MODELING WTI PRICES WITH MARKOV CHAINS

CONN VALUATION SERVICES LTD.

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 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.

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.

outputobtainedmustrepresentonerenditionofwheretheWTIpricepathcouldgo.ThecaveathereisthattheMCMCmodelwillproduceaninfinitenumberofiterations–andthefinancialanalyststillmustexercisecare/integritysoasnottocherrypickpreferredoutcomes.

GRAPH1

OVERVIEWOFTHEMETHODOLOGY

Asexplainedinthefirstpaper,wesetouttofirsttestifWTIhistoricpricemovementsdoexhibitthepropertiesofaMarkovchain(andtheydo)and,ifourinvestigationsupportedthishypothesis,weweregoingtousetheresultingMarkovprobabilitiestoconstructamodelthatsoughttopredictthefuturedirection(i.e.steps,betheyUp,DownorSame)ofthepricemovementswithoutactuallyconsideringthedollarsizeofthosemovements.Decidinghowlarge,indollarterms,eachofthesechangesinstatewillbetheprimaryfocusofthispaper.Wewilloftenrefertothedollarsizeofthischangeinstateasthedailyorweeklydelta.

100.00

150.00

200.00

250.00

300.00

350.00

Nominal USD

$ per BBL

EIA 2014 WTI Outllook ‐ Nominal Dollars

Reference Case High Oil Price Case Low Oil Price Case

Source: EIA Annual Energy Outlook 2014

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):

1 2 3 4 5 6 7 8 9 10 11

Price Level

Time Period

11 Period Change of States

DSDUUSUDUD

GRAPH3

Andcomparingthistothegraphofactualweek‐endingpricesthatoccurredbetweenthesedates,itbecomesreadilyapparentastothedifficultieswewillencounterinattemptingtomodelthe‘step’sizeordelta:

GRAPH4

1986 ‐ 2013 Weekly Change of States ‐ Actual Data

Weekly Change of States

$ per BBL

WTI Actual Weekly Price Data

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.

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.

‐5.00

‐4.58

‐4.16

‐3.74

‐3.32

‐2.90

‐2.48

‐2.06

‐1.64

‐1.22

‐0.80

‐0.38

Nominal Dollar Daily Price Change

Histogram of Daily Price Changes: 1986 ‐ 2013

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.

‐7‐6.43

‐5.86

‐5.29

‐4.72

‐4.15

‐3.58

‐3.01

‐2.44

‐1.87

‐1.3

‐0.73

‐0.16

Nominal Dollar Weekly Price Change

Histogram of Weekly Price Changes: 1986 ‐ 2013

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.

Graph5

GRAPH6

Nominal USD

$ per Barrel

Daily MCMC WTI Price Paths

MCMC Trial 1 MCMC Trial 2 MCMC Trial 3 MCMC Trial 4 MCMC Trial 5

May16,2014

May16,2016

May16,2018

May16,2020

May16,2022

May16,2024

May16,2026

May16,2028

May16,2030

May16,2032

May16,2034

May16,2036

May16,2038

May16,2040

May16,2042

Nominal USD

$ per Barrel

Weekly MCMC WTI Price Paths

MCMC Trial 1 MCMC Trial 2 MCMC Trial 3 MCMC Trial 4 MCMC Trial 5

Ascanbeseenabove,theMarkovChainMonteCarloapproachisunbiasedandproducesawidearrayofpossiblepricepaths(indeed,virtuallyinfinite).Sinceitisimpossibletoactuallypredictthefuture,eachoutcomepresentsaplausiblerepresentationofwhereWTIpricesmaygo.Andeachoffersthestudiousfinancialanalystvaluableinsightintotheviability/profitabilityofanygivenenergyproject.ThisMCMCmodelcan,purelybyrandomchance,alsosimulatetheEIAReferenceCasequiteclosely:9

GRAPH7

9However,thegoalofthemodelisnottoproduceresultsthatconformtoapre‐existingestimate.Graph7isincludedhereonlytoshowthatthehighly‐engineeredEIAestimatescanbeapproximatelyduplicatedbyarandompathMCMC.TheTrial7abovewasdiscoveredbyfirstrunning200iterationsoftheMCMCandthenapplyingamathematicalalgorithmtodiscoverwhichofthe200wasclosesttotheEIAReferenceCase.Aclosermatchwouldprobablyhavebeenfoundif1,000or10,000iterationshadfirstbeenproduced.

Nominal USD

$ per Barrel

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.

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.

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:

GRAPH8A

GRAPH9A

GRAPH8B

GRAPH9B

Graph10A

GRAPH11A

GRAPH10B

GRAPH11B

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.

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.

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’.

160USD

$ per BBL

WTI Actual Weekly Price Data

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

A1‐GRAPH3

Visually,itappearsthattheMarkovChainMonteCarlo(MCMC)simulatedpricepathissufficientlycloseenoughtotheactualhistoricalWTIpricepaththatwewouldassumethemodelingexperimenttobeasuccess.Amoreanalyticalapproachwouldbetoexaminethedifferentialsofthemonthlyaverages:

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

$ Per Barrel

1986 ‐ 2013 Actual WTI Prices vs. Simulated Path

Actual Weekly History Simulated MCMC Price Path

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

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,

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

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

GRAPHA2‐2

GRAPHA2‐3

19‐M

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ay‐41

Nominal USD

$ per Barrel

MCMC Siumulated WTI Daily Price Path

MCMC Limited Price Path EIA Ceiling EIA Floor MCMC Unlimited Price Path

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

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).

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

Unlimited "D" Delta

Rand-between

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.

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}

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:

A3GRAPH1

Evenwithsucharestrictedmodel,usingverynarrowrangeofMarkovProbabilitiesandPriceDeltas,avarietyofoutcomescanbeobserved:

A3GRAPH2

$98.00

$100.00

$102.00

$104.00

$106.00

$108.00

1 2 3 4 5 6 7 8 9 10

$ per Barrel

Simplified Demonstration MCMC WTI Price Path

Limited MCMC WTI Price Path Unlimited MCMC WTI Price Path

A3GRAPH3

MODELING WTI PRICES WITH MARKOV CHAINS

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