AMODELFORHEDGINGLOADANDPRICERISKINTHETEXASELECTRICITYMARKETMichael Coulon(correspondingauthor)Address: ORFEDepartmentPrincetonUniversityPrincetonNJ08544,USATelephone: 6092585894Fax: 6092583791Email: mcoulon@princeton.eduWarrenB. PowellAddress: ORFEDepartmentPrincetonUniversityPrincetonNJ08544,USAEmail: powell@princeton.eduRonnieSircarAddress: ORFEDepartmentPrincetonUniversityPrincetonNJ08544,USAEmail: sircar@princeton.edu0AMODELFORHEDGINGLOADANDPRICERISKINTHETEXASELECTRICITYMARKETAbstract. Energy companies with commitments to meet customers daily electricity demands facethe problemofhedgingloadandpricerisk. We proposeajointmodelforloadandpricedynamics,whichismotivatedbythegoaloffacilitatingoptimalhedgingdecisions,whilealsointuitivelycap-turingthekeyfeaturesof theelectricitymarket. Drivenbythreestochasticfactorsincludingtheloadprocess, ourpowerpricemodel allowsforthecalculationof closed-formpricingformulasforforwardsandsomeoptions,productsoftenusedforhedgingpurposes. Makinguseoftheseresults,weillustrateinasimpleexamplethehedgingbenetof theseinstruments, whilealsoevaluatingtheperformanceofthemodelwhenttedtotheTexaselectricitymarket.Keywords: electricitymarket;structuralmodel;spikes;forwardprices;spreadoptions;hedgingJELClassicationNumbers: C60,C80, G12,G13,Q401. IntroductionInrecentyears, theuseof nancial products, suchasfuturesandoptions, byretail supplierstohedge electricity price anddemandspikes has grown. The occurrence of spikes inelectric-itymarkets, aswell astheirrelationshiptoloads(energydemands)whichhavestrongseasonalcomponents, requiresnon-standardnancial models. Ontheotherhand, havingcontinuous-timestochasticmodelsbuiltaroundBrownianmotion,asistypicalforunderstandingoptionsinnan-cialmarkets,allowsforconvenientpricingformulaswhichcanreducethesimulationburden onanoptimizationprogramforhedgingrisk.Themodelweproposeaimstocapturetheuniquefeaturesandcomplexdependencestructureof electricitypriceandloaddynamicswhileretainingenoughmathematical tractabilitytoallowforsuchpricingresults. Inparticular, weincludeasstatevariablesthekeyfactorswhichdriveelectricityprices,suchas fuel price (naturalgasinparticular),loaditself,and aproxy forcapacityavailable. Weexpresspowerspotpriceasaparametricfunctionof underlyingfactors, includinganadditional regime todescribetheriskofextremepricespikes, whicharemostlikelytooccurwhendemandisrelativelyhigh,forexampleduringtimesofunexpectedlyhightemperatures. Wealsomodelperiodicityandseasonalityinloadandpriceatvarioustimehorizonstoreecthourlypatterns, weekends, andalsoannual eects. Despitetherichdependencestructureembeddedinthemodel, convenientformulasforderivativespricesareavailable, facilitatingthecalibrationtomarketdataandthemodelsapplicationtohedgingproblems.Wechoose toanalyze datafrom the US electricitymarket in Texas,oftenreferred to as ERCOT(ElectricReliabilityCouncilofTexas),afterthenameoftheISO(IndependentSystemOperator)whichmanagestheTexasInterconnectionpowergrid. AlongwiththeEasternandWesternIn-terconnections, itisoneof thethreemainelectricitygridsintheUSandservesover20millioncustomers. Asinmanyelectricitymarketsaroundtheworld, deregulationinTexasoccurredap-proximatelytenyearsago. Sincethen, thehighlyvolatileandquitedramaticbehaviourofpriceshas drawnmuchattentiontothechallenges of electricitypricemodeling. GiventhegrowthofintermittentwindenergyinTexasandthestatessusceptibilitytoheatwavesandotherextremeweather,features suchas price spikes areparticularly important forthe ERCOT market. A strongDate:August2012.1reminderof thiswasprovidedbytheheatwaveof earlyAugust2011, whenthetotal loadhitarecord level of 68.4 GWh, and day-ahead prices for peak afternoon hours reached their cap of $3000perMWhonseveral consecutivedays. Suchextremeeventsmaybeevenmoredramaticinthefuture,followingarecentdecisionbyERCOTtoincreasethecapto$4500eectiveAugust2012andtoincreaseittoashighas$9000by2015.Muchof theliteratureonquantitative models for electricityprices has focusedonextendingtraditional nanceapproachestoaccount for thesespikes, for examplethroughjumpprocesses.Suchapproachestypicallybeginbyspecifyingastochasticprocessdirectlyfortheelectricityspotprice, possibly incorporating several unobservable factors, seasonalfunctions and sometimes multi-ple regimes (typicallylasting just a few hours, so one should not interpret the terminologyregimeto mean a lasting paradigm shift). An early single-factor model by [8] uses a jump diusion process,while[16]separatesthejumpsfromthediusioninatwo-factorversiontoaccountforveryrapidrecovery from price spikes. In [14], the authors instead propose a threshold level above which jumpsbecomenegativetorecoverfromspikes,whileseveralauthors(cf,[24,12])haveinsteadsuggestedregime-switchingmodelstohandle suddenspikesandrapidrecoveries. In[4],ageneralframeworkbasedonsumsof Levyprocesses is advocated, whichcanallowfor someconvenient resultsforforwardprices, whilein[23] theauthorsapplymultivariateLevysemistationaryprocessestotheEEXmarketinEurope. Anotheralternativeistheuseof heavy-taileddistributionssuchastheCauchydistribution,aspresentedin[18]andappliedtotwoUSmarkets,PJMandERCOT.Whiletheaboveworksdierextensivelyinboththeir motivations andmathematical details,theyall sharethecharacteristicof takingspotelectricitypricesas thestartingpointfor asto-chasticmodel,thus placingthem in the categoryof reduced-form models. Whilesuchapproachesmay be successful forcapturing price spikes and overallprice distributions, they rarely capture thecomplicateddependencestructurebetweenprice,loadandotherfactors,whichisequallyvitalforhedgingpurposesinpractice. Hence, weinsteadfavor thecategoryoftenknownas structuralmodels, asreviewedforexampleintherecentsurveypaperof [6]. Insuchamodel, powerpriceiswrittenasafunctionof several underlyingsupplyanddemandfactors, anditsdynamicsarethereforenot specieddirectlythroughanSDE(stochasticdierential equation), butproducedindirectlyasaresultof thedynamicschosenfor thefactors. EarlyworkbyBarlow[3] treateddemand astheonly driving factor,before variousauthors extendedthisbranchof theliteraturetoincludefactorssuchasfuelprices[20,7],capacitychanges[5,10],orboth[11,2].Abenetof thestructural approachis that it makes useof readilyavailable informationonfundamentals suchasmarketloadandinsomecasessupply sideinformationlikegenerationcosts.However, formathematical tractability, itstopsshortofafulldescriptionofall thedetailsofthepricesettingmechanismsuchasoperationalandtransmissionconstraints,insteadsimplyapprox-imatingtheshapeof theelectricitystack. Nonetheless, itreectskeyfeaturesof loadandpricedynamics,suchastheobservationthattimesofhighloadaremorelikelytoproducepricespikes,for example when the highest cost and least ecient units are forced to run to satisfy demand. Thiscloserelationshipbetweenloadandpriceisimportantforenergycompaniestounderstandwhenhedgingtheriskofeitherphysicalassetownershiportheirobligationstoserveretailcustomersatpredeterminedpricelevels. However, therelationshipbetweenpriceandloadisblurredbyeectssuchasoutages,transmissionproblemsandotherconstraintsorshockswhichcansometimespro-ducepricespikesevenatperiodsofloworaveragedemand. Suchcomplicationsoftheelectricitygridcreateachallengeforstructural modelsthatrelyonaclearandconsistentrelationshipbe-tweenpriceand load. Adding additionalunobservable factorssuchasjump processes is acommonreduced-formsolutiontosuchobstacles,butlessinthespiritofthestructuralapproach.2Wethereforeproposeamodelwhichbuildsonthestructuralapproachesmentionedabove, butalsoincorporates someideasfromthereduced-formliteratureinorder toobtainabetter ttotheERCOTmarket. Inparticular, weextendthetypical stack-basedmethodology(e.g., as in[20, 11, 1]) to include a spike regime, in which the price-to-loadrelationship adjusts to reect suchtimes of extreme market conditions. Within each regime, the power price is lognormal, but we showthatthemixingof theselognormalscanproducetheheavy-tailedpricedensitiesobservedinthemarket. The probabilityofbeinginthespikeregimeisalsoassumedtobe load-dependent,yetweretaintheimportantadvantageofclosed-formsolutionsforforwardandoptionprices, exploitingconvenient properties of multivariate Gaussian distributions. Section 2 introduces the model, whileSection3presentstheresultsforforwards, aswell asparameter estimationandcalibration. InSection4,wepresenttherelatedclosed-formoptionpricingresultsandtheninSection5study anapplicationofthemodeltohedginganobligationtoservecustomerload. FinallyweconcludeinSection6.2. Model&MotivationThe electricity price model consists of several separate pieces, correspondingtoeachof theunderlyingstochastic factors followedbytheir linkwithspot power price. Inthis section, weaddresseachoftheseinturn,andintroducetheparametersandnotation.2.1. Load. The primary short-termdriverforelectricitypricesisload,whichisthestartingpointof ouranalysis. Laterwewill incorporate the longer-term eectsof fuel prices, specicallythroughnatural gasprices. Figure1ashowsthestrikingseasonal variationindailyaverageload. Infact0 10 20 30 40 30 60 !an-03 !ul-03 !an-06 !ul-06 !an-07 !ul-07 !an-08 !ul-08 !an-09 !ul-09 !an-10 !ul-10 !an-11 !ul-11 L8CC1 ually Average Load (kMWh) (a)HistoricaldailyaverageERCOTloads0 3 10 13 20 23 30 0 30 100 130 200 230 300 !an-03 !ul-03 !an-06 !ul-06 !an-07 !ul-07 !an-08 !ul-08 !an-09 !ul-09 !an-10 !ul-10 !an-11 !ul-11 Cas prlce ($/m81u) ower prlce ($/MWh) ower (L8CC1 dally avg) naLural Cas (Penry Pub) (b) Historical daily average electricityandgaspricesFigure1. ERCOTloadandelectricitypricesover2005-11. Notethatwechoosetoplotdailyaveragepowerpricesonlyfortherange$0to$300. However,duringtheperiod2005-11,therewereninedayswithaveragesabove$300(includingveinAugust2011),andtwodaysinSept2008withaveragesjustbelowzero(duetoseveral hourlyvaluesbelow-$200).asFigure2shows,theseasonal patternvariessignicantlyhourtohourthroughouttheday. Forexample, hour8hasbothasummerandawinterpeak, whilehour16onlyhasasummerpeakandamuchgreaterpeaktotroughratio. Therearealsoperiodicitiescausedbyweekendswhenbusinessesareclosed.3Werstde-seasonalizetheERCOTloadLt:Lt= S(t) +Lt,wheretheseasonalcomponent(estimatedusinghourlydata)isgivenbyS(t) = a1(h) +a2(h) cos(2t +a3(h)) +a4(h) cos(4t +a5(h)) +a6(h)t +a7(h)1we.Here his thehour, and 1weisanindicatorvariableforweekends;a2toa5arethe seasonalcompo-nents,a6picksuptheupwardtrendvisibleinFigure1a, anda7capturesthedropindemandonweekends. Figure2showsthettedseasonalcomponentsforhours8and16.ThenwettheresidualloadLttoanOrnstein-Uhlenbeck(OU)model:dLt= LLtdt +LdW(L)t.0 10 20 30 40 30 60 !an-03 !ul-03 !an-06 !ul-06 !an-07 !ul-07 !an-08 !ul-08 !an-09 !ul-09 !an-10 !ul-10 !an-11 !ul-11 L8CC1 Load (kMWh) Pour 8 Cbserved Pour 8 Seasonal llL (weekdays) Pour 8 Seasonal llL (weekends) (a)Hour80 10 20 30 40 30 60 70 !an-03 !ul-03 !an-06 !ul-06 !an-07 !ul-07 !an-08 !ul-08 !an-09 !ul-09 !an-10 !ul-10 !an-11 !ul-11 L8CC1 Load (kMWh) Pour 16 Cbserved Pour 16 Seasonal llL (weekdays) Pour 16 Seasonal llL (weekends) (b)Hour16Figure2. ERCOTloadover2005-11forchosenhours,alongwithttedseasonalityfunctions.2.2. Structural ElectricityModel. FromFigure1b, weobservethatelectricitypricesexhibithighvolatilityandnumerousspikes, andtheyseemtouctuatearoundalevel drivenbynaturalgasprices. Weusethewell-knownone-factorSchwartzmodel[21]inwhichthegaspriceGtistheexponentialofanOUprocess:d log Gt= G(mGlog Gt) dt +GdW(G)t,whereW(L)andW(G)areindependent.Wealsointroduceanadditional factorXwhichproxiesfortheeectof capacityoutagesandgridcongestion,andisgivenbyXt= SX(t) +Xt,withseasonalcomponenttreatedsimilarlytothatofload:SX(t) = b1(h) +b2(h) cos(2t +b3(h)) +b4(h) cos(4t +b5(h)).TheprocessXtfollowsd Xt= XXtdt +X dW(X)t,4where theBrownianmotionsW(X)andW(L)arecorrelatedwithparameter. NotethatL andXareassumedtobemean-zeroOUprocesses, sincetheirmeanlevelsareincorporatedintoS(t)andSX(t)respectively.Whileweposeourmodel incontinuoustimetotakeadvantageoftheconvenientpropertiesofBrownian motion, we note that in reality spot power is observed at discrete times, only once per hour(inmostmarkets). See,forinstance,thediscussionin[4,Section1.5]. Thereforeanyspikeisonlyobservedatanhourlyfrequency,andsoweusethefollowingregime-switchingmodelforPtwhichis driven by a sequence of independent random variables mkdened for times tk T = {t1, t2, . . .},thesetcontainingthestartof everyhour. Ateachtk, thevalueofmk {1, 2}isdeterminedbyanindependentcoinipwhoseprobabilitiesdependonthecurrentloadLtk:(1) mk=___1 withprobability1 ps_Ltkss_2 withprobabilityps_Ltkss_,where ps, s and s are positive constants and () is the standard Gaussian cumulative distributionfunction(cdf). Then,foreachtimet,(2) Pt= Gt exp(mk+mkLt +mkXt) fortk t < tk+1, k N.Hence, mkdetermines each hour whether we are in the normal regime (with parameters 1, 1, 1)or thespike regime (withparameters 2, 2, 2). Note that theparameter 1is aredundantparameterinthemodelttingsinceXtisanunobservedvariable. Thus 1couldbesetequalto1ifdesired,butwechoosetokeepitfornotational symmetry,andhencemoreconvenientformulaslater. Theparameterpsrepresentsthemaximumspikeprobability(ie,asLt ),whilesandscontrol theprecisedependenceonload. Asensiblechoicefortheparameterssandsisthemean and standard deviation of the stationary distribution of deseasonalizedloadLt, such that theprobabilityofaspikeisthenlinearinthequantileofload:(3) s= 0, s=L2LThismodelhasthreestochasticfactors,gasGt,loadLtandtheadditionalfactorXt,aswellasaregimeswitchingmechanismtocapturespikes. Wecommentoneachofthese:The gas multiplies afunctionwhichcanbe interpretedas approximating the range ofgenerator heat rates inthemarket. Thismultiplicative structurehasbeenproposedbyother authors (cf. [20, 13, 7]) and reects the fact that fuel costs are typically the dominantdriveroftheproductioncostcurve,whichinturndeterminesbidlevels.Theempirical relationshipbetweenpriceandloadistypicallyconvexandoftenmodelledwithanexponential function(c.f.[22, 10, 19]), as less ecient generators are usedonlyat peakdemandtimes, producingsignicantlyhigher prices. Oftendrivenprimarilybytemperature, demand is well-known to be mean-reverting (around seasonal levels) and oftenchosentobeGaussianasproposedhere.The additional factor Xt is not strictly equal to market capacity, but represents all additionalfactors includingmost notablychanges incapacityavailable andshort-termoutages orcongestion-relatedevents. In this way, we capture the additional volatilitypresent in powerpriceswhichcannotbeexplainedbyloadandgaspriceuctuationsalone.Finally, animportantfeatureof themodel isitssecondregime, whichoccurswithsomespikeprobability, andallows ustoreproducetheextremelyheavy-tailednatureof thepricedistributions. Althoughbothregimeshavethesameexponential form, thechoiceof2, 2, 2allowsforasteeperandpotentiallymorevolatilepricetoloadrelationship. Inaddition, as spikes are observed to increase in likelihood as load increases (but do also occur5ato-peaktimes),weallowthespikeprobabilitytodependonLt,andinparticulartobelinearinthequantileofthedeseasonalizedloaddistribution.Noticethattheresultingmodel forspotpowerpricesisinfactamixtureof lognormal distribu-tions, due to the choice of Gaussian processes for Xt, Ltand log Gt. This convenient form will havebenetsforthepricingofforwardsandoptions,asweshalldiscusslater.Wenotethatvarious straightforwardextensionstotheset-uparepossible. For example, wemight add a third regime if we wish to include negativespikes (and negativeprices), as is discussedfurther in [6]. The regime probabilities could also be allowed to be piecewiselinear in load quantile,for example for markets in which spikes are only observed to occur for demand above some thresholdlevel. Furthermore, in order to obtain longer duration spikes, the transitions between regimes couldbedrivenbyaMarkovchaininwhichtheprevioushoursstateaectsthespikeprobabilitythishour. However, giventheimportantload-dependencebuiltintotheseprobabilities,thisextensionwouldcomeatthecostof lessconvenient forwardandoptionpricingresults, withlittlebenetifwecaremoreaboutthefrequencyofspikesthantheirexacttimingorduration. Finally, whilethecurrentvalueofloadprovidessomeinformationaboutfuturespikeprobability,wemightalsoincorporate additional forward-looking information about supply or demand (e.g., weather forecastsoroutageschedules)intotheregimeprobabilities,alongthelinesof[9]. However, forourcurrentpurposes, thesimplerframeworksuces.2.3. EmpiricalEvidence. Empirical evidence from ERCOT provides justication for the form ofthe model proposed in(2)above. Firstly,aspowerprices inERCOT aremostoftensetby naturalgas generators at the margin, the co-movementof gas and electricityprices overlong time horizonsisstriking, asobservedearlierinFigure1b. Thisappearsconsistentwiththemultiplicativerela-tionship proposed above, and further evidence is provided by Figure 3a which plots monthly averagepowerpricesagainstgaspricesfortheperiod2005-11. Asgasisthemostslowlymovingfactorinthemodel(seeparameterestimateslater),takingmonthlyaverageshighlightsthisrelationshipmorethanothers. Thepointsinthescatterplotarefairlywell tbyastraightlinethroughtheorigin(withslopeapproximately8, correspondingintuitivelytotheheatrateinMWh/mmBTUof thetypicalmarginalunit),althoughexceptionalmonths canoccur,mostnotablythe previouslymentionedcaseofAugust2011,responsiblefortheoutlierintheupperleftcorner.Next,Figure3billustratestherelationshipbetweenpriceandloadinERCOT viaascatterplotforall hoursintheyear2011. Byplottingonlyoneyearof datahere, therelationshipisclearersincegaspricesmovedrelativelyslowlyduring2011. Inaddition, plottinglogpricesallowsustoincludethehugespikes(eg,uptothepricecapof$3000)whilestillillustratingclearlythestrongrelationshipinthetypical priceregion(betweenabout$20and$60). Thisrelationshipappearsclose tolinear (as suggestedinthemodel) for thevast majorityof datapoints, althoughit isworthnotingthatonthislogplotasignicantlowertail of pricesnearzeroisvisibleevenafterremovingallprices below$1. However,our focus inthis model isnot ontheseo-peakprice dropsbutinsteadonthedramaticpositivespikes. Weobservethatthemajorityofthesespikes,aswellas thelargest spikes, tendtooccur at times of veryhighdemand, but that somewhat smallerspikes dooccur at times of lower tomediumdemand. Figure3cillustrates this samepointbyusingaroughdenitionof aspikeas apricethreetimes theaveragemonthlyvalueinagiventime period. The plotshows thatthe probability of aspike does indeedappear toincreaseroughlylinearly (apart from the last data point) in the quantile of demand, as suggested by the model in (2).It is important to understand that any one of the relationships between Pt and a single underlyingfactor will of course be weakened by the volatility of the other factors driving prices. For example, inFigure 3d, we repeat the scatterplot of Figure 3b, but with 2008data added to the 2011 data (and60 20 40 60 80 100 120 140 031013 LlecLrlclLy rlce (monLhly avg) naLural Cas rlce (monLhly avg) (a)Monthlyaveragegasvspower0 1 2 3 4 3 6 7 8 200003000040000300006000070000 Log rlce ($) uemand (MWh) (b)Hourlypricevsloadfor20110 2 4 6 8 10 0.030.20.330.30.630.80.93 robablllLy of `splke' Cuanule of uemand (c)Probabilityofspikevsload2 3 4 3 6 7 8 200003000040000300006000070000 Log rlce ($) uemand (MWh) 2008 2011 (d)Hourlypricevsloadfor2008and2011Figure 3. Relationship between power price and gas price (top left), between powerprice and load (right column), and between spike probability and load (bottom left).the priceaxis shiftedslightly). Recallfrom Figure 1bthat 2008was atimeof veryhighgasprices,andmoregenerallyrecordpricelevels throughout commoditymarkets. As aresult, theentirecloudof 2008pointsintheprice-loadscatterplotisshiftedsignicantlyupwardsrelativetothe2011points. Foreachoftheyears,thelinearrelationshipbetweenlogpriceandloadisreasonablystrong,but ifcombinedtogetherwouldbe muchweaker,causingdicultiesforastructuralmodelbasedonly onload. Thisevidencethushighlightstheimportanceof includinggaspricesinthemodel, bothinordertobetterreproducemovementsobservedinhistorical data, andtocapturethis additional risk in future electricityprice distributions. In Figure 4,we now plot the price ratioPt/Gtagainst load Lt, thus avoidingthe issue discussed above. Although we do not see the highestspikesinthisplot,wedoobservethatthemodelstwoexponentialfunctions (oneforeachregime)can provide a reasonable t to the data. We shall refer back to this plot when describing parameterestimationinthenextsection.72 2.5 3 3.5 4 4.5 5 5.5 6 6.5x 104100102030405060708090100loadpower / gas price ratio datamean of fitted curves (X=0)1 standard deviation (X=1)Figure4. Fittingresult-solidlinesillustrateexponential price-loadrelationshipwhenX=0(at themean), whiledottedlines represent onestandarddeviationbands (X= 1). They-axishasbeentruncatedforillustrationpurposes: minandmaxvaluesare-38and450respectively.3. ParameterEstimation&ForwardCurveCalibrationCalibrating a model to market datais oftena multi-stepprocedure, as will be the case here. Webeginwithhistorical dataforelectricityspotprices, gaspricesandload, constructingestimationprocedures for both the parameters driving the factor dynamics and those determining the shape ofthe electricitystack(the link betweenelectricityprice andits drivers). Next,we turn toelectricityforwardcontracts, derivingexplicit formulaefor prices whichcanthenbeusedtocalibratethemodel toobservedforwardquotes. It is important that aspot pricemodel correctlypriceallavailableforwardcontractsinthemarketbeforebeingusedtopriceothercontractsortotackleotherproblems.3.1. Parameter EstimationfromHistory. All model parameters areestimatedusingsevenyearsof data(covering2005-2011) forday-aheadERCOTpriceandload, aswell asHenryHubspotgasprices. Maximumlikelihoodestimation(MLE)isusedthroughout, andtheestimationprocedureisdividedintothefollowingsteps:Wejointlyestimateall parameterswhichappearin(2)givenobservedvaluesofloadandgasprice,1andundertheassumptionthatXtisastandardGaussianrandomvariableateveryt. ThisassumptionisreasonablesinceweexpectXtobeaveryfastmean-revertingprocess,reectingshort-termnoise. Parameters1and2providetheappropriatescaling1Aswechoosenottoincludearegimewhichallowsfornegativeprices(orpricesverynear0),werstremoveallhourlydatapointsforwhichPt/Gt 0.1(only132outofover61,000). Inordertoimprovethetintheregionsweareinterestedin,wealsoiterativelyexcludedownwardspikesofmorethanthreestandarddeviationsfromthemeanofthe normalregime,whileacknowledgingthat improvements wouldbeneededtomoreaccuratelycapturethe pricedistributionnearzero.8111222ps0.915 2.79e-05 0.237 0.453 6.11e-05 0.741 0.129Table1. ParametersrelatingtothepowerpricefunctionLoad NaturalGas Capacity/NoiseLLGmGGXX92.59 53932 1.069 1.664 0.611 1517 66.07 -0.113Table2. ParametersrelatingtothestochasticprocessesforGt,LtandXtforthisprocess. Thelikelihoodfunction L(1, 1, 1, 2, 2, 2, ps)thatwemaximizeisL =

t_ps_Ltss_122exp__log(Pt/Gt) 22Lt2_2_(4)+_1 ps_Ltss__112exp__log(Pt/Gt) 11Lt1_2__.Next, giventheseparametersforthepowerpricefunction(seeTable1), weusehistoricalday-aheadelectricitypricedatatobackoutthehistoricaltimeseriesforXtimpliedbythemodel,thustreatingthisfactorasanunobservedresidualprocessfromthemodel.For bothLandX, we ndseasonality parameters byminimizing the sumof squareddierencesbetweenhistoricalvaluesandthefunctionsS(t)andSX(t)respectively.2TheOUprocessparametersforLtandXtareestimatedbystandardmaximumlikelihoodestimationusing the timeseriesof hourly residuals afterdeseasonalization. This procedureisdonejointlyforLtandXtinordertoaccuratelyestimatetheircorrelation. Finally, fornaturalgas,werepeatthesameMLEprocedureusing(thelogarithmof)HenryHubspotprices,butobservedatadailylevelinsteadofhourly.Note that for L and X, we allow the seasonality parameters to vary from hour to hour, but applythe OU t to the single hourly time series of residuals since there is only one random factordrivingeachoftheseprocessesinthemodel. Table2showstheresultsofttingtheOUprocessesforallthreefactors. Itisimportanttoobservethatthethreefactorsmoveonverydierenttimescales,withgasveryslowlymean-revertingover months(G=1.07), loadmeanrevertingover severaldays(L=92.6)andXtmeanrevertingonanintra-daytimescale(X=1517). InTable1, weobservethat2> 1,implyingthatinthespikeregimetheexponentialrelationshipbetweenpriceand loadis signicantlysteeper, as expectedin order toproduce extremespikes. In addition,since2> 1,thespikeregimeisalsomorevolatilethanthenormalpriceregime, asisreectedinthescatter plot of Figure 3b. Note that ps= 0.129 implies that we visit the spike regime approximately6.5%ofthetimeonaverage(ps/2), sincetheprobabilityateachtvariesbetween0and12.9%asLtchanges. Finally,the seasonal parameters for both S(t) and SX(t) are listed in Table 3 and varysignicantlyfromhourtohour,aswasillustratedearlierinFigure2.Figure 4 shows the results of tting the power price function, by plotting the ratio Pt/GtagainstLtforall hoursinthedatasetandthensuperimposingthettedexponential curves. Sincetheprice axis has been truncated from above,we can now see clearly the strong link between price andloadinthenormalregime,whichiswellcapturedbythefunctionexp(1 +1Lt),representedby2Whenimplementingthesefunctions, weusetheconventionthattequalsthecalendartimeinyears, suchthatforexamplethedatasetbeginsonJan1st,2005, whent = 2005.9Hour Parameters forLtParameters forXta1a2a3a4a5a6a7b1b2b3b4b51 30502 5881 2.989 -4019 3.023 0.00140 -1 0.092 0.044 2.984 0.322 2.6212 29090 5015 2.957 -3905 3.011 0.00196 60 -0.196 0.119 4.086 0.274 3.0253 28267 4270 2.934 -3833 3.003 0.00214 180 -0.419 0.081 6.062 0.208 3.1114 27884 3681 2.911 -3794 3.023 0.00209 391 -0.553 0.244 6.545 0.130 3.4085 28145 3128 2.883 -3816 3.006 0.00229 872 -0.464 0.425 6.482 0.113 3.2236 29796 2438 2.796 -3896 3.011 0.00257 2174 -0.162 0.538 6.482 0.189 3.2207 32880 1496 2.616 -3609 2.928 0.00286 4498 -0.376 0.578 6.512 0.258 3.4678 34331 1145 2.526 -3521 2.959 0.00337 5160 -0.146 0.736 6.332 0.248 3.5379 34823 2275 2.892 -3877 3.044 0.00673 3974 -0.155 0.477 6.376 0.217 2.95710 35929 4097 2.968 -3951 3.063 0.00643 3229 0.080 0.392 6.288 0.185 2.80711 37231 6247 2.995 -4161 3.014 0.00528 2993 0.212 0.326 6.141 0.151 2.98512 38383 8337 2.998 -4244 2.962 0.00574 2911 0.135 0.316 6.163 0.166 2.81313 39276 10125 2.998 -4466 2.926 0.00532 2891 0.124 0.123 5.570 0.253 3.14514 40324 11730 3.000 -4488 2.883 0.00550 3195 0.180 0.085 4.418 0.356 3.24215 41143 13073 2.998 -4367 2.854 0.00570 3389 0.204 0.205 3.459 0.455 3.20216 41696 13943 3.008 -4193 2.842 0.00578 3471 0.193 0.328 3.406 0.557 3.25017 42001 14182 3.011 -4105 2.854 0.00593 3469 0.250 0.282 3.374 0.585 3.13618 42091 13447 2.997 -4162 2.970 0.00627 3165 0.253 0.122 6.491 0.454 2.74219 41940 11490 2.964 -4823 3.085 0.00633 2804 0.300 0.669 6.295 0.221 2.07420 41091 9845 2.961 -4675 3.013 0.00826 2515 0.156 0.710 6.332 0.179 3.05821 40425 9189 2.978 -4114 2.953 0.00940 2377 0.104 0.501 5.925 0.365 3.42922 39048 8785 3.029 -4234 3.016 0.00989 2055 -0.230 0.338 5.736 0.168 3.26823 36200 7844 3.045 -4177 3.055 0.01103 1436 0.363 0.367 5.710 0.245 3.12424 33053 6820 3.028 -4083 3.064 0.01543 901 0.025 0.170 5.673 0.331 2.836Table3. Parametersforloadandcapacityseasonality(ie,forfunctionsS(t)andSX(t))thelowersolidline. Theuppersolidlineshowsthefunctionexp(2 +2Lt)forthespikeregimewhilethedottedlinesaroundeachsolidlineplotexp(1+ 1Lt 1)andexp(2+ 2Lt 2),indicatingonestandarddeviationmovementsinXt.Notethatsincewedonotconstrainanyparametersintheoptimization, thespike valuemaywell turnouttobelowerthanthenormal valueinsomecases, especiallyforlowloadandlargenegativeXt, since2>1. For example, at thefar left of Figure4, theXt= 1dottedlineforspikesisalmost$1belowthatof thenormal regime. Asaresult, thevalueforps=12.9%islargerthanwemightexpectformaximumspikeprobability, particularlygivenFigure3c. Whileonaverageps/2=6.5%ofsimulatedpointswillbespikes, someofthesewillbeverylowvalueswhichdonotappearasvisiblespikesinatimeseries, butinsteadmixwiththenormal points.While this may be disadvantageous from the perspective of intuition, the tting of a mixture of twolognormals inevitably requires some overlap,and our priority is an appropriate overall distribution.Finally,atrst glanceitmayappearthatthereisawidervariationinpricesformoderateLtthanforhighLt, butthisisdeceptivegiventhehighdensityofpointsinthemiddleregion, aswell asthemanyspikesbeyondthetopoftheplot.3Indeed, thestandarddeviationoflog(Pt/Gt)givenLtisquitestablefordierentLtranges,lendingsupport tothemodel.3In fact,Lt [30000, 40000]for nearly50% ofthe data points,with about one quarter oneither side ofthis range.Callingthesethreebinslow, medium,andhighdemandcases,wecountthattheproportionofpointsabovethehighestdottedlineofFigure4is0.58%, 0.90%, and1.27%respectivelyforeachbin,increasingslightlyaswemightexpect given our load-dependent spike probabilities. If instead we count the proportionabove the spike regime mean,weget 5.39%, 3.60%and3.08%respectively. ThedecreaseherecanbeexplainedbythevisibleoverlapbetweenregimesforlowerLt,wherebythenormalregimecanalsoproducehighenoughvaluestocount.103.2. PowerForwardPrices. Forwardandfuturescontractsareof utmostimportanceinelec-tricitymarketsastheyareoftentradedinmuchhighervolumesthanthespotandarecrucialforthehedgingneedsofbothproducersandconsumersofelectricity. Theycomeinmanyvariationsin dierent markets,including characteristicssuch as physical versus nancial settlementand peakversusopeakversusbaseloaddelivery. Inall cases, powerforwards(orfuturessinceweusethetermsinterchangeablyhere)havedeliveryperiods forwhichtheelectricityisboughtorsoldinadvance. Therefore, instead of a single maturity T, the delivery is typically spread (uniformly) overamonth(orseasonoryear)longperiod,whichcanberepresentedinsteadbyaninterval[T1, T2].Inthecaseofphysicaldelivery, aDecember2012baseloadcontractforexamplespeciesdeliveryof(say)1MWofpowerduringeveryhourinthemonthofDecember2012.Sinceforwardcontractshavenoupfrontpayment, standardno-arbitrageargumentsimplythattheforwardpriceF(t, [T1, T2])attimet < T1< T2isgivenbyF(t, [T1, T2]) =1(T2T1)/t

T[T1,T2]EQt[PT],wheretisthelengthoftimestepinthedeliveryperiod(ie, onehour)andEQtdenotesthetimetconditional expectationundertherisk-neutral pricingmeasureQ. Weassumefornowthattherisk-neutralmeasureisknowninthemarket,orcanbeuniquelydeterminedfromavailabletradedinstruments.AsLandXarenotthemselvestraded, werequireanassumptionabouttheirmarketpricesofriskinordertowritedowntheirdynamicsunderQ. Eitheraconstantordeterministicmarketpriceof riskimpliesthattheprocessesremainOUprocessesunderQ, withunchangedspeedofmeanreversionandvolatility. Inotherwords,onlytheirmeanlevelschange,andsounderQ,dLt= L(mLLt) dt +LdW(L)t,d Xt= X(mX(t) Xt) dt +X dW(X)t, (5)where W(X)and W(L)are nowQ-Brownianmotionscorrelatedwithparameter. WealsoassumethatunderQ,thepowerpricePtisstillgivenby(2),asitwasunderthephysicalmeasureP.Notice that we assume aconstant meanlevel mLfor the loadprocess, but insteadatime-dependentmeanlevel mX(t)fortheadditional factorX. Thischoiceallowsforaconvenientap-proachtocalibratingthemodeltoobservedmarketforwardquotes,anaturalrstpriorityofanyspotpricemodelbeforetacklingotherderivativepricingorhedgingproblems. AnalogouslytotheHull-Whiteapproachtoyieldcurvecalibration(see[17]),anappropriatelychosentime-dependentmean-reversionlevelallowsthemodelcorrectlypriceforwardsofallmaturities. Furthermore,thiscalibrationprocedure is alsoa wayof identifying a risk-neutral measureQ whichis consistentwithnoarbitrageinthemarket. Inparticular,sincewetypicallyobserveonlyoneforwardcontractforeach calendar month (ignoring the issue of peak and o-peak forwards), we suggest choosing mX(t)tobe apiecewiseconstantfunctionwithonejump per month. Eachconstantcanbe foundsucces-sivelyusingtheforwardpriceforthenextavailablematurity. Theavailabilityoftheclosed-formexpressionforforwardpricesincomingpagesgreatlyfacilitatesandspeedsupthisprocedure.For simplicity, we nowconsider the forwardprice Fp(t, T) for ahypothetical contract withdelivery at a single maturity T. Since Fp(t, T) =EQt[PT], the expression above is simply an averageoverforward contractsfor single hours. In order to calculateFp(t, T) in our model, we rst requirethe conditionaldistributionofXTgivenLTunder therisk-neutral measureQ. NotethatgivenXt11andLt,XTandLTarebivariateGaussianwithparameters {X, X, L, L, },where_XTLT_ N__XL_,_2XXLXL2L__andmeansandvariancesarealltimedependent (althoughwewilltypicallysuppress (t, T)toeasenotation):X(t, T) =XteX(Tt)+X_TtmX(u)eX(Tu)du, (6)2X(t, T) =2X2X_1 e2X(Tt)_, (7)L(t, T) =LteL(Tt)+mL_1 eL(Tt)_, (8)2L(t, T) =2L2L_1 e2L(Tt)_, (9)(t, T) =_1XL_XLX+L_1 e(X+L)(Tt)_. (10)HencetheconditionaldistributionofXTgivenLTisXT|LT N_SX(T) +X+XL(LT L), (1 2)2X_.NotealsothattheforwardpriceofgasisdenotedFg(t, T)andsatisesFg(t, T) =EQt[GT]. Thentheforwardpriceofpowerisfoundasfollows:Fp(t, T) = EQt_EQt[PT|LT]_= EQt_GTexp (1 +1LT) EQt[exp(1XT)|LT]_1 ps_LT ss__+ GTexp (2 +2LT) EQt[exp(2XT)|LT] ps_LT ss__= Fg(t, T)EQt_exp(k1 +l1LT)_1 ps_LT ss__+ exp(k2 +l2LT) ps_LT ss__wherekiandlifori {1, 2}arefunctionsoftimegivenbyki(t, T) = i +iS(T) +i_SX(T) +X XLL +12i(1 2)2X_, (11)li(t, T) = i +iXL. (12)Thisexpectationisthencomputedusingthefollowingrelationship(forconstantsa,b,c):(13)_ecx_a +bx_12e12x2dx = e12c2_a +bcb2+ 1_.Thisisaspecialcaseofaformulain[7,6]whichconvenientlyrewritestheintegraloftheproductofanexponential, aunivariateGaussiandensityandGaussiancdfasasimplebivariateGaussiancdf (whichcollapsesagaintoaunivariatecdf whenintegratingover(, )asabove). Inthis12case,weobtainthefollowingclosed-formexpression:Fp(t, T) = Fg(t, T)__exp_k1 +l1L +12l212L___1 ps__Ls +l12L_2L +2s____(14)+ exp_k2 +l2L +12l222L_ps__Ls +l22L_2L +2s____.Given (3), for large enough time to maturity (T t), whenLTis well approximated by its stationarydistribution(ie,formorethanaweekorsosinceL= 92.6),(14)canbewrittenasFp(t, T) = Fg(t, T)_exp_k1 +l1mL +l212L4L__1 ps_l1L4L__+ exp_k2 +l2mL +l222L4L_ps_l2L4L__.0 50 100 150 20000.050.10.150.20.250.30.35price binsprobability history 2 AMsimulation 2 AMhistory 2 PMsimulation 2 PM(a)Price0 1 2 3 4 5 6 7 8x 10600.050.10.150.20.250.30.35price times load binsprobability history 2 AMsimulation 2 AMhistory 2 PMsimulation 2 PM(b)PricetimesLoadFigure5. HistogramsforPt(left)andPt Lt(right): modelsimulationvs. his-toricaldata3.3. ModelPerformance. The histograms in Figure 5 provide some evidence of the models abil-itytocapturehourlypricedistributionsaccurately. Theseplotsweregeneratedbysimulating100pathsofsevenyearsofpriceandloaddynamics,whilealwaysusingthehistoricalgaspricesfromthesevenyearsofdata(2005-2011). Asnaturalgaspriceisaslowlymovingfactorinthemodel,itsdynamicsover asingleseven-year periodcanvarywidely. Therefore, inordertoisolatetheload-to-pricerelationshipandobtainamoremeaningfulcomparisonwithhistoricaldata,wehavechosentoxthepathofgaspricemovementstomatchwithhistory.For illustration purposes, we plot one representative peak hour (2pm) and one representative o-peak hour (2am), which can be observed to dier signicantly. Nonetheless, despite being driven bythesamesourcesofrandomness,simulatedtrajectoriesfor2amand2pmcapturethesedierencesquite wellthrough theseasonalityfunctions S(t)and SX(t). In additiontosimply analyzingprice,therightplotshowshistogramsforpricetimesload,whichwillbeparticularlyimportantlaterinourhedgingexample. The gures indicatethatthecorrelationstructurebetweenpriceandloadiswellreproducedbythemodel.134. OptionPricingFormulasAsnotedabove, theavailabilityofanexplicitformulaforFp(t, T),theforwardpriceofpower,isasignicantadvantageofthemodelproposedinSection2. Wenowshowthatincertaincases,optionpricesarealsoavailableinclosed-form. Weconsidercall optionsonbothelectricityaloneand on the spread between electricityand gas, and note that analogous results are availablefor putoptions. Finally,wemakeuseofthesederivativepricingresults inasimpleexampleofhedging,inwhichaload-servingutilityrmwishestominimizethevarianceofhisrevenuesbyhedgingwithforwardsoroptions.4.1. OptionsonSpotPower. Webeginwithasimplecall optionwithstrikeKandmaturityTontheelectricityspotpricePT. Weassumeaconstantinterestrater. TheoptionpriceVptattimetcanbecalculatedasfollows,rstconditioningonLTaswasthecaseforforwardprices:Vpt= er(Tt)EQt_(PT K)+= er(Tt)EQt_EQt[(PT K)+|LT]_= er(Tt)EQt_EQt_(GTexp (1 +1LT+1XT) K)+|LT_1 ps_LT ss__+EQt_(GTexp (1 +1LT+1XT) K)+|LTps_LT ss__Eachof theinnerexpectationscorrespondstopricingacall optiononalognormal asset, andthuscanbefoundusingthefollowingvariationofBlacksformula: IfY N_Y, 2Y_,thenE__eYK_+_= eY +122Y_Y+2Y log KY_K_Y log KY_Hence,theresultingintegrandcontainstheproductofBlacksformulaandtheregimeprobabilityfunctions, bothof whichincludeunivariateGaussiancdfsof linear functionsofLT, aGaussianrandom variable. Hence,for the outer expectationwe require integratingovera bivariate Gaussiancdf (withcorrelationzerointhiscase), andmakeuseof thefollowingresult, whichisanalogousto(13)butinhigherdimensionsandisaspecial caseof aformulain[6] involvingthetrivariateGaussiancdf:(15)_ecx2_a1+b1x, a2+b2x; _e12x22dx = e12c22_a1 +b1c_b21 + 1,a2 +b2c_b22 + 1;b1b2 +_(b21 + 1)(b22 + 1)_wherea1, a2, b1, b2, c, areconstantsand2(, ; )isthestandardbivariateGaussiancumulativedistributionfunctionwithcorrelation.AfterafewlinesofalgebraweeventuallyobtainVpt= er(Tt)_Fg(t, T) exp_k1 +l1L +12l212L__d+1_K_d1_+ ps2

i=1(1)i_Fg(t, T) exp_ki +liL +12l2i2L_2_d+i, g+i; i_K2_di, gi; i___(16)14whered+i=ki +liL +12l2i2L + log(Fg(t, T)/K) +122i_2i+l2i2L, di= d+i2i+l2i2L_2i+l2i2L,g+i=Ls +li2L_2s+2L, gi= g+ili2L_2s+2L,i=li2L_(2i+l2i2L)(2s+2L), 2i=2G2G_1 e2G(Tt)_+2i (1 2)2X.Asthebivariatenormal cdfiseasytoimplementusingabuiltinfunctioninmostmathematicalprogrammingsoftware(justliketheunivariatecdf), theoptionpricesgivenby(16)veryfasttocompute,eventhoughtheformulaslookalittlecomplicated. Thisisofgreatbenetforexampleif weconsider aportfolioof optionsonmanydierent individual hoursduringamonth-longoryear-longperiod.4.2. SpreadOptionsonSpotPowerandGas. Now,considerinsteadaspread(orexchange)optiononpowerandgas, withpayo(PT hGT)+, wherehisaconstant. Notethatthispayoissometimesusedtoapproximatetherevenuefromowningagas-redpowerplantattimeTwithheat rate h and negligiblenon-fuel operating costs. See [7] for an example of powerplant valuationbased on this idea. In the current model, the calculation of the price Vp,gtof such an option proceedsvery similarly to that shown above, and is in fact slightly simpler, since gas price GTis independentandcanbeimmediatelyfactoredoutofthepayofunction:Vp,gt= er(Tt)EQt_EQt[(PT hGT)+|LT]_= er(Tt)EQt_GTEQt_exp (1 +1LT+1XT) h)+|LT_1 ps_LT ss__+ GTEQt_exp (1 +1LT+1XT) h)+|LTps_LT ss__.EventuallyweobtainVp,gt= er(Tt)Fg(t, T)_exp_k1 +l1L +12l212L__d+1_h_d1_(17)+ps2

i=1(1)i_exp_ki +liL +12l2i2L_2_d+i, g+i;i_h2_di, gi; i___,whered+i=ki +liL +12l2i2Llog(h) +12 2i_ 2i+l2i2L,di=d+i 2i+l2i2L_ 2i+l2i2L, g+i=Ls +li2L_2s+2L, gi= g+ili2L_2s+2L,i=li2L_( 2i+l2i2L)(2s+2L), 2i= 2i (1 2)2X.154.3. Options onForwards. Themajorityof optionstradedincommoditymarkets(andespe-cially on exchanges) reference forward prices as their underlying instead of spot prices. For example,onemaywanttopriceacall optionwithpayo(Fp(T, Tf) K)+atT, oraspreadoptionwithpayo(Fp(T, Tf) hFg(T, Tf))+, wheretheforwardshavematurityTfgreaterthantheoptionmaturityT. Inpracticeamodicationmayoftenberequiredtothispayoduetotheforwardsdeliveryperiod, especiallyforoptionsoncalendar-monthforwards. Inthispaperweignorethiscomplication,andconcentrateinsteadonthesimplerpayostructure,forwhichsomeclosed-formexpressionsarefeasible. Notethatthiscategoryincludesforexampleanoptionwhichallowsthebuyertodecideonedayinadvanceofdeliverywhethertoexerciseornot(soTf T= 1/365).Animportantobservationwhenvaluingoptionsonforwardsinourstructuralmodelisthatthetimescales of thedynamics of our factors varywidelybetweenGt(slowlymean-revertingovermanymonths), Lt(fairlyrapidlymean-revertingover several days) andXt(extremelyrapidlymean-revertingatanintra-daylevel). This isimportantforoptionsonforwards,becausethetime(Tf T)betweenforwardandoptionmaturitydictateswhichdistributionsattimeTwillbeim-portant. Forexample, if Tf T islargeenoughthatXTfconditional ontimeT informationisalreadywell-approximatedbyitsstationarydistribution,thentheforwardpriceFp(T, Tf)canbeassumedindependentofXT,greatlysimplifyingcalculations. Eventhoughmostexchange-tradedoptionsmatureonlyafewbusinessdaysbeforethebeginningofthedeliveryperiodoftheunder-lyingforwardcontract,givenourparameterestimateofX= 1517,thisisalreadylongenoughtomaketheaboveassumptionandhelptoobtainsomeclosed-formresults.Inthecaseofoptionsonelectricityforwards,noclosed-formexpressionisavailable,butpricingbysimulationisstraightforwardandfastthankstoourexpressionforforwardsin(14). Ontheotherhand, foroptionsonspreadsbetweenpowerandgasforwards, Fg(T, Tf)canbepulledoutofboththepayofunctionandtheexpectation, pavingthewayforarathercomplicatedformulafor the optionprice. The result is in the spirit of Geskesclosed-form formula for compound equityoptions[15], sinceitrequiresrstcalculatingavalueof loadLTabovewhichtheoptionwill beexercised.LTthenappearsinsideabivariateGaussiancdfintheresult,aspresentedinAppendixA.5. RevenueHedgingDespitethegrowthofinterestinrecentyearsfromnancial institutions, alargeproportionofthetradinginelectricityforwardsandoptions iscarriedoutbyproducersandconsumers, whohedgeagainstbothupwardsanddownwardspriceswings. Inparticular,weconsiderherethecaseofaretailpowerutilitycompany,whichisclassiedasaconsumer,sinceitmustbuypowerfromthewholesalemarketinordertosatisfyretail customerneeds. Althoughthereal consumersareindividuals and businesses, these end-users are typically subject to a xed (or slowly-varying) priceperMWh,andhencethepowercompanycannotsimplypassontheriskofspotpricemovementstoitscustomers. Instead, itfacesthechoiceof waitingandbuyingtherequiredamountof elec-tricityeachhour from the spot market,orhedging its obligationsinadvancethrough thepurchaseof forwards, optionsorsomecombinationof thetwo. Insomecases, utilitycompaniesalsoowngenerationassetswhichactasanalternativehedgefortheirobligations.In this section,we shall consider a highly simplied versionof this hedging decision,in which weareonlyconcernedaboutdeliveryoverasingledayinthefuture, andwemakeasinglehedgingdecision today. In other words, this is a statichedge, so we do not consider dynamically re-hedgingthroughtimeasmaturityapproaches. Furthermore, weassumethatthermsloadisperfectlycorrelatedwiththetotal ERCOTmarketloadLt, implyingthatthermdeliverstosomexedfraction of the market, say w [0, 1]. (In one case study, we have found correlations of around 90%16betweenrmloadandmarketload,withw 0.5%.) Setting= 1/365andpfixedtobethepriceitchargesitscustomersperMWh,therms revenueRTovertheone-dayperiod[T, T+]inthefutureisgivenbyRT=24

j=1wLTj_pfixedPTj_, whereTj= T+j24.Weallowtradinginthefollowingcontractsinordertohedgethermsrisk:Forwardcontractswithdeliveryonaparticularhourj,withpriceFp(t, Tj)Calloptionsontheseforwards,withpayoatTj ofVpTj= (Fp(Tj , Tj) K(j))+Spark spread options on forwards, with payoVp,gTj= (Fp(Tj , Tj) hFg(Tj , Tj))+Calloptionsonthehourlyspotpowerprice,withpayoVpTj=_PTj K(j)_+Sparkspreadoptionsonspotenergyprices,withpayoVp,gTj=_PTj hGTj_+,where handK(j) are the heat rates andstrikes speciedinthe optioncontracts. We allowthehourlystrikestovaryduetothelargepricevariationthroughtheday, andinthebasecaseconsider onlyat-the-money(ATM) options for all hours. Inthebasecase, wealsoset t tobeJanuary1st, 2013, andmaturityT =t+1 to be one year later, as well as xingconstantsr = 2%, pfixed= 50, w= 1andchoosingcurrentvaluesXt=Lt= 0andlog(Gt) = mG.Notethatwedonotconsiderforwardswithdeliveryperiods here,butinsteadforwardsonindi-vidualhours,withpricesgivenexplicitlyby(14)inourmodel(andsimilarlyforoptionsonsinglehourspotprices, by(16)and(17)). Asaresultthespreadoptionsonforwardsthatweconsiderhavepricesalsogiveninclosed-formin(18)intheappendix. Wecommentthatwhileexchangetraded forward contractsdo not reference specic hours, it is possible to trade such hourly forwardcontracts over thecounter (OTC) for examplethroughcustom-madeso-calledshape productswhichtailortormsexpectedintra-daydemandprole. Foroptionsonforwards,itispossibletocreateandtradeallsortsofdierentproductsbyvaryingmaturitiesanddeliveryperiods,butforsimplicity here we choose only options which have maturity one day before the delivery hour of theforwardcontract. Inotherwords, thebuyermustdecideatthebeginningofonedaywhethertobuypoweratstrikeK(j)foreachhourj {1, . . . , 24}ofthefollowingday.As our example proposes a statichedge, weare not attemptingto eliminateall risk. Instead, weaimtoassessthedegreetowhicheachofthepowerpricederivativeslistedabovecanreducetheriskinthedistributionofrevenueRT,whichdependsonloadtimesprice. Recallthatsincepriceisitselfafunctionofloadinthismodel,derivativesonpriceareimplicitlyalsoderivativesonloaditself. As a measure of risk reduction, we take the common objective of variance minimization. ThevarianceofthermsprotsatTcanbewrittenasVar_RT N

n=1nU(n)T_where the1, . . . , Nrepresentthe quantitiespurchased of Navailablehedgingproducts withpay-osU(1)T, . . . , U(N)T. Formostofouranalysis, wechooseN=1tocompareproductsonebyone.TheeectivenessofeachhedgeisdeterminedbythecovarianceofRTandU(n)T.Inaddition,forsimplicityweassumethatthereisnoriskpremiumpricedintoforwardsorop-tions: mL=mX=0in(5)(andtherisk-neutralmeasureQcoincideswiththephysicalmeasureP). Asaconsequence, themeanofourprotdistributionisessentiallyunaectedbythenumberofforwardsandoptionsthatwepurchase. Therefore,thepricesU(1)t, . . . , U(N)toftheproductsdo17notappearinourobjectivefunctiongivenabove. Whilethisisofcourseunrealistic, itallowsustofocus solelyonthevarianceaspectofthehedging problem,insteadofrequiring amoreinvolvedmean-variance analysis which would yield dierent conclusions depending on the risk premia of thedierentproductstradedinthemarket.0 2 4 6 8 10x 10400.511.522.533.5x 107number of contracts purchasedstandard deviation of revenues forwardscalls on forwardsspreads on forwardscalls on spotspreads on spot(a)Basecase(1yr,ATM)0 2 4 6 8 10x 10400.511.522.533.5x 107number of contracts purchasedstandard deviation of revenues forwardscalls on forwardsspreads on forwardscalls on spotspreads on spot(b)OTMoptions0 2 4 6 8 10x 10400.511.522.53x 107number of contracts purchasedstandard deviation of revenues forwardscalls on forwardsspreads on forwardscalls on spotspreads on spot(c)1monthoptions0 2 4 6 8 10x 1040.500.511.522.53x 107Value at Risk (95%) forwardscalls on forwardsspreads on forwardscalls on spotspreads on spot(d)5%quantileplot(basecase)Figure6. Comparison of variance reduction from dierent hedging strategies (for-wardsvs. various options) under several maturityandstrikescenarios. Bottomrightplotinsteadcomparesrevenuequantile(95%valueatrisk).Inordertoimplementthishedgingproblem, wesimulate200,000pathsofloadandprice, cal-culatingrevenuefor eachpathunder dierent hedgingstrategies. InFigure6, wecanseethesubstantial benet obtainedfromusingderivative products tohedge risk. Inall cases we canclearly observe an optimal hedge quantity on the plot,above which any further purchases increasestheriskoftheportfolioinsteadofhedging. However, thereissignicantvariationinperformanceacrossproducts. Whentakenaloneforwardcontractsarethemosteectiveatvariancereduction.Thisisperhapsnotsurprisingsincetheyshrinktheentiredistribution,insteadoffocusingononetailofthedistributionlikeoptionsdo. Nonetheless,duetotheriskofpowerpricespikes,therev-enuedistributionissignicantlynegativelyskewed, andhencepurchasingevenout-of-the-money(OTM) calloptionscansignicantlyreducevariance. (Thoughnot plotted,thevariancereduction18frombuyingputsisofcoursemuchless.)The next most eective hedge after forward contracts is options on spot power, although it shouldbenotedthatthesearerarelytradedinpractice. Spreadoptionsonspotpowerandgasprovideslightlyless variancereduction, sincetheyessentiallyisolateandhedgeonlytheriskof LtandXt, notthenatural gasrisk. Althoughsuchoptionsarealsorarelytradedasnancial contracts,theprotsfromageneratingasset(ie, apowerplant)arecommonlyapproximated(byignoringoperational and transmission constraints) as spread options (for example, see [7]). Hence,buying anaturalgaspowerplantcanbequiteaneectivehedgeforapowerutilitycompanyinTexas,andmoreover, inpracticeitisimportanttoaccountforanygenerationassetswhenmakinghedgingdecisions. Inaddition, oneshouldconsiderthepurchasesof natural gasrequiredtooperatetheplant,whichwouldosetthegas priceexposure inpowerprices andpotentiallyprovide additionalvariance reduction. Finally we note that options on forwards provide substantially less variance re-duction than options on spot since they do not provide protectionagainstindividual hourly spikes.Inotherwords, decidingonedayinadvanceonwhichhourstoexerciseyouroptionisof courselesseectivethandecidingafterwitnessingthespike!012x 1042.533.54x 10412345678910x 106number of optionsnumber of forwardsstandard deviation of profits(a)Surfaceplot2.52.52.52.53333333.53.53.53.53.53.53.53.544444444.54.54.54.54.5555555.55.55.56666.56.5777.58number of forwardsnumber of options0 2000 4000 6000 8000 10000 12000 14000 16000 180002.42.62.833.23.43.63.84x 104(b)Contourplot(allcontourlabels106)Figure7. Variancereductionwhentradinginbothoptionsandforwards.Whiletherankingofproductsbyhedgingeectivenessandtheotherfeaturesdiscussedabovehold true throughout the four subplots of Figure 6, comparing the dierent scenarios in the subplotsrevealsotherinterestinginsights. WhereasthebasecaseplottedinFigure6acorresponds totrad-inginATMoptions,Figure6bconsidersOTMoptions,withstrikes50%higherthanintheATMcase. The eectivenessof variancereductionviaoptionsissomewhatreduced, sinceonlyasmallerportion of the tail of the distribution is aected. Since deep in-the-money call options behave muchlikeforwardcontracts (except withanupfrontpremium), it isnotsurprisingthat higherstrikecallsmoveusfurtherawayfromthehedgingabilityof forwards. Next, inFigure6c, weshortenthematuritytimeTfromoneyeartoonemonthandobserveanarrowingofthegapbetweencalloptionsandspreadoptionsintheplots. Thisislogical sincethevarianceofgaspricechangesismuchlessoveronemonththanoveroneyear,whileforloadandothernoisethereislittlechange.Hencetheonefactorleftunhedgedbyspreadoptionsislesssignicantonemonthbeforedelivery.Finally, inthelastplot(Figure6d), wenotethatautilitycompanymayinsteadbeinterestedinrisk measuresotherthanvariance,suchasthequantileoftherevenuedistribution,orequivalently19theValueatRisk of thelossdistribution, acommonriskmanagementtool. Plottingthe95%ValueatRisk(VaR), weobservethattherelationshipbetweenthedierenthedgingproductsissimilartothevariancereductioncase,withslightvariationsintheoptimalpurchasequantities.In practice, energy companies typically own large portfolios of both physical and nancial assets,which require continual monitoring and risk management. By combining several dierent products,suchasforwardsandoptions,evengreatervariancereductionmaybepossiblethaninoursimplesingle product discussion above. We illustrate this in Figure 7, where purchases of both forwards andspot spread options are allowed. Forclarity,we present the same results both as a surface plot andacontourplot. Wecanobservethataminimum inthevariancesurfaceappearssomewhereinthemiddle, implying that simultaneously trading in the two products can provide further risk reductionthantradingineitherproduct alone. However,eveninthestaticcase,ndinganoptimalhedgingportfolio across many maturities and many product types may require a sophisticated optimizationalgorithm,andhenceweleavethisforfurtherwork.6. ConclusionInthis paper wehaveproposedandttedamodel for theTexas electricitymarket whichiswell suitedtoaddressingcommongoalsof energycompanies, suchasthepricingandhedgingofportfoliosof assetsandload-servingobligations. Aswehaveseen, thecomplexitiesof electricitymarketsrequireusingmodelswithnumerousdierentcomponents, designedtocapturefeaturesranging from high volatility and dramatic price spikes to subtle periodicities in load, price and theirdependencestructure. Byproposingastructural model withtwopriceregimes, webenetfromthecleanprice-to-loadrelationshipof simplestackmodels, whilestill reproducingthenoisyandcharacteristicallyspikeyhourlypriceseriesoftenobservedinhistorical data. Althoughelectricitymarketscanvarywidelyfromregiontoregionandmayrequiretailor-mademodels, wesuggestthat our framework is especially suited to spike-prone markets driven predominantly by a single fueltype, as is the case for ERCOT. Simulated spot prices from the model are found to match well withobservedhistorical prices. Moreover, weobtainclosed-formsolutionsforforwardprices, whichisof greatbenet forbothcalibrationand toreduce thecomputationalburden whichcansometimesaccompanyhigh-dimensionalpricingandhedgingapplications. Closed-formpricesforoptionsarealsoavailableinseveral cases, includingforspreadoptions, productsof particularimportanceinallenergymarkets. Lastly, throughasimplerevenuehedgingexample,wehaveillustratedoneofthemanypracticalusesofajointpriceandloadmodelforpower,animportantriskmanagementtoolforanyrmexposedtothemanyintricaciesanduncertaintiesofmodernenergymarkets.20AppendixA. PricingResultforSpreadOptionsonForwardsWebeginbyintroducingconvenientshorthandnotationforthetime-dependentfunctionsgivenin(6)-(12). Sincewearenowpricinganoptiononaforward,therearethreerelevanttimepoints(t