2t27fWtRewritingthisindi11erentialformandfurtherapplicationof - PDF document

1 lnKt�lnK0=�2 2t+fWt:Rewr
lnKt�lnK0=�2 2t+fWt:Rewritingthisindierentialform,andfurtherapplicationofIt^o'slemmacompletestheproof. Corollary3. Inthethree-factormarketmodel,thewealthoftheslidingMWhstrategysatisesdKt Kt=3Xi=1idfWit:Theproofisstraightforward,andcloselyfollowsthatabove. Proposition4. GivenaseriesofobservationsfZtgrt=1,Zt2RmandmatricesfHtgrt=1,Ht2Rmn,denote:Xt=argminXt2RnrXt=1(Zt�HtXt)2;t=Zt�HtXt;R=cov(t);Q=cov(Xt):Nowconsiderastate-spacemodeloftheform:Zt=HtXt+t;Xt=d+"t;whereX;d2Rn,"ti:i:d:N(0;Q),andti:i:d:N(0;R).Further,denotebyfbXtgrt=1theKalmanlterestimatorofthestatesequencefXtgrt=1giventheobservationsequencefZtgrt=1.Then,Xt=bXt;t=1;:::;r: Proof. Theequivalencefollowsfromthefollowingsetofequations:bXt=E[XtjZ1;:::;Zt]=E[XtjZt]=cov(Xt;Zt)var(Zt)�1Zt=cov(Xt;Zt)var(Zt)�1Zt=cov((H0tHt)�1H0tZt;Zt)var(Zt)�1Zt=(H0tHt)�1H0tvar(Zt)var(Zt)�1Zt=Xt;wherethesecondequalityfollowsfromtheindependencestructureofthestate-spacemodel,thethirdisastandardresultfrommultivariateregressiontheory,thefourthisimpliedbythechoiceofmatricesQandR,andthefthandthelastbythefactthatXtisaleast-squaresestimator. AAppendix:TechnicalDerivations Denition1. LetK(n;t)bethewealthattimetofatradingstrategyinwhichwestartwithunitwealth,andforeachi=1;:::;n,weinvestallthewealthattimei�1 nttobuytheelectricityfuturescontractmaturingattimei nt,andholdthatcontractuntilmaturity.WedeneKt(thewealthoftheslidingMWh

2 tradingstrategy)tobeKt=limn!1K(n;t): Pro
tradingstrategy)tobeKt=limn!1K(n;t): Proposition2. ThewealthoftheslidingMWhtradingstrategyintheone-factormarketmodel(seeSection3)satisesthefollowingSDE:dKt Kt=dfWt: Proof. Letusxatimet.ItiseasytoseethatK(n;t) K(n;0)=nYi=1F(i nt;i nt) F(i�1 nt;i nt);andtakinglogarithmsonbothsides:lnK(n;t)�lnK(n;0)=nXi=1lnFi nt;i nt�lnFi�1 nt;i nt:(A.1)BydirectapplicationofIt^o'slemma,weobtainlnFi nt;i nt�lnFi�1 nt;i nt=�2 41�e�2t=n+Zi nti�1 nte�(it=n�s)dfWs:Substitutinginto(A.1)andrearranginggiveslnK(n;t)�lnK(n;0)=�2 4n1�e�2t=n+Zt0h(n;s)dfWs;(A.2)wherefors2[i�1 nt;i nt],h(n;s)=e�it=n�s,withi=1;:::;n.Itisclearthatlimn!1n(1�e�2t=n)=2t:(A.3)Sincelimn!1Rt0jh(n;s)�1jds=0,wehavebyelementarypropertiesoftheIt^ointegral(seeforexampleKaratzasandShreve(1997))thatlimn!1Zt0h(n;s)dfWs=fWt:(A.4)Takingthelimitin(A.2),andsubstitutingfrom(A.3)and(A.4),weget PWNo-peak Component Eigenvalue Varianceprop. Cumulativeprop. Comp1 0.016681 0:556604 0.556604 Comp2 0.007961 0:265644 0.822248 Comp3 0.001567 0:052291 0:874539 Comp4 0.001323 0.044151 0.91869 Comp5 0.001092 0.036435 0.955126 Comp6 0.000784 0.026149 0.981274 Comp7 0.000561 0.018726 1 Table14:Summaryofprincipalcomponentsanalysis:Eigenvaluesandproportion(cu-mulative)ofvariationexplainedbyeachcomponent(latter100gives%),PWNo-peak Variable Vector1 Vector2 Vector3 PWN D1OP �0:9

3 95653 �0:08988 �0:010718 PWN W1OP
95653 �0:08988 �0:010718 PWN W1OP �0:08926 0:993245 �0:008249 PWN M1OP �0:007825 0.067204 0:389266 PWN M2OP 0.006119 �0:026976 0:716039 PWN Q1OP �0:02271 0.00765 0:44968 PWN Q2OP �0:009645 �0:002484 0:234895 PWN Y1OP �0:000947 �0:008835 0:279623 Table15:Summaryofprincipalcomponentsanalysis:Eigenvectorscorrespondingtothethreelargesteigenvalues(productsexplainedinSection4),PWNo-peak PWNpeak Component Eigenvalue Varianceprop. Cumulativeprop. Comp1 0.019741 0:731021 0.731021 Comp2 0.005354 0:198251 0.929272 Comp3 0.00091 0:03371 0:962982 Comp4 0.000404 0.014959 0.977941 Comp5 0.000259 0.009577 0.987518 Comp6 0.000189 0.007007 0.994526 Comp7 0.000148 0.005474 1 Table12:Summaryofprincipalcomponentsanalysis:Eigenvaluesandproportion(cu-mulative)ofvariationexplainedbyeachcomponent(latter100gives%),PWNpeak Variable Vector1 Vector2 Vector3 PWN D1P �0:99371 �0:106958 0.030252 PWN W1P �0:091029 0:94107 0:308796 PWN M1P �0:054421 0:264436 �0:576325 PWN M2P �0:022684 0.142541 �0:505413 PWN Q1P �0:023795 0.10449 �0:522919 PWN Q2P �0:010231 0.041393 �0:179128 PWN Y1P �0:010353 0.008408 �0:102983 Table13:Summaryofprincipalcomponentsanalysis:Eigenvectorscorrespondingtothethreelargesteigenvalues(productsexplainedinSection4),PWNpeak APXo-peak Component Eigenvalue Varianceprop. Cumulativeprop. Comp1 0.064229 0:506172 0.506172 Comp2 0.044631 0:351728 0.8579 Comp3 0.007344 0:057874 0:915774 Comp4 0.004438 0.034973 0.950747 Comp5 0.001817 0.01432 0.965

4 067 Comp6 0.001415 0.011149 0.976215 Com
067 Comp6 0.001415 0.011149 0.976215 Comp7 0.001101 0.008673 0.984888 Comp8 0.000876 0.006904 0.991792 Comp9 0.000719 0.005662 0.997454 Comp10 0.000323 0.002546 1 Table10:Summaryofprincipalcomponentsanalysis:Eigenvaluesandproportion(cu-mulative)ofvariationexplainedbyeachcomponent(latter100gives%),APXo-peak Variable Vector1 Vector2 Vector3 APX D1OP �0:970464 �0:238448 0.027776 APX W1OP �0:23858 0:9696 �0:011324 APX M1OP �0:028456 �0:017072 �0:975946 APX M2OP 0.016435 �0:004221 �0:211689 APX Q1OP 0.002488 �0:005064 �0:031079 APX Q2OP �0:01105 0.016967 �0:006574 APX Q3OP 0.001478 0.007865 �0:018573 APX Q4OP 0.002619 0.005862 �0:01399 APX Y1OP 0.007075 �0:047706 �0:003176 APX Y2OP �0:003301 �0:004917 �0:016072 Table11:Summaryofprincipalcomponentsanalysis:Eigenvectorscorrespondingtothethreelargesteigenvalues(productsexplainedinSection4),APXo-peak APXpeak Component Eigenvalue Varianceprop. Cumulativeprop. Comp1 0.045176 0:782812 0.782812 Comp2 0.009612 0:166558 0.94937 Comp3 0.00139 0:024092 0:973462 Comp4 0.000596 0.010326 0.983788 Comp5 0.000359 0.006222 0.99001 Comp6 0.000198 0.003426 0.993437 Comp7 0.000143 0.002486 0.995923 Comp8 0.000129 0.002244 0.998167 Comp9 0.0000643 0.001114 0.999281 Comp10 0.0000415 0.000719 1 Table8:Summaryofprincipalcomponentsanalysis:Eigenvaluesandproportion(cu-mulative)ofvariationexplainedbyeachcomponent(latter100gives%),APXpeak Variable Vector1 Vector2 Vector3 APX D1P �0:995422 �0:094211 �0:000321 A

5 PX W1P �0:093404 0:985785 0.139115 AP
PX W1P �0:093404 0:985785 0.139115 APX M1P �0:008911 0:11254 �0:780492 APX M2P �0:001495 0:065061 �0:50361 APX Q1P �0:006029 0.037752 �0:25663 APX Q2P �0:009766 0.01901 �0:161852 APX Q3P �0:012391 0.012024 �0:11023 APX Q4P �0:006009 0.008375 �0:078122 APX Y1P 0.001691 0.018686 �0:067913 APX Y2P �0:002178 0.010199 �0:054176 Table9:Summaryofprincipalcomponentsanalysis:Eigenvectorscorrespondingtothethreelargesteigenvalues(productsexplainedinSection4),APXpeak EEXo-peak Component Eigenvalue Varianceprop. Cumulativeprop. Comp1 0.019391 0:687414 0.687414 Comp2 0.003903 0:138377 0.825792 Comp3 0.001868 0:06622 0:892012 Comp4 0.000697 0.024717 0.916728 Comp5 0.000421 0.014909 0.931637 Comp6 0.000386 0.013695 0.945332 Comp7 0.000304 0.010762 0.956094 Comp8 0.000271 0.009612 0.965706 Comp9 0.000237 0.008397 0.974103 Comp10 0.000176 0.006232 0.980335 Comp11 0.000143 0.005058 0.985393 Comp12 0.000121 0.004306 0.989699 Comp13 0.000108 0.003841 0.99354 Comp14 0.000105 0.003724 0.997264 Comp15 0.0000772 0.002736 1 Table6:Summaryofprincipalcomponentsanalysis:Eigenvaluesandproportion(cu-mulative)ofvariationexplainedbyeachcomponent(latter100gives%),EEXo-peak Variable Vector1 Vector2 Vector3 EEX D1OP 0:995054 �0:089808 �0:03918 EEX W1OP 0.094304 0:965934 0:181016 EEX M1OP 0.015469 0.06809 0:262071 EEX M2OP 0.017821 �0:020692 0:366053 EEX M3OP 0.009045 �0:065764 0:316446 EEX M4OP 0.004256 �0:103131 0:303805 EEX M5OP �0:002925 �0:038423 0:204193 EEX M6

6 OP �0:001416 �0:053985 0:168283 EE
OP �0:001416 �0:053985 0:168283 EEX Q1OP 0.00344 �0:093916 0:417884 EEX Q2OP �0:00329 �0:045614 0:238909 EEX Q3OP 0.003313 0.016259 0.115969 EEX Q4OP 0.011723 �0:007383 0:201498 EEX Y1OP 0.003141 �0:110891 0:306071 EEX Y2OP 0.002942 �0:074536 0:200298 EEX Y3OP 0.010787 �0:073603 0:284638 Table7:Summaryofprincipalcomponentsanalysis:Eigenvectorscorrespondingtothethreelargesteigenvalues(productsexplainedinSection4),EEXo-peak EEXpeak Component Eigenvalue Varianceprop. Cumulativeprop. Comp1 0.028918 0:806624 0.806624 Comp2 0.004474 0:124803 0.931427 Comp3 0.000979 0:02732 0:958747 Comp4 0.000655 0.018279 0.977026 Comp5 0.000244 0.006817 0.983843 Comp6 0.000123 0.00343 0.987273 Comp7 0.0000992 0.002768 0.990041 Comp8 0.0000778 0.002171 0.992213 Comp9 0.0000669 0.001866 0.994079 Comp10 0.0000578 0.001611 0.99569 Comp11 0.0000541 0.001508 0.997198 Comp12 0.0000302 0.000844 0.998042 Comp13 0.0000278 0.000775 0.998817 Comp14 0.0000218 0.000608 0.999425 Comp15 0.0000206 0.000575 1 Table4:Summaryofprincipalcomponentsanalysis:Eigenvaluesandproportion(cu-mulative)ofvariationexplainedbyeachcomponent(latter100gives%),EEXpeak Variable Vector1 Vector2 Vector3 EEX D1P 0:991675 0.126394 �0:014239 EEX W1P 0.126809 �0:937227 0:187542 EEX M1P 0.020925 �0:25609 �0:368589 EEX M2P 0.005448 �0:154066 �0:333433 EEX M3P 0.002623 �0:078789 �0:270971 EEX M4P 0.000761 �0:006772 �0:287515 EEX M5P �0:00296 �0:011587 �0:2266 EEX M6P 0.00069 �0:009759 �0:189085 EEX Q1

7 P 0.0022 �0:062955 �0:354689 EEX Q
P 0.0022 �0:062955 �0:354689 EEX Q2P �0:000912 0.00522 �0:239806 EEX Q3P �0:000996 �0:045609 �0:091033 EEX Q4P �0:001788 �0:036149 �0:105014 EEX Y1P 0.000929 0.015394 �0:232396 EEX Y2P �0:002161 0.028415 �0:191589 EEX Y3P �0:000804 0.037517 �0:425509 Table5:Summaryofprincipalcomponentsanalysis:Eigenvectorscorrespondingtothethreelargesteigenvalues(productsexplainedinSection4),EEXpeak Figure13:Riskpremiumterm-structure(%/day)(time-to-mat.days):EEXpeak Figure14:Riskpremiumterm-structure(%/day)(time-to-mat.days):EEXo-peak Figure15:Riskpremiumterm-structure(%/day)(time-to-mat.days):APXpeak Figure16:Riskpremiumterm-structure(%/day)(time-to-mat.days):APXo-peak Figure17:Riskpremiumterm-structure(%/day)(time-to-mat.days):PWNpeak Figure18:Riskpremiumterm-structure(%/day)(time-to-mat.days):PWNo-peak Figure7:Kernel-tinstantaneousriskpremiumbR(
)(100gives%):EEXpeak Figure8:Kernel-tinstantaneousriskpremiumbR(
)(100gives%):EEXo-peak Figure9:Kernel-tinstantaneousriskpremiumbR(
)(100gives%):APXpeak Figure10:Kernel-tinstantaneousriskpremiumbR(
)(100gives%):APXo-peak Figure11:Kernel-tinstantaneousriskpremiumbR(
)(100gives%):PWNpeak Figure12:Kernel-tinstantaneousriskpremiumbR(
)(100gives%):PWNo-peak Figure1:Logarithmicwealthprocess(lnK)againstobservationnumber:EEXpeak Figure2:Logarithmicwealthprocess(lnK)againstobservationnumber:EEXo-peak Figure3:Logarithmicwealthprocess(lnK)against

8 observationnumber:APXpeak Figure4:Logari
observationnumber:APXpeak Figure4:Logarithmicwealthprocess(lnK)againstobservationnumber:APXo-peak Figure5:Logarithmicwealthprocess(lnK)againstobservationnumber:PWNpeak Figure6:Logarithmicwealthprocess(lnK)againstobservationnumber:PWNo-peak Abadir,K.M.,andS.Lawford(2004):\Optimalasymmetrickernels,"EconomicsLet-ters,83,61-68.Bessembinder,H.,andM.L.Lemmon(2002):\Equilibriumpricingandoptimalhedginginelectricityforwardmarkets,"JournalofFinance,57,1347-1382.Clewlow,L.,andC.Strickland(1999):\Valuingenergyoptionsinaonefactormodelttedtoforwardprices,"QuantitativeFinanceResearchPaperno.10,UniversityofTechnology,Sydney.Clewlow,L.,andC.Strickland(2000):EnergyDerivatives:PricingandRiskMan-agement.London:LacimaPublications.Cortazar,G.,Schwartz,E.S.,andL.Naranjo(2003):\Termstructureestimationinlow-frequencytransactionmarkets:AKalmanlterapproachwithincompletepanel-data,"WorkingPaper6-03,AndersonSchoolofManagement,UCLA.Culot,M.,Gon,V.,Lawford,S.,deMenten,S.,andY.Smeers(2006):\Ananejumpdiusionmodelforelectricity,"mimeo,Electrabel.Hinz,J.,vonGrafenstein,L.,Verschuere,M.,andM.Wilhelm(2005):\Pricingelectric-ityriskbyinterestratemethods,"QuantitativeFinance,5,49-60.Karakatsani,N.V.,andD.W.Bunn(2005):\Diurnalreversalsofelectricityforwardpremia,"mimeo,DepartmentofDecisionSciences,LondonBusinessSchool.Karatzas,I.,andS.E.Shreve(1997):BrownianMotionandStochasticCalculus.Berlin:Springer.Koekebakker,S.,andF.Ollmar(2005):\ForwardcurvedynamicsintheNordicelec-tricitymarket,"ManagerialFinance,31,74-95.Longsta&

9 #11;,F.A.,andA.W.Wang(2004):\Electricity
#11;,F.A.,andA.W.Wang(2004):\Electricityforwardprices:Ahigh-frequencyempiricalanalysis,"JournalofFinance,59,1877-1900.Schwartz,E.S.,andJ.E.Smith(2000):\Short-termvariationsandlong-termdynamicsincommodityprices,"ManagementScience,46,893-911.Villaplana,P.(2003):\Pricingpowerderivatives:Atwo-factorjump-diusionap-proach,"WorkingPaper03-18,DepartmentofIndustrialEconomics,UniversityCarlosIII,Madrid. 6ConclusionsInthispaper,wehaveinvestigatedthepresenceandstructureofriskpremiainforwardpricesonthreemajorcontinentalEuropeanmarkets{German,FrenchandDutch.WeconrmpreviousnonparametricresultsobtainedintheliteratureontheAmericanPJM,andUKmarkets,andshowthattheshort-termforwardpricesarenotsimplytheexpectationofthespotprices.Wefurtherlinkthepresenceofariskpremiumwiththepropertiesoftheforwardpricedynamicsatalltime-horizons,anduseittoextendtheriskpremiumanalysisbeyondtheveryshort-term.Takingintoaccountalltheavailablehistoricaldata,wediscoverthepresenceofsignicantriskpremiainthelong-termaswellastheshort-term.Weinfertheshapeoftheriskpremiumterm-structure,i.e.thedependenceoftheriskpremiumontimetomaturityoftheforwardcontract.Wendthattheterm-structureisinagreementwiththetheoreticalmodeloftheelectricitymarketdevelopedbyBessembinderandLemmon(2002).Itre ectsthechangingbalanceoftwoforcesthatdeterminetheriskpremium,namelythesensitivitytoskewnessofthespotprice,andthevariabilityofthespotprice.Asthetimetomaturityincreases,thein uenceofskewnessbecomesrelativelylessimportantcomparedtothevariability,andsotheriskpremiumdecreases

10 . EEX APX PWN Long peak o-peak peak
. EEX APX PWN Long peak o-peak peak o-peak peak o-peak b 0:00115 0:00093 �0:00068 4:9610�5 0:00071 0:00133 (p-value) 0:05702 0:08980 0:60414 0:98622 0:34245 0:32479 b 0:01702 0:01285 0:03379 0:07489 0:01967 0:03519 b 0:005 0:00001 0:01 0:005 0:002 0:005 half-life 138:63 69314:72 69:31 138:63 346:57 138:63 Medium peak o-peak peak o-peak peak o-peak b 0:00079 �0:00309 0:06126 0:02036 0:00229 �0:00115 (p-value) 0:90052 0:73460 0:11878 0:86526 0:60256 0:92568 b 0:17721 0:21353 1:01947 3:13130 0:11403 0:32171 b 0:1 0:15 0:5 0:25 0:1 0:15 half-life 6:93 4:62 1:39 2:77 6:93 4:62 Short peak o-peak peak o-peak peak o-peak b �0:02257 0:02948 �0:08119 �0:13910 �0:01429 0:01392 (p-value) 0:01785 0:00240 0:04543 0:35910 0:03374 0:28698 b 0:26831 0:22699 1:05358 3:95803 0:17472 0:3412 b 1:5 1:3 1:2 1:7 1:4 1:7 half-life 0:46 0:53 0:58 0:41 0:50 0:41 Table3:Estimatedparametersofthethree-factormodel.SeeSection5.1andequa-tions(5.1)-(5.3)forfurtherdetails.Theparameters(i)areestimatedthroughStep1(Section5.1),jointlywithdYit,theunobservedfactors,fori=1;2;3.The'sarethensortedbyincreasingmagnitude,togivethe\long-term"(small),medium-termandshort-term(large)riskfactors.Parametersandcorrespondingtoeachoftheriskfactorsarethenunambiguouslyestimatedusingleastsquares,throughStep2(Sec-tion5.1),andthep-valuefollowsdirectly.Thecontinuous-timehalf-lifeiscalculatedas(ln2)=,andisreportedindays.Re

11 sultsarereportedforpeakando-peakhou
sultsarereportedforpeakando-peakhours,ontheEEX,APXandPWNpowermarkets. risk-factoruncertaintyandthecorrespondingpositiveriskpremium,therebycausingandownwarddriftinforwards.Thepriceskewnesspremiumcausedbyunexpectedpricespikes,andrepresentedbytheshort-termrisk-factorinourmodel,remainspresentinthepeakpricesuntiljustbeforedeliveryandthenquicklydisappears,hencethepositivedrift.Itisnotpresentintheo-peakpricesasthosearenotpronetospikes(theonlyexceptionmaybetheDutchmarket,whereweseeevidenceofanegativeriskpremium,evenintheshort-termo-peakprices{thismayeitherbeaspuriousresult,orcanbeexplainedbyrelativelyhighskewnessofAPXo-peakspotprices).AssuggestedbyFigures7{12,theinstantaneousriskpremiaseemtoevolveovertimeinallthreemarkets.Inparticular,andastobeexpected,theyaresmallerasmarketsbecomemorematureandattractmorespeculators.Wealsoobservethattheriskpremiumexhibitssimilarbehaviouracrossthemarkets.Itwouldbeinterestingtoextendourterm-structureanalysistoanon-constantriskpremiumsetting,andtofurthercompareriskpremiaacrosscontinentalmarkets.Weleavethisforfuturework. dYit=idt+idWit;i=1;2;3; (5.1) dKt Kt=3Xi=1dYit; (5.2) dF(t;T1;T2) F(t;T1;T2)=3Xi=1c(t;T1;T2;i)dYit: (5.3) Afterdiscretization,thesystem(5.1){(5.3)canbeviewedasastate-spacemodelwithstateandobservationvectors(
Y1t;
Y2t;
Y3t),(
Kt=Kt;
F(t;T1;T2)=F(t;T1;T2)).Inprinciple,thiscanbeestimatedusingtheKalmanlter,modiedtoaccountformiss-ingdata(sincewedonotobservealltheforwardpricesonallquotationdays),andcou-pledwithaparameter

12 spacesearchalgorithmasinCortazaretal.(20
spacesearchalgorithmasinCortazaretal.(2003).Kalmanlterestimationis,however,stronglydependentonthechoiceoftheobservationerrorcovari-ancematrix,whichinourcasehasverylargedimensionandisnoteasilyparameterized.Inaddition,theparameterspaceislarge(parameters(1;2;3;1;2;3;1;2;3)),anditisnosimplemattertondaglobaloptimum.Toovercomethisproblem,wede-signatwo-stepleastsquaresestimationprocedureinthespiritoftheone-factormodelestimation. Step1. FinddYitandithatminimize(where123)Xt( dKt Kt�3Xi=1dYit!2+X[T1;T2] dF(t;T1;T2) F(t;T1;T2)�3Xi=1c(t;T1;T2;i)dYit!2): Step2. GivendYitfromStep1,obtainleastsquaresestimatesofiandi.Intuitively,themethodrstnds(1;2;3)andcorresponding(
Y1t;
Y2t;
Y3t)tominimizethemodelpricingerror,andthenestimates(1;2;3;1;2;3)asthemeanandstandarddeviationof(
Y1t;
Y2t;
Y3t).IntheAppendix,wedemonstratethatthisestimatorisinfactequivalenttotheKalmanlter.ParameterestimatesforeachofthemarketsarelistedinTable3.Ofparticularnote,weseethatthehalf-livesassociatedwiththeshort-termfactor(peakproducts)areallroughlyhalfaday(and2),andsotheone-factormodeloverstatestheshort-termrateofreversion(seethediscussioninSection3.1).5.2Term-structureandtemporaldynamicsofriskpremiaUsingtheestimationresultsfromTable3,wecanvisualizetheterm-structureoftheriskpremia;thatis,thedependenceoftheinstantaneousdriftintheforwardpricesonthetimetomaturity.Figures13{18plotthesefor

13 thethreemarketsunderstudy.Theresultsobta
thethreemarketsunderstudy.TheresultsobtainedareinverygoodagreementwiththetheoreticalmodelofBessembinderandLemmon(2002),whopredictnegativeriskpremiacausedbythepresenceofskewnessinthespotpricedistribution,andpositiveriskpremiaduetothelevelofvolatilityinthespotprice.Asthetimetomaturitydecreases,sodoesthelong-termandmedium-term Wt=(W1t;W2t;W3t).Multi-factormodelshavebeenproposedbyseveralauthorsasanappropriateset-upforenergycommodities(seee.g.Culotetal.(2006),KoekebakkerandOllmar(2005)andSchwartzandSmith(2000)fordiscussion).Inthisframework,thespotpriceSt=exp(t+X1t+X2t+X3t),andunderthephysicalmeasurePthethreeriskfactorssatisfydXit=�iXitdt+idWit;i=1;2;3:Theforwardcontractsundertherisk-neutralmeasurefollowdF(t;T) F(t;T)=1e�1(T�t)dfW1t+2e�2(T�t)dfW2t+3e�3(T�t)dfW3t;andbythesamereasoningasabove,therisk-neutralslidingMWhstrategywealthpro-cesssatisesdKt Kt=1dfW1t+2dfW2t+3dfW3t;andunderthephysicalmeasure:dKt Kt=(1+2+3)dt+1dW1t+2dW2t+3dW3t:TheimpliedhistoricalforwarddynamicsarethendF(t;T) F(t;T)=(1e�1(T�t)+2e�2(T�t)+3e�3(T�t))dt+1e�1(T�t)dW1t+2e�2(T�t)dW2t+3e�3(T�t)dW3t:Thisformulaeectivelyallowsustouseallofthehistoricalforwardpricestoestimatethepotentialriskpremiumintheforwardmarketatvarioustimehorizons.Itisalsonoteworthythattheformuladoesnotinvolvethefunctiontappearinginthespotpriceevolutionequation.Hence,

14 ourestimationisrobustwithrespecttothesea
ourestimationisrobustwithrespecttotheseasonalityofpowerprices,whichcanberathercomplexanddiculttomodel.5.1EstimationAsforthesimpleone-factormodelabove,weapproximatethemarketquotedforwardprices F(t;T1;T2)tofacilitateestimation.BydirectapplicationofIt^o'slemma,weobtaindF(t;T1;T2) F(t;T1;T2)=3Xi=1c(t;T1;T2;i)(idt+idWit):ThestochasticprocessesY1t;Y2t;Y3tfollowtheSDE'sdYit=idt+idWit;i=1;2;3,andwecanrewritethesystemtobeestimatedas: EEX APX PWN peak o-peak peak o-peak peak o-peak b �0:04211 0:01325 �0:04656 0:00732 �0:02232 0:00545 (p-value) 1:2210�8 0:01851 1:9610�8 0:38620 3:4310�5 0:19290 b 0:19849 0:16361 0:21905 0:24384 0:14428 0:12141 b 2 0:202 2 0:278 1:59 0:164 half-life 0:35 3:43 0:35 2:49 0:44 4:23 Table2:Estimatedparametersoftheone-factormodel.SeeSection3.1andequations(3.2)and(3.3)forfurtherdetails.Theparametersandareestimatedbyleastsquares(thep-valuefollowsdirectly),andbynonlinearleastsquares.Thecontinuous-timehalf-lifeiscalculatedas(ln2)=,andisreportedindays.Resultsarereportedforpeakando-peakhours,ontheEEX,APXandPWNpowermarkets. peakando-peakhours.Sincetheeigenvectorsassociatedwithdistincteigenvaluesofanormalsquarematrix(0=0)areorthogonal,wemayreducethedimensionof(dlnF=F)bysortingeigenvaluesindecreasingorderofmagnitude,andthenselect-ingonlythoseeigenvectorswhichcontribute\substantially"toexplainingtheobservedvariationindlnF=F.Thisgivesaguidetodeterminingtheappropriatenumber

15 ofrisk-factorsthatareneeded.Resultsoneig
ofrisk-factorsthatareneeded.ResultsoneigenvaluesandeigenvectorsaregiveninTables4{15,whereelementsofeigenvectorsthatexceed0:15inabsolutevaluearehighlighted.Notationally,werefertoforwardproductsas(e.g.)EEX D1P,i.e.theday-aheadEEXpeakforward.OtherproductsareW#(week),M#(month),Q#(quarter)andY#(year),where#denotesaperiodrelativetotoday,e.g.W1(forwardcoveringnextweek),andQ2(forwardcoveringquarterafternext).Peakando-peakproductsaredenotedbyPandOP.ForEEX,APXandPWNpeakhours,weseethatthreelargestprincipalcomponentsexplain95:9%,97:3%and96:3%ofthevariationrespectively.ForEEX,APXandPWNo-peakhours,thevaluesarereducedto89:2%,91:6%,and87:5%.Theeigenvectorsassociatedwiththethreelargestprincipalcomponentshaveausefulinterpretation.Forinstance,onEEXpeakhours(Table5),weseethattherstprincipalcomponentcorrespondstoshort-termeects,throughtheday-aheadforward.Thesec-ondandthirdprincipalcomponentscorrespondtomedium-termeects(week-aheadandone-month-aheadandtwo-month-aheadforwards)andlonger-termeects(week-aheaduptothree-year-aheadforwards),respectively.SimilarresultsareseenforbothAPXandPWNpeakhours,andforo-peakhours,althoughtheimpactofthesecondandthirdprincipalcomponentsatlongertimehorizonsisthenreduced.Weconcludethatathree-factormodelisasensibleimprovementoverthesimpleone-factormodeloutlinedabove(atleastwhenconsideringriskpremia),wherethethreefactorscorrespondtoshort-term,medium-term,andlonger-termdrivingforces.Wedevelopthisbelow.5Multi-FactorMarketModelWeextendthesimplemarketmodelofSection3byintroducingt

16 hreerisk-factorsthatdrivetheelectricitys
hreerisk-factorsthatdrivetheelectricityspotprice.Again,westartwithalteredprobabilityspace( ;F;Ft;P)butnowtheltrationFtisgeneratedby3-dimensionalWienerprocess F(t;T1;T2)=1 T2�T1ZT2T1F(t;s)ds:Forthesakeofcomputationallytractableestimation,weapproximatetheforwardpriceF(t;T1;T2)byassumingthatthepricesoftheforwardcontractsforasingletimedeliverywithintheperiod[T1;T2]areapproximatelyequal,i.e.fors;u2[T1;T2]:F(t;s)F(t;u).ApplicationofIt^o'slemmaandrearrangingthengivesdF(t;T1;T2) F(t;T1;T2)=c(t;T1;T2;)(dt+dWt);c(t;T1;T2;)=e�(T2�t)(1�e(T2�T1+1)) (T2�T1)(1�e);wherec(
)isderivedbyinniteseriesexpansionsof(T2�T1)�1PT2T=T1exp(�(T�t)),thatarisesin(dF(t;T1;T2)=F(t;T1;T2))=(dKt=Kt).Directly,weobtainthefollowingsystem:dKt Kt=dt+dWt; (3.2) dF(t;T1;T2)=F(t;T1;T2) dKt=Kt=c(t;T1;T2;): (3.3) Theparametersandareestimatedbyleastsquareson(3.2),andisesti-matedbyminimizingthemeansquarederrorofthedierencebetweentheleftandrighthandsidesof(3.3).Notethat(3.2)maybewritten,afteruseofIt^o'slemma,asdlnKt=(�2=2)dt+dWt.Discretizing(withoneunitoftimebetweenconsecutiveobservations)gives
lnKt=(�2=2)+"t,where"tisN(0,1).Estimationofandfollowsbymaximumlikelihood(leastsquares).TheestimationresultsobtainedfortheGerman,FrenchandDutchmarketsareinTable2.TheyareverysimilartotheinitialnonparametricanalysisinSection2(indeed,compareTables1and2),whichisunsurprisingsinceaone-factormodelonly

17 givesenough exibilitytomodelthestrongsho
givesenough exibilitytomodelthestrongshort-termriskpremium.Thenumericalestimationofinc(t;T1;T2;)isconstrainedsuchthat2[0;2],duetoa atlikelihoodsurfacefor�2.FortheEEXandAPXpeakproducts,b=2,withcorrespondinghalf-life0:35days.Theone-factormodelillustratesthefocusontheshort-termriskpremium,andthemaininsightisthatthehalf-livesonEEX,APXandPWNpeakproductsarealllessthanhalfaday.Thisanalysisisextendedthroughthethree-factormodelbelow,wherewedidnotfacethesamenumericalproblems,andtheestimationisunconstrained.4PrincipalComponentsAnalysisFollowingClewlowandStrickland(2000),weperformaprincipalcomponentsanalysisontheempirical(symmetric)covariancematrixofdlnF=Fforeachmarket,andfor physicalmeasure.WedenotebySt=exp(Xt)thespotpriceofelectricityfordeliveryattimet.Thetimeevolutionofthelog-spotpricefollowsanOrnstein-Uhlenbeckprocesswithconstantspeedofreversion,instantaneousvolatilityandtime-varyingmeanlevelt,satisfyingthestochasticdierentialequation(SDE)dXt=(t�Xt)+dWt;;�0:ThepriceofaforwardcontractattimetfordeliveryattimeTtwillbedenotedF(t;T).Weassumethatateachtimet,aforwardcontractwitheverymaturityTtcanbetraded.AsshownbyClewlowandStrickland(1999),intheabsenceofarbitragethedynamicsoftheforwardpriceF(t;T)underthe(unique)risk-neutralmeasureQequivalenttoP,satisfytheSDEdF(t;T) F(t;T)=e�(T�t)dfWt;(3.1)wherefWtdenotesaWienerprocessunderQ.Therisk-neutraldynamicsofthewealthKtgeneratedbythecontinuoustimeequivalentoftheslidingMWh

18 strategyinthismodelareshownintheAppendix
strategyinthismodelareshownintheAppendixtofollowdKt Kt=dfWt:Asexpected,thewealthprocessofthistradingstrategyisaQ-martingale.However,ourpreliminaryempiricalobservationsshowthatunderthehistoricalprobabilitymeasureP,thewealthprocessKthasa(negative)drift.So,underPweobservethatdKt Kt=dt+dWt:FromtheCameron-Martin-Girsanovtheorem,wehavethatdfWt=dWt+(=)dt,andbysubstitutinginto(3.1)weobtaintheimpliedforwarddynamicsunderthephysicalmeasure:dF(t;T) F(t;T)=e�(T�t)dt+e�(T�t)dWt:Weconcludethatinthesimpleone-factormean-revertingmarketmodeltheempiri-callyobserveddriftintheslidingMWhtradingstrategyshouldimplyadrift(decreasingwithtimetomaturity)intheforwardcontracts.InSection5,wedirectlyextendtheaboveargumenttoamorerealisticmarketmodel.3.1EstimationInpractice,electricitymarketsquoteforwardsfordeliveryoveraperiodoftime(forinstance,aweek,month,quarter,oryear).Let F(t;T1;T2)bethepriceofaforwardcontractquotedattimetfordeliveryovertheperiod[T1;T2].Toexcludearbitragewemusthave s.d.[R]aretheestimatedmeanandstandarddeviation,andisthet�statisticthatE[R]=0.Clearly,thereisaverysignicantnegativeaverageshort-termriskpremiumforallmarkets,duringpeakloadhours,overthesampleperiod.Weobserveapositiveaverageshort-termriskpremiumonEEXo-peakhours,thatissignicantatthe5%level.ThereisnosignicantaverageriskpremiumduringeitherAPXorPWNo-peakhours. EEX APX PWN peak o-peak peak o-peak peak o-peak D 721 846 698 833 717 842 E[R] �0:04211 0:01325 �0:0465

19 6 0:00732 �0:02232 0:00545 s.d.[R] 0:
6 0:00732 �0:02232 0:00545 s.d.[R] 0:19849 0:16361 0:21905 0:24384 0:14428 0:12141  �5:70 2:36 �5:62 0:87 �4:14 1:30 Table1:SummarydescriptivestatisticsforR(i):=lnS(i)�lnF(i);i2[1;D],whereR(
)istheone-dayriskpremium,S(
)andF(
)thespotandforwardpricesrespectively,itheday(observationnumber),andDthetotalnumberofobservationsinthesample.Thesamplemean(andcorrespondingt-statistic,)andsamplestandarddeviation,arereportedforpeakando-peakhours,ontheEEX,APXandPWNpowermarkets. NotethattheslopeoflnK(d)representstheinstantaneousshort-term(one-day)riskpremium,whichcanbeapproximatedondaydby
lnK(d)=
d,whichequalslnK(d)�lnK(d�1)=lnS(d)�lnF(d)=R(d),given
k:=1�Lk,withLthelagoperator,andk=1.Further,(1=2)[lnK(d)�lnK(d�2)]=(1=2)[R(d)+R(d�1)],ifk=2,andanm�stepbackwardsmovingaverageofR(d)whenk=m.Amoreaccurateapproximation,whichisconvenientfromagraphicalviewpoint,isgivenbyconstructingaunivariateNadaraya-WatsonkernelregressionestimatorofR(i)oni(days).ThekernelestimatorofR(x)ateverypointxisgivenbybR(x)=argmin DXi=1(R(i)� )2K((x�i)=h);whereDisthenumberofobservationsinthesample,and isalocally-tconstant.Thebandwidthhcontrolsthedegreeofsmoothing.Itisxedacrossthesample,andisselectedbythe\rule-of-thumb"h=0:15(maxfig�minfig)=0:15(D�1).ThekernelweightingfunctionK(u)ischosentobetheGaussiandensityfunction(2)�1=2exp(�u2=2).Foranintroductiontokerneltechniques,seeAbadirandLaw-ford(2004)andreferencestherein.TheresultsareplottedinFigures7{12,andgiveacl

20 earindicationoftime-varyingtrendsinthein
earindicationoftime-varyingtrendsintheinstantaneousshort-termriskpremia.3Single-FactorMarketModelHere,weintroducetheconceptoftheriskpremiuminastandardtheoreticalmodelofthespotandforwardelectricitymarket,thathasbeenanalyzedby,interalia,ClewlowandStrickland(1999).Denealteredprobabilityspace( ;F;Ft;P),whereFtisa(null-setaugmented)ltrationgeneratedbyaone-dimensionalWienerprocessWt,andPthe 2EmpiricalMotivationToillustrate,weexaminetheperformanceofahypotheticalslidingMWhtradingstrat-egy(discussedinHinzetal.(2005))oneachofthemarkets,forbothpeakloadando-peakhours,andconsidershort-termforwards(namely,day-aheadover-the-counterprices,aspublishedbyPlatts),andspotprices(exchangeaverageclearingprice).WestartwithaninitialcapitalK(0),andrepeatthefollowingstrategyoneachdayi2[1;D]: BuyonForward Ondayi,investallcurrentcapitalK(i�1),inordertobuypowerontheover-the-countermarket(payingtheover-the-counterpriceF(i)). SellonSpot Alsoondayi,resellthepoweronthespotexchange(receivingtheex-changeclearingpriceS(i)),endingwithanewamountofcapitalK(i).Forinstance,ifK(0)=10e,andspotandforwardpricesareS(1)=5e/MWhandF(1)=4e/MWhrespectively,thenwewillbuy:=K(0)=F(1)=2:5MWhontheforwardmarket,andreceiveS(1)=12:5efromsaleonthespotmarket.TheaccumulatedcapitalattheendofdaydDfrompursuingthisstrategy,assumingnowthatK(0)=1,isgivenbyK(d)=di=1S(i)=F(i),sothatlnK(d)=Pdi=1flnS(i)�lnF(i)g.Apositive(negative)averageriskpremiumover[1;d]correspondstolnK(d)greater(less)thanzero,i.e.spotpricegreater(less)thanforwardprice,o

21 naverage.TheaccumulatedlogcapitallnK(d)o
naverage.TheaccumulatedlogcapitallnK(d)overtime,usingactualmarketdata,isplottedinFigures1{6,forEEX,APXandPWNpeakloadando-peakhours.ThespotpriceS(i)isthemean(conditionalonpeakoro-peakhour)exchangeclearingpriceforphysicaldeliveryofpowerondayi.Spotpricesandvolumesoneachofthemarketsaredeterminedinadvancebyatwo-sidedblindauction,organizedbytheindividualexchanges.Untilthemorningofthedaypriortodelivery,marketparticipantsmaycontinuouslyproposeprice/quantitybid/sellcombinationsusinganelectronicsystem,foreachhourofthedeliveryday.Thebidsareenteredintoasealedorderbook,anduponclosureofthebiddingphase,areaggregatedtogivemarketdemandandsupplycurvesforthefollowingday.Theintersectionofeachofthesecurvesgivesthemarketclearingpriceandvolumebyhour.Peakhoursaremarket-specic,andaregivenas:[0700-2300)onAPX,[0800-2000)onEEXand[0800-2000)onPWN.Forfurtherdetails,seetheexchangewebsiteswww.apx.nl,www.eex.deandwww.powernext.fr.TheforwardpriceF(i)isdeterminedonanover-the-counter(bilateral)market,ondayi�1,alsofordeliveryondayi.Thesampleperiodsarefrom01/2001to08/2005,notingthattherearenoforwardquotationsonweekends.Returnsarecalculatedrelativetothenexttradingday.Whileexchangesgive24hourlyspotprices,setondayi�1fordeliveryoni,Platts(seewww.platts.comfordetails)givesbase,peakando-peak atdeliveryforwardprices,wherepeak/o-peakhoursarethesameontheexchangesandtheover-the-countermarket.ThePlattspricesareavolume-weightedmeanofallover-the-counterdealsfordeliveryondayiexecutedduringdayi�1.Therearenogeographicalmismatchesbe

22 tweentheexchangeandover-the-countermarke
tweentheexchangeandover-the-countermarkets(asmayariseonAmericanmarkets),norarethereanytimingmismatches(sincevirtuallyallover-the-counterdealsareexecutedbeforethespotexchangeclosesondayi�1).SummarydescriptivestatisticsforR(i):=lnS(i)�lnF(i)aregiveninTable1,whereDisthenumberofobservations(notnecessarilyconsecutivedays),E[R]and 1IntroductionEarlymodelsofelectricitymarketpricesthatwereinspiredbythenancialliteraturegenerallyarguedthat(discounted)forwardpricesshouldequalcurrentspotprices.Thisno-arbitragereasoningwasbasedonabuy-and-holdstrategythatworkswellintypicalnancialmarketsyetbreaksdowninelectricitymarketsduetothenon-storablenatureofthetradedcommodity.Inasense,electricityforwardsarenotstrictlyspeakingderivatives,becausetheirvalueisnotafunctionofanothertradedasset.However,theyarebasictradablesintheelectricitymarket,andthepresenceofaproperlydenedriskpremiumintheirpricescannotberuledout.Inthispaper,westudythepresenceofriskpremiainthreeofthemostliquidconti-nentalEuropeanelectricitymarkets:Germany(EEX),France(Powernext:PWN)andtheNetherlands(APX).Ouranalysisfollowsandextendstheresultsofrecentworkbyseveralauthors.LongstaandWang(2004)andKarakatsaniandBunn(2005)bothperformnonparametricanalysisofvery-short-termforwardprices,andspotprices,ontheAmericanPJM,andUKelectricitymarkets,andndevidenceofsignicantriskpre-mia.Villaplana(2003)calibratesatwo-factormean-revertingmodelwithjumps,usinghistoricalelectricityspotpricesonthePJMmarketandthencomparesthetheoreticalpriceoftheforwardwiththatquotedo

23 nthemarket,toobtaintheriskpremiuminterms
nthemarket,toobtaintheriskpremiumintermsofthemarketpriceofriskoftheunderlyingrisk-factors.Theformerapproachisunabletoprovideinformationontheriskpremiumoverlongertimehorizons,whilethelatterissensitivetoestimationofseasonalitypatternsinelectricityspotpriceseries(robustestimationofseasonalityistypicallydiculttoperforminelectricitymarkets,seee.g.Culotetal.(2006)fordiscussion).Ourmethodcombinestheadvantagesofthetwoapproachestoprovideestimatesofriskpremiaatvarioustimehorizons,whilebeingrobusttospotpriceseasonality.Moreover,weextendthepreviousworkandshowanapparenttime-evolutionoftheriskpremia,andaclearreductioninmagnitudewithprogressivematurityofthemarkets.WefocusonmajorEuropeanmarkets-German,FrenchandDutch-andconsiderseparatelypeakando-peakprices,asthepropertiesofriskpremiahavebeenshowntodierempiricallyforpeakando-peak(seeKarakatsaniandBunn(2005)).ThetheoreticalequilibriummodelofBessembinderandLemmon(2002)alsopredictsdierentbehaviourforpeakando-peakriskpremiaduetodierentdemandlevelsinthetwoperiods.ThespotpricedatausedinthispaperistakenfromEEX,PWNandAPX,whiletheforwarddataistakenfromPlatts,anindependentenergymarketdatapublishingcompany(furtherdetailsaregivenbelow).Therestofthepaperisorganizedasfollows.InSection2,weperformanonpara-metricanalysisalongthelinesofLongstaandWang(2004)thatshowssignicantriskpremiainshort-termforwardsinallthreemarkets.Werecasttheanalysisinady-namictradingframeworkthatallowsustodemonstratethetime-evolutionoftheriskpremium.Section3introducesasimpleforw

24 ardmarketmodelandlinksthetheoreticalnoti
ardmarketmodelandlinksthetheoreticalnotionofthemarketpriceofrisktotheresultsofSection2.InSections4and5,wemotivatetheuseofathree-factorforwardmarketmodelforrealmarketprices,showthatourmainempiricalresultsconrmsignicantriskpremiainforwardprices,andoeravisualizationoftheriskpremiumterm-structure.Section6concludes. RiskPremiainElectricityForwardPricesPavelDiko,SteveLawford,andValerieLimpensAbstractWeinvestigatethepresenceofsignicantelectricityforwardriskpremia,usingdatafromthreemajorcontinentalEuropeanenergymarkets-German,DutchandFrench.Weintroducetheriskpremiumintheframeworkofastandardelectricityspot/forwardunobservedfactormodel,andderivetheimpliedforwardpricebehaviour.Wethenassesstheterm-structureandtime-evolutionoftheriskpremiaforeachofthemarkets. Correspondingauthor:PavelDiko.Address:Electrabel,avenueEinstein2a,1348Louvain-la-Neuve,Belgium.Email:pavel.diko@electrabel.com.Diko,LawfordandLimpensarewiththeStrategy,ResearchandDevelopmentteam,ElectrabelSA.WeacknowledgesupportfromElectra-belSA,JacquelineBoucherandAndr´eBihain.Allremainingerrorsareourown.ThispaperwascompiledusingMiktex,andnumericalresultswereobtainedusingEviewsandPython. StudiesinNonlinearDynamics&Econometrics Volume10,Issue32006Article7NONLINEARANALYSISOFELECTRICITYPRICES RiskPremiainElectricityForwardPricesPavelDikoSteveLawfordyValerieLimpenszElectrabelS.A.,pavel.diko@electrabel.comyElectrabelS.A.,steve.lawford@electrabel.comzElectrabelS.A.,valerie.limpens@electrabel.comCopyrightc 2006TheBerkeleyElectronicPress.All