The price of power The valuation of power and weather deriva

发布时间：2024-11-18

ThePriceofPower:TheValuationofPowerandWeatherDerivatives

CraigPirrong

UniversityofHouston

Houston,TX77204

713-743-4466

cpirrong@uh.edu

MartinJermakyan

http://

December7,2005

Abstract.Pricingcontingentclaimsonpowerpresentsnumerouschal-lengesdueto(1)thenonlinearityofpowerpriceprocesses,and(2)time-dependentvariationsinprices.Weproposeandimplementamodelinwhichthespotpriceofpowerisafunctionoftwostatevariables:demand(loadortemperature)andfuelprice.Inthismodel,anypowerderivativepricemustsatisfyaPDEwithboundaryconditionsthatre ectcapacitylimitsandthenon-linearrelationbetweenloadandthespotpriceofpower.Moreover,sincepowerisnon-storableanddemandisnotatradedasset,http://inginverseproblemtechniquesandpowerforwardpricesfromthePJMmarket,wesolveforthismarketpriceofriskfunction.During1999-2001,theupwardbiasintheforwardpricewasaslargeas$50/MWhforsomedaysinJuly.By2005,thelargestestimatedupwardbiashadfallento$19/MWh.Theselargebiasesareplausiblyduetotheextremerightskewnessofpowerprices;thisinducesleftskewnessinthepayo toshortforwardpositions,andalargeriskpremiumisrequiredtoinducetraderstosellpowerforwards.Thisriskpremiumsuggeststhatthepowermarketisnotfullyintegratedwiththebroader nancialmarkets.

1Introduction

Pricingcontingentclaimsonpowerpresentsnumerousdi culties.Thepriceprocessforpowerishighlynon-standard,andisnotwellcapturedbypriceprocessmodelscommonlyemployedtopriceinterestrateorequityderiva-tives.Electricity“spot”pricesexhibitextremenon-linearities.Thevolatilityofpowerpricesdisplaysextremevariationsoverrelativelyshorttimeperiods.Furthermore,powerpricesexhibitsubstantialmeanreversionandseasonal-ity.Noreducedform,low-dimensionpriceprocessmodelcanreadilycapturethesefeatures.Finally,andperhapsmostimportant,thenon-storabilityofpowercreatesnon-hedgeablerisks.Thus,preferencefreepricinginthestyleofBlack-Scholesisnotpossibleforpower.

Toaddresstheseproblems,thisarticlepresentsanequilibriummodeltopricepowercontingentclaims.Thismodelutilizesanunderlyingdemandvariableafuelpriceasthestatevariables.Thedemandvariablecanbeoutput(referredtoas“load”)ortemperature.Thepriceofpoweratthematurityofthecontingentclaimisrelatedtothestatevariablesthroughaterminalpricingfunction.Thispricingfunctionestablishesthepayo ofthecontingentclaim,andthusprovidesoneoftheboundaryconditionsrequiredtovalueit.Givenaspeci cationofthedynamicsofthestatevariablesandtherelevantboundaryconditions,conventionalPDEsolutionmethodscanbeusedtovaluethecontingentclaim.

Sincetherisksassociatedwiththedemandstatevariablearenothedge-able,anyvaluationdependsonthemarketpriceofriskassociatedwiththisvariable.Weallowthemarketpriceofrisktobeafunctionofload.Given

thisfunction,itisrelativelystraightforwardtosolvethe“direct”problemofvaluingpowerforwardsandoptions.However,sincethemarketpriceofriskfunctionisnotknown,itmustbeinferredfrommarketprices(analogouslytodetermininganimpliedvolatilityorvolatilitysurface).Weuseinverseproblemmethodstoinferthisfunctionfromobservedforwardprices.Thissolutionforthemarketpriceofriskfunctioncanthenbeusedtopriceanyotherpowercontingentclaimnotusedtocalibratetheriskprice.

WeimplementthismethodologytovaluepowerforwardpricesinthePennsylvania-NewJersey-Maryland(“PJM”)market.Theresultsofthisanalysisarestriking.First,giventerminalpricingfunctionderivedfromeithergenerators’bidsintoPJMoreconometricestimates,we ndthatthemarketpriceofriskfordeliveryduringthesummersof1999-2005islarge,andrepresentsasubstantialfractionofthequotedforwardpriceofpower.Inparticular,thisriskpremiumwasaslargeas$50/MWhfordeliveryinJuly2000(representingasmuchas50percentoftheforwardprice),andremainedashighas$19/MWh(ornearly30percentoftheforwardprice)fordeliveryinJuly2005.Second,thismarketpriceofriskfunctionexhibitslargeseasonalities.ThemarketpriceofriskpeaksinJulyandAugust,andissubstantiallysmallerduringtheremainderoftheyear.1

Theseresultsimplythatthemarketpriceofriskfunctioniskeytopricingpowerderivatives.Demandandcostfundamentalsin uenceforwardandoptionprices,butthemarketpriceofriskisquantitativelyveryimportant

Indeed,insomeyearsthereisdownwardbiasinforwardpricesfordeliveriesduringshouldermonths.Bessembinder-Lemon(2002)presentamodelinwhichpricescanbeupwardbiasedfordeliveriesinhighdemandperiodsanddownwardbiasedinlowdemandperiods.1

indeterminingtheforwardpriceofpower,atleastinthecurrentimmaturestateofthewholesalepowermarket.Ignoringthisriskpremiumwillhaveseriouse ectswhenattemptingtovaluepowercontingentclaims,includinginvestmentsinpowergenerationandtransmissioncapacity.

Inadditiontopricingpowerderivatives,theapproachadvancedinthisarticlecanbereadilyextendedtopriceclaimswithpayo sthatdependonpowervolume(i.e.,loadsensitiveclaims)andweather.Indeed,theequilib-riumapproachprovidesanaturalwayofvaluingandhedgingpowerprice,load,andweathersensitiveclaimsinsingleuni edframework.Moretradi-tionalapproachestoderivativevaluationcannotreadilydoso.

Theremainderofthisarticleisorganizedasfollows.Section2presentsanequilibriummodelofpowerderivativespricing.WeimplementthismodelforthePJMmarket;anappendixbrie ydescribestheoperationofthismarket.Section3presentsamethodforestimatingtheseasonallytime-varyingmeanofthedemandprocessrequiredtosolvethevaluationPDE,andimplementsitusingPJMdata.Section4analyzesthemethodsforestimatingtheterminalpricingfunctionsrequiredtoestimateboundaryconditionsusedinsolutionofthePDE.Section5employsinversemethodstosolveforthemarketpriceofriskfunctionandpresentsevidenceonthesizeofthemarketpriceofriskforPJM.Thissectionalsodiscussestheimplicationsofthese ndings.Section6showshowtointegratevaluationofweatherandpowerderivatives.Section7summarizesthearticle.

2AnEquilibriumPricingApproach

Thetraditionalapproachinderivativespricingistowritedownastochasticprocessforthepriceoftheassetorcommodityunderlyingthecontingentclaim.Thisapproachposesdi cultiesinthepowermarketbecauseoftheextremenon-linearitiesandseasonalitiesinthepriceofpower.Thesefeaturesmakeitimpracticaltowritedowna“reducedform”powerpriceprocessthatistractibleandwhichcapturesthesalientfeaturesofpowerpricedynamics.Figure1depictshourlypowerpricesforthePJMmarketfor2001-2003.Anexaminationofthis gureillustratesthecharacteristicsthatanypowerpricedynamicsmodelmustsolve.Lineardi usionmodelsofthetypeunder-lyingtheBlack-ScholesmodelclearlycannotcapturethebehaviordepictedinFigure1;thereisnotendencyofpricestowanderasatraditionalrandomwalkmodelimplies.Pricestendtovibratearoundaparticularlevel(ap-proximately$20permegawatthour)butsometimesjumpupwards,attimesreachinglevelsof$1000/MWh.

Toaddresstheinherentnon-linearitiesinpowerpricesillustratedinFig-ure1,someresearchershaveproposedmodelsthatincludeajumpcomponentinpowerprices.Thispresentsotherdi culties.Forexample,asimplejumpmodellikethatproposedbyMerton(1973)isinadequatebecauseinthatmodelthee ectofajumpispermanent,whereasFigure1showsthatjumpsinelectricitypricesreversethemselvesrapidly.

Moreover,thetraditionaljumpmodelimpliesthatpricescaneitherjumpupordown,whereasinelectricitymarketspricesjumpupandthendeclinesoonafter.BarzandJohnson(1999)incorporatemeanreversionandexpo-

nentiallydistributed(andhencepositive)jumpstoaddressthesedi culties.However,thismodelpresumesthatbigshockstopowerpricesdampoutatthesamerateassmallpricemoves.Thisisimplausibleinsomepowermarkets.GemanandRoncoroni(2006)presentamodelthateasesthiscon-straint,butinwhich,conditionalonthepricespikingupwardbeyondathresholdlevel,(a)themagnitudeofthesucceedingdownjumpisindepen-dentofthemagnitudeoftheprecedingupjump,and(b)thenextjumpisnecessarilyadownjump(i.e.,successiveupjumpsareprecludedoncethepricebreachesthethreshold).Moreover,inthismodeltheintensityofthejumpprocessdoesnotdependonwhetherajumphasrecentlyoccurred.Theseareallproblematicfeatures.Barone-AdesiandGigli(2002)addresstheproblemthrougharegimeshiftingmodel.However,thismodeldoesnotpermitsuccessiveupjumps,andconstrainingdownjumpstofollowupjumpsmakesthemodelnon-Markovian.Villaplana(2004)easestheconstraintbyspecifyingapriceprocessthatisthesumoftwoprocesses,onecontinuous,theotherwithjumps,thatexhibitdi erentspeedsofmeanreversion.Theresultingpriceprocessisnon-Markovian,whichmakesitdi culttouseforcontingentclaimvaluation.

Estimationofjump-typemodelsalsoposesdi culties.Inparticular,areasonablejumpmodelshouldallowforseasonalityinpricesandajumpintensityandmagnitudethatarealsoseasonalwithlargejumpsmorelikelywhendemandishighthanwhendemandislow.GiventhenatureofdemandintheUS,thisimpliesthatlargejumpsaremostlikelytooccurduringthesummermonths.Moreover,changesincapacityanddemandgrowthwilla ectthejumpintensityandmagnitude.Estimatingsuchamodelonthe

limitedtimeseriesdataavailablepresentsextremechallenges.GemanandRoncoroni(2006)allowssuchafeature,butmostothermodelsdonot;fur-thermore,duetothecomputationalintensityoftheproblem,evenGeman-Roncoronimustspecifytheparametersofthenon-homogeneousjumpinten-sityfunctionbasedonaprioriconsiderationsinsteadestimatingitfromthedata.Fittingregimeshiftingmodelsisalsoproblematic,especiallyiftheyarenon-Markovianasisnecessarytomakethemarealisticcharacterizationofpowerprices(Geman,2005).

Evenifjumpmodelscanaccuratelycharacterizethebehaviorofelectricitypricesunderthe“truemeasure,”theyposeacutedi cultiesasthebasisforthevaluationofpowercontingentclaims.Jumpriskisnothedgeable,andhencethepowermarketisincomplete.2Arealisticjumpmodelthatallowsformultiplejumpmagnitudes(andpreferablyacontinuumofjumpsizes)requiresmultiplemarketriskpricesforvaluationpurposes;acontinuumofjumpsizesnecessitatesacontinuumofriskpricefunctionstodeterminetheequivalentmeasurethatisrelevantforvaluationpurposes.Moreover,thesefunctionsmaybetimevarying.Thehighdimensionalityoftheresultingvaluationproblemvastlycomplicatesthepricingofpowercontingentclaims.Indeed,themoresophisticatedthespotpricemodel(withGeman-Roncoronibeingtherichest),themorecomplicatedthetaskofdeterminingthemarketpriceofriskfunctions.

Therearealsodi cultiesinapplyingjumpmodelstothevaluationofvolumetricsensitiveclaims.Forexample,autilitythatwantstohedgeits

Themarketwouldbeincompleteevenifpowerpriceswerecontinuous(asispossibleinthemodelpresentedbelow)becausepowerisnon-storable.Non-storabilitymakesitimpossibletoholdahedging“position”inspotpower.2

revenuesmustmodelboththepriceprocessandthevolumeprocess.Theremustbesomelinkagebetweenthesetwoprocesses.Graftingavolumepro-cesstoanalreadycomplexpriceprocessisproblematic,especiallywhenonerecognizesthatthereislikelytobeacomplexpatternofcorrelationbetweenload,jumpintensity,andjumpmagnitude.

Relatedly,therelationbetweenfuelpricesandpowerpricesisofparticularinteresttopractitioners.Forinstance,the“sparkspread”betweenpowerandfuelpricesdeterminesthepro tabilityofoperatingapowerplant.Therela-tionbetweenfuelandpowerpricesisgovernedbytheprocessoftransformingfuelinputsintopoweroutputs.Thisprocesscangeneratestate-dependentcorrelationsbetweeninputandoutputpricesthatisverydi culttocaptureusingexogenouslyspeci edpowerandfuelpriceprocesses.

Toaddresstheselimitationsoftraditionalderivativepricingapproachesinpowermarketvaluation,weproposeinsteadanapproachbasedontheeconomicsofpowerproductionandconsumption.Inthisapproach,powerpricesareafunctionoftwostatevariables.Thesetwostatevariablescap-turethemajordriversofelectricityprices,arereadilyobservedduetothetransparencyoffundamentalsinthepowermarket,andresultinamodelofsu cientlylowdimensiontobetractible.

The rststatevariableisademandvariable.Tooperationalizeit,weemploytwoalternativede nitions.The rstmeasureofthedemandstateisload.Thesecondistemperature.Sinceloadandtemperaturearesocloselyrelated,theseinterpretationsareessentiallyequivalent.Tosimplifythediscussion,http://teronwediscusshowuseofweatherasthestatevariablepermitsuni edvaluation

andhedgingofpowerprice,powervolume,andweathersensitiveclaims.Ananalysisofthedynamicsofloadfrommanymarketsrevealsthatthisvariableisverywellbehaved.Loadisseasonal,withpeaksinthesummerandwinterforEastern,Midwestern,andSouthernpowermarkets.Moreover,loadforeachofthevariousNationalElectricityReliabilityCoun-cil(“NERC”)regionsisnearlyhomoskedastic.ThereislittleevidenceofGARCH-typebehaviorinload.Finally,loadexhibitsstrongmeanreversion.Thatis,deviationsofloadfromitsseasonally-varyingmeantendtoreversefairlyrapidly.

Wetreatloadasacontrolledprocess.De ningloadasqt,notethatqt≤X,whereXisphysicalcapacityofthegeneratingandtransmissionsystem.3Ifloadexceedsthissystemcapacity,thesystemmayfail,impos-ingsubstantialcostsonpowerusers.Theoperatorsofelectricpowersystems(suchastheindependentsystemoperatorinthePJMregionwediscusslater)monitorloadandintervenetoreducepowerusagewhenloadapproacheslev-elsthatthreatenthephysicalreliabilityofthesystem.4Undercertaintechni-

Thischaracterizationimplicitlyassumesthatphysicalcapacityisconstant.Investmentinnewcapacity,plannedmaintenance,andrandomgenerationandtransmissionoutagescausevariationsincapacity.ThisframeworkisreadilyadaptedtoaddressthisissuebyinterpretingqtascapacityutilizationandsettingX=1.Capacityutilizationcanvaryinresponsetochangesinloadandchangesincapacity.Thisapproachincorporatesthee ectofoutages,demandchanges,andsecularcapacitygrowthonprices.Theonlyobstacletoimplementationofthisapproachisthatdataoncapacityavailabilityisnotreadilyaccessible.Inongoingresearchweareinvestigatingtreatingcapacityasalatentprocess,andusingBayesianeconometrictechniquestoextractinformationaboutthecapacityprocessfromobservedrealtimepricesandload.Theanalysisofprice-loadrelationsinsection3impliesthatloadvariationsexplainmostpeakloadpricevariationsinPJMprices,http://forinformationonemergencyproceduresinPJM.3

calconditions(whichareassumedtoholdherein),theargumentsofHarrisonandTaksar(1983)implythatunderthesecircumstancesthecontrolledloadprocesswillbeare ectedBrownianmotion.5Formally,theloadwillsolvethefollowingSDE:

dqt=αq(qt,t)qtdt+σqqtdut dLut(1)

6whereLutistheso-called“localtime”oftheloadonthecapacityboundary.

uTheprocessLutisincreasing(i.e.,dLt>0)ifandonlyifqt=X,with

dLt=0otherwise.Thatis,qtisre ectedatX.

Thedependenceofthedrifttermαq(qt,t)oncalendartimetre ectsthefactthatoutputdriftvariessystematicallybothseasonallyandwithintheday.Moreover,thedependenceofthedriftonqtallowsformeanreversion.Onespeci cationthatcapturesthesefeaturesis:

αq(qt,t)=μ(t)+k[lnqt θq(t)](2)

Inthisexpression,lnqtrevertstoatime-varyingmeanθq(t).θq(t)canbespeci edasasumofsinetermstore ectseasonal,predictablevariationsinelectricityoutput.Alternatively,itcanberepresentedasafunctionofcalen-dartime ttedusingnon-parametriceconometrictechniques.Theparameterk≤0measuresthespeedofmeanreversion;thelarger|k|,themorerapidthereversalofloadshocks.Thefunctionμ(t)=dθq(t)/dtrepresentstheportionofloaddriftthatdependsonlyontime(particularlytimeofday).

Theconditionsare(1)thereexistsa“penaltyfunction”h(q)thatisconvexinsomeinterval,butisin niteoutsidetheinterval,and(2)intheabsenceofanycontrol,qwouldevolveasthesolutiontodq=μdt+σdW.Thepenaltyfunctioncanbeinterpretedasthecostassociatedwithlargeloads.Ifq>X,thesystemmayfail,resultinginhugecosts.WethankHeberFarnsworthformakingusawareoftheHarrison-Taksarapproach.6ThisisanexampleofaSkorokhodEquation.5

Forinstance,givenlnqt θq(t),loadtendstorisefromaround3AMto5PMandthenfallfrom5PMto3AMonsummerdays.

Theloadvolatilityσqin(1)isrepresentedasaconstant,butitcandependonqtandt.Thereissomeempiricalevidenceofslightseasonalityinthevarianceofqt.

Thesecondstatevariableisafuelprice.Forsomeregionsofthecountry,naturalgasisthemarginalfuel.Inotherregions,coalisthemarginalfuel.Insomeregions,naturalgasisthemarginalfuelsometimesandcoalisthemarginalfuelatothers.Weabstractfromthesecomplicationsandspecifytheprocessforthemarginalfuelprice.Theprocessfortheforwardpriceofthemarginalfuelis:

dft,T=αf(ft,T,t)+σf(ft,T,t)dztft,T(3)

whereft,TisthepriceoffuelfordeliveryondateTasoftanddzisastandardBrownianmotion.NotethatfT,TisthespotpriceoffuelondateT.

Theprocesses{qt,ft,T,t≥0}solve(1)and(3)underthe“true”prob-abilitymeasureP.Topricepowercontingentclaims,weneedto ndanequivalentmeasureQunderwhichde atedpricesforclaimswithpayo sthatdependonqtandft,Taremartingales.SincePandQmustsharesetsofmeasure0,qtmustre ectatXunderQasitdoesunderP.Therefore,underQ,qtsolvestheSDE:

udqt=[αq(qt,t) σqλ(qt,t)]qtdt+σqqtdu t dLt

Inthisexpressionλ(qt,t)isthemarketpriceofriskfunctionanddu tisaQmartingale.Sincefuelisatradedasset,undertheequivalentmeasure

dft,T/ft,T=σfdzt,wheredztisaQmartingale.Thechangeinthedrift

functionsisduetothechangeinmeasure.

De nethediscountfactorYt=exp( t0rsds)wherersisthe(assumeddeterministic)interestrateattimes.(Laterweassumethattheinterestrateisaconstantr.)UnderQ,theevolutionofade atedpowerpricecontingentclaimCis:

YtCt=Y0C0+ t

0CsdYs+ t

0YsdCs

Inthisexpression,http://ingIto’slemma,thiscanberewrittenas:

YtCt=C0+ t

0 CYs(AC+ rsCs)ds+ s t0 C C [du+dz] qs fs t0Ys CudL qs

whereAisanoperatorsuchthat:

AC= C[αq(qt,t) σqλ(qt,t)]qt qt

2C22 2C22 2C+.52σqqt+.52σfft,T+σfσqρqfqtft,T. qt ft,T qt ft,T(4)

Forthede atedpriceofthepowercontingentclaimtobeaQmartingale,itmustbethecasethat:

and

E[ t0Ys(AC+ t

0 C rsCs)ds]=0 s CudL]=0 qsYs

forallt.Since(1)Yt>0,and(2)dLut>0onlywhenqt=X,withaconstantinterestrater,wecanrewritetheseconditionsas:

AC+ C rC=0 t

13(5)

and

C=0whenqt=X q(6)

Itisobviousthat(5)and(6)aresu cienttoensurethatCisamartingaleunderQ;itispossibletoshowthattheseconditionsarenecessaryaswell.Expression(5)canberewrittenasthefundamentalvaluationPDE:7

C C+rC=[αq(qt,t) σqλ(qt,t)]qt t qt

2C22 2C22 2C+.52σqqt+.52σfft,T+σfσqρqfqtft,T qt ft,T qt ft,T

relevantPDEis:

Ft,T

τ= Ft,T[αq(qt,t) σqλ(qt,t)]qt qt222 t,T t,T t,T2222+.52qtσq+.52σfft,T+qtft,Tσfσqρqf qt ft,T qt ft,T(7)Foraforwardcontract,afterchangingthetimevariabletoτ=T t,the(8)

whereFt,TisthepriceattfordeliveryofoneunitofpoweratT>t.

Expression(6)isaboundaryconditionoftheNeumanntype.Thisbound-aryconditionisduetothere ectingbarrierthatisinherentinthephysicalcapacityconstraintsinthepowermarket.8Theconditionhasanintuitiveinterpretation.Ifloadisattheupperboundary,itwillfallalmostcertainly.Ifthederivativeofthecontingentclaimwithrespecttoloadisnon-zeroattheboundary,arbitrageispossible.Forinstance,ifthepartialderivativeispositive,sellingthecontingentclaimcannotgeneratealossandalmostcertainlygeneratesapro t.

Throughachangeofvariables(tonaturallogarithmsofthestatevariables)thisequa-tioncanbetransformedtoonewithconstantcoe cientsonthesecond-orderterms.8Ifthereisalowerboundonload(aminimumloadconstraint)thereexistsanotherlocaltimeprocessandanotherNeumann-typeboundarycondition.7

In(7)-(8),thereisamarketpriceofriskfunctionλ(qt,t).ThevaluationPDEmustcontainamarketpriceofriskbecauseloadisnotatradedclaimandhenceloadriskisnothedgeable.Accuratevaluationofapowercontin-gentclaimthereforedependsonaccuratespeci cationandestimationoftheλ(qt,t)function.

Valuationofapowercontingentclaim(“PCC”)alsorequiresspeci cationofinitialboundaryconditionsthatlinkthestatevariables(loadandthefuelprice)andpowerpricesattheexpirationofaPCC.Inmostcases,thebuyerofaPCCobtainstheobligationtopurchasea xedamountofpower(e.g.,25megawatts)oversomeperiod,suchaseverypeakhourofaparticularbusinessdayoreverypeakhourduringaparticularmonth.Similarly,thesellerofaPCCisobligatedtodelivera xedamountofpoweroversometimeperiod.Therefore,thepayo toaforwardcontractatexpirationis:

F(0)= t

t δ(s)P (q(s),f(s),s)ds(9)

whereFistheforwardprice,q(s)isloadattimes,f(s)isthefuelspotpriceats,δ(s)isafunctionthatequals1iftheforwardcontractrequiresdeliveryofpoweratsand0otherwise,P (.)isafunctionthatgivestheinstantaneouspriceofpowerasafunctionofloadandfuelprice,t isthebeginningofthedeliveryperiodundertheforwardcontract,andt istheendofthedeliveryperiod.Inwords(9)statesthatthepayo totheforwardequalsthevalueofthepower,measuredbythespotprice,receivedoverthedeliveryperiod.Forinstance,iftheforwardisamonthlyforwardcontractforthedeliveryof1megawattofpowerduringeachpeakhourinthemonth,δ(s)willequal1ifsfallsbetween6AMand10PMonaweekdayduringthatmonth,and

willequal0otherwise.

EconomicconsiderationssuggestthatthepricefunctionP (.)isincreas-ingandconvexinq;section4providesevidenceinsupportofthisconjecture.Asloadincreases,producersmustemployprogressivelylesse cientgener-atingunitstoserviceit.Thespotpricefunctionshouldalsobeafunctionofcalendartime,withhigherprices(givenload)inspringandfallmonthsthaninsummermonthsduetothefactthatutilitiesscheduletheirroutinemaintenancetocoincidewiththeseasonaldemand“shoulders.”

http://ingIto’slemma,

dP =Φ(qt,ft,f,t)dt+Pq σqqtdut+Pf σfft,fdzt

with

Φ(qt,ft,f,t)=Pq αq(qt,t)qt+Pf αf(ft,t,t)ft,t

2222 +.5Pqqσqqt+.5Pf fσfft,f+Pqfqtft,fσqσfρqf(10)

whereρqfisthecorrelationbetweenqtandft,T;thiscorrelationmaydependonqt,ft,T,andt.9ThevolatilityoftheinstantaneouspriceinthissetupistimevaryingbecauseP isaconvex,increasingfunctionofq.Speci cally,thevarianceis

22222(qt,ft,t,t)=Pq 2σqqt+Pf 2ft,tσf+2Pf Pq qtft,tρqfσqσf.σP

9(11)ThespotpriceprocessiscontinuousifP hascontinuous rstderivatives.Nonetheless,themarketisstillincompletesinceqtisnottraded.Moreover,whenoutputnearscapacity andhencePqbecomesverylarge,thepricecanappeartoexhibitlargejumpsevenifpricesareobservedathighfrequency(e.g.,hourly).Thespotpriceprocessisalsolikelytobedis-continuousduetodiscontinuitiesingenerators’bidstosellpower.Thesebidsarestepfunctions.

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