The price of power The valuation of power and weather deriva

Abstract. Pricing contingent claims on power presents numerous chal-lenges due to (1) the nonlinearity of power price processes, and (2) time-dependent variations in prices. We propose and implement a model in which the spot price of power is a function of

ThePriceofPower:TheValuationofPowerandWeatherDerivatives

CraigPirrong

UniversityofHouston

Houston,TX77204

713-743-4466

cpirrong@uh.edu

MartinJermakyan

http://

December7,2005

1

Abstract. Pricing contingent claims on power presents numerous chal-lenges due to (1) the nonlinearity of power price processes, and (2) time-dependent variations in prices. We propose and implement a model in which the spot price of power is a function of

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.

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Abstract. Pricing contingent claims on power presents numerous chal-lenges due to (1) the nonlinearity of power price processes, and (2) time-dependent variations in prices. We propose and implement a model in which the spot price of power is a function of

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

3

Abstract. Pricing contingent claims on power presents numerous chal-lenges due to (1) the nonlinearity of power price processes, and (2) time-dependent variations in prices. We propose and implement a model in which the spot price of power is a function of

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

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Abstract. Pricing contingent claims on power presents numerous chal-lenges due to (1) the nonlinearity of power price processes, and (2) time-dependent variations in prices. We propose and implement a model in which the spot price of power is a function of

indeterminingtheforwardpriceofpower,atleastinthecurrentimmaturestateofthewholesalepowermarket.Ignoringthisriskpremiumwillhaveseriouse ectswhenattemptingtovaluepowercontingentclaims,includinginvestmentsinpowergenerationandtransmissioncapacity.

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