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MODELINGWTI PRICES WITHMARKOVCHAINS
ByRichardR.ConnCMA,MBA,CPA,ABV,ERP
Thispaperisacontinuationofatwo‐partseries.ThefirstpaperisentitledDoWTIOilPricesFollowaMarkovChain?InthatinitialworkallthepreliminarydiscussionsurroundingMarkovprobabilities,limits,steadystatesandthecharacteristicsofthe1986through2013WTIpricemovementswereinvestigated.ItwaslearnedthatboththedailyandweeklyWTIpricedatadidexhibitstrongMarkovcharacteristicsandbothquicklyachievedthesteady‐statesthatfollow(whereD=DownState,S=StaySame,andU=UpState):
TABLE1
Daily Data – Steady State
D S U
D 47.44% 47.44% 47.44%
S 1.74% 1.74% 1.74%
U 50.82% 50.82% 50.82%
100.00% 100.00% 100.00%
TABLE2
Weekly Data - Steady State
D S U D 46.827% 46.827% 46.827% S 0.406% 0.406% 0.406% U 52.767% 52.767% 52.767%
100.00% 100.00% 100.00%
AllthatislefttobedoneisconstructaMarkovChainMonteCarlo(MCMC)modelthateffectivelyemploystheseprobabilities.ThereweresomeshortcomingsalsoidentifiedintheWTIdata.Forexample,thereisagreatdealofvariationintheyear‐to‐yearannualpricevolatility.And,theposthocWTIdatadoesnotadheretotheMarkovassumptionoftemporalhomogeneity.Onthecontrary,onanindividualyear‐onlybasisthereisagooddealofvariationintheMarkovprobabilities.Finally,whileithasyettobediscussed,therearesomeboundaryissuesthatmayrequirefiltersand/orlimitstobeincorporatedintoourMonteCarlomodels.
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PURPOSEOFTHEMODEL
Beforestartingtheconstructiondiscussion,itmakessensetospeakabouthowthemodelmightultimatelybeemployed.Forexample,thismodelwouldbequiteusefultoananalystattemptingtopriceaderivativewhereintheunderlyingistheWTIspotrateatsomegivendateinthefuture.ItwouldalsobeusefulforNPVmodelingofconventionaloilandoilsandsprojectsthatareafunctionofthelong‐termWTI.1AnotherpossibleuseofthemodelwouldbetoobservetheresultingrangeoffutureWTIvolatilitiesthatresultunderthevariouspossiblescenarios.
TheproximatecauseofWTIpricemovementsisnot,asthemodelassumes,purelyrandom.Onthecontrary,WTIpricesareundoubtedlythefunctionofamyriadofmacroeconomic,political,geotechnicalandsocioeconomicdriversthatareallinterrelatedandincomprehensiblycomplex.TheEIA,forexample,attemptstopredictfutureWTIprices(andnowBrentLight)viaverysophisticatedeconometricmodelswithhundreds(perhapsthousands)ofinterdependentinputvariables.Undoubtedlythisisthemostscientificallycorrectandrigorousapproach.Buttheproblemwitheconometricmodelsisthattheybecomesocomplex,andmodelingoftheinterdependenciesandtimingoffutureeventsbecomesotenuous,thatthefinaloutputcanlargelybeseentobewhollyrandom.Acaseinpointwouldbetoestimatewhen/ifanotherpoliticalconflict(suchasthe1991GulfWar)mightreoccurandwhatimpactthatmighthaveonWTI.Afurtherexamplewouldbetospeculateuponwhenalternatetechnologiesmightarisethatlessentheworlddemandforcrudeoil.EventheEIA,withallofitsresourcesandsophistication,makesnopretenseatbeingabletopreciselyestimatethefuture.Whileitdoesannuallypublisha‘ReferenceCase’predictionforWTIgoingforwardseveraldecades,italsodoesincludeaLowandHighOilPricecasethatreflectsthescopeofpossibilitiesthatmayoccuriftheunderlyingReferenceCaseassumptionsdonotcometopass.Inthe“AnnualEnergyOutlook2014:withprojectionsto2040”,theEIAaverageHighOilpriceamountsto$223/BBL2andonly$89/BBLfortheLowOilPrice–anaveragedifferenceof$139/BBL(seeGraph1).Thisallowsforaconsiderablerangeofcontingentoutcomes.SowhynotjustusetheEIAestimates–whybothertoconstructaMCMCmodel?ThereasonrelatestobeingabletoproduceamultitudeofplausibleWTIoutcomesthatmayoccuranywherewithintheboundariesoftheEIAHigh/Lowrange.Oftenitisusefultorecastseveral(perhapsthousands)ofreiterationsoftheWTIpricepathwithinthetimespanofinterest.Thisallowstheanalysttotestjusthowsensitivehis/herprojectNPVistodifferentWTIoutcomes.Generallyspeaking,however,wewouldexpecttheMCMCoutputtofallwithintheHigh/LowestimatesoftheEIA.AnyexcursionsbeyondtheseboundariesmeansthatrandomchancehastakenthemodeloutsideoftheeconomicconsiderationsthattheEIAenvisionedandshouldprobablybeconsideredanunsupportableoutlier.Perhapsthemostimportantfunctionofthemodelistoprovideobjectivity.Thefinancialanalystiscertainlyfreetomakehis/herownpredictionofthelong‐termWTIpricepath.Buttheconcernwillalwaysremainastowhetherhe/sheisinflictingapersonalbiasintheestimate.Incontrast,providedtheinputvariablesandconstraintsoftheMCMCmodelarerationallysupported,the1Inthiscase,itislikelythatamonthlyoryearlyaverageWTIpricewillbegeneratedfromthemodeloutput.FewNPVmodelswouldbesopreciseastorequiredailyorweeklyestimates.2Forthe2014to2040periodinclusive.
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outputobtainedmustrepresentonerenditionofwheretheWTIpricepathcouldgo.ThecaveathereisthattheMCMCmodelwillproduceaninfinitenumberofiterations–andthefinancialanalyststillmustexercisecare/integritysoasnottocherrypickpreferredoutcomes.
GRAPH1
OVERVIEWOFTHEMETHODOLOGY
Asexplainedinthefirstpaper,wesetouttofirsttestifWTIhistoricpricemovementsdoexhibitthepropertiesofaMarkovchain(andtheydo)and,ifourinvestigationsupportedthishypothesis,weweregoingtousetheresultingMarkovprobabilitiestoconstructamodelthatsoughttopredictthefuturedirection(i.e.steps,betheyUp,DownorSame)ofthepricemovementswithoutactuallyconsideringthedollarsizeofthosemovements.Decidinghowlarge,indollarterms,eachofthesechangesinstatewillbetheprimaryfocusofthispaper.Wewilloftenrefertothedollarsizeofthischangeinstateasthedailyorweeklydelta.
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EIA 2014 WTI Outllook ‐ Nominal Dollars
Reference Case High Oil Price Case Low Oil Price Case
Source: EIA Annual Energy Outlook 2014
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Further,thesymbolismthatwasemployedwasD,S,UforDown,SameandUprespectively.Therefore,astringor‘chain’ofpricemovementsof,forexample,DSSUrepresentedafiveperiodobservationwhereinthemovementfromthefirsttosecondperiodisrepresentedbytheleftmost“D”finallythefifthperiodendedona“U”Uppricerelativetothepreviousperiod’sprice.
Asanexample,wecouldgraphicallyrepresentthechainofDSDUUSUDUDasfollows:
GRAPH2
Visually,itbecomesreadilyapparentthat,afterthe10pricechanges,theendingpriceatthecloseofperiod11isthesameasthestaringpriceatperiod1.Moreover,thevolatilityofapricechange,excluding‘S’periods,areallconstant:thepricelevelchangesbyanequal‘One’levelperperiod.Further,withsuchagraphicrepresentation,itbecomeseasytodeterminewhatthepricelevelwouldbeatanygivenperiodendbetween1and11(e.g.PriceLevel“3”attheendofperiod4,andLevel“6”inperiods8and10).Forlackofabetterdescriptor,weshallrefertographssuchasGraph2above,whichonlyreflectthehistoricalorprojectedD/S/Ustateshiftsdevoidofpricingdataasa“D/S/UStateSpace”.
Graph3belowprovidesavisualrepresentationoftheactualD/S/UStateSpacefortheweeklydatafromJan.1986throughDec.2013inclusive(1,461periods):
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DSDUUSUDUD
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GRAPH3
Andcomparingthistothegraphofactualweek‐endingpricesthatoccurredbetweenthesedates,itbecomesreadilyapparentastothedifficultieswewillencounterinattemptingtomodelthe‘step’sizeordelta:
GRAPH4
1986 ‐ 2013 Weekly Change of States ‐ Actual Data
Weekly Change of States
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WTI Actual Weekly Price Data
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TherelativeslopeofGraph3doesnotdramaticallychangethroughoutthe26years(savethethreemainperiodsofnegativeslope)–thetendencyalwaysfavoursanUPmovementatapproximatelythesamerateoffrequency.However,fromGraph4wecanseethatpricesremainedrelativelycenteredaroundthe$20/BBLrangefortheentireperiodof1986totheendof2001(Indeed,theaveragepriceperbarrelduringthistimespanis$20.20).Incontrast,fortheperiodofJanuary2004,throughDecemberof2013inclusive,pricelevelstartsat$33andconcludesat$99–andtheaveragepricethroughoutthattimespanis$76.35.Sinceweknowthatthegeneraldegreeofvolatilitypercentdidnotmigratetoanewlevelovertheobservationperiod(seeAppendix1,A1‐Table1),thismustmeanthattheabsolutenominaldollarvalueofthevariancehasincreasedovertheyears.Thedatadoesvalidatethishypothesis.Theaveragedollarvalueoftheabsolutechangeinweek‐over‐weekWTIpricesfortheperiodofJanuary1986throughDecember2001wasonly$0.66/BBL.Thiscontraststoanaverageof$2.12/BBLfortheperiodofJanuary2004throughDecember2013.
InAppendix1wetakethechangingmagnitudeofthepastpricedeltasintoaccountwhenweusetheMCMCmodelto‘fit’asimulatedpricepathtothehistoricaldata.Theendresultisthat,undertherightassumptions,themodeldoessimulatepastpricemovementsquiteclosely.Totheextentthatthisgivesusgreaterconfidenceinusingthemodeltopredictfuturepricemovements,thenAppendix1canbethoughtofasaworthyexercise.Butourrealgoalhereistocomeupwithaplausiblepredictionofthefuture–notthepast.
THEVOLATILITYOFVOLATILITY
OneofthefactorsconfoundingthepredictionofWTIpricesisvolatility.And,itisnotjustthattraditionallythevolatilityofoilpriceshasbeenhigh–comparedwithsomeothercommodities,forexample.Itisalsoduetothefactthatthevolatilityofoilpricesdoesnotremainconstantthroughouttime(seePaperI,Figure4)3.Thatis,therateofvolatilityisvariableonayear‐over‐yearbasis.
ConsiderthatthereareactuallytwocomponentstoWTI‘volatility’.Thefirst,aswehaveidentifiedinPaperI,isthefrequencyof‘D’and‘U’movements.Obviously,themorefrequentDandUshiftsarerelativetoSevents,thenthemorevolatileWTIpriceswillbe.Secondly,therelativesizeoftheDandUshifts(thedelta‐eitherintermsof%changeorabsolutedollarvaluechange)willalsohaveamajorimpactupontheoverallWTIpricevolatility.If,forexample,theaveragedailyUshiftwasonly$.01/BBL/day,thenthiswouldbemuchlessvolatilethatiftheaveragewere$0.50/BBL/day.
Withrespecttothedailydata,forexample,thefollowingstatisticsapplytotheday‐over‐dayrateofchange:
3Curiously,thevolatilityofWTIisnotseasonal.Comparingtheaveragerateofvolatilityofthepast26JanuarysisnotsignificantlydifferentthattheaverageFebruaryvolatilityorMarch…etc.Theaverageofthemonth‐to‐monthvolatilityremainsrelativelyconstant.
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TABLE3
Day-over-Day rate of Change U Deltas D Deltas
Min 0.010% -40.640%4
Max 19.151% -0.010%Mean 1.712% -1.795%
Median 1.261% -1.266%Mode 2.985% -0.238%
SuchawiderangeintherateofchangeindicatesthattherelianceuponanysingleaverageorfixedamountwillnotrealisticallycapturethetruevariabilityinWTIoilprice.WhatweneedtoincorporateinourMonteCarlomodelisarateofvolatilitythatisitselfsubjecttorandomvariability.Itwillbeuseful,therefore,toknowthefrequencyanddistributionoftheratesofchangeoftheperioddelta.
THEDISTRIBUTIONOFHISTORICWTIPRICECHANGES
MuchcanbelearnedbyexaminingthefrequencyhistogramsofthenominaldollarWTIpricechanges:
GRAPH55
4The40.6%decreaseindailypricerelatestotheJan.16,1991startoftheGulfWarandthiswasthelargestsingle‐daychangeinWTIpriceovertheentire26yearhistory.5Thedailypricedeltarangeof‐$5.00to+$5.00isasimplification.Theactualdailydeltainthedatarangesbetween‐$14.86and+$18.56buttheseextremesareinfrequentandhavebeeneliminatedfromthemodeloutcomesonthebasisthattheyareoutliers.
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Nominal Dollar Daily Price Change
Histogram of Daily Price Changes: 1986 ‐ 2013
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GRAPH66
Boththedailyandweeklydataareapproximatelynormallydistributedwithmeans$0.01and$0.05respectively.Separatingthedatasetsintotwogroups,thosebelow$0.00andthoseabovewillprovideuswithprobabilitydistributionsassociatedwiththeDandUtransactions.Forexample,withrespecttothenominaldollarsizeofadailyDevent,thefollowingfrequenciesareobservedinthehistoricaldata:
TABLE4
Range of Daily Price Change
Frequency of Changes
Probability of Occurrence
‐$5.00 ~ ‐$4.51 24 1.70%
‐$4.50 ~ ‐$4.01 9 0.64%
‐$4.00 ~ ‐$3.51 17 1.21%
‐$3.50 ~ ‐$3.01 22 1.56%
‐$3.00 ~ ‐$2.51 32 2.27%
‐$2.50 ~ ‐$2.01 59 4.18%
‐$2.00 ~ ‐$1.51 99 7.02%
‐$1.50 ~ ‐$1.01 178 12.62%
‐$1.00 ~ ‐$.51 283 20.07%
‐$0.50 ~ ‐$.01 687 48.72%
1,410 100.00%
6Theweeklypricedeltarangeof‐$7.00to+$7.00isasimplification.Theactualweeklydeltainthedatarangesbetween‐$14.53and+$13.93buttheseextremesareinfrequentandhavebeeneliminatedfromthemodeloutcomesonthebasisthattheyareoutliers.
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Nominal Dollar Weekly Price Change
Histogram of Weekly Price Changes: 1986 ‐ 2013
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Therefore,inordertoincorporatetheeverchangingvolatilityoftheWTIpricechanges,wewouldwishtoconstructarandomelementinthemodelthatwould,giventhefactthata“D”movementhasalreadybeenpredicted,pricethatdownwarddeltabetween‐$5.00and‐$4.51,1.7%ofthetime,andbetween‐$4.50and‐$4.01,0.64%ofthetime…and,finallybetween‐$0.50and‐$0.01,48.27%ofthetime.7
Insummary,themodelfirstrandomlyselectswhetherthetransactionwillbeaD,SorUmovement(accordingtotheprobabilitiessetoutinTables1and2above).Then,havingdeterminedthedirectionalmovementofthepricepath,it‘prices’thosemovementsrandomly,butinaccordancewiththeprobabilitiesofthedeltadollaramountsthathavebeenobservedinactualWTIhistoricaldatainthe1986to2013period.Atthispoint,somemightquestionthattheuseoftheoldpricedeltadatawillnotreflectfuturenominaldollars.Specifically,anargumentmaybemadethatrelyinguponthepastpricedeltasdoesnotallowforfutureinflation.Thisisnotentirelycorrect.Thishistoricdatadoesinclude26yearsofinflation.Thefrequencyofthe$0.01priceshiftsthatoccurredin1986isaddedtothefrequencyofthe$0.01priceshiftsthathappenedin2013–inspiteofthefactthattherealvalueofthesechanges,intermsofcurrentdaypurchasingpower,isentirelydissimilar.Theactualissueathandhereis:Willexpectedrateofinflationoverthenext26yearsbesignificantlydifferentfromthatofthepast26?And,eventhoughthehistoricaldeltadatadoesinclude26yearsofoilpriceinflation:Canitalsoberepresentativeof52yearsofinflation?Thisisasubtletythatwillbeleftforsubsequentconsideration.8
TheactualconstructionofasmallscaleExcel®modelwillbedemonstratedinAppendix3.Fornow,withoutconsiderationofanyfurthercomplexities,twographicalexamples(onefortheDailydatamodel,onefortheWeekly)follow.Ineachcase,themodelhasbeenrunfivetimesinordertoshowthecontrastofpathsthatcanbegenerated.
7TheactualmodelusesprobabilitiesassociatedwithspecificpricepointsasexplainedinAppendix2.Theabove$0.50/deltaexampleiscitedjusttosimplifyexposition.A$0.50possiblepriceshiftforeveryDorUtransactionwouldaddtoomuchvolatilityintheactualmodelingandisnotrealisticofwhatactuallyhappensintheWTIMarket.8Therealissueisevenmoresubtlethatthis.Thisisbecause,whenwestarttheMCMCmodelprojection,webeginthepricepathusingacurrentdaypriceperbarrel(approximately$100/BBLatthetimeofwriting).Therefore,themodelbeginsbuildingon2014dollars,butusingaconglomerationof1986through2013dollardeltas.Iftheoverallrateofinflationoverthepast26yearsroughlyequatestothenext26,thenthemodelwillstilldoanacceptablejobeventhoughtheactualdeltashavenotbeentimeadjustedtoreflectcurrentdaydollars.
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Graph5
GRAPH6
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Daily MCMC WTI Price Paths
MCMC Trial 1 MCMC Trial 2 MCMC Trial 3 MCMC Trial 4 MCMC Trial 5
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Weekly MCMC WTI Price Paths
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Ascanbeseenabove,theMarkovChainMonteCarloapproachisunbiasedandproducesawidearrayofpossiblepricepaths(indeed,virtuallyinfinite).Sinceitisimpossibletoactuallypredictthefuture,eachoutcomepresentsaplausiblerepresentationofwhereWTIpricesmaygo.Andeachoffersthestudiousfinancialanalystvaluableinsightintotheviability/profitabilityofanygivenenergyproject.ThisMCMCmodelcan,purelybyrandomchance,alsosimulatetheEIAReferenceCasequiteclosely:9
GRAPH7
9However,thegoalofthemodelisnottoproduceresultsthatconformtoapre‐existingestimate.Graph7isincludedhereonlytoshowthatthehighly‐engineeredEIAestimatescanbeapproximatelyduplicatedbyarandompathMCMC.TheTrial7abovewasdiscoveredbyfirstrunning200iterationsoftheMCMCandthenapplyingamathematicalalgorithmtodiscoverwhichofthe200wasclosesttotheEIAReferenceCase.Aclosermatchwouldprobablyhavebeenfoundif1,000or10,000iterationshadfirstbeenproduced.
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MCMC Comparison to EIA 2014 Reference Case
EIA Reference Case Trial 7
Daily MCMC Data: BurgundyLine is the average annual per barrel dollar amount from the 'Trial 7' Daily MCMC Data.
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THENEEDFORBOUNDARYLIMITS
Themodel’sWTIpricepathisrandomlygeneratedwhichmeansthatextremeoutcomesarerarely,butoccasionallygenerated.Thisisadesirablequality,asthisalsohappenstoreflectactualreality.Indeed,ifrealityseldomeverdeviatedfromtheexpectednorm,therewouldbeverylittlepointtofinancialanalysisingeneral.However,theMCMCWTImodelaswehavethusfardiscusseditcan,inrareinstances,producefinanciallyimpossibleresults.Itcan,forexample,generatenegativeoilpricepredictions.WhiletheMarkovprobabilitiesareweightedtowardsacontinuingUshiftinstates,randomvariationwilloccasionallydriveastringofDshiftstopredictWTIpriceslessthan$0/BBL.AndweknowthiscouldneverreflectactualMarketconditions.Infact,itisreasonabletospeculatethat,asfarascanbeconceivedundercurrentlyforeseeableglobaleconomicconditions,WTIpricesoverthenext26yearswouldprobablyneverdescendbelow$X/BBL.Astowhatthat$X/BBLamountexactlyiswouldbesubjectofmuchdebateandspeculationamongsttheeconomicexperts.Aspreviouslysuggested,onepossibleinterpretationofreasonableboundaryconditionswouldbetheEIALowOilPrice(afloorboundary)andHighOilPrice(aceilingboundary)cases.
WhyshoulditbenecessarytoactuallyprogramboundarylimitsintotheMCMCmodel?Afterall,ifproducinga7,000daypricepathisassimpleaspressinga‘recalc’key–whynotsimplydiscardthosefewtrialswherenegativepriceshavebeenincurred?Inotherwords,whynotputtheonusofdiscerningbetweenthe‘possible’andimpossible’trialsonthemodeluser?Thisissuebecomesquitephilosophicalbecauserelianceupontheusertodecidebetween‘good’and‘bad’pricepathsintroducesthepossibilityofselectionbiasintotheprocess.Therefore,itisdesirabletohard‐coderealisticboundarylimitsintothemodel,ifpossible.Regardlessofwhichcontrolsareactuallycodedintothemodel,thereisnosubstitutefortheintegrityofthefinancialanalyst.Thefirstrequiredboundarylimitisuncontroversial–toeliminatetheeconomicallyimpossiblefromthemodeloutcome(e.g.prohibitnegativeprices–wewillrefertothisasahardlimit).Andsecondly,limitsthatreduceormitigatethepossibilityofpricesrandomlytransitioningbeyondreasonableboundaries(wewillrefertotheseassoftlimits).Suchanapproachwillrelievetheuserfromhavingtoexercisejudgementaboutthepathproducedandthereinmaintainhis/herobjectivity.
WhiletheactualformulasemployedwillbedescribedinAppendix2,wewillbrieflysummarizethemethodologyofthelimitshere.Thehardlimitiseasy:anystatethatoccurswheretheestimatedWTIpriceislessthanzeroissimplyreplacedwithazero.Thesoftlimits(bothfloorandceiling),testthecurrentstateforproximitytotheapplicablelimitatthattime.If,forexample,aUtransitionhasbeendetectedandtheresultsaredeterminedtobethatthepriorday’spricelevelhasexceededthecurrentlyapplicableceilingpriceforthattimeperiod,thentherandomlyselectedUdeltadollaramountisreducedbyacertainamount.Thepricelimitsareprogressive–thatis,thefartherabovetheceilingtheactualunlimitedpricepathis,theproportionatelysmalleristhepricedeltaallowedtotravelintheoffendingdirection.ThisincreasingrateofUdollarreductioncontinuesinordertomitigate(butnotentirelyeliminate)therandompossibilityofobservingapricepaththatunrealisticallyexceedstheupperpricelimit.TheMarkovD/S/Uprobabilitiesarenotaltered,however,anditislefttotherandomoccurrenceoftheDtransactionstobringan‘above‐ceiling’pathbackintounlimitedterritory.
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ThefloorlimitworksinthesamemannerdetectingDtransactionsthatarebelowthefloorlimit.Thisprogrammingworksto‘soften’,butnoteliminatethepossibilityofbelowlimitpricepaths–whichiswhyahardlimitprohibitingnegativepricesisstillnecessary.
Forthepurposeofthismodeling,andtheexamplesshownbelow,theEIA2014LowOilPricehasbeenusedforthesoftlimitfloorpriceandtheEIA2014HighOilPricehasbeenusedforthesoftlimitceilingprice.SincetheEIAdoesnotproduceweeklyordailypriceestimates,the2020(forexample)HighOilPriceservesastheceilinglimitforallthedaily2020pricepaths,andthe2024(forexample)LowOilPriceservesasthefloorlimitforallthe2024daily2024pricepaths,etc.
Havingsufficientlydiscussedthenecessityforupperandlowerboundarylimits,afewexamplesofthe‘limited’pricepathsareproducedbelow.TheEIA2014High(greenline)andLow(redline)PriceCeilings/Floorshavebeenincludedsothatthereadermayseehowtherandompathsvaryrelativetotheboundaries:
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GRAPH8A
GRAPH9A
GRAPH8B
GRAPH9B
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Graph10A
GRAPH11A
GRAPH10B
GRAPH11B
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CONCLUSIONS
Thepast26yearsofpricehistorystronglyindicatesthatWTIpricesdofollowaMarkovchain.WehavedeterminedtheMarkovprobabilitiesforboththedailyandweeklyposthocdataandappliedtheseinaMarkovChainMonteCarlo(MCMC)simulation.However,theMarkovprobabilitiesofpastWTIpricemovementsdonotstrictlyadheretotheassumptionoftimehomogeneity.Anadded(and,undoubtedlyrelated)complexityisthatWTIpricevolatilityfluctuatesintheshorttomediumterm.Inordertodealwiththeseissuesandderiveamorerealisticestimationoffutureoilprices,arandomelementwasincorporatedintotheMCMCmodelthatwouldallowthepriceshiftordeltaofanygivenUorDchangeofstatetovaryinaccordancewiththeactualfrequencydistributionobservedoverthepast26years.Theresultofthisdouble‐stochasticmodelingisarealisticestimatorofWTIpricemovementsthatcan,underthecorrectassumptions,closelysimulateactualhistoricalpricepaths.
Randomchancedoesoccasionallyallowthemodeltoprojectbeyondreasonablyboundariesand,asaresult,certainlimitshavebeenincorporatedattheboundaries.Theselimits‘soften’theimpactofpricevolatility(butdoesnotinanywayaltertheMarkovchange‐of‐statesprobabilities)atthepricefloorandceiling.TheEIA2014EnergyOutlookLowOilPricehasbeenusedasareasonablepricefloorand,similarly,theHighOilPricehasbeenincorporatedasthepriceceiling.TheMCMCmodeldoesallowthesimulatedpricepathtobreachtheselimits(astheEIAhasnoabsolutemonopolyuponpredictingthefuture),buttheprobabilitythatthepathwillbecomeincreasinglyfartherabovetheceilingorsignificantlyfartherbelowthefloordeclineswitheachmovementawayfromtheboundary.
TheresultingMCMCmodeliseasyandefficienttoconstructandparsimoniousinitsdesign.ItyieldsanunbiasedandobjectivesimulationofwherefutureWTIoilpricescouldgo,andinthisregardshouldbeofgreatusetothoseofuswhospendalotofourtimeassessingthefutureviabilityof oilandrelatedenergyprojects.ItprovidesalogicalalternativetoaGBM(GeometricBrownianMotion)predictionapproachand,inmanywayswouldbeeasiertoexplaintoanon‐financialaudience.
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APPENDIX1–SIMULATINGPASTWTIPRICEPATHSWITHMCMC
Contrarytotheforward‐lookingpurposeofthepaperingeneral,thegoalofthisappendixwillbetotestwhetherwecanroughlyreproducethepastWTIpricemovementsfortheperiodof1986through2013inclusive.Thereasoninghereisalongthelinesof:IfwecanusetheMarkovprobabilitiesinconjunctionwithpricingvariablesthatwereparticulartothathistoricperiodandapproximatelyreplicatethe1986to2013WTIpriceline,thenthatservesassomedegreeofvalidationtotherealismofthemodellogic.
Itbearsrepeating,however,thatthepurposeofthemodelisnotsimplytoreproducethepast–therewouldbenobenefitindoingso.Thisappendixwillonlybeusefulifitservestoincreasetheuser’sconfidenceinpredictingfutureWTIpricepaths.
OurmodusoperandiwillbetosimulatethepastbyapplyingtheMarkovprobabilitiestoaJanuary3,1986WTIstartingpriceandusingtheMonteCarloapproachtoforecastingapossiblepricepathuptoDec.31,2013.ThissynthesizedrecreationoftheWTIpricelinewillthenbegraphedagainsttheACTUALWTIPriceData(forconvenience,onlytheWeeklydatawillbeusedinthisappendix).Ifthesynthesizedpastvisuallyappearstobeareasonablyclosefacilityofthepast26yearsofactualWTIpricemovements–thenwewilldeclaretheMarkovchainapproachasuccess.IfusingtheMarkovprobabilitieswecansimulatethepast–thenthesametechniqueshouldbeatrustedmeansofpredictinghowtheforwardWTIpricepathmayunfold.
‘FITTING’ALINETOTHEPAST–DIFFERENTASSUMPTIONSREQUIRED
InthebodyofthepaperourprimaryconcernwaswhetherthepricingvariablewasindicativeofwherefutureWTIpriceswerelikelytogo.Thisapproachwillbesuboptimalifappliedhere–evenifwecouldpsychologicallyadjustourperspectivestoviewtheWTIworldfromthevantagepointofearly1986lookingforward–itisunlikelythatwecouldhavecorrectlypredictedtheuniqueeconomicfactorsthatimpingeduponWTIpricesoverthefollowing26years.Themainreasonwhyisbecausetherewasasea‐changeintheaveragepriceofWTIthatbeganapproximatelyatthestartof2003.Priortothattime,WTImaintainedarelativelytightrangearound$20/BBLfrom1986through2002.1Since2004,WTIhasdisplayedamuchwiderpricerange–butnotnecessarilyahigherannualizedvolatilitypercentage.Onthecontrary,WTIpricesareapproximatelyjustasvolatilenow,inthe$70to$120/BBLrangeastheywereinthe1980’sand90’satthe$20/BBLrange(seePaperI,Figure4).2
1TheoneexceptionisinJanuary1991wherethestartoftheGulfWarinthemiddleeastcausedshort‐termedpricestoinflateto$40and,asaresult,theJanuary1991WTIsingle‐monthpricevolatilityremainsthehighestinrecenthistory–highereventhanthatofDecember2008.2Ifthegeneraldegreeofvolatilityoverthe2004~2013yearshadbeenincreasingcomparedwiththe1986~2002years,wewouldhaveassumedtheriseinWTIpricestobetheresultofincreasedvolatility(becauseanUstateismoreprobablethanaDstate).Itisunusualthatthisisnotthecase–thereforewemustattributetheriseinaveragepricetoanincreaseintheabsolutedollarvalueofthestate‐to‐stateshifts.
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A1‐GRAPH1
A1‐TABLE1
Period Average
WTI Price/BBL
Annualized Volatility
for Period
Average 'U'
Change in Period
Average 'D'
Change in Period
1986 ~ 1998 $ 19.06 32.17% $ 0.578 -$ 0.631
1999 ~ 2003 $ 26.52 32.39% $ 0.906 -$ 0.974
2004 ~ 20073 $ 58.96 25.14% $ 1.718 -$ 1.641
2008 ~ 2013 $ 87.96 33.30% $ 2.478 -$ 2.722
A1‐Table1aboveindicatesthattherehavebeenroughlyfourdifferentWTIpricingstratainthepast26years.Now,someofthenominaldollarchangesfromoneperiodtothenextwouldcertainlybeattributabletogeneralinflation.However,mostoftheobservedchanges(bothintermsoftheAvg.$/BBLandtheAvg.week‐over‐weekUandDchange)istheresultofarealchangeinoilpricing.Notethat,whileoverallWTIvolatilitydiddeclineto25%intheboomperiodof2004~2007,overall,WTIvolatilityhasremainedstablefortheentire26yearperiod.
TheactualD/S/UStateSpace4forthe1986~2013periodisasfollows:
3ItisinterestingtonotethattheaverageDStateweeklychangeinpriceisconsistentlygreater(inabsoluteterms)thantheUStatechangeexceptfortheboomyearsof2004through2007.4D/S/UStateSpaceiseffectivelyjustthehistoricsequenceofD,S&Ustateshiftsdevoidofpricingdata.Or,itcanbeassumedthatallthepricechangesthroughtimeareallthesameandequaltoageneric‘unitone’.
0
20
40
60
80
100
120
140
160USD
$ per BBL
WTI Actual Weekly Price Data
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A1‐GRAPH2
ContrastingtheaboveA1‐Graph2tothepreviousA1‐Graph1wearestruckbythedifferenceinshapes.InGraph2,therelativeupwardsloperemainsconstanteveninthose1986~1998yearswhenweknowthataverageWTIpricewas,atthattime,relativelyconstant.Thisisyetanotherindicatorthattheabsolutedollarvalueofthestateshiftshasincreasedovertheyears.Wecouldattempttofindoneoverallrepresentative‘U’dollaramountandone‘D’dollaramountthatwasrepresentativeofallfourdifferentstratawithinthepast26yearsofWTIpricehistory.However,inapplyingthesetwoamountstotheactualD/U/SStateSpace,whatwewouldprobablyfindisthatsimulatedpricesfortheearlyyearswouldrisehigherandfasterthantherelativelystagnantWTIpriceofthe1986~1998period.Conversely,theactualWTIpricewouldprobablyout‐pacethesimulatedpricepathfortheyears2008~2013whenactualDandUshiftswereattheirhighestunitamounts.Thebetter,andperhapseasiestoverallsolution,wouldbetousetheactualfourDandUaverageamountsobservedthroughouttime(assetoutinA1‐Table1above)foreachofthefourrespectiveperiods.HavingdonethatandjuxtaposingtheActualWTIPricePath(blueline)againstthesimulatedone(burgundyline),weget:
1986 ‐ 2013 Weekly Change of States ‐ Actual Data
Weekly Change of States
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A1‐GRAPH3
Visually,itappearsthattheMarkovChainMonteCarlo(MCMC)simulatedpricepathissufficientlycloseenoughtotheactualhistoricalWTIpricepaththatwewouldassumethemodelingexperimenttobeasuccess.Amoreanalyticalapproachwouldbetoexaminethedifferentialsofthemonthlyaverages:
0
20
40
60
80
100
120
140
160
Jan 03, 1986
Jan 03, 1988
Jan 03, 1990
Jan 03, 1992
Jan 03, 1994
Jan 03, 1996
Jan 03, 1998
Jan 03, 2000
Jan 03, 2002
Jan 03, 2004
Jan 03, 2006
Jan 03, 2008
Jan 03, 2010
Jan 03, 2012
Jan 03, 2014
USD
$ Per Barrel
1986 ‐ 2013 Actual WTI Prices vs. Simulated Path
Actual Weekly History Simulated MCMC Price Path
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A1‐GRAPH4
Thelargedifferentialsof2008and2009are,ofcourse,theresultoftheglobalfinancialcrisis.ItwasnotexpectedthatthisMarkovmodelwouldhavethecapacitytoanticipatethesetypesofeconomicanomalies,sonoimportanceisattachedtoseeingsuchlargevariances.The95%confidenceintervalforthe1986through2007differentialsperiodrunsbetweennegative$12.87topositive$16.81andonly22ofthe264monthlyobservationsexceedtheselimits.
CONCLUSIONS:ThesimulatedpricepathissufficientlyrepresentativeoftheactualWTIpricehistory.Inordertoachievetheseresults,afour‐tierpricestratumwasadoptedspecificallytoaccommodatetheuniquechangesthattheWTIpricelevelshaveundergoneinthepast26years.
$(60.00)
$(40.00)
$(20.00)
$‐
$20.00
$40.00
$60.001986‐01
1987‐01
1988‐01
1989‐01
1990‐01
1991‐01
1992‐01
1993‐01
1994‐01
1995‐01
1996‐01
1997‐01
1998‐01
1999‐01
2000‐01
2001‐01
2002‐01
2003‐01
2004‐01
2005‐01
2006‐01
2007‐01
2008‐01
2009‐01
2010‐01
2011‐01
2012‐01
2013‐01
Per BBL Differential
Simulated WTI Prices vs. Actual WTI Prices
Simulated minus Actual
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APPENDIX2–BOUNDARYLIMITS
Asdiscussedinthebodyofthepaper,randomchancewilloccasionallydrivetheMCMCpricepathstounreasonableoutcomes.Whenthishappensinreallife,thelawsofsupplyanddemandengagetomitigatetherunawayprice.Thatis,whenWTIpricesbegintoascendbeyondwhatthemarginalbuyeriswillingtopay,hereduceshisquantityconsumptionuntilsuchtimeaspricesdescendbacktoacceptablelevels.Conversely,whenWTIpricesfalltheenergyconsumerincreasestheirquantitypurchasedwhichultimatelyhastheimpactofdrivingpricesbackupwards.
WhileoursimpleMCMCmodeldoesnotincorporateeconometricvariables(e.g.therearenoinputsforquantitiesdemandedorsupplied),wecanapproximatethepricemitigatingeffectsofdemandandsupplyboundariesviatheuseofsimpleformulas.TheEIALowOilPricewillbeusedasapricefloorandHighOilPriceforapriceceiling.Aspreviouslyexplained,however,thesewillonlyserveas“soft”limits–thesimulatedpricepathswillbeallowedtodescendbeloworclimbabovetheEIAlimits.Butthenatureoftheformulaswillserveasaprogressivelygreaterdeterrentthefartherawayfromtheboundarythesimulatedpricepathbecomes.
VARIABLESUSED:
Cy=EIAHighOilPriceinyear“y”;servesassoftpriceceiling
Fy=EIALowOilPriceinyear“y”;servesassoftpricefloor
P=UnlimitedPrice
Ln=LimitedPricefortheperiod
D≡“DownState”hasbeenrandomlyindicatedbyMarkovProbabilities
U≡“UpState”hasbeenrandomlyindicatedbyMarkovProbabilities
d=downdollaramountrandomlyselectedbaseduponhistoricWTIfrequencies
dL=dscaledtoalesseramountviathelimitingformula
u=updollaramountrandomlyselectedbaseduponhistoricWTIfrequencies
uL=uscaledtoalesseramountviathelimitingformula
PRICECEILINGFORMULA:
Given:L(n–1)>CyandU,uL=[1+ln(Cy/L(n–1))]u
Inwords:Giventhatthepreviousday’slimitedpricepath,L(n–1),hasexceededthereferenceceiling(i.e.Cy)andthataUstateshifthasbeenrandomlyselectedforthecurrentday’spricemovement,
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theotherwiserandomlyselectedupricedeltawillbereducedbyoneplusthenaturallogarithmoftheratioofthecurrentpriceceilingdividedbyyesterdayslimitedprice.
Anexamplewillbestservetoshowhowtheceilingboundaryworks:
Cy=$100,theeffectiveEIAHighOilPriceforthisdayis$100/BBL
L(n–1)=$120,yesterday’slimitedpricepathwas$120/BBLwhichexceedsthepriceceiling
u=$1.00therandomlyselectedUdeltafortoday,beforeapplicationofthelimitwouldhavebeentomovetoday’sendingWTIprice$1.00higherthanyesterday’sendingprice.
Therefore:
uL=[1+ln(100/120)]x$1.00=$0.82
Now,asaresultofthelimitingformula,today’sendingWTIwillonlyincrease$0.82overyesterday’spriceinsteadoftherandomlyplanned$1.00.Notethatthelimitingformulaisprogressive:thehigherthatL(n–1)isaboveCy,thegreaterthereductiontou.Forexample,ifallthesamevariablesappliedexceptthatyesterday’slimitedpricewas$160,theresultantuLwouldonlybe$0.53.Hadyesterday’slimitedpricebeen$200,thentoday’suLwouldonlybe$0.31
PRICEFLOORFORMULA:
Thepricefloorformulaisreallyjustthereciprocaloftheceilingformula:
Given:L(n–1)<FyandD,dL=[1+ln(L(n–1)/Fy)]d
Inwords:Giventhatyesterday’spricelimitedWTIpriceislessthanthecurrentfloor(i.e.Fy)andthataDstateshifthasbeenrandomlyplannedfortoday’spricemovement,theotherwiserandomlyselecteddpricedeltawillbereducedbyoneplusthenaturallogarithmoftheratioofyesterday’slimitedpricedividedbythecurrentfloorprice.
Forexample:
Fy=$80,theeffectiveEIALowOilPriceforthisdayis$80/BBL
L(n–1)=$60,yesterday’slimitedpricepathwas$60/BBLwhichisbelowthepricefloor
d=‐$1.00,therandomlyselectedDdeltafortoday,beforeapplicationofthelimitwouldhavebeentomovetoday’sendingWTIprice$1.00lowerthanyesterday’sendingprice.
Therefore:
dL=[1+ln(60/80)](‐1.00)=‐$0.71
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Ratherthantherandomlyselected$1.00dropinWTIprice,thelimitedpricewillonlydescend$0.71/BBL.Notethat,hadallthesamefactorsbeeninplay,exceptthatyesterday’slimitingpricewasonly$40,thenthecurrentdaydLwouldhaveonlybeen‐$0.31
Theeffectivenessofthelimitingformulascanvisuallybequicklyappreciatedifthelimitedpricepathisshowncontemporaneouslywiththeunlimitedpricepath(i.e.P).Inthegraphsbelow,theBlueLinerepresentedthelimitedpricepathandtheGrayLineshowswhattheMCMCwouldhavepredictedintheabsenceofanyboundarylimits.TheEIAboundaries(GreenLine=HighOilPrice,RedLine=LowOilPrice)havebeenincludedtoshowatwhichpointthelimitingformulasbeginactingupontheBlueLimitedPricePath.
GRAPHA2‐1
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GRAPHA2‐2
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GRAPHA2‐3
0
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400
19‐M
ay‐14
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ay‐40
19‐M
ay‐41
Nominal USD
$ per Barrel
MCMC Siumulated WTI Daily Price Path
MCMC Limited Price Path EIA Ceiling EIA Floor MCMC Unlimited Price Path
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FREQUENCYOFEXACTNOMINALDOLLARDELTAS
Contrarytothesimplifiedexplanationofferedinthebodyofthepaper,theselectionofthenominaldollardeltaisnotbaseduponcategories.Forgreaterprecision,themodelreferencestheentirepopulationofdollarchangesthathaveoccurredoverthepast26yearsandusesthatfrequencytodeterminewhatprobabilitythatthesamenominalchangecouldoccurintheMCMCmodel.Forexample,inthedailyDtransactionhistory,amoveof‐$0.01occurred63timesinasampleof3363Dshifts,therefore,the‐$0.01hasa63/3363chanceofoccurringintheMCMCgiventhataDtransactionhasbeenselected(i.e.thedeltaprobabilitiesareconditional).
Thedataistoovoluminoustopresenthere,butanexcerptoftheoccurrencesofdailypricedeltasoccurringwiththerangeof‐$0.10to+$0.10is:
Daily Price Delta
# of Occurrences
‐$ 0.10 75
‐$ 0.09 55
‐$ 0.08 51
‐$ 0.07 51
‐$ 0.06 52
‐$ 0.05 72
‐$ 0.04 54
‐$ 0.03 47
‐$ 0.02 54
‐$ 0.01 63
$ ‐ 124
$ 0.01 36
$ 0.02 51
$ 0.03 58
$ 0.04 64
$ 0.05 73
$ 0.06 58
$ 0.07 51
$ 0.08 51
$ 0.09 53
$ 0.10 83
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APPENDIX3–SIMPLIFIEDEXCEL®MODEL
ThepurposeofthisappendixwillbetoprovideasimplifiedexampleofhowsuchadiscreettimeMCMCmodelcouldbeconstructedusingMicrosoft’sExcel®.TheexamplethatfollowswillrefrainfromusinganyVBAcodingandshouldbeeasilymasteredbythoseusershavingonlyanelementaryunderstandingofthemostcommonlyusedExcel®formulas.Inallinstancesthegoalwillbetomaintainsimplicityandefficiencyinhowthemodelworks.
Foreaseofexposition,allreferenceswillbetotheWeeklyWTIdatamovements.
RANDBETWEEN:TheheartoftheWTIMCMCmodelasdescribedinthebodyofthepaperrelatestotwointer‐relatedstochasticfunctions.Thefirstistherandomselectionofthedirectionofthepricemovement;eitherD,SorU.Thesecondisrelatedtotherandomlyselecteddollaramountofthemovementorpricedelta.BoththesestochasticoutcomeswillbesimulatedviatheRANDBETWEENfunction.Incaseswherethefrequencydistributionoftherandomprocessweareattemptingtosimulateisneithernormalnoruniform,itisconvenienttouseRANDBETWEENbecausevirtuallyanydistributioncanberepresented.
D/S/UStateShifts:Forexample,forsimplicityassumethatthehistoricweeklystateshiftswererepresentedbyafrequencydistributionof4/12th,1/12thand7/12threspectively.ThiswouldallowustosetacolumnwhereRANDBETWEEN(1,12)representedtheuniformrandomoccurrenceofnumbersbetween1and12inclusive.ColumnBinWorksheet1belowrepresentsthisoutput.Therowsrepresentdiscreetweek‐endingWTIperiods(over10pricemovements,therefore,actuallyrepresenting11weeksoftime).
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A3WORKSHEET1
A B C D E F G H I J K L M
1 Starting WTI Spot Price $ 100.00
2 Week
Ending #
Rand-between 1 ~ 12
D / S / U : -1 / 0 / 1
Rand-between
1 ~ 4
Unlimited "D" Delta
Rand-between
1 ~ 7
Unlimited "U" Delta
Unlimited MCMC
Price Path
EIA Price Floor for Period
EIA Price Ceiling for
Period
Limited "D" Delta
Limited "U" Delta
Limited MCMC
Price Path
3 1 9 1 3 -0.750 3 1.000 $ 101.000 $ 99.90 $ 100.250 -0.750 1.000 0 $ 101.00
4 2 10 1 4 -0.750 7 1.250 $ 102.250 $ 99.90 $ 100.250 -0.750 1.2410 $ 102.24
5 3 9 1 1 -0.500 4 1.000 $ 103.250 $ 99.90 $ 100.250 -0.500 0.9800 $ 103.22
6 4 11 1 3 -0.750 5 1.000 $ 104.250 $ 100.00 $ 100.150 -0.750 0.9700 $ 104.19
7 5 10 1 4 -0.750 1 0.500 $ 104.750 $ 100.00 $ 100.150 -0.750 0.4800 $ 104.67
8 6 7 1 2 -0.750 3 1.000 $ 105.750 $ 100.00 $ 100.150 -0.750 0.9560 $ 105.63
9 7 1 -1 1 -0.500 7 1.250 $ 105.250 $ 100.00 $ 100.150 -0.500 1.1830 $ 105.13
10 8 5 0 3 -0.750 4 1.000 $ 105.250 $ 100.05 $ 100.250 -0.750 0.9520 $ 105.13
11 9 3 -1 4 -0.750 6 1.250 $ 104.500 $ 100.05 $ 100.250 -0.750 1.1910 $ 104.38
12 10 12 1 3 -0.750 4 1.000 $ 105.500 $ 100.05 $ 100.250 -0.750 0.9600 $ 105.34
ColumnCfunctionsinamannertoassigneitheraD,SorUeventtotheColumnBresults.RatherthanusingtheTextdescriptorsofD,SandU,itismoreconvenienttorepresenttheseeventswiththenumbers‐1,0and1respectively.Therefore,theStateSpaceoverthis10weekobservationis{UUUUUUDSDU}.TheformulaincellC4,forexample,is=if(B4<5,‐1,if(B4=5,0,1))Notethatthisaccomplishesthedesiredprobabilityfrequencythatwehaddesired:a“D”state(numbers1to4inclusive)willrandomlyoccur4/12th’softhetimeinColumnB;a“S”istriggeredbytheoccurrenceofa5,whichstatisticallyRANDBETWEENwillselect1/12thofthetimeforeveryiterationofthemodel,and,finally,anynumberoccurringgreaterthan5(namely,6to12inclusive),inducesa“U”statetooccuratthedesired7/12thprobability.
Further,forsimplicity,letusassumethatthereareonlytwopossible“D”deltas:‐$0.50and‐$0.75.Further,wewillpresumefromtheinspectionofpreviousactualposthocdata,thatthereisa1/4thprobabilitythatthe‐$0.50pricedeclinehappening,a3/4thprobabilitythatthe‐$0.75weeklydeclineoccurs.Therefore,ColumnDisyetanotherindependentRANDBETWEENfunctionof=RANDBETWEEN(1,4).BasedupontherandomnumberselectedinColumnD,ColumnEusesVLOOKUPtoreferencetheseparateD_DeltaTable(seeA3‐Table1below)inordertofindtheappropriatepricedeclineamount.
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A3TABLE1
D_Delta Table
Frequency D Delta $
1 ‐0.50
2 ‐0.75
3 ‐0.75
4 ‐0.75
ThedesiredDDeltaprobabilitiesareachievedbythefactthatinColumnD,eachofthenumbers1,2,3,4hasanequal1/4thchanceofbeingselectedonRECALC(F9).Thereisoneoccurrenceof‐$0.50inTableD_Deltaand3occurrencesof‐$0.75.Therefore,theprobabilitiesofselectionare1/4thand3/4threspectively.TheformulaincellE4,forexample,is=VLOOKUP(D4,D_Delta,2,False)where“D_Delta”isthenamedrangeforthetwodatacolumnsintheD_DeltaTable.TheVLOOKUPfunctionherenotestheexistenceofa4incellD4,thenequatestheE4valuetothecorresponding‐$.075fromtheD_DeltaTable(whichitalsowouldhavedone,ofcourse,ifthecellD4valuewasa2or3).
Similarly,withrespecttothe“U”deltavalues,weareassumingthereareonlythreeofthoseintheposthocactualdata:+$0.50,+$1.00and+$1.25Thesehavebeenobservedinthefrequencyof2/7thofthetimefor+$0.50;3/7thofthetimefor$1.00and2/7thofthetimefor$1.25.1
A3TABLE2
U_Delta Table
Frequency U Delta $
1 0.50
2 0.50
3 1.00
4 1.00
5 1.00
6 1.25
7 1.25
TheU_DeltaTableshowsthat$0.50hastheprobabilityofbeingselected2/7thofthetime;$1.003/7thofthetimeand;$1.25hastheprobabilityofbeingselected2/7thofthetime.Accordingly,theRANDBETWEENformulainColumnFofWorksheet1is:=RANDBETWEEN(1,7).ThisdrivestheVLOOKUPformulainColumnGwhich,forcellG4,is:=VLOOKUP(F4,U_Delta,2,False)where“U_Delta”isnamedrangeforthetwocolumnsofdataintheU_DeltaTable.
1Thestatisticallyastutewillnowrecognizethat,inordertohavethisspecificfrequencydistribution,ahistoricalposthocobservationof12weeklypricechangesmusthaveoccurredandthatthesubsetofthesepricechanges,re‐arrangedtobeinascendingorder,is:{‐$0.50,‐0$.75,‐0$.75,‐0$.75,$0.00,$0.50,$0.50,$1.00,$1.00,$1.00,$1.25,$1.25}
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ColumnHinWorksheet1isthepresentationoftheUnlimited(i.e.thesoftpricefloorandpriceceilinghaveyettobeapplied)MarkovChainMonteCarloWTIPricePath.Thelogicofthesecellsissimple:ifa“D”transaction(i.e.isa‐1)hasbeenindicatedinColumnC,thentakethecurrentvaluefromColumnEandaddthattothepriorweek’sWTIprice.Conversely,an“S”stateisindicated(i.e.isa0);justbringforwardlastweek’sWTIprice.Finally,ifColumnCindicatesa“U”changeofstate(i.e.isa+1);thentakethecurrentvalueinColumnGandaddthattothepriorweek’sWTIprice.TheformulaforcellH4is:=if(C4=‐1,H3+E4,if(C4=0,H3,H3+G4))
JustwiththesefewsimpleformulaswehavebeenabletoconstructanefficientworkingMCMCmodelthatprojects10weeksofWTIspotprices(subject,ofcourse,tothesimplifiedMarkovprobabilitiesandverynarrowrangeofdeltaswehaveemployedfordemonstrationpurposesonly).Expansionofthismodeltocoveravirtuallyinfinitetermintothefuturewouldmerelyrequirecopying‐and‐pastingthelastrowasmanytimesasdesired.
Onlythepricefloorandceilinglimitsstillneedtobecreated.Again,forsimplicity,wehaveassumedanunrealisticallynarrowEIAfloor‐to‐ceilingrangeincolumnsIandJrespectively.Thisrangeisasmallas$0.15/BBLinsomeweeks–butthishasbeenspecificallyselectedinordertodemonstratehowthepricefloors/ceilingsimpactthecalculationoftheLimitedMCMCPricePath(inColumnM).
ColumnKcalculatesthevalueofthe‘limited’or‘scaled’DdeltaintheeventthatthesimulatedWTIpricepathdescendsbelowthefloorpriceindicatedinColumnI.TheformulaforcellK4,forexample,is:=if(M3<I4,1+ln(M3/I4),1)*E4Thishastheeffectthat,intheeventthatthepreviousweek’slimitedWTIspotpriceisbelowtherecommendedfloor,therandomlyselectedDdeltaamountinE4willbereducedbyafactorof[1+ln(M3/I4)].However,ifthepricepathisstillabovethefloor,theDdeltaamountinE4remainsunaltered.
ColumnLcalculatesthelimitedUdeltaandfunctionsinasimilarmannerasColumnK.TheformulaforcellL4is:=if(M3>J4,1+ln(J4/M3),1)*G4RandomchancehasprovidedanexampleatWeek‐ending#2wherethepreviousWTIlimitedpriceof$101.00isindeedabovethecurrentceilingof$100.25.Therefore,theotherwiseexpectedUdeltaof$1.25(incellG4)hasbeenscaleddownbytheamountof[1+ln(100.25/101.00)]x$1.25=.992546x$1.25=$1.241.Attheendofweek#2,theunlimitedpricepathsettlesonavalueof$102.25whereasthelimitedpathisreducedto$102.24(theparametervariablesarenotveryrealistic,andonlyservetodemonstratehowthepricefloor/ceilinglimitswork).
Finally,ColumnMisthesummationoftheentiremodel.ItfunctionsinasimilarfashionasColumnHdoes.Specifically,ifthestateshiftindicatedinColumnCisaD(i.e.‐1),thenthepriorweekslimitedpriceisaddedtothecurrentvalueinColumnK.Conversely,ifColumnCindicatesanSshift(i.e.is0),thenthepriorweek’slimitedWTIpriceissimplybroughtforward.Finally,ifColumnCindicatesaUtransaction(i.e.is+1),thenthepriorweek’slimitedWTIpriceisaddedtothecurrentcolumnL.TheformulaforcellM4is:=if(C4=‐1,M3+K4,if(C4=0,M3,M3+L4))
ThegraphoftheabovecolumnsHandMis:
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Evenwithsucharestrictedmodel,usingverynarrowrangeofMarkovProbabilitiesandPriceDeltas,avarietyofoutcomescanbeobserved:
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$98.00
$100.00
$102.00
$104.00
$106.00
$108.00
1 2 3 4 5 6 7 8 9 10
USD
$ per Barrel
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Simplified Demonstration MCMC WTI Price Path
Limited MCMC WTI Price Path Unlimited MCMC WTI Price Path
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