB 3

4 Inthesepapers,Williamsonhasshownthatcreditrationingcanariseevenintheabsenceofadverseselectionor

moralhazardproblemsiflendersmustincurcoststomonitorborrowers.5

Notethateq.(3)isequivalenttotheexpectedvalueof eq.(4a)oftheStiglitzandWeisspaper;however,eq.(3)ofthispaperhighlightsseparatelyrevenuesandcostsoftheprofitfunction.


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Notethatevenifthecollateralciszero,theborrowermayincuralossbecauseoftheopportunity

costs(notexplicitlyspecifiedineq.(3))ofinvestinghisowncapitalintheproject.Expectationsin

themoregeneralmodeldodependontheactionofeachtypeofborrowers.Assumenowthat,atthebeginning

6

ofperiod2,EYN EYN : buttheVarYNisnotnecessarilythesame.Considernowthecaseoftheadverseselection(agentsaredifferentaprioributcannotchange

theirbehaviorexpost,opportunistically).Assumethattherearesomeborrowerswho,given

fYN, c andsome(initial)valueofRNB ,finditnotworthwhiletoinvestintheprojectENB 0.Sinceexpectedvaluesarethesame,thismeansthatthevarianceoftheproject,whichisdifferentamongborrowers,isthesourceoftheprofitnegativityofsomeborrowers.Oneshowsin

appendixAthatahighervariance7oftheYNimpliesalowervalueofthedebt,namely:EminYN, DRNB.Thismeansthatonlyborrowerswithahighvariancearewillingtoacceptthedebtcontract.Itfollowsthattheprofitsofthebankarenow8:

ENB EminYN, DRNBcE minYN DRNB , 0

YN DRNB : dH RND 4

Becauseofthedifferentintegrationinterval,theseprofitsarelessthantheprofitsconsideredin

Eq.(2).Inordertomaintainthedesirablelevelofprofit,banksareinducedtoraiseRNB whichincreases9EminYN, DRNB andcausesanewroundofadverseselection.Alltheseadjustmentsoccurinperiod2;inequilibrium(attheendofperiod2), marketfailureispossibleas,for

example,creditrationing.

Moralhazardworksinasimilarway:aftergettingthedebt,someborrowers finditprofitableto

changetheir(original)projectinordertohave(positive)profits10,hencechanging projectswitha

6Oneassumesthatatthebeginningofperiod2,isnotyetaffectedbythepriceoftherelevantmarket.

7NotethatsinceYislognormallydistributed,themeanpreservingproperty(asinStiglitzandWeiss(1981))doesnot

apply.8Attheendofperiod2whenisaffectedbytheinitialprice(interestrate)oftherelevantmarket.

9

SeeAppendixB.10Since,atthebeginningofperiod2,theywerenotnecessarilypositive.


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highervariance causesexpectedlossesforthebankwhich,ifitisabletoforeseethat,shouldexit

thismarket;thisagaincausesthemarketfailure.

Notethatifborrowersarecharacterized(orcanaffect) therevenuecomponentoftheirprofit

functionsinsteadoftheircostcomponent,asintheabovesituation,afavorableselectionwould

bepossibleandthemechanismbehindadverseselectionandmoralhazardwouldnotwork.Thisis

thesituationwhereborrowersevaluatebuyinganOFcontract.Inthiscase,theprofitofthe

borroweris

EN E YN QmmaxEYN/RN K,0CORQ 5

whereYN. istheincomeatperiodNasafunctionoftheinvestedcapital.Notenowthat EYNdoesdependontheoutcomeoftheOFinperiod2;asanticipated,thefavorableselectionapplies

heresincetheborrowermustevaluatethepossibilitythattheoptionendsupoutofmoneyor

thatitsrevenuewheninvestedintheprojectisnotenoughtocoverthecostsCOR.HenceborrowerswhohaveabadevaluationofEYNdonotaffecttheprofitofthebankapartfromthefactthattheydonotbuytheOFcontract.

The

comparison

with

the

costs

of

a

T-

D

and

an

O-

F

contract

ThissectioncomparesthepriceofthisTDcontract(withcostlystateverification)withthepriceof

theOFcontract.TheaimistoshowthattheOFischeaper.

TheoptimalbankingbehaviorwhenbiddingaT-Dcontract

OnehasalreadyarguedthatwhenthebankdoesnotreceivethepromisedpaymentDRNB itwantstoincurinverificationcoststoseewhethertheinvestorischeating.Itcanbeshownthat,under

this

kind

of

contract,

when

the

gross

payment

is

smaller

than

DRNB

then

it

equals

11

YN;

see

Townsend(1979)andGaleandHellwig(1985). Hence,theprofitfunctionofthebankcanbe

specifiedasineq.(2):

11

Onlyforthesakeofsimplicity,hereonehasassumedthatthereisonlyonetypeofborrower(thatremainsunknown)andthatthecollateralciszero.


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B EminYN, DRNBCE , RND; (6)Note,firstofall,that:

EminYN, DRNB YNfYNdYN DRNB 1 fYNdYN (7)

E , fYNdYN

fYNdYN

fYNdYN

(8)

Assumethatthebank,inthiscontext,maximizesitsprofitwithrespecttoD12.So,bytheLeibniz

rule,

one

can

find

the

following

derivatives:

RNB fDRNB ;,

RNB fYNdYN ;SeeAppendixBforfurtherdetails.Resuming,thefirstorderconditionsare:

RNB fYNdYN CRNB fDRNB R 0;RNB FFDRNB CRNB fDRNB R 0;FDRNB C f DRNB 1 ; (9)ConsiderthefollowingTaylorSeriesapproximationsofthefirstorder:

FDRNB FEYN fEYNDRNB EYN;fDRNB fEYN fEYNDRNB EYN;Substitutingthemineq.(9)oneobtains:

12 Thisisthevariableonechoosesinacompetitivemarket.


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FEYN fEYNDRNB EYNC f EYNC f EYNDRNB EYN 1

DRNB EYN FEYNC f EYN

D

Define FEYNC f EYN; fEYNC f EYN

D EYNRNB 1 RNB

RRNB 10

Eq.(10)describesthesupplycurveofthebank.Thepossibilityofcreditrationingcanbeseenby

thefactthatDisnotanincreasingfunctionofRNB .InordertofindtheequilibriumRNB inthecompetitivemarket,onehasjusttosettheprofitfunctionequationtozero.Firstnotethat:

B EminYN, DRNBCE , RND EminYN C RND,DRNB RND

EminYN CRD,DRNB RND YNfYNdYN CRND fYNdYN

1 DRNB RNfYNdYN

YNfYNdYN CRNDDRNB RND fYNdYN

1

fYNdYN FDRNB FEYN fEYNDRNBEYN

YNfYNdYN YNfYNdYN YNfYNdYN

Since YNfYNdYN EYN then

YNfYNdYN EYN 1

EYN

Thecompetitivemarketcondition

B

0implies:


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B EYN CDRNBFEYN fEYNDRNBEYN 1 0

EYN C DRNB FEYN fEYNDRNBEYN 1 (11)

Ontherightside:CDRNBFEYN fEYNDRNBEYN 1 CFEYN DRNBFEYN CDRNB fEYN DRNBfEYN

CEYNfEYNEYNDRNB fEYN 1 DRNBfEYN DRNB FEYN CfEYNEYNfEYN CFEYN

CEYNfEYN

1

Call FEYN fEYNCEYN; CFEYN CEYNfEYN 1Then,therighthandsideof(11)becomes:

DRNB fEYN DRNB (12)Therighthandsideisaparabolicfunction.Althougheq.(11)describes animplicitfunctionofRNB ,onecanconsiderthefollowingstrategy.Inordertofindviablesolutionsoftheabovefunctionone

shouldtheoreticallyfindtheintersectionvaluesbetweenthefunctionsoftherightandleftsides

(pointsAandBforGraph3).Indeed,oneisonlyinterestedinthesmallestpositivesolutionas

competitionshouldpushallthesolutionstothesmallestone.

However,itiseasytoseethatpointQisalwayslessthanA(aslongitispositive),henceonecan

considerQasalowerboundofthevalueA.

Q A B

EYN

Graph3


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15

Hence,inordertofindpointQonehastosolve:

DRNB fEYN DRNB EYN 0; thesmallestsolutionis:

RNB C 1D 4fEYN EYN2fEYN 0 13

Thisistheinterestrateconsistentwiththezeroprofitcondition13. Combiningeq.(10)andeq.

(13)onecanfindtheequilibriuminterestrateRNB onlyasafunctionoftheunderlying parametersandYN.

EYN

RNB EYN

0 14

Eq.(14)alsoshowstheconditionswherebythebankinginterestrateishigherthantheriskfree

marketinterestrate.

Theoptimal

banking

behavior

when

bidding

an

O

-Fcontract

Theprofitfunction,evaluatedinperiod2,inthecaseoftheoptiononfuturesis14:

EB QCOR QmEEYN E EYNQmminEYN, K KEYN K 15

EBQ COR EYNm

1mE EYN

minEYN, K KEYN K 0 16

13Theconditionwherebythesolutionisrealis: 4fEYN EYN 0.Inordertosatisfythiscondition,it

issufficienttoshowthat 0;cFEYN EYNfEYN 1; FEYN EYNfEYN ;

FEYN

EYNfEY N.

14misthemultipleoftheoption,suchthatm>0.


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CO EYNRm 1

RmE EYNminEYN, K KEYN K 17

Thispriceequalstoboththemarginalandtheaveragecost,henceitalsoensureszeroprofits.

Thecomparisonofthecostsofthetwocontracts

Assumethatbothcontractslastforoneperiodonly(theTDcontractlastsfromperiod2toperiod

3andtheOFcontractfromperiod1toperiod2).Notethatthecomparisonofthetwocontracts

shouldbebasedonthesequantities:

RB 1

/ 1 1 18

havingdefined E EYN, ;thequantitiesandWaremultipliedand dividedby(respectively)inordertocomparethemtothemoneyflowsofthelefthandsideof(18).Consideringeq.(17), therighthandsideisequalto 1.Hence,byeq.(18)onecanconcludethat,innonperversecases,thecostoftheTDcontractis

always

greater

than

the

O

F

contract

15

.

However,itisnotenoughtoconcludethattheOFismoreefficientsincethepriceisnottheonly

variablethatagentsmayevaluate.Infact,substitutingadebtcontractwiththeOFcontract,the

agentchangesa(costlier)fundingofacertainamountwitharandomfunding(theuncertaintyis,

infact, internalized).Thisistrueeveniftheagenthasapositiveevaluationofhis/herfuture

income.AriskaverseconsumerwouldnotnecessarilyfinditoptimaltoswitchtotheOFcontract.

15

NotethatRB Rbecauseoftheriskinessofthebankingloan;hencetheTDcontractisalwaysmoreexpensivethantheOFcontractinthecaseofaflattermstructureofinterestrate(R R.


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Theoptimalchoiceofanonrisk-neutralconsumerinthecaseofadverse

selectionandmoralhazard.

Theconsumermaximizes hisutilityfunctionbychoosingthecompositionofaportfolioofTD and

OFwhoseshareofTDisxandtheshareofOFis(1x).Inthisway,oneshowsthecondition

whereby theconsumerchooseseithercontract. Assume,withoutlossofgenerality,thatN=3and

thatthecollateralciszerobutthereareverificationcostsC.

Forthesakeofexposition,define:

min minY, DRB C, (19)

max maxEY/R K,0 (20)Utilityandbudgetconstraintsarethen(whereU(.)isawellbehavedconcaveutilityfunction, isadiscountfactorandC istheconsumptionstream):U E UC s.t.: (21)

1; (22)

1 max 0; (23)whereistheinitialwealth.Theinequalityconstraintstemsfromthefactthatthefirstfactor(thesavinginperiod2)cannotbenegativebecauseotherwisetheagentwouldbeallowedto

borrowattherate. min; 1 1 max; 1 1max;Define 1 1max min; (24)


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Forthesakeofexposition,onesubstitutes throughthebudgetconstraintofeq.(24)andthenonesetsuptheLagragianwherethemultiplierofinequalityconstraintofeq.(23)issetexplicitly16.

ThederivativeofthisLagrangianwithrespecttoxis:

; where istheLagrangemultiplieroftheconstraintofeq.(26)with 0.Thisimplies:EUC maxD min ; (25)EUCD min EUCmax ; (26)Disregard fornowtheterm.Eq.(26)showsthattheoptimalsolutionxmustequalthemarginalnetbenefitsofbothcontracts.Considernowanonequilibriumsituation;if,forexample,

thelefthandsidewouldbelargerthantherighthandside,itwouldbeoptimaltoincreasethe

shareoftheTDandhencetoincreasex.Consequently,onehastofindtheconditionwherebythe

righthandsideisalwayslargerthanthelefthandsidesothattheoptimalsolutionisx 0,namely,onlytheOFcontractischosen. Alsonotethatsincexischoseninperiod1,theexpected

valueofperiod1mustbeappliedtobothsidesofequations.Thatis:

EUCmax EUCD min ; (27)EUCmax EUC EUCD EUCmin;covUCmax EUCEmax EUC EUCD covUCmin EUCEmin; (28)Afterrearrangingandnotingthat17

covUC, min EUCcovC,min (29)covUC,max EUCcovC,max (30)

16Foranintroductionto theKuhnTuckerconditionsforamaximizationwithaninequalityconstraintseeVarian,

1992.17

ThisapproximationisobtainedbytheTaylorseriesexpansion;sinceoneisonlyinterestedinshowingtheinequality,thisapproximationservesonlyforclarificationpurposes.


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EUCcovC, max EUCEmax EUCD Emin EUCcovC, min ; (31)Alternatively,onecouldspecifyaprocessforminandmaxinordertofindtherelevant

momentsasinCochrane2001.Assumenowthatthisrepresentativeconsumerhasgood

expectationsaboutY.Morepreciselyassumethat, Zsufficientlylarge, itholds:EY/R KvarY 32

Thismeansthat:

max

maxEY/R K, 0

EY/R K (33)

min minY, DRB c, DRB (34)Eq.(34)thenbecomes:

EUCcovC, EY/R EUCE EY/R K EUCD EDRB EUCcovC, DRB ; (35)notingthat covC, DRB 0andthat,covC, EY covY, EY EY EYEY EEY

EY EYEY EY EYEY YEY EYEY EYEY 0

havingassumedthattheonlycorrelationbetweenCandEYworksthroughYitself.Simplifyingfurther:

Q m EY/R DRB ; (36)


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Sincetherighthandsideisalwaysnegative18,eq.(36)becomes

EY/R (37)

Notethatthisconditionisalwayssatisfiedprovidedthattheagentislongabouthisorherfuture

income:theexpectedpayoffoftheOFislargerthanitsopportunitycost.Thisimpliesthata

favorableselectionoccursthatcausesa screeningdevicebetweenconsumers.Recallthatthis

screeningdevice,whenthebankofferstheTDcontract, worsenstheexpectedprofitofthebank

aggregatedonthepooloftheconsumers.Infact, changessinceconsumerswhowouldlikeaTD(theyhaveinsufficientpositiveexpectationsonY,inthesensedefinedabove)causeadeclineintheexpectedrevenueEminYN, DRNB.Inotherwords,whenthisscreeningdeviceisapplied,oneisabletostatethattheTDcontract

betweenthebankandtheconsumerwithgoodexpectationsisParetoinefficient,because:

thebankisindifferentaboutoffering eithercontractsincebothcontractsareconstructedsuchthattheyensurethezeroprofitcondition;

theconsumerwithgoodexpectations(asdefinedabove)onthefutureincomealwaysprefersthe OFcontract.

Itfollowsthattwoseparateequilibriumsarepossible:

intheOFmarket,whereassumptionsofthewelfaretheoremsapply; intheTDmarket,wherethebankeithercanchargeahigherinterestrate

RNB conditionalontheconsumersdemandcurveoritfindsitnotoptimaltostayinthismarketasitsprofitremainsanotmonotonicallyincreasingfunctionofRNB .

Notethatriskofbankruptcyhasbeeninternalizedintheprofitfunctionsoftheborrowersthrough

theOFcontract.Inthisperspective,theOFisasolutionofanegativeexternalitycreatedby

borrowerswithhiddenbadexpectations.

18ThisisbecauseRB Randbecause 0.Indeed,onecanshowthat 0(strictly)becauseotherwisethe

inequalityconstraintwouldnotbebinding(seeVarian,1992); this,inturn,isnotoptimalbecausetheagentwouldborrowattherateRBandinvestattherateR.


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Conclusions

Throughoutthepaper,onehasshownthatanoptiononfuturesmaysolvethemarketinefficiency

(orfailure)duetoasymmetryofinformationamongthemarketparticipants.Ithasbeenexplained

thatthefundamentalreasonforthisbeingpossibleisthatborrowerswhobuythisoptionmustbe

longinthesensethattheyhaveenoughgoodexpectationsoftheirfutureprofit(correlatedwith

thepayoffoftheoption).Thismeansthat(rational)dishonestborrowerswillnotusetheir

privateinformationagainstthemselves.Anotherwaytoseethisisthat,underasymmetric

information,theselectionbetweenpotentialborrowersisadverseinthesensethatafterthe

selectiononlyborrowerswithbadexpectationsoftheirprofitsremainwillingtoobtainthe

funds;thisselectionworksthroughthecostssideoftheborrowersprofitfunction.Withthe

optiononfuturescontract,theselectionisfavorablebecauseitworksthroughtherevenues

sideoftheborrowersprofitfunction.Thisalsomeansthattheriskofbankruptcyinmodelswith

adverseselectionandmoralhazardarebasicallyshiftedfromtheborrowertothelender;

conversely,withtheoptiononfuturescontract,thisriskisreinternalisedintheprofitfunction

ofthelender. Inotherwords,withasymmetricinformation,therearenegative externalitiesin

thecreditmarketandhencethemarketbecomesParetoinefficient,whichisnotthecasewiththe

optiononfuturesmarket.

References

[1] Bernanke,B.GertlerM.,AgencyCosts,NetWorth,andBusinessFluctuations,TheAmericanEconomicReview,Vol79,No.1(Mar1989),pp1431.

[2] Cochrane,J.H.,AssetPricing,2001,PrincetonUniversityPress.[3] JaffeeD.M., RussellT.,ImperfectInformation,Uncertainty,andCreditRationing, The

QuarterlyJournalofEconomics,Vol.90,No.4(Nov.,1976),pp.651666.

[4] GaleD.,HellwigM.,IncentivecompatibleDebtContracts:TheOnePeriodProblem.TheReviewofEconomicStudies,vol.52,No.4(Oct.1985),pp.647663.

[5] HullJ.C.,Options,FuturesandotherDerivates,2009,PrenticeHall.[6] MoschiniG.,LapanH.,TheHedgingRoleofOptionsandFuturesunderjointPrice,Basis,

andProductionRisk,InternationalEconomicReview,Vol.36,No.4(Nov.1995)pp.1025

1049.


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[7] Romer,D.H.,AdvancedMacroeconomics,NewYork,McGrawHill,2001.[8] StiglitzJ.E.,WeissA.,CreditRationinginMarketswithImperfectInformation,The

AmericanEconomicReview,Vol.71,No.N(Jun.,1981),pp.N9N410.

[9]

Townsend,

R.M.,

Optimal

Contracts

and

Competitive

Markets

with

costly

State

Verification.JournalofEconomicTheory21,265293(1979).

[10]Varian,H.R.,MicroeconomicAnalysis,1992,W.W.Norton&Company,Inc.[11]WalshC.E.,MonetaryPolicyandTheory,2003,TheMITPress.[12]WilliamsonS.D.,CostlyMonitoring,FinancialIntermediationandEquilibriumCredit

Rationing,JournalofMonetaryEconomics,18(2),Sept.1986,pp.159179.

[13]WilliamsonS.D.,CostlyMonitoring,LoanContracts,andEquilibriumCreditRationing,TheQuarterlyJournalofEconomics,102(1)Feb.1987a,pp.135145.

[14]WilliamsonS.D.,FinancialIntermediation,BusinessFailures,andrealBusinessCycles,JournalofPoliticalEconomy,95(6)Dec.1987b,pp.135145.

AppendixA:

SinceYNislogarithmically distributed, then EYN e.Fixing EYN EYN oneobtains lnEYN .Atthesametime,itcomesdirectlyfromthedefinitionofthevariancethat:VarYN EYN e 1;thismeansthatanincreaseinVarYNimpliesanincreaseinwhichimpliesadecreasein. Nownotethatthemedianise.TheincreaseinVarYNimpliesthattheprobabilitymassofYNconcentratesleftwardssincethemedianisalsodecreased.HencetheprobabilitythatYN RNB isincreased.Thisimpliesthat YNfYNdYN EYN|YN RNB PrYN RNB increasesandDRNB fYNdYN decreases.Bydefinition,allthisimpliesadecreaseinEminYN, DRNB.AppendixB:

RNBDRNB fRNB 0 0 fYNdYN

;

0 RNBDRNB fRNB RNB fYNdYN

.

ThederivationoftheaboveintegralswereperformedtroughtheLeibnizrule.Theconditions

wherebythisrulecanbeappliedare:


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thefunctionundertheintegralsigniscontinuousalongwithitsfirstderivative; thefunctionsoftheendpointsarecontinuousandderivable.

Theseconditionsarealwayssatisfied.Furthermore,thelatterintegralisdefinedasimpropersince

theupperendpointis.ThisimpliesthatoneshouldalsoverifyitsuniformconvergenceinordertoapplytheLeibnizrule.Tothispurpose,itissufficienttoshowthat DRNB fYNdYN whichisveryeasytoprove.

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Optimal Contract Under Asymmetric Information - the Role of Options on Futures

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