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Efficient Optimization Algorithms for Pricing Energy Derivatives and Standard Vanilla Options

Permanent Link: http://ufdc.ufl.edu/UFE0022492/00001

Material Information

Title: Efficient Optimization Algorithms for Pricing Energy Derivatives and Standard Vanilla Options
Physical Description: 1 online resource (136 p.)
Language: english
Creator: Ryabchenko, Valeriy
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: agreement, boundary, contracts, derivatives, efficient, energy, exercise, exotic, hedging, linear, optimal, optimization, options, pricing, programming, quadratic, spark, spread, strategy, tolling
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Our first study researched a problem of scheduling operational flexibility of electricity generating facilities. Recently, a number of new approaches based on stochastic dynamic programming techniques have been suggested in the academic literature. Although these approaches are flexible in terms of incorporating various operational constraints, they are computationally inefficient when considering problems with relatively large horizons. Here we suggest a simple framework that is computationally efficient as well as numerically robust when dealing with large horizon problems. We show that the optimal dispatch policy can be characterized through a set of optimal exercise boundaries and also theoretically derive the shape properties of the boundaries. The problem of finding the optimal exercise boundaries is then reduced to solving a simple linear programming problem. The suggested approach is flexible in incorporating various real world operational constraints. We compare the computational performance of the suggested scheme with alternative dynamic programming based methods. Our second study considered a regression approach to pricing European options in an incomplete market. The algorithm replicates an option by a portfolio consisting of the underlying security and a risk-free bond. We apply linear regression framework and quadratic programming with linear constraints (input = sample paths of underlying security; output = table of option prices as a function of time and price of the underlying security). We populate the model with historical prices of the underlying security (possibly massaged to the present volatility) or with Monte Carlo simulated prices. Risk neutral processes or probabilities are not needed in this framework.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Valeriy Ryabchenko.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Uryasev, Stanislav.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022492:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022492/00001

Material Information

Title: Efficient Optimization Algorithms for Pricing Energy Derivatives and Standard Vanilla Options
Physical Description: 1 online resource (136 p.)
Language: english
Creator: Ryabchenko, Valeriy
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: agreement, boundary, contracts, derivatives, efficient, energy, exercise, exotic, hedging, linear, optimal, optimization, options, pricing, programming, quadratic, spark, spread, strategy, tolling
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Our first study researched a problem of scheduling operational flexibility of electricity generating facilities. Recently, a number of new approaches based on stochastic dynamic programming techniques have been suggested in the academic literature. Although these approaches are flexible in terms of incorporating various operational constraints, they are computationally inefficient when considering problems with relatively large horizons. Here we suggest a simple framework that is computationally efficient as well as numerically robust when dealing with large horizon problems. We show that the optimal dispatch policy can be characterized through a set of optimal exercise boundaries and also theoretically derive the shape properties of the boundaries. The problem of finding the optimal exercise boundaries is then reduced to solving a simple linear programming problem. The suggested approach is flexible in incorporating various real world operational constraints. We compare the computational performance of the suggested scheme with alternative dynamic programming based methods. Our second study considered a regression approach to pricing European options in an incomplete market. The algorithm replicates an option by a portfolio consisting of the underlying security and a risk-free bond. We apply linear regression framework and quadratic programming with linear constraints (input = sample paths of underlying security; output = table of option prices as a function of time and price of the underlying security). We populate the model with historical prices of the underlying security (possibly massaged to the present volatility) or with Monte Carlo simulated prices. Risk neutral processes or probabilities are not needed in this framework.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Valeriy Ryabchenko.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Uryasev, Stanislav.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022492:00001


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IwouldliketothankmyadvisorProf.StanUryasevforintroducingmetotheareaofnancialengineering,forhispassionforthescienceandformakingthefouryearsIspentattheUniversityofFloridaoneofthemostinterestingtimesinmylife.Ilearnedalotfromhisdeterminationandhisabilityofconstantlearningandselfdevelopment.IwouldalsoliketoexpressmydeepappreciationtoProf.FaridAitSahlia.Iwanttothankhimforhisconstantwillingnesstohelp,andforthecountlesshourshespentwithmediscussinginnumerableproblems.Igainedalotfromhistruedevotiontothesciencefromhisdeepunderstandingofnumerouseldsinmathematicsandnancialengineering.Iwouldliketogivemanythankstomyothercommitteemembers(Prof.PanosPardalosandProf.LiqingYan)forallthehelptheyprovidedmethroughtheyears.IwanttothankandsayalotofwarmwordstoProf.BelovYuriyAnatoljevichandProf.RublyovBogdanVladislavovichwhoweremyadvisorswhenIwasstudyingatKyivNationalTarasShevchenkoUniversity.IwanttothankmycollaboratorandfriendSergeySarykalin.Ienjoyedtheyearsweworkedtogether.Iwouldalsoliketosaywordsofthankstoallmyfriendsfortheirhelp,supportandthetimewespenttogether.IwouldliketosaymyspecialthankstomyschoolmathteacherVasiliyVasiljevichHopachenko.Hewastheonewhointroducedmetothescienceofmath,fosteredinmeloveandadmirationforthesubject,instilledinmetheabilitytovaluebrilliantmathematicalideasandthinkbeyondstandardlimits.Itisnotpossibletoexpressinwordsmythankstomyfamilyandmygirlfriendfortheirtrueloveandconstantsupport.TheymademystayandworkintheUnitedStatescomfortableandfruitful. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 2PRICINGSCHEDULINGFLEXIBILITYOFELECTRICITYGENERATIONFACILITIES ..................................... 14 2.1Introduction ................................... 14 2.2ProblemDescriptionandNotation ...................... 17 2.3StochasticOptimalControlProblem ..................... 20 2.4OptimalOperatingStrategy .......................... 23 2.5OptimalExerciseBoundary .......................... 34 2.6FindingOptimalStationaryExerciseBoundaries ............... 36 2.6.1OptimizationonaGrid ......................... 36 2.6.2OptimizationwithTwoOptimalStationaryExerciseBoundaries .. 38 2.6.3OptimizationwithOneOptimalStationaryExerciseBoundary ... 40 2.6.4HeuristicwithTwoStationaryExerciseBoundaries ......... 50 2.7TimeDependentOptimalExerciseBoundaries ................ 58 2.8SetupwithNoRamp-UpPeriod ........................ 60 2.9NumericalCaseStudy ............................. 69 2.9.1LongHorizonCaseStudy ........................ 70 2.9.2ShortHorizonCaseStudy ....................... 73 2.9.3PriceDynamicswithJumps ...................... 77 2.9.4SamplePathSimulation ........................ 78 2.9.5PriceDynamicsSelectionandProcessCalibration .......... 78 2.10Summary .................................... 79 3PRICINGEUROPEANOPTIONSBYNUMERICALREPLICATION ..... 91 3.1Introduction ................................... 91 3.2FrameworkandNotations ........................... 95 3.2.1PortfolioDynamicsandSquaredError ................ 95 3.2.2HedgingStrategy ............................ 97 3.3AlgorithmforPricingOptions ......................... 99 3.3.1OptimizationProblem ......................... 100 3.3.2FinancialInterpretationoftheObjective ............... 103 3.3.3Constraints ............................... 103 5

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............................ 104 3.4JusticationofConstraintsonOptionValuesandStockPositions ..... 105 3.4.1ConstraintsforPutOptions ...................... 105 3.4.2JusticationofConstraintsonOptionValues ............. 106 3.4.3JusticationofConstraintsonStockPosition ............. 114 3.5CaseStudy ................................... 117 3.5.1PricingEuropeanOptionsontheStockFollowingtheGeometricBrownianMotion ............................ 118 3.5.2PricingEuropeanOptionsontheS&P500Index ........... 118 3.5.3DiscussionofResults .......................... 119 3.6ConclusionsandFutureResearch ....................... 122 REFERENCES ....................................... 132 BIOGRAPHICALSKETCH ................................ 136 6

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Table page 2-1ParametersofgeometricOrnstein-Uhlenbeckprocessesforthelonghorizoncasestudy. ......................................... 86 2-2Prices(inmilliondollars)of10-yeartollingagreementcontracts:calculatedusingthealgorithmwithan\average"exerciseboundaryversustheexactvalues. ... 86 2-3Prices(inmilliondollars)of10-yeartollingagreementcontracts:calculatedusingtheheuristicalgorithmwithtwoexerciseboundariesversustheexactvaluesandversusthepricesgivenbythealgorithmwithan\average"exerciseboundary. 86 2-4Prices(inmilliondollars)of10-yeartollingagreementcontracts:calculatedonan80x80gridversuscalculatedona50x50grid. .................. 87 2-5Computationaltimes(inseconds)ofpricinga10-yeartollingagreementcontractusing1000samplepathsanda50x50grid. ..................... 87 2-6ParametersofgeometricOrnstein-Uhlenbeckprocessesfortheshorthorizoncasestudy. ......................................... 87 2-7Prices(inmilliondollars)of1-yeartollingagreementcontractscalculatedusingan\average"exerciseboundary. ........................... 88 2-8Prices(inmilliondollars)of1-yeartollingagreementcontractscalculatedusingtheheuristicwithtwoexerciseboundaries. ..................... 88 2-9Computationaltimes(inmilliseconds)ofpricinga1-yeartollingagreementcontractusing500samplepathsanda50x50grid. ...................... 88 2-10Prices(inmilliondollars)of1-yeartollingagreementcontractsfortheidealizedpowerplantsetupcalculatedusingMonteCarlosimulationwith1000samplepathsand50independentruns. ........................... 89 2-11Prices(inmilliondollars)of1-yeartollingagreementcontractscalculatedusingtheheuristicalgorithmwithtwoexerciseboundariesversusthepricescomputedfortheidealizedpowerplantsetupwithoutcosts. ................. 89 2-12Prices(inmilliondollars)of1-yeartollingagreementcontractscalculatedusingthealgorithmwithtime-dependentexerciseboundaries. .............. 89 2-13Prices(inmilliondollars)of1-yeartollingagreementcontractsunderthejumpdiusionenergypricedynamicsandcalculatedusingtheheuristicalgorithmwithtwoexerciseboundaries. ............................ 90 2-14Prices(inmilliondollars)of1-yeartollingagreementcontractsunderthejumpdiusionenergypricedynamicsandcalculatedusingtheheuristicalgorithmwithtwoexerciseboundaries. ............................ 90 7

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............................. 127 3-2TheS&P500optionsdataset. ........................... 128 3-3PricingoptionsonS&P500index:100paths ................... 129 3-4PricingoptionsonS&P500index:20paths .................... 130 3-5SummaryofcashowdistributionsforobitainedhedgingstrategiespresentedonFigures 3-5 3-6 3-7 ,and 3-8 ............................ 131 3-6Calculationtimesofthepricingalgorithm. ..................... 131 3-7Numericalvaluesofinexionpointsofthestockpositionasafunctionofthestockpriceforsomeoptions. ............................. 131 8

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Figure page 2-1Optimalexerciseboundaries. ............................. 81 2-2Optimaloperatingpolicywhencurrentstateoftheplantis\o". ........ 81 2-3Gridonalogarithmicplane. ............................. 82 2-4Interpolationonagrid. ................................ 82 2-5Heuristicforndingtwooptimalexerciseboundaries. ............... 83 2-6Distancebetweenapointandthe\o"boundary. ................. 83 2-7Distancebetweenapointandthe\on"boundary. ................. 84 2-8Asequenceoftimecutsforpoints0t1
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Ourrststudyresearchedaproblemofschedulingoperationalexibilityofelectricitygeneratingfacilities.Recently,anumberofnewapproachesbasedonstochasticdynamicprogrammingtechniqueshavebeensuggestedintheacademicliterature.Althoughtheseapproachesareexibleintermsofincorporatingvariousoperationalconstraints,theyarecomputationallyinecientwhenconsideringproblemswithrelativelylargehorizons.Herewesuggestasimpleframeworkthatiscomputationallyecientaswellasnumericallyrobustwhendealingwithlargehorizonproblems.Weshowthattheoptimaldispatchpolicycanbecharacterizedthroughasetofoptimalexerciseboundariesandalsotheoreticallyderivetheshapepropertiesoftheboundaries.Theproblemofndingtheoptimalexerciseboundariesisthenreducedtosolvingasimplelinearprogrammingproblem.Thesuggestedapproachisexibleinincorporatingvariousrealworldoperationalconstraints.Wecomparethecomputationalperformanceofthesuggestedschemewithalternativedynamicprogrammingbasedmethods. OursecondstudyconsideredaregressionapproachtopricingEuropeanoptionsinanincompletemarket.Thealgorithmreplicatesanoptionbyaportfolioconsistingoftheunderlyingsecurityandarisk-freebond.Weapplylinearregressionframeworkandquadraticprogrammingwithlinearconstraints(input=samplepathsofunderlyingsecurity;output=tableofoptionpricesasafunctionoftimeandpriceoftheunderlyingsecurity).Wepopulatethemodelwithhistoricalpricesoftheunderlyingsecurity(possibly 10

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Weintroducednewecientoptimizationapproachestosolvingtwofundamentalproblemsofderivativespricing:PricingoftollingagreementcontractsandpricingofEuropeanoptions.Theproblemofpricingtollingagreementcontractshasbecomepopularafterenergymarketsde-regulationin1970s.Intherecentyearstheproblemhasdrawnagreatamountofresearchers'attentionafterthetollingcontractsstartedactivelytradingontheexchanges.Allthesuggestedapproachestopricingthetollingagreementcontractsinheritatleastoneofthefollowingshortcomingstoagreaterorsmallerextent.Theyeithermakeexcessivelysimplisticassumptionsregardingtheoperationalconstraintsortheunderlyingpricedynamics,ortheyhavepoorcomputationaleciencyandstabilitycharacteristics.PricingEuropeanoptionsisoneoftheoldestandthemostheavilyresearchedprobleminnancialengineering.Agreatamountofvarioustechniqueshasbeensuggestedthroughtheyearsforsolvingtheproblem.Themostcommondrawbacksofthesuggestedmethodsarerestrictiveassumptionsregardingtheunderlyingpricedynamics,alargenumberofsamplepathsneeded,absenceofeasytousehedgingstrategyandpoorcomputationaleciency. Themainadvantagesofthedevelopedalgorithmsoverthecorrespondingcounterpartsalreadyexistingintheacademicliteraturearethefollowing:Aminimalnumberofassumptionsregardingtheunderlyingpricedynamicsneeded,computationaleciencyandnumericalstabilityoftheoptimizationprocedures,asmallnumberofsamplepathsneeded.Inbothcaseswereducethecorrespondingproblemstosolvingasimplecontinuousoptimizationproblem.UnlikethetraditionalapproachesusedinderivativespricingsuchasdynamicprogrammingorMonteCarlosimulation,theproposedapproachallowstopreservethecomputationaleciencyandnumericalrobustnessevenwhendealingwithlargehorizoncontracts.Theapproachessuggestedforsolvingbothproblemssharethecommonmodelingidea.Theoptimaloperatingstrategyintherstproblemand 12

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InChapter2weconsideraproblemofpricingtollingagreementcontracts.Anothernameforthisproblemispricingschedulingexibilityofelectricitygenerationfacilities.Thisisafundamentalprobleminpricingexoticenergyderivatives.Theproblemfallsintoaclassofmultiplestoppingtimeproblemsandisextremelychallengingtosolve.Astandardwayofsolvingthiskindofproblemsisbasedontheideasofstochasticdynamicprogramming.Althoughexibleinincorporatingvariousoperationalconstraintstheapproachesbasedonthedynamicprogrammingideashaveaproblemknownasacurseofdimensionality.Thedynamicprogrammingalgorithmsdonothaveattractivecomputationalcharacteristicsandusuallyarenotapplicableatallforproblemswithlargehorizons.Wedevelopedanalgorithmthatreducestheconsideredproblemtosolvingasinglelinearprogrammingproblem.Thesuggestednumericaltechniquehasstrongeciencyandstabilitypropertiesandisexibleinincorporatingvariousoperationalconstraints. inChapter3wesolveaproblemofpricingEuropeanoptions.Thisisaclassicalprobleminthetheoryofoptionspricing.WeconsideranincompletemarketsetupandsuggestareplicationbasedalgorithmforpricingEuropeanoptions.Wereducetheconsideredproblemtosolvingasimplequadraticprogrammingproblem.Theintroducedalgorithmhasattractivecomputationaleciencycharacteristicsandiscapableofproducingaccuratepriceestimateswithasmallnumberofsamplepaths.Theframeworkisbuiltongeneralassumptionsregardingtheunderlyingpricedynamics.Theconductedexperimentsusingrealmarketdatashowedtheabilityofthealgorithmtocatchthemarket'svolatilitysmile. 13

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Historically,tollingagreementcontractshavebecomepopularsincede-regulationofenergymarketsinthe1970s.Whenthecontractsjustappeared,practitionerstriedtoapplystandardatthetimediscountcashowmethods.Almostimmediately,theyrealizedthatthediscountcashowapproachisnotsuitableinthehighlyvolatileenvironmentofenergyandfuelprices.Whenresearchersrstbecameinterestedinthisproblemtheytriedtomakeuseofawell-developedintuitionofoptionpricing.Morespecically,theytriedtorepresentaschedulingoperationalexibilityofenergygeneratingfacilitiesasasequenceofsocalledsparkspreadoptionsownedbyamanager.Asparkspreadoptionisanoptiongivingitsholderataspeciedtimeinthefuturetherighttoexerciseaprotequaltoanon-negativepartofthedierencebetweenthepriceofenergyandthepriceoffuelmultipliedonacoecientcalledheatrate.ApplicationofthisideacanbefoundinDeng,JohnsonandSogomonian(1998),andEydelandandWolyniec(2003).AniceoverviewofmethodsonspreadoptionspricingisgiveninCarmonaandDurrleman(2003). 14

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SimilarlytoCarmonaandLudkovski(2008),anumberofauthorstriedtoapplyadynamicoptimalcontrol(alsocalledanoptimalswitching)settingtosolvetheoptimalschedulingexibilityproblem.Theyattemptedtoderiveaclosed-formsolutionoftheproblem.Inordertosolvetheproblem,theyhavetomakevarioussimplifyingassumptionssuchasaninnitehorizonandgeometricBrownianmotionwithconstantcoecientpricedynamics.AreadercannddevelopmentsofthisapproachinDixit(1989),BrekkeandOksendal(1994),BayraktarandEgami(2007),PhamandLyVath(2007),andreferencestherein.AmoregeneralsetupisconsideredinHamadeneandJeanblanc(2007),yettheresultobtainedinthispaperistootheoretical,anditisnotclearhowonecancreateanecientnumericalalgorithmonitsbasis. Inthisworkwesuggestasimple,robustandcomputationallyecientoptimizationframeworkthatproducesaccurateestimatesforthepricesoftollingagreementcontracts.Theoptimizationprocedurealsoprovidesaneasy-to-useoptimaldispatchstrategydenedbyasetofoptimalexerciseboundaries.Atthenalend,theoptimizationprocedurereducestosolvingalinearprogrammingproblemwithanumberofvariablesandconstraintsindependentofthenumberofsamplepathsorthetimehorizonusedinthemodel.Extremecomputationaleciencyoflinearprogrammingoptimizationtechniquesisthekeyfactorofcomputationaleciencyofourframework.Thesuggestedapproachiscapableofdealingwithcontracthorizonsaslargeas20yearsandlonger.Therobustnessoftheframework,resultingfromthemodelingapproachandexplicitincorporationofshapepropertiesofoptimalexerciseboundariesintothemodel,allowsustoobtainstablecontractpriceestimatesusingonlyonethousandsamplepathswithinthemodel.Thisremarkablefeatureofthealgorithmprovidesapossibilityofusinghistorical 16

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InSection2,weprovideadescriptionoftheconsideredpricingtollingagreementsproblemandintroducethenotationthatisutilizedthroughouttheconsequentsections.InSection3,wederiveastochasticoptimalcontroloptimizationproblemthatamanagerofthepowerplanthastosolveinordertondtheoptimaloperatingpolicyand,consequently,tondthepriceofthecorrespondingtollingagreementcontract.InSection4,weexaminethepropertiesofoptimaloperatingstrategiesandprovetheoreticalresults.Theresultsofthissectionunderlyandjustifyouroptimizationframeworkdevelopedinthefollowingsections.InSection5,weintroduceanotionofoptimalexerciseboundariesandformulateanoptimaloperatingpolicyintermsofoptimalexerciseboundaries.Section6describesouralgorithmforndingthetimeindependentoptimalexerciseboundariesandthecorrespondingpriceofthetollingagreementcontract.InSection7,wegeneralizeourapproachtothecasewithtimedependentoptimalexerciseboundaries.InSection8,weconsideraspecialcaseoftheproblemwhenthereisnotheramp-upperiodconstraintinthemodel.Inthissectionwere-deriveourmaintheoreticalresultsforthenewsetting.Section9providesresultsofnumericalexperimentsforarealworldpowerplantsetup.Weinvestigatevariouscomputationalaspectsofthealgorithmandcomparetheperformanceofouralgorithmwithadynamicprogrammingbasedoptimizationapproach.Section10concludesourworkprovidingasummaryoftheresultsandcommentsregardingthesuggestedframework. 17

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>0): andinthelowcapacitymodetheplanthasaheatrateequaltoH 18

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: Q 60=5 3,andtheratio H 14>3 5(acommoncaseiswhen H Tonalizethelistofourassumptionsweneedtospecifythefollowing.Tobringtheplantonlineinadditiontoramp-upcosts,theoperatoralsoincursxedstartupcostsCs(Cs0): 19

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respectively,andaoneperiodriskfreeinterestrateasr:

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Westartfromintroducingthestochasticframework.Let(;F;F=(Fi);P)beastochasticbasis,whereFisaltration(F1F2:::Fi:::FNF).WeassumethatPisarisk-neutralprobabilitymeasure.Wealsoassumethat(Pi)and(Gi)areMarkovianprocessesdenedontheprobabilityspaceaboveandareadaptedtotheltrationF.Assumethatthecurrenttimeperiodis1.Letthevector(1;:::;N)bethevectorrepresentingtherenter'scurrentswitchingdecision,1,andhisfutureswitchingdecisions,2;:::;N.Wedenote,0,astheinitialstateofthepowerplant.Anyoftheiscantakeoneoftwovalues,0or1,meaningthattheplantisonwheni=1andisowheni=0.BecauseweconsiderMarkovianpriceprocessonly,therenter'sdecisionattimeidependsonlyonthecurrentstateoftheplant,i1,andthevectorofcurrentgasandenergyprices,(Gi;Pi).Therefore,wecanrepresentiasthefollowingfunction:i(i1;Gi;Pi).Lookingfromthetimeperiod1,theswitchingdecisionsattimeperiods2;3;:::;Narerandomstochasticcontrolstakingvalues0or1,andonly1isadeterministic01controlvariable. Thepowerplantisonline(producingenergy)atthebeginningofatimeperiodionlyifithasbeeninthe"on"stateduringtheprecedingtimeperiod.Therefore,theplantis 21

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Inthesubsequentsectionswealsomakeuseofthenotation:fi(i1;i)=[ii1]+^Cis+[i1i]+Cd: Tondthevalueofthetollingcontractandtheoptimalswitchingdecision1attime1,therenterhastomaximizetheexpectedcumulativeprotS(1;N)overthesetofalladmissibleFi-measurablestochasticcontrolsi(w).Inotherwords,sheneedstosolvethestochasticoptimizationproblem: Let(1;2(w);:::;N(w))beanoptimalsolutionoftheaboveproblem.ThevalueofthecontractistheoptimalobjectivevalueJ1N,andtheoptimalswitchingdecisionattime1istheoptimalsolution1.Similarlytotheproblemabove,wecanwriteanoptimizationproblemfortimej: TheupperindexinJjNj+1andPjNj+1denotesthestartingtimeperiod,andthelowerindexdenotesthetotalnumberofperiodsintheoptimizationproblem.TondanoptimaloperatingdecisionattimejtherenterhastosolvetheproblemPjNj+1andtakethe 22

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Using( 2{3 )andtheBellmanprincipleofoptimalitywecanderiveaBellmanequationforJ1N(0):J1N(0)=sup1;:::;N01M1f1(0;1)+E"NXi=2(ii1Mifi(i1;i))er(i1)jF1#!==max101M1f1(0;1)+erE"sup2;:::;NE"NXi=2(ii1Mifi(i1;i))er(i2)jF2#jF1#!==max101M1f1(0;1)+erEJ2N1(1)jF1: UsingthederivedBellmanequationwecanformulatethenecessaryandsucientconditionsfor1=1tobeoptimalintheproblemsP1N(0)andP1N(1),correspondingly.

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2{6 )itowsthat1=1isoptimalinP1N(1)ifandonlyif:M1f1(1;1)+erEJ2N1(1)jF1f1(1;0)+erEJ2N1(0)jF1,,EJ2N1(1)jF1EJ2N1(0)jF1er(CdM1): 2{6 )itowsthat1=1isoptimalinP1N(0)ifandonlyif:f1(0;1)+erEJ2N1(1)jF1f1(0;0)+erEJ2N1(0)jF1,,EJ2N1(1)jF1EJ2N1(0)jF1er^C1s: 1;N:Mi+^Cis+Cd0: 1;N:

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1).First,weverifythestatementforN=1.InthiscasetheproblemP1N(0)takestheform:max1^C1s1: 2).AssumethestatementistrueforN=K1(K>1).Letusprovethetheorem'sresultforN=K.Using( 2{6 ):P1K(0):max1f1(0;1)+erEJ2K1(1)P1K(1):max1M11f1(1;1)+erEJ2K1(1) 1 ithastobesatised: Usingthesamelemma,thenecessaryandsucientconditionfor11=1tobeoptimalinP1K(1)is FromLemma 2 : ^C1sCdM1:(2{9) Applying( 2{9 )to( 2{7 )wecanget( 2{8 ),henceitistruethat11=1isoptimalinP1K(1).Thus,wecanalwaysclaim: ForanyrealizedvaluesofG2andP2letusconsidertheproblem:P2K1:sup2;:::;KE[S(2;K)jF2]:

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2;K:(K1)i(w;1)(K1)i(w;0):(2{11) From( 2{10 )and( 2{11 ): 2;K:(K1)i(w;11)(K1)i(w;01):(2{12) Fromtheprincipleofoptimalityitcanbeconcluded: Finally,from( 2{12 ),( 2{13 )and( 2{14 ): 2;K:1i(w)0i(w)(2{15) Combining( 2{15 )with( 2{10 ),wegetthetheorem'sstatement.Therefore,theinductionstepisshownandthiscompletestheproofofthetheorem.2 1;N:

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1;N:M0i+^Ci0sMi+^Cis: 1;Nwecanmakethederivationsbelow. IfQ > QPi Finally,because^Cis^Ci0s=L(GiG0i),andusing( 2{16 )wecometo:M0i+^Ci0sMi+^Cis: 27

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1 2{6 )andapplyingLemma 2 wehaveJ1N(1)J1N(0)=Cd)M1+^C1sJ1N(1)J1N(0)Cd: 2{6 ),wehaveJ1N(1)J1N(0)=M1+^C1s)M1+^C1sJ1N(1)J1N(0)Cd: Because11=0issuboptimalinP1N(1),wegettheinequality: From( 2{17 )and( 2{18 )wegettheassertionofthelemmaforcase2).Hence,thelemmaisproved.2 3 .Let1,0,10and00beoptimalsolutionsoftheproblemsP1N(1),P1N(0),P10N(1),andP10N(0),correspondingly.8N>0thefollowingstatementsaretrue:1)J10N(1)J10N(0)J1N(1)J1N(0);2)01001;and11101: 28

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2{19 ),wederive WecontinuetheproofconditioningontheoptimalsolutionofJ1K. a)Assume01=1isoptimal,thenfromTheorem 1 weget11=1.Using( 2{6 ): FromLemma 1 ,01=1implies Combining( 2{20 )with( 2{22 )andusingG01G1:EJ20K1(1)jF1EJ20K1(0)jF1er^C1ser^C10s: 1 ,thelastinequalityisequivalenttooptimalityof001=1inP10K(0).UsingTheorem 1 wealsoget101=1optimalinP10K(1).Withthehelpof( 2{6 )wecannowcomputethedierence: SincetheconditionofLemma 3 issatised,thenusingthelemmaandtakingintoaccount( 2{21 )and( 2{23 ):J10K(1)J10K(0)J1K(1)J1K(0): b)Assume11=0isoptimalinP1K(1).Automatically,fromTheorem 1 wehavethat01=0isoptimalinP1K(0).Using( 2{6 ): 30

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4 and( 2{24 ):J10K(1)J10K(0)C0d=Cd=J1K(1)J1K(0): c)Thelastcaseleftiswhen11=1and01=0.Using( 2{6 )wecancompute: Fromtheoptimalityof11=1andLemma 1 : Combining( 2{20 ),( 2{26 ),andtheconditionthatM01M1: Using( 2{27 )andLemma 1 again,wendthat101=1isoptimalinP10K(1).Therefore,themonotonicityofoptimalsolutionsisshown.Itislefttoshowtheinequalitypartoftheinductionhypothesis. If001=0isoptimal,thenusing( 2{6 ),( 2{20 ),M01M1and( 2{25 ):J10K(1)J10K(0)=M01+erEJ20K1(1)jF1EJ20K1(0)jF1M1+erEJ2K1(1)jF1EJ2K1(0)jF1=J1K(1)J1K(0): 2{6 ): 31

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3 andLemma 4 to( 2{28 ),wegetJ10K(1)J10K(0)M1+^C1sJ1K(1)J1K(0): 2 .1. thentheoptimalsolutionsofP1NandP10Nhavethefollowingproperties:1)01001;2)11101: 32

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22dt;dXt 22dt: 2{29 ),andsettingSt=lnXtand=b2 Therefore,S(t)isastandardarithmeticOrnstein-Uhlenbeckprocess.Thesolutionto( 2{30 )is If(Pt)and(P0t)followthesamegeometricOrnstein-Uhlenbeckdynamics,thenfromtheconditionP0P00and( 2{31 )wehave Analogously,if(Gt)and(G0t)havethesamegeometricOrnstein-UhlenbeckdynamicsthenfromG0G00: ForaprocessXtfollowingthegeometricBrownianMotionwehave 22(s))ds+Rt0(s)dWs:(2{35) Using( 2{35 )andthecorollary'sassumptionswecanderive( 2{33 )and( 2{34 ). Wecanalsoallowjumpswithindependentintensitiesandmagnitudesinthepricedynamics.Indeed,ifweaddajumpcomponentdJtothedynamicsequationofaprice 33

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2{33 )-( 2{34 )likerelationsaresatisedforX(t)thentheyarealsosatisedfortheprocessX0(t). Sincefromthecorollary'sassumptionstheconditions( 2{33 )and( 2{34 )canbederived,thenapplyingTheorem 2 wegettheassertionofthecorollary.2 0;1considerthesetsbelow:RN;i0=f(XG;XP)ji1=0g;RN;i1=f(XG;XP)ji1=1g: 34

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2 ,itfollowsthattheoptimalexerciseboundarieshavetobemonotonic.Monotonicityofaboundaryhastobeunderstoodinthefollowingsense.Ifapoint(X1G;X1P)belongstothe0-optimalexercisesetforsomestateoftheplant,thenanypoint(X2G;X2P),suchthatX1PX2PandX1GX2G,hastobelongtothe0-optimalexercisesetforthesamestateoftheplantaswell.Fromthemonotonicityoftheboundaryitalsofollowsthatthe0and1-optimalexercisesetsareconnected.Wedenotetheoptimalexerciseboundariesforthe\o"andthe\on"statesoftheplantasOB0andOB1correspondingly.FromTheorem 1 wealsohavethattheboundaryOB0hastobealwaysnolowerthantheboundaryOB1.Figure 2-1 summarizespropertiesoftheoptimalexerciseboundaries.Foranyperiodoftimetheproblemofndingtheoptimalexerciseboundariesisequivalenttotheoperator'soptimalswitchingproblem.Withthehelpofoptimalexerciseboundariesitiseasytoformulatetherenter'soptimalswitchingpolicy.Ifthecurrentpoint(XG;XP),representingcurrentenergyandgasprices,liesabovetheoptimalexerciseboundaryforthecurrentstateoftheplant,thentheoptimaloperator'sdecisionistoturnonthepowerplant(orleaveitworkingifitscurrentstateis\on").Ifthepricepointisbelowtheoptimalexerciseboundary,thenitisoptimaltoturndownthepowerplant(orleaveitnotworkingifitscurrentstateis\o").Iftheoptimaloperatingdecisionistosettheplantinthe\on"state,thentheoptimalcapacityregimeisdeterminedbyinstantaneousgainsoftheregimebecauseswitchingbetweendierentcapacityregimesiscostless.Figure 2-2 explainstheoptimaloperatingbehavioroftheoperatorifthecurrentstateoftheplantis\o"(sheneedstousetheboundaryOB0inthiscase).Findingtheoptimalexerciseboundariesforeverytimeperiodisatimelyandresourceconsumingprocess.Toovercomethisdiculty,wesuggesttomakeuseoftheheuristicargumentbelow.Ifatollingagreementcontracthasarelativelylargetimehorizon(oneyearshouldbelongenough),thenitisnotnecessarytobuildtheoptimalexerciseboundariesforeverytime 35

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2.6.1OptimizationonaGrid 36

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Letf(Gji;Pji)gj=1;:::;NSi=1;:::;NbeasetofsamplepathsgeneratedusinggeometricOrnstein-Uhlenbeckdynamics,whereasbeforeNisthetotalnumberoftimeperiodsinthecontract,andNSisthetotalnumberofsamplepaths.AlsoletPmax=max(i;j)lnPji,Gmax=max(i;j)lnGji,Pmin=min(i;j)lnPjiandGmin=min(i;j)lnGji.Onthe(lnXG;lnXP)-planeconsiderarectanglewiththetopleftvertexhavingcoordinates(Gmin;Pmax)andthebottomrightvertexhavingcoordinates(Gmax;Pmin).InsidetheconstructedrectangleletusbuildauniformgridwithNHhorizontalnodesandNVverticalnodes,seeFigure 2-3 .Wenumeratehorizontallinesofthegridfrom1toNVwithnumbersincreasingtothetop,andnumerateverticallinesofthegridfrom1toNHwithnumbersincreasingtotheright.Theobtaineduniformgridisourdiscretizationoftheinitial(lnXG;lnXP)-plane.Belowwedevelopalgorithmsforndingoptimalexerciseboundariesonthediscretized(lnXG;lnXP)-plane.Atanynode(i;j)oftheconstructedgridweassignapairof0-1variables(i;j;i;j);i= 1;NV;j= 1;NH,seeFigure 2-3 .Theinterpretationforthesevariablesisthefollowing:Ifatagridpoint(i;j)i;j=1,thentheplanepointcorrespondingtothenode(i;j)liesaboveSOB0.Ifi;j=0,thentheplanepointcorrespondingtothenode(i;j)liesbelowSOB0.Asimilarinterpretationistruefori;js,theonlydierenceisthatthevariablesi;jdeneSOB1.Hence,theproblemofndinganoptimalSOB0andSOB1isreducedtondingoptimalvaluesfori;jsandi;js.SinceSOB0andSOB1havethemonotonicitypropertyandSOB0liesaboveSOB1,weneedtoimposesomeconstraintsoni;jsandi;jsinordertosatisfyoptimalexerciseboundaries 37

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Sincetheenergyandgaspriceprocessesarecontinuous,andthe(lnXG;lnXP)-planeisdiscrete,weneedtochooseaninterpolationruleforassigningapairof(i;j;i;j)foreachsamplepathpoint(Gji;Pji).WepicktheclosestgridnodeintheEuclidiansenseasourinterpolationrule.Let(Gji;Pji)beasamplepathpoint.Applyingthelogarithmtransformation,wegeta(lnGji;lnPji)pointonthelogarithmicplane.Let(m;n)betheclosestto(lnGji;lnPji)gridnodeonthe(lnXG;lnXP)-plane,thenweassign(m;n;m;n)to(Gji;Pji)asthecorrespondingsetofexerciseboundaryvariables.Tosimplifythenotation,foreachsamplepathpoint(Gji;Pji)wedenotethecorrespondingpairofexerciseboundaryvariablesas(ji;ji),seeFigure 2-4 .Usingtheassignedvariablesitiseasytoformulatetheoptimaloperatingpolicyonthesamplepaths: 1)Ifatapoint(Gji;Pji)thecurrentstateoftheplantis\o", (2{39) thenitisoptimaltoturnontheplant,ji=1:2)Ifatapoint(Gji;Pji)thecurrentstateoftheplantis\on", (2{40) thenitisoptimaltoturnotheplant,ji=0: 2{39 )-( 2{40 )weseethatdependingonthecurrentstateoftheplant,atanypoint(Gji;Pji)theoptimalswitchingruleiseitherjiorji.Forthesakeofsimplicity,weintroduceanauxiliarysetofvariablesfjigj=1;:::;NSi=1;:::;N.Withthenewvariables,theoptimaldecisionruleatanypoint(Gji;Pji)isji.Tofollowthelogicintroducedby( 2{39 )-( 2{40 ) 38

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(2{41) Thelogicbehindtheseconstraintsisstraightforward.Atanypoint(Gji;Pji)thecurrentstateoftheplantisdeterminedbyavariableji1.Wheneverji1=0,meaningthattheplantiscurrently\o",theoptimalswitchingruleat(Gji;Pji)isji.Wheneverji1=1,meaningthattheplantiscurrently\on",theoptimalswitchingruleat(Gji;Pji)isji.Similarlyto( 2{1 ),wecanconstructaprotfunctionforeachsamplepathj(j=1;:::;NS)andeachperiodi(i=1;:::;N):ji=er(i1)jiji1Mji[jiji1]+Cjs;i[ji1ji]+Cd; Q; H;Q 2{41 )-( 2{42 );optimalexerciseboundariesshapeconstraints( 2{36 )-( 2{38 );i;j;i;j2f0;1g;i=1;:::;NV;j=1;:::;NH:

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40

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wecanrewrite(P1B)inthefollowingequivalentform:(P1B0):maxi;j1 41

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(P1BL):maxi;j1 whereconstraints( 2{45 )-( 2{48 )ensurethatji=[jiji1]+andji=[ji1ji]+.Thereisonemajorproblemwithwriting(P1BL)intheformgivenabove.Althoughwegotalinearproblem,thenotationstatesthatweneedtointroduceNNSauxiliaryvariables.Itiseasytoshowthatthereisalinearizationof(P1B)thatrequiresintroducingnomorethan(N2VN2H)newvariables.Inthiscase,thetotalnumberofvariablesintheproblemdependsonlyonthenumberofnodesinthegridandnotonthenumberofsamplepathsortimeperiods.Tomake(P1B)linearweneedtointroduceanauxiliaryvariableforeachofthepairs(ji;ji1)and(ji1;ji).Hence,thetotalnumberofauxiliaryvariablesneededequalstothetotalnumberofdierentorderedpairs(ji;ji1)and(ji1;ji).Rememberingthenotationintroducedearlier,bywritingvariablesjiweassumethefollowingmappingfortheindexes(ji)(wedenoteitby(ji)becauseitcorrespondstovariables):(ji):f1;:::;Ngf1;:::;NSg!f1;:::;NHgf1;:::;NVg: 42

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Withthehelpof(P1BL)wecanndasingleoptimalstationaryexerciseboundary,but,ingeneral,(P1BL)maynotbeeasytosolvebecauseitisamixedintegerprogrammingproblem.Thus,wehavetoworkonndingabetterformulation.Letuswritethefollowingchainofderivations:8i=1;:::;N;j=1;:::;NS:Mji+Cjs;i=max : TheconditionCsK 43

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2{52 ))andCd0.Nowwehaveeverythingsettoformulatethemainresultofthissection.

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1;m;i6=j:xi1;:::;xili\nxj1;:::;xjljo=?;m[i=1xi1;:::;xili=nk[p=1fxip;xjpg: 1;m9p:1pli;9j:1jksuchthat:xip=xj:

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3

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1)Onlyji=0ispresentinthesystem. 2)Onlyji=jiji1ispresentinthesystem. 3)Bothji=0andji=jiji1arepresentinthesystem. Incase1)weeliminatethevariablevjifromthesystemandthenewlyobtainedsystemstillhastohaveauniquesolution.Togetthesolutionoftheinitialsystemwejustneedtoaddji=0tothesolutionoftheobtainedsystem. Incase2)wealsoeliminatethevariablejiandthecorrespondingequationfromthesystem.Theresultantsystemhasonelessvariableandonelessequationthantheinitialsystem.Astheinitialsystemhasauniquesolution,thenewsystemhastoinheritthisproperty.Letji=jiandji1=ji1beapartofthesolutionofthenewsystem.Wecangetthesolutionoftheinitialsystembyaddingtothesolutionoftheobtainedsmallersystemji=jiji1. Incase3)weperformsimilarsteps.Weeliminatethevariablejiandthetwocorrespondingequations.Theonlydierenceinthiscaseisthatwealsoaddanewconstraintji=ji1(thisequationcanbeeasilyderivedfromthetwoequationswithji)tothesystem.Asinthepreviouscasetheobtainedsystemhasonelessvariable,onelessconstraint,anditinheritsthepropertythatithastohaveauniquesolution.Thesolutionoftheinitialsystemcanbeobtainedbyaddingji=0tothesolutionofthenewlyconstructedsystem. 48

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5 withC=f0;1gtothesystemwithi;jsonly,wendthatitsuniquesolutionhasa0-1representation.Fromthevariableeliminationprocedurewealsogetthatji;ji2f1;0;1g,butsinceji=1andji=1arenotfeasiblein(L1)wegettheresultthatthevertexesofthefeasibleregioncanonlyhave0-1coordinates.Thelemmaisproved.2 49

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2-5 ): 50

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2)Assumethattheboundaryfoundonthepreviousstageisanoptimal\on"boundaryandndanoptimal\o"boundarythatisparalleltoit(ndanoptimalsizeupoftheupshift). Usingthenotationintroducedearlierinthissection:pi=8>>>>>><>>>>>>:pi;ifthecurrentstateoftheplantis\o"andi6=0;pi;ifthecurrentstateoftheplantis\on"andi6=0;0;ifi=0;assumingtheinitialstateoftheplantis\o": 1;N;p= 1;NS:pi=iXk=1pkiYl=k+1(plpl): 1)Fori=1theformulasimpliestothefollowing:p1=p1.Thelastidentityistruebythedenitionofpisincetheinitialstateoftheplantis\o". 2)Assumetheformulaistruefor8i(K1)andletusproveitfori=K.FromthedenitionofpiwehavepK=(1pK1)pK+pK1pK=pK+pK1(pKpK):

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1;NVandj= 1;NHareknown.Theheuristicapproachsuggeststondan\on"boundaryasanoptimalboundarythatcanbeobtainedbyshiftingthefound\o"boundarydownwards.Therefore,thetaskistondanoptimalsizeoftheshift0.Let0beanarbitraryshiftsize.Foreachpoint(Gji;Pji)letusalsodeneji,averticaldistancefromapoint(lnGji;lnPji)tothe\o"boundary.Let(m;n)betheclosestgridnodetothepoint(lnGji;lnPji)onthelogarithmicplane.Sinceweusetheclosestgridnodeasaninterpolationruleonthegrid,weapproximatejiwithavalueequaltothedistancebetweenthe\o"boundaryandthenode(m;n).Letin0bethesmallestindexinthenthcolumn,suchthatin0;n=1,and4vbethedistancebetweentwoadjacentverticalnodes.Then,ji=y(in0)4v 2-6 .Becauseweconsideronly0,tomakethealgorithmcomputationallymoreecientwemakethefollowingadjustmenttothecomputationofji.Wecomputejiasdenedabove,andifweobtainji<0thenwesetji=(>0).Anypositivevaluecanbetakenasthevalueof.IncaseswhenNV;n=0,meaningthatthereareno1sinthenthcolumn,wesetji=+1.Sincethe\on"boundaryisobtainedfromthe\o"boundarybyashiftofsize,thefollowingistrue: 52

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FromLemma 6 ,wehave 1;N;j= 1;NS:ji=8>>>>>><>>>>>>:Qil=kji+1jl;ifkjii:(2{54) Applying( 2{53 )to( 2{54 ): 1;N;j= 1;NS:ji=8>>>>>><>>>>>>:Qil=kji+1sgn[jl]+=sgn[i;j]+;ifkjii;(2{55) 53

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1;N;j= 1;NS:i;j=8>>>>>><>>>>>>:maxl= 2{55 )canberewritten: 1;N;j= 1;NS:ji=sgn[i;j]+:(2{56) Tondanoptimalweneedtosolveaversionof(P2B)withouttheboundaryshapeconstraints.Wedonotneedtheshapeconstraintsbecausethe\o"boundaryisalreadyknown,andtheshapeofthe\on"boundaryiscompletelydeterminedbytheshapeofthe\o"boundary(sincewearelookingforaparallelboundary).Summarizing,weneedtosolvetheproblem:max1 256 )constraints: 2{43 )thelastproblemcanbetransformedintoanequivalentproblem:(H1):max1 256 )constraints: 2{56 )itcanbederived: [jiji1]+=8>><>>:0;ifi;ji1;j;sgn[i;j]+sgn[i1;j]+;otherwise.(2{57) 54

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From( 2{56 ),( 2{57 ),and( 2{58 )itfollowsthat(H1)reducestoaproblemofthetype: max0MXi=1aisgn[i]+;(2{59) forsomeconstantsai2R;i0,andintegerM>0.Thelastproblemiseasilysolvable.Withoutlossofgeneralitywemayassume012:::M(wealwayscanmakeachangeofvariablesifnecessary).Weneedtomakethefollowingcalculations:S0=0;Si=iXj=1aj;i= 1;M;i=argmaxi= 0;MfSig: 1;NV;j= 1;NHareknown.Theheuristicsuggestsndingan\o"boundaryasanoptimalboundarythatcanbeobtainedbyshiftingthe\on"boundaryupwards.Thus,thetaskistondanoptimalsizeoftheshift0.Todothisweneedtofollowstepssimilartothestepsweperformedattherststage.Again,weneedtodeneadistancebetweenapointonthelogarithmicplaneandthe\on"boundary.Consideranarbitrarysamplepoint(Gji;Pji).Let(m;n)betheclosestgridnodetothepoint(lnGji;lnPji)onthelogarithmicplane,in0bethelargestindexinthenthcolumn,suchthatin0;n=0,and4vbethedistancebetweentwoadjacentverticalnodes.Then,adistancebetween(lnGji;lnPji)andthe\on"boundaryis 55

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2-7 .Asattheststate,wemakeasimilaradjustmentoftheformulaabove.Ifwehaveji<0thenwesetji=,wherecanbeanypositiveconstant.Incaseswhenthereareno0sinthenthcolumn,wesetji=+1.Let0beanarbitraryshiftsize.Then,8i= 1;N;j= 1;NSitistruethat: Now,weneedtoprovethelemmabelow. 1;N;j= 1;NS:ji=sgn[i;j]+; 1;NS.Weprovethelemmausinganinductionbyi. 1)Ifi=1thenusing( 2{60 ):j1=j1=sgn[j1]+:

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7 establishesaresultsimilarto( 2{56 ).Fromtheproofofthelemmawecandeduceanexplicitformulaforcomputingi;j:8i>1;8j:i;j=8>><>>:;ifji=0;maxfi1;j;jig;otherwise. Fori=1wehave8j:1;j=j1: 2{59 ): max0MXi=1aisgn[i]+;(2{61) forsomeconstantsai2R;i0,andsomeintegerM>0.Withoutlossofgeneralitywemayassume12:::M0(wealwayscanmakeachangeofvariablesifnecessary).Weneedtomakethefollowingcalculations:S0=0;Si=iXj=1aj;i= 1;M;i=argmaxi= 0;MfSig:

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58

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2-8 .Nowwecanusealgorithmsfromtheprevioussectiontobuildseparateexerciseboundariesforeachtimecut,(tk).Weneedtofollowthesamestepsasbefore.Themaindierenceinthiscaseisthatdierentsamplepathpointsmaybeprojectedontodierentplanes(cuts).Weapplytheclosestdistancecriteriuminmappingsamplepathpointsontothecuts.Consideranarbitrarysamplepathpoint,(Gji;Pji).Letanindexicorrespondtoatimemoment,Ti.Hence,inourthreedimensionalspacethesamplepathpoint,(Gji;Pji),hascoordinates,(lnGji;lnPji;Ti).Inthiscase,weproject(perpendicularly)thepoint(lnGji;lnPji;Ti)ontoacut(tk),suchthatk=argminl= 1;njTitlj.Nowletusconsideracasewhenwewanttobuildaseparate\average"exerciseboundaryineach(tk)(itislessclearifthesuggestedheuristicforndingseparate\on"and\o"boundariesremainsecientinthetime-dependentcase).Asbefore,attherststep,ineachplane(tk)weneedtobuildauniformgridinasimilarwayaswedidbefore.Then,oneachgridweneedtointroducevariableski;jsdeningan\average"exerciseboundary.Finally,usingagivensetofsamplepaths(Gji;Pji);i= 1;N;j= 1;NSwecanbuildandsolveanoptimizationproblemsimilarto(L1).Asaresult,weobtainndierentexerciseboundaries,oneforeach(tk),seeFigure 2-9 .Thepointst1;:::;tndonothavenecessarilytobeuniformly 59

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Oneofthekeyresultsforthesetupwitharamp-upperiodisTheorem 2 .Nowwehavetoderiveasimilarresultinasettingwithnoramp-upperiod.WestartbyrestatingsomeoftheauxiliaryresultsweneededtoproveTheorem 2 60

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1 )Thenecessaryandsucientconditionfor1=1tobeoptimalinP1N(1)isEJ2N1(1)jF1EJ2N1(0)jF1er(CdM1): 1 .2 1 )Let1=(11;:::;1N)and0=(01;:::;0N)beoptimalsolutionsoftheproblemsP1N(1)andP1N(0),correspondingly.8N>0thefollowingrelationshold:8w2:1i(w)0i(w);i= 1;N: 1).First,weverifythestatementforN=1.InthiscasetheproblemP1N(0)takestheform:max1M11^C1s1:

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2{62 ):P1K(0):max1M11f1(0;1)+erEJ2K1(1)P1K(1):max1M11f1(1;1)+erEJ2K1(1) 8 ithastobesatised: Usingthesamelemma,thenecessaryandsucientconditionfor11=1tobeoptimalinP1K(1)is Takingintoaccount^C1sM1CdM1andusing( 2{63 )wecanget( 2{64 ).Hence,11=1isoptimalinP1K(1),thus,wecanalwaysclaim: ThefollowingresultcanbeshowninacompletelysimilarfashionasitwasshownintheproofofTheorem 1 : 2;K:1i(w)0i(w):(2{66) Combining( 2{66 )with( 2{65 )wegetthetheorem'sstatementforN=K.Therefore,theinduction'sstepisshown,andthiscompletestheproofofthetheorem.2 4 )8N>0thefollowinginequalitieshold:1)J1N(1)J1N(0)^C1s;2)J1N(1)J1N(0)Cd: 62

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4 2{62 ),wehaveJ1N(1)J1N(0)=Cd)^C1sJ1N(1)J1N(0)Cd: 2{62 ),wehaveJ1N(1)J1N(0)=^C1s)^C1sJ1N(1)J1N(0)Cd: Since11=0issuboptimalinP1N(1),wegettheinequality: From( 2{67 )and( 2{68 )wegettheassertionofthelemmaforcase2).Hence,thelemmaisproved.2 2 2 )Considertwopairsofenergyandfuelpriceprocessesf(Gi);(Pi)gandf(G0i);(P0i)g,satisfyingtheconditionofLemma 3 .Let1,0,10,and00beoptimalsolutionsoftheproblemsP1N(1),P1N(0),P10N(1),andP10N(0),correspondingly.8N>0thefollowingstatementsaretrue:1)J10N(1)J10N(0)J1N(1)J1N(0);2)01001;and11101: 63

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a)01=11=1:Thisimplies:M1^C1s)M01M1^C1s=^C10s: 5 wealsohave101=1.Nowwecancompute:J11(1)J11(0)=^C1s=^C10s=J101(1)J101(0): 9 :J101(1)J101(0)C0d=Cd=J11(1)J11(0):

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SinceG2;P2;G02,andP02arerandomattime1,bytakingtheexpectationsofbothsidesin( 2{69 )wederive WecontinuetheproofconditioningontheoptimalsolutionofJ1K. a)Assume01=1isoptimal,thenfromTheorem 4 weget11=1.Using( 2{62 ): 65

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8 ,01=1implies: Combining( 2{70 )with( 2{72 ),andusingG01G1:EJ20K1(1)jF1EJ20K1(0)jF1er^C1sM1er^C1sM01=er^C10sM01: 8 ,thelastinequalityisequivalenttooptimalityof001=1inP10K(0).UsingTheorem 4 wealsoget101=1optimalinP10K(1).Withthehelpof( 2{62 )nowwecancomputethedierence: Using( 2{71 )and( 2{73 ):J10K(1)J10K(0)=J1K(1)J1K(0): b)Assume11=0isoptimalinP1K(1).AutomaticallyfromTheorem 4 wehavethat01=0isoptimalinP1K(0).Using( 2{62 ): FromLemma 9 and( 2{74 ):J10K(1)J10K(0)C0d=Cd=J1K(1)J1K(0): c)Thelastcaseleftiswhen11=1and01=0.Using( 2{62 )wecancompute: 66

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8 : Combining( 2{70 )and( 2{76 )andtheconditionM01M1: Using( 2{77 )andLemma 8 again,wendthat101=1isoptimalinP10K(1).Therefore,themonotonicityofoptimalsolutionsisshown.Itislefttoshowtheinequalitypartoftheinductionhypothesis. If001=0isoptimal,thenusing( 2{62 ),( 2{70 ),M01M1,and( 2{75 ):J10K(1)J10K(0)=M01+erEJ20K1(1)jF1EJ20K1(0)jF1M1+erEJ2K1(1)jF1EJ2K1(0)jF1=J1K(1)J1K(0): 2{62 ): ApplyingLemma 9 to( 2{78 ),wegetJ10K(1)J10K(0)=^C10s=^C1sJ1K(1)J1K(0): 2 .TheverysamecorollarycanbestatedforTheorem 5 .Hence,thepropertiesoftheoptimalexerciseboundariesforthecasewithnoramp-upperiod 67

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3 .Thisisthekeytheoremthatreducestheproblemofndingasingleoptimalstationaryexerciseboundarytosolvingalinearprogrammingproblem. 68

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3 .Inthecaseswhenweneedtondtwostationaryexerciseboundariesortimedependentoptimalexerciseboundarieswecanapplythesametechniquesthatwedevelopedforthemodelwithnoramp-upperiod.Summarizing,inthissectionweshowedhowtheresultsobtainedforthemodelwiththeramp-upperiodoperationalconstraintcanbeeasilymodiedtobeappliedtothemodelwithnoramp-upperiod. 69

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Q=10016=1600MWperperiod;Q 70

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2-1 .Havingdenedallthemodelparameters,wecanproceedwithpricinga10-yeartollingagreementcontract.Belowweprovidenumericalresultsforsetupswithfourdierentpairsofheatratelevels 2-2 summarizesresultsofthealgorithmwithone\average"exerciseboundaryandcorrespondingresultsoftheexactalgorithmpresentedinDengandOren(2003).Table 2-3 summarizesresultsobtainedbyapplyingtheheuristicalgorithmusingthe\average"boundaryfoundbytheinitialalgorithm.Inthistable,wealsocomparetheobtainedresultswiththeexactpricesfromDengandOren(2003).Inaddition,weshowtheadvantageofusingtheheuristicalgorithmbycomparingtheheuristicresultswiththeresultsoftheinitialalgorithm. FromTable 2-2 ,weseethattheinitialalgorithmwithone\average"exerciseboundaryprovidesgoodestimatesforcontractpricesinsetupswiththemostecientpowerplants( 2-3 ,wepresentcomputationalresults 71

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Theresultsshowthatwithonly1000samplepathsand20independentrunsweareabletoachievelowlevelsofthestandarddeviation.Thelargeststandarddeviationreportedinthetablesreachesonly1:4%(when Uptonowwedidnotmentionanythingaboutthegranularityofagridusedintheexperiments.Robustnesswithrespecttothegridgranularityisoneofthemostimportantpropertiesofanyalgorithmdenedonagrid.Lackofthistypeofrobustnessmakesthealgorithmtoosensitivetotheparticularchoiceofthegrid,andtheobtainedexperimentalresultscannotbeconsideredreliable.Theresultsabovewereobtainedfromrunningthealgorithmsonagridwith50verticaland50horizontalnodes.Totesttherobustnessofthereportedresultsweconductedanalogouscasestudieswithagridhaving80verticaland80horizontalnodes.TheresultsoftheexperimentswithtwoexerciseboundariesarepresentedinTable 2-4 .Asweseefromthetable,increasingthegranularityofthegridhasverylittleinuenceonthealgorithmresults.Hence,theconductednumericalexperiments 72

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Toestimatethetimeeciencyofthealgorithm,inTable 2-5 wereportcomputationaltimeofonerunoftheoptimizationalgorithmndingan\average"boundaryforthesetupwith1000samplepathsanda50x50grid.Thechoiceof InFigure 2-10 ,weshowcloseapproximationsofexerciseboundariesfoundforthepowerplantsetupwith 73

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Q=15012=1800MWperperiod;Q K=K 2-6 .Belowweprovidenumericalresultsofpricinga1-yeartollingagreementcontractundersetupswithfourdierentpairsofheatratelevels, 74

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2-7 and 2-8 containtheestimatesof1-yearcontractpricescalculatedusingthealgorithmwithone\average"exerciseboundaryandtheheuristicwithtwoexerciseboundaries,correspondingly.Wecanseethattheresultsreportedinthetablesarealmostidentical,whichmeansthattheheuristicalgorithmprovidesalmostnoadvantagecomparedtothebasealgorithmwithoneboundary.Thisndingcanbeexplainedbylowswitchingcostsinthemodelandasmallnumberofswitchesbetweendierentregimesrequiredbytheoptimaloperatingstrategy.Toestimatethetimeeciencyofthealgorithm,inTable 2-9 wereportthecomputationaltimeofonerunoftheoptimizationalgorithmforndingan\average"boundaryforthesetupwith500samplepathsanda50x50grid.Thechoiceof OurinitialreasonforchoosingtheparticularpowerplantsetupwastocomparetheresultsobtainedbyouralgorithmswithresultsreportedinDengandXia(2005).Havingdonethecasestudy,wesawthattheresultsfromDengandXia(2005)aresignicantlyhigherthantheresultsproducedbyouralgorithms.Thedierenceintheresultstakesanextremevalueinthesetupswith 75

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2-10 ,wereporttheresultsoftheconductedMonteCarlosimulations.Togenerateoneestimateweused1000samplepathsand50independentruns.Analyzingtheresults,oneshouldexpectthecontractpricesfortheidealizedpowerplantsetuptobetheupperboundsforthecontractpricesforthepowerplantsetupwithoperationalcosts.WefoundthattheresultsreportedinDengandXia(2005)signicantlyexceedtheresultscomputedforthesetupwithoutcosts.Theremaybeanumberofpotentialexplanationsforthisnding.Thesimplestexplanationmaybethetyposinthespecicationofthepowerplantsetupparametersorthetyposinthedocumentationofthenumericalresults.Otherpotentialexplanationscomefromthemodellingapproach.DengandXiauseasimpleEulerschemetogeneratesamplepathsgivenbythegeometricOrnstein-Uhlenbeckdynamics.AnumberofresearcherswarnedaboutpotentialproblemswithusingtheEulerscheme.Discretizationerrorsassociatedwiththisschememayleadtovariouskindsofcomputationalinstabilities.OnecannddiscussionsofthistopicinGlasserman(2004).DengandXiaalsousetheTsitsiklisandVanRoyform(TsitsiklisandVanRoy(2001))oftheregressionalgorithmforthedynamicprogrammingequations.InCarmonaandLudkovski(2008)theauthorsmentionthatempiricalevidenceshowspoorconvergencepropertiesoftheTsitsiklisandVarRoyformoftheregression. ContrarytotheresultsfromDengandXia(2005),thepriceestimatescomputedusingouralgorithmsneverexceedthecontractpriceestimatesfortheidealizedsetup.Theresultsforthepowerplantsetupwithoutcostsaretheupperboundsforthecontractpricesfortheoriginalsetupwithcosts.Therefore,consideringtheaveragesreportedinTable 2-10 ascontractpricesforthecostlesssetupandtheaveragesdocumentedinTable 2-8 ascontractpricesfortheoriginalsetupwithcostscomputedusingourheuristicwithtwoboundaries,wecanestimateupperboundsforthepricingerrors.WeexhibittheseupperboundestimatesinTable 2-11 .AnalyzingtheresultsfromTable 2-11 ,wecandrawaconclusionthatforalltheheatratesbutone( 76

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Wenishthissubsectionbyinvestigatingtheeectofusingtime-dependentexerciseboundaries.Forthesepurposesweconductacasestudyusingthealgorithmwithfourtime-dependentexerciseboundaries.Thisalgorithmisageneralizationofthealgorithmwithonestationaryexerciseboundaryandwasdescribedintheearliersections.InTable 2-12 ,wereportnumericalresultsoftheconductedcasestudy.Asseenbefore,weused30runsand500samplepathstogenerateoneestimate.Weconsidertimecutsatthefollowingpointsonthetimeaxis:1 2,3 4,7 8,and15 16. LookingatTable 2-12 andTable 2-7 wecanndlittleornodierenceinthereportedresults.Therefore,thenumericalexperimentshowsthatoneyearisalargeenoughhorizontojustifytheuseofstationaryexerciseboundariesinsteadoftime-dependentexerciseboundaries. 2-13 ,weshowtheresultsofthenumericalexperiment.Asseenbefore,weused 77

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2-14 ,weshowtheresultsofanexperimentwith2000samplepathsand30runsperestimate.Asweseefromthetable,thevarianceofcontractpriceestimateswassignicantlyreduced. 78

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Theintroducedalgorithmreducestheproblemofpricingtollingagreementcontractstosolvingasinglelinearprogrammingproblem.Theexistenceofcomputationallyecienttechniquesforsolvinglinearprogrammingproblemsistheprimaryreasonfortheextremecomputationaleciencyofouralgorithm.Theapproachesavailableintheliteratureareeitherincapableofincorporatingmostofthepowerplantoperationalconstraints,orsuggestalgorithmsthatarecomputationallyinecientwhendealingwithlargehorizons.Themostsuccessfulalgorithmsintroducedintheliteraturereducetheproblemofpricingtollingagreementcontractstoastochasticdynamicprogrammingproblem.TheyalsorequiretheuseoftheLongsta-SchwartzortheTsitsiklis-VanRoyformoftheregressionforcomputingconditionalexpectationswhenneeded.Thecurseofdimensionality,themajordrawbackofalldynamicprogrammingbasedalgorithms,andconvergenceproblemsassociatedwiththementionedregressionsarethemainobstaclesforusingthesealgorithmstopricetollingagreementcontractswithevenmoderatelylargehorizons.Thealgorithmdevelopedinthisresearch,toourknowledge,istheonlyalgorithmthatremainsecientwhendealingwithcontracthorizonsof10yearsandlarger(practically 79

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Thedevelopedalgorithmiscomputationallystableandiscapableofproducinglowvariancecontractpriceestimatesevenwitharelativelysmallnumberofsamplepaths.Itisalsorobustwithrespecttothegranularityofthegridonwhichtheoptimalexerciseboundariesaredened.Alltheseimportantpropertiesofthealgorithmweresupportedbynumerouscasestudieswithvariouscontracthorizonsandpowerplantsetups.Oneofthecrucialfactorsofthenumericalstabilityofthealgorithmisanimplicitincorporationoftheshapepropertiesoftheoptimalexerciseboundariesintothelinearoptimizationproblemviaasetofmonotonicityconstraintsongridvariables.Alltheoptimalexerciseboundaryshapepropertiesweretheoreticallyjustiedandthecorrespondingresultswereformulatedastheorems. 80

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Optimalexerciseboundaries. Figure2-2. Optimaloperatingpolicywhencurrentstateoftheplantis\o". 81

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Gridonalogarithmicplane. Figure2-4. Interpolationonagrid. 82

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Heuristicforndingtwooptimalexerciseboundaries. Figure2-6. Distancebetweenapointandthe\o"boundary. 83

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Distancebetweenapointandthe\on"boundary. Figure2-8. Asequenceoftimecutsforpoints0t1
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\Averageboundary"inacut(tk). Figure2-10. Exerciseboundariesforthepowerplantsetupwith 85

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ParametersofgeometricOrnstein-Uhlenbeckprocessesforthelonghorizoncasestudy. Note:Initialpriceofgas=$3:16,initialpriceofenergy=$21:7. Table2-2. Prices(inmilliondollars)of10-yeartollingagreementcontracts:calculatedusingthealgorithmwithan\average"exerciseboundaryversustheexactvalues. 7.539.700.3440.82.78.530.500.3132.125.09.522.980.2224.827.410.516.770.2418.8811.2 Note: Table2-3. Prices(inmilliondollars)of10-yeartollingagreementcontracts:calculatedusingtheheuristicalgorithmwithtwoexerciseboundariesversustheexactvaluesandversusthepricesgivenbythealgorithmwithan\average"exerciseboundary. 7.540.420.250.91.88.531.640.291.53.79.524.150.282.75.110.518.280.183.29.0 Note: 2-2 -Average)/ExactfromTable 2-2 ,1BoundaryDi.(%)=(Average-AveragefromTable 2-2 )/AveragefromTable 2-2 86

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Prices(inmilliondollars)of10-yeartollingagreementcontracts:calculatedonan80x80gridversuscalculatedona50x50grid. 7.540.420.2540.300.208.531.640.2931.570.269.524.150.2824.270.2210.518.280.1818.270.21 Note: Table2-5. Computationaltimes(inseconds)ofpricinga10-yeartollingagreementcontractusing1000samplepathsanda50x50grid. \Average"Boundary\On"Boundary\O"Boundary FormingProblemSolvingProblem7622 Note:\Average"Boundary=timeofndingtheoptimal\average"boundary,FormingProblem=timeofconstructinganoptimizationproblemforndingtheoptimal\average"boundary,SolvingProblem=timeofsolvingalinearprogrammingproblemdeterminingtheoptimal\average"boundary,\On"Boundary=timeoftheheuristicndingtheoptimaldownshiftdeterminingthesuboptimal\on"boundary,\O"Boundary=timeoftheheuristicndingtheoptimalupshiftdeterminingthesuboptimal\o"boundary. Table2-6. ParametersofgeometricOrnstein-Uhlenbeckprocessesfortheshorthorizoncasestudy. Note:Initialpriceofgas=$3,initialpriceofenergy=$34:7. 87

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Prices(inmilliondollars)of1-yeartollingagreementcontractscalculatedusingan\average"exerciseboundary. 7.515.620.168.013.750.1510.54.530.1213.50.090.03 Note: Table2-8. Prices(inmilliondollars)of1-yeartollingagreementcontractscalculatedusingtheheuristicwithtwoexerciseboundaries. 7.515.620.168.013.750.1510.54.550.1113.50.110.03 Note: Table2-9. Computationaltimes(inmilliseconds)ofpricinga1-yeartollingagreementcontractusing500samplepathsanda50x50grid. \Average"Boundary\On"Boundary\O"Boundary FormingProblemSolvingProblem547625280280 Note:\Average"Boundary=timeofndingtheoptimal\average"boundary,\FormingProblem"=timeofconstructinganoptimizationproblemforndingtheoptimal\average"boundary,\SolvingProblem"=timeofsolvingalinearprogrammingproblemdeterminingtheoptimal\average"boundary,\On"Boundary=timeoftheheuristicndingtheoptimaldownshiftdeterminingthesuboptimal\on"boundary,\O"Boundary=timeoftheheuristicndingtheoptimalupshiftdeterminingthesuboptimal\o"boundary. 88

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Prices(inmilliondollars)of1-yeartollingagreementcontractsfortheidealizedpowerplantsetupcalculatedusingMonteCarlosimulationwith1000samplepathsand50independentruns. 7.515.860.0058.013.930.00410.54.640.00513.50.180.001 Note: Table2-11. Prices(inmilliondollars)of1-yeartollingagreementcontractscalculatedusingtheheuristicalgorithmwithtwoexerciseboundariesversusthepricescomputedfortheidealizedpowerplantsetupwithoutcosts. 7.515.6215.861.58.513.7513.931.39.54.554.642.010.50.110.1863.6 Note: 2-8 ,PricewithoutCosts=AveragefromTable 2-10 ,ErrorUpperBound(%)=upperboundforthepricingerrorproducedbytheheuristicwithtwoboundaries((PricewithoutCosts-PricewithCosts)/PricewithCosts). Table2-12. Prices(inmilliondollars)of1-yeartollingagreementcontractscalculatedusingthealgorithmwithtime-dependentexerciseboundaries. 7.515.650.178.013.710.2010.54.570.1413.50.120.04 Note: 89

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Prices(inmilliondollars)of1-yeartollingagreementcontractsunderthejumpdiusionenergypricedynamicsandcalculatedusingtheheuristicalgorithmwithtwoexerciseboundaries. 7.538.350.788.036.280.7610.526.660.8113.517.520.78 Note: Table2-14. Prices(inmilliondollars)of1-yeartollingagreementcontractsunderthejumpdiusionenergypricedynamicsandcalculatedusingtheheuristicalgorithmwithtwoexerciseboundaries. 7.538.140.378.036.170.4010.526.680.4013.517.290.39 Note: 90

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Below,werefertooptionpricingmethodsdirectlyrelatedtoouralgorithm.AlthoughthispaperconsidersEuropeanoptions,somerelatedpapersconsiderAmericanoptions. Replicationoftheoptionpricebyaportfolioofsimplerassets,usuallyoftheunderlyingstockandarisk-freebond,canincorporatevariousmarketfrictions,suchastransactioncostsandtradingrestrictions.Forincompletemarkets,replication-basedmodelsarereducedtolinear,quadratic,orstochasticprogrammingproblems,see,forinstance,BouchaudandPotters(2000),Bertsimasetal.(2001),DemboandRosen(1999),Colemanetal.(2004),NaikandUppal(1994),Dennis(2001),DempsterandThompson(2001),Edirisingheetal.(1993),FedotovandMikhailov(2001),King(2002),andWuandSen(2000). 91

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Anothergroupofmethods,whicharebasedonasignicantlydierentprinciple,incorporatesknownpropertiesoftheshapeoftheoptionpriceintothestatisticalanalysisofmarketdata.Ait-SahaliaandDuarte(2003)incorporatemonotonicandconvexpropertiesofEuropeanoptionpricewithrespecttothestrikepriceintoapolynomialregressionofoptionprices.Inouralgorithmwelimitthesetoffeasiblehedgingstrategies,imposingconstraintsonthehedgingportfoliovalueandthestockposition.ThepropertiesoftheoptionpriceandthestockpositionandboundsontheoptionpricehasbeenstudiedboththeoreticallyandempiricallybyMerton(1973),PerrakisandRyan(1984),Ritchken(1985),BertsimasandPopescu(1999),GotohandKonno(2002),andLevi(85).Inthispaper,wemodelstockandbondpositionsonatwo-dimensionalgridandimposeconstraintsonthegridvariables.Theseconstraintsfollowundersomegeneralassumptionsfromnon-arbitrageconsiderations.SomeoftheseconstraintsaretakenfromMerton(1973). Monte-CarlomethodsforpricingoptionsarepioneeredbyBoyle(1977).Theyarewidelyusedinoptionspricing:Joyetal.(1996),BroadieandGlasserman(2004),LongstaandSchwartz(2001),Carriere(1996),TsitsiklisandVanRoy(2001).ForasurveyofliteratureinthisareaseeBoyle(1997)andGlasserman(2004).Regression-basedapproachesintheframeworkofMonte-CarlosimulationwereconsideredforpricingAmericanoptionsbyCarriere(1996),LongstaandSchwartz(2001),TsitsiklisandVanRoy(1999,2001).BroadieandGlasserman(2004)proposedstochasticmeshmethodwhichcombinedmodellingonadiscretemeshwithMonte-Carlosimulation.Glasserman(2004),showedthatregression-basedapproachesarespecialcasesofthestochasticmeshmethod. Thealgorithmusesthehedgingportfoliotoapproximatethepriceoftheoption.Weaimedatmakingthehedgingstrategyclosetoreal-lifetrading.Theactualtradingoccurs 92

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Thepricingalgorithmdescribedinthispapercombinesthefeaturesoftheaboveapproachesinthefollowingway.Weconstructahedgingportfolioconsistingoftheunderlyingstockandarisk-freebondanduseitsvalueasanapproximationtotheoptionprice.Weaimedatmakingthehedgingstrategyclosetoreal-lifetrading.Theactualtradingoccursatdiscretetimesandisnotself-nancingatre-balancingpoints.Theshortageofmoneyshouldbecoveredatanydiscretepoint.Largeshortagesareundesirableatanytimemoment,evenifself-nancingispresent.Weconsidernon-self-nancinghedgingstrategies.Externalnancingoftheportfolioorwithdrawalisallowedatanyre-balancingpoint.Weuseasetofsamplepathstomodeltheunderlyingstockbehavior.Thepositioninthestockandtheamountofmoneyinvestedinthebond(hedgingvariables)aremodelledonnodesofadiscretegridintimeandthestockprice.Twomatricesdeningstockandbondpositionsongridnodescompletelydeterminethehedgingportfolioonanypricepathoftheunderlyingstock.Also,theydetermineamountsofmoneyaddedto/takenfromtheportfolioatre-balancingpoints.Thesumofsquaresofsuchadditions/subtractionsofmoneyonapathisreferredtoasthesquarederroronapath. Thepricingproblemisreducedtoquadraticminimizationwithconstraints.Theobjectiveistheaveragedquadraticerroroverallsamplepaths;thefreevariablesarestockandbondpositionsdenedineverynodeofthegrid.Theconstraints,limitingthefeasiblesetofhedgingstrategies,restricttheportfoliovaluesestimatingtheoptionpriceandstockpositions.Werequiredthattheaverageoftotalexternalnancingoverallpathsequalstozero.Thismakesthestrategy"self-nancingonaverage".Weincorporatedmonotonic,convex,andsomeotherpropertiesofoptionpricesfollowingfromthedenitionofanoption,anon-arbitrageassumption,andsomeothergeneralassumptionsaboutthe 93

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Weperformedtwonumericaltestsofthealgorithm.First,wepricedoptionsonthestockfollowingthegeometricBrownianmotion.StockpriceismodelledbyMonte-Carlosample-paths.CalculatedoptionpricesarecomparedwiththeknownpricesgivenbytheBlack-Scholesformula.Second,wepricedoptionsonS&P500Indexandcomparedtheresultswithactualmarketprices.Bothnumericaltestsdemonstratedreasonableaccuracyofthepricingalgorithmwitharelativelysmallnumberofsample-paths(consideredcasesinclude100and20sample-paths).Wecalculatedoptionpricesbothindollarsandintheimpliedvolatilityformat.TheimpliedvolatilitymatchesreasonablywelltheconstantvolatilityforoptionsintheBlack-Scholessetting.TheimpliedvolatilityforS&P500indexoptions(pricedwith100sample-paths)trackstheactualmarketvolatilitysmile. Theadvantageofusingthesquarederrorasanobjectivecanbeseenfromthepracticalperspective.Althoughweallowsomeexternalnancingoftheportfolioalongthepath,theminimizationofthesquarederrorensuresthatlargeshortagesofmoneywillnotoccuratanypointoftimeiftheobtainedhedgingstrategyispracticallyimplemented. Anotheradvantageofusingthesquarederroristhatthealgorithmproducesahedgingstrategysuchthatthesumofmoneyaddedto/takenfromthehedgingportfolioonanypathisclosetozero.Also,theobtainedhedgingstrategyrequireszeroaverageexternalnancingoverallpaths.Thisjustiesconsideringtheinitialvalueofthehedgingportfolioasapriceofanoption.Weusethenotionof"apriceofanoptioninthepracticalsetting"whichisthepriceatraderagreestobuy/selltheoption.IntheexampleofpricingoptionsonthestockfollowingthegeometricBrownianmotionthealgorithmndshedgingstrategywhichdeliversrequestedoptionpaymentsatexpirationwithhigh 94

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Weassumeanincompletemarketinthispaper.Weusetheportfoliooftwoinstruments-theunderlyingstockandabond-toapproximatetheoptionpriceandconsidermanysamplepathstomodelthestockpriceprocess.Asaconsequence,thevalueofthehedgingportfoliomaynotbeequaltotheoptionpayoatexpirationonsomesamplepaths.Also,thealgorithmisdistribution-free,whichmakesitapplicabletoawiderangeofunderlyingstockprocesses.Therefore,thealgorithmcanbeusedintheframeworkofanincompletemarket. Usefulnessofouralgorithmshouldbeviewedfromtheperspectiveofpracticaloptionspricing.Commonlyusedmethodsofoptionspricingaretime-continuousmodelsassumingspecictypeoftheunderlyingstockprocess.Iftheprocessisknown,thesemethodsprovideaccuratepricing.Ifthestockprocesscannotbeclearlyidentied,thechoiceofthestockprocessandcalibrationoftheprocesstotmarketdatamayentailsignicantmodellingerror.Ouralgorithmissuperiorinthiscase.Itisdistribution-freeandisbasedonrealisticassumptions,suchasdiscretetradingandnon-self-nancinghedgingstrategy. Anotheradvantageofouralgorithmislowback-testingerrors.Time-continuousmodelsdonotaccountforerrorsofimplementationonhistoricalpaths.Theobjectiveinouralgorithmistominimizetheback-testingerrorsonhistoricalpaths.Therefore,thealgorithmhasaveryattractiveback-testingperformance.Thisfeatureisnotsharedbyanyoftime-continuousmodels. 3.2.1PortfolioDynamicsandSquaredError

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Thepriceoftheoptionattimetjisapproximatedbythepricecjofahedgingportfolioconsistingoftheunderlyingstockandarisk-freebond.Thehedgingportfolioisrebalancedattimestj,j=1;:::;N1.Supposethatatthetimetj1thehedgingportfolioconsistsofuj1sharesofthestockandvj1dollarsinvestedinthebond tobenon-zero.Thevalueajistheexcess/shortfallofthemoneyinthehedgingportfolioduringtheinterval[tj1;tj].Inotherwords,ajistheamountofmoneyaddedto(ifaj0)orsubtractedfrom(ifaj<0)theportfolioduringtheinterval[tj1;tj].Thus,theinow/outowofmoneyto/fromthehedgingportfolioisallowed. Werequirethatthevalueofthehedgingportfolioatexpirationbeequaltotheoptionpayoh(SN),uNSN+vN=h(SN);whereh(S)=8>><>>:maxf0;SXgforcalloptions;maxf0;XSgforputoptions. Thenon-self-nancingportfoliodynamicsisgivenby wheretheportfoliovalueattimetjiscj=ujSj+vj;j=0;:::;N. 96

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tomeasurethedegreeof\non-self-nancity".Thereasonsforchoosingthisparticularmeasurewillbedescribedlateron. 97

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[Ukj]=266666664U10U11:::U1NU20U21:::U2N............UK0UK1:::UKN377777775;[Vkj]=266666664V10V11:::V1NV20V21:::V2N............VK0VK1:::VKN377777775(3{4) arereferredtoasahedgingstrategy.Thesematricesdeneportfoliomanagementdecisionsonthediscretesetofthegridnodes.Inordertosetthosedecisionsonanypath,notnecessarilygoingthroughgridpoints,approximationrulesaredened. Wemodelthestockpricedynamicsbyasetofsamplepaths whereS0istheinitialprice.Letvariablesupjandvpjdenethecompositionofthehedgingportfolioonpathpattimetj,wherep=1;:::;P,j=0;:::;N.ThesevariablesareapproximatedbythegridvariablesUkjandVkjasfollows.SupposethatfS0;Sp1;:::;SpNgisarealizationofthestockprice,whereSpjdenotesthepriceofthestockattimetjonpathp,j=0;:::;N,p=1;:::;P.Letupjandvpjdenotetheamountsofthestockandthebond,respectively,heldinthehedgingportfolioattimetjonpathp.VariablesupjandvpjarelinearlyapproximatedbythegridvariablesUkjandVkjasfollows wherepj=lnSpjln~Sk(j;p) 3{1 ),wedenetheexcess/shortageofmoneyinthehedgingportfolioonpathpattimetjbyapj=upj+1Spj+1+vpj+1(upjSpj+1+(1+r)vpj):

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WedenetheaveragesquarederrorEonthesetofpaths( 3{5 )asanaverageofsquarederrorsEpoverallsamplepaths( 3{5 ) E=1 Thematrices[Ukj]and[Vkj]andtheapproximationrule( 3{6 )specifythecompositionofthehedgingportfolioasafunctionoftimeandthestockprice.Foranygivenstockpricepathonecanndthecorrespondingportfoliomanagementdecisionsf(uj;vj)jj=0;:::;N1g,thevalueoftheportfoliocj=Sjuj+vjatanytimetj,j=0;:::;N,andtheassociatedsquarederror. Thevalueofanoptioninquestionisassumedtobeequaltotheinitialvalueofthehedgingportfolio.Firstcolumnsofmatrices[Ukj]and[Vkj],namelythevariablesUk0andVk0,k=1;:::;K;determinetheinitialvalueoftheportfolio.Ifoneoftheinitialgridnodes,forexamplenode(0;~k);correspondstothestockpriceS0,thenthepriceoftheoptionisgivenbyU~k0S0+V~k0:Iftheinitialpoint(t=0;S=S0)ofthestockprocessfallsbetweentheinitialgridnodes(0;k),k=1;:::;K,thenapproximationformula( 3{6 )withj=0andSp0=S0isusedtondtheinitialcomposition(u0;v0)oftheportfolio.Then,thepriceoftheoptionisfoundasu0S0+v0. 99

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minE=1 3{6 ),constraints( 3{10 )-( 3{18 )(denedbelow)forcalloptions,orconstraints( 3{19 )-( 3{27 )(denedbelow)forputoptions,freevariables:Ukj;Vkj;j=0;:::;N;k=1;:::;K: 3{9 )istheaveragesquarederroronthesetofpaths( 3{5 ).Therstconstraintrequiresthattheaveragevalueoftotalexternalnancingoverallpathsequalstozero.Thesecondconstraintequatesthevalueoftheportfolioandtheoptionpayoatexpiration.FreevariablesinthisproblemarethegridvariablesUkjandVkj;thepathvariablesupjandvpjintheobjectiveareexpressedintermsofthegridvariablesusingapproximation( 3{6 ).Thetotalnumberoffreevariablesintheproblemisdeterminedbythesizeofthegridandisindependentofthenumberofsample-paths.Aftersolvingtheoptimizationproblem,theoptionvalueattimetjforthestockpriceSjisdenedbyujSj+vj,whereujandvjarefoundfrommatrices[Ukj]and[Vkj],respectively,usingapproximationrules( 3{6 ).Thepriceoftheoptionistheinitialvalueofthehedgingportfolio,calculatedasu0S0+v0. Thefollowingconstraints( 3{10 )-( 3{18 )forcalloptionsor( 3{19 )-( 3{27 )forputoptionsimposerestrictionsontheshapeoftheoptionvaluefunctionandontheposition 100

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Below,weconsidertheconstraintsforEuropeancalloptions.Theconstraintsforputoptionsaregiveninthenextsection,togetherwithproofsoftheconstraints.Mostoftheconstraintsarejustiedinaquitegeneralsetting.Weassumenon-arbitrageandmake5additionalassumptions.Proofsoftwoconstraintsonthestockposition(horizontalmonotinicityandconvexity)inthegeneralsettingwillbeaddressedinsubsequentpapers.InthispaperwevalidatetheseinequalitiesintheBlack-Scholescase. ThenotationCkjstandsfortheoptionvalueinthenode(j;k)ofthegrid,Ckj=Ukj~Skj+Vkj: Thisconstraintsboundsensitivityofanoptionpricetochangesinthestockprice. 3{10 )coincideswiththeimmediateexercisevalueofanAmericanoptionhavingthecurrentstockprice~SkjandthestrikepriceXer(Ttj):

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~Skj Horizontalmonotonicity.Thepriceofanoptionisadecreasingfunctionoftime. 0Ukj1;j=0;:::;N;k=1;:::;K:(3{15) 102

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(1k+1j)Uk+2j+k+1jUkjUk+1j;ifk>^k;(1k1j)Uk2j+k1jUkjUk1j;ifk^k;whereljissuchthat~Slj=lj~Sl1j+(1lj)~Sl+1j;l=(k+1);(k1): 3.3.4 ). Theexpectedhedgingerrorisanestimateof\non-self-nancity"ofthehedgingstrategy.Thepricingalgorithmseeksastrategyascloseaspossibletoaself-nancingone,satisfyingtheimposedconstraints.Ontheotherhand,fromatrader'sviewpoint,theshortageofmoneyatanyportfoliore-balancingpointcausestheriskassociatedwiththehedgingstrategy.Theaveragesquarederrorcanbeviewedasanestimatorofthisriskonthesetofpathsconsideredintheproblem. Thereareotherwaystomeasuretheriskassociatedwithahedgingstrategy.Forexample,Bertsimasetal.(2001)considersaself-nancingdynamicsofahedgingportfolioandminimizesthesquaredreplicationerroratexpiration.Inthecontextofourframework,dierentestimatorsofriskcanbeusedasobjectivefunctionsintheoptimizationproblem( 3{9 )and,therefore,producedierentresults.However,consideringotherobjectivesisbeyondthescopeofthispaper. 3{10 )-( 3{14 )forcalloptionsand( 3{19 )-( 3{23 )for 103

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3{5 )speciesthebehavioroftheunderlyingstock.Forthisreason,theapproachisdistribution-freeandcanbeappliedtopricinganyEuropeanoptionindependentlyofthepropertiesoftheunderlyingstockpriceprocess.Also,asshowninsection5presentingnumericalresults,theinclusionofconstraintstoproblem( 3{9 )makesthealgorithmquiterobusttothesizeofinputdata. Thegridstructureisconvenientforimposingtheconstraints,sincetheycanbestatedaslinearinequalitiesonthegridvariablesUkjandVkj.Animportantpropertyofthealgorithmisthatthenumberofthevariablesinproblem( 3{9 )isdeterminedbythesizeofthegridandisindependentofthenumberofsamplepaths. 3{16 )-( 3{18 )requiringmonotonicityandconcavityofthestockpositionwithrespecttothestockpriceandmonotonicityofthestockpositionwithrespecttotime(constraints( 3{25 )-( 3{27 )forputoptionsarepresentedinthenextsection).Thegoalistolimitthevariabilityofthestockpositionwithrespecttotimeandstockprice,whichwouldleadtosmallertransactioncostsofimplementingahedgingstrategy.Theminimizationoftheaveragesquarederrorisanothersourceofimproving\smoothness"ofahedgingstrategywithrespecttotime.Theaveragesquarederrorpenalizesallshortages/excessesapjofmoneyalongthepaths,whichtendstoattenthevaluesapjovertime.Thisalsoimprovesthe\smoothness"ofthestockpositionswithrespecttotime. 104

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3.4.1ConstraintsforPutOptions 3{9 )forpricingEuropeanputoptions. Verticalmonotonicity. Horizontalmonotonicity.

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3{9 )weusedthefollowingconstraintsholdingforoptionsinquiteageneralcase.Weassumenon-arbitrageandmaketechnicalassumptions1-5(usedbyMerton(1973)forderivingpropertiesofcallandputoptionvalues.SomeoftheconsideredpropertiesofoptionvaluesareprovedbyMerton(1973).Otherinequalitiesareprovedbytheauthors. Therestofthesectionisorganizedasfollows.First,weformulateandproveinequalities( 3{10 )-( 3{14 )forcalloptions.Someoftheconsideredpropertiesofoptionvaluesarenotincludedintheconstraintsoftheoptimizationproblem( 3{9 ),theyareusedinproofsofsomeofconstraints( 3{10 )-( 3{14 ).Inparticular,weakandstrongscaling 106

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Second,weconsiderinequalities( 3{19 )-( 3{23 )forputoptions.Weprovideproofsofverticalandhorizontaloptionpricemonotonicity;proofsofotherinequalitiesaresimilartothoseforcalloptions. Weusethefollowingnotations.C(St;T;X)andP(St;T;X)denotepricesofcallandputoptions,respectively,withstrikeX,expirationT,whenthestockpriceattimetisSt.Whenappropriate,weuseshorternotationsCtandPttorefertotheseoptions. SimilartoMerton(1973),wemakethefollowingassumptionstoderiveinequalities( 3{10 )-( 3{14 )and( 3{19 )-( 3{23 ). Belowaretheproofsofinequalities( 3{10 )-( 3{14 ). 1."Immediateexercise"constraints.Merton(1973)Ct[StXer(Tt)]+: 107

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Foranyk>0considertwostockpriceprocessesS(t)andkS(t).Fortheseprocesses,thefollowinginequalityisvalidC(kSt;T;kX)=kC(St;T;X);whereStisthevalueoftheprocessS(t)attimet. Underassumptions4and5,thecalloptionpriceC(S;T;X)ishomogeneousofdegreeoneinthestockpricepershareandexerciseprice.Inotherwords,ifC(S;T;X)andC(kS;T;kX)areoptionpricesonstockswithinitialpricesSandkSandstrikesXandkX,respectively,thenC(kS;T;kX)=kC(S;T;X): NowconsideranoptionCwiththestrikeX1writtenononeshareofthestock1.DenoteitspricebyC1(S1;T;X1):OptionsAandChaveequalinitialpricesS1=1 108

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ForanyX1,X2suchthat0X1X2,thefollowinginequalityholdsC(St;T;X1)C(St;T;X2)+(X2X1)er(Tt): 109

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X1;T;X):BysettingX1=S1 LetC(t;S;T;X)denotethepriceofaEuropeancalloptionwithinitialtimet;initialpriceattimetequaltoS;timetomaturityT;andstrikeX:Undertheassumptions1,2and3foranyt,u,t
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multiplyingbothsidesofthepreviousinequalitybyS3givesS3C(1;T;X3)S1C(1;T;X1)+(1)S2C(1;T;X2):Further,usingtheweakscalingproperty,wegetC(S3;T;S3X3)C(S1;T;S1X1)+(1)C(S2;T;S2X2):UsingdenitionsofX1andX2andexpandingS3X3asS3(X1+(1)X2)=S3X0B@ S1+1 S21CA==S3X0B@S1 S3+1 S31CA=X;

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1.\Immediateexercise"constraints. Foranyk>0,considertwostockpriceprocessesS(t)andkS(t).Fortheseprocessesthefollowinginequalityholds:P1(kSt;T;kX)=kP2(St;T;X);whereP1andP2areoptionsontherstandthesecondstocksrespectively. Undertheassumptions4and5,putoptionvalueP(S;T;X)ishomogeneousofdegreeoneinthestockpriceandthestrikeprice,i.e.,foranyk>0;P(kS;T;kX)=kP(S;T;X): 112

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Underassumptions1,2,and3,foranyinitialtimestandu,t
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3{15 )-( 3{18 )and( 3{24 )-( 3{27 )onthestockposition.Stockpositionboundsandverticalmonotonicityareproveninthegeneralcase(i.e.underassumptions1-5andthenon-arbitrageassumption);horizontalmonotonicityandconvexityarejustiedundertheassumptionthatthestockprocessfollowsthegeometricBrownianmotion. ThenotationC(S;T;X)(P(S;T;X))standsforthepriceofacall(put)optionwiththeinitialpriceS,timetoexpirationT,andthestrikepriceX.Thecorrespondingpositioninthestock(forbothcallandputoptions)isdenotedbyU(S;T;X). First,wepresenttheproofsofinequalities( 3{15 )-( 3{18 )forcalloptions. 1.Verticalmonotonicity(Calloptions). 0U(S;T;X)1SincetheoptionpriceC(S;t;X)isanincreasingfunctionofthestockpriceS,itfollowsthatU(S;t;X)=C0s(S;t;X)0. NowweneedtoprovethatU(S;t;X)1.WewillassumethatthereexistssuchSthatC0s(S)forsome>1andwillshowthatthisassumptioncontradictstheineqiality

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3)Horizontalmonotonicity(Calloptions) 2dZ;(3{29) andd1andd2aregivenbyexpressionsd1=1 2p 2p

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2)+ln(S X)2 X 2: X:F(S)0(implyingU0t(S;T;X)0)whenSLandF(S)0(implyingU0t(S;T;X)0)whenSL,whereL=XeT(r+2=2): 3-7 ). TheError(%)columncontainserrorsofapproximatinginexionpointsbystrikeprices.Theseerrorsdonotexceed3%forabroadrangeofparameters.Weconcludethatinexionpointscanbeapproximatedbystrikepricesforoptionsconsideredinthecasestudy. 3{24 )-( 3{27 )forputoptions. 116

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117

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3-1 3-3 ,and 3-4 report\relative"valuesofstrikesandoptionprices,i.e.strikesandpricesdividedbytheinitialstockprice.Pricesofoptionsarealsogivenintheimpliedvolatilityformat,i.e.,foractualandcalculatedpriceswefoundthevolatilityimpliedbytheBlack-Scholesformula. 3-1 Table1showsquitereasonableperformanceofthealgorithm:theerrorsintheprice(Err(%),Table 3-1 )arelessthan2%formostofcalculatedputandcalloptions. Also,itcanbeseenthatthevolatilityisquiteatforbothcallandputoptions.Theerrorofimpliedvolatilitydoesnotexceed2%formostcallandputoptions(Vol.Err(%),Table 3-1 ).Thevolatilityerrorslightlyincreasesforout-of-the-moneyputsandin-the-moneycalls. 3-2 .Theactualmarketpriceofanoptionisassumedtobetheaverageofitsbidandaskprices.ThepriceoftheS&P500indexwasmodelledbyhistoricalsample-paths.Non-overlappingpathsoftheindexweretakenfromthehistoricaldatasetandnormalizedsuchthatallpathshavethesameinitialpriceS0.Then,thesetofpathswas\massaged"tochangethespreadofpathsuntiltheoptionwiththeclosesttoat-the-moneystrikeispricedcorrectly.Thissetofpathswiththeadjustedvolatilitywasusedtopriceoptionswiththeremainingstrikes. Table 3-3 displaystheresultsofpricingusing100historicalsample-paths.Thepricingerror(seeErr(%),Table 3-3 )isaround1:0%forallcallandputoptionsandincreases 118

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3-4 showsthatin-the-moneyS&P500indexoptionscanbepricedquiteaccuratelywith20sample-paths.)Atthesametime,themethodisexibleenoughtotakeadvantageofspecicfeaturesofhistoricalsample-paths.WhenappliedtoS&P500indexoptions,thealgorithmwasabletomatchthevolatilitysmilereasonablywell(Figures 3-1 3-2 ).Atthesametime,theimpliedvolatilityofoptionscalculatedintheBlack-Scholessettingisreasonablyat(Figures 3-3 3-4 ).Therefore,onecanconcludethattheinformationcausingthevolatilitysmileiscontainedinthehistoricalsample-paths.Thisobservationisinaccordancewiththepriorknownfactthatthenon-normalityofassetpricedistributionisoneofcausesofthevolatilitysmile. Figures 3-5 3-5 3-7 ,and 3-8 presentdistributionsoftotalexternalnancing(PNj=1apjerj)onsamplepathsanddistributionsofdiscountedmoneyinows/outows(apjerj)atre-balancingpointsforBlack-ScholesandSPXcalloptions.WesummarizestatisticalpropertiesofthesedistributionsinTable( 3-5 ). Figures 3-5 3-6 3-7 ,and 3-8 alsoshowthattheobtainedpricessatisfythenon-arbitragecondition.Withrespecttopricingasingleoption,thenon-arbitrageconditionisunderstoodinthefollowingsense.Iftheinitialvalueofthehedgingportfolioisconsideredasapriceoftheoption,thenatexpirationthecorrespondinghedgingstrategyshouldoutperformtheoptionpayoonsomesamplepaths,andunderperformtheoptionpayoonsomeothersamplepaths.Otherwise,thefreemoneycanbeobtainedbyshortingtheoptionandbuyingthehedgingportfolioorviseversa.Thealgorithmproducesthepriceoftheoptionsatisfyingthenon-arbitrageconditioninthissense.Thevalueofexternal 119

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Thepricingproblemisreducedtoquadraticprogramming,whichisquiteecientfromthecomputationalstandpoint.ForthegridconsistingofProws(thestockpriceaxis)andNcolumns(thetimeaxis),thenumberofvariablesintheproblem( 3{9 )is2PNandthenumberofconstraintsisO(NK),regardlessofthenumberofsamplepaths.Table 3-6 presentscalculationtimesfordierentsizesofthegridwithCPLEX9.0quadraticprogrammingsolveronPentium4,1.7GHz,1GBRAMcomputer. Inordertocompareouralgorithmwithexistingpricingmethods,weneedtoconsideroptionspricingfromthepracticalperspective.Pricingofactuallytradedoptionsincludesthreesteps. Mostcommonlyusedapproachforpracticalpricingofoptionsistimecontinuousmethodswithaspecicunderlyingstockprocess(Black-Scholesmodel,stochasticvolatilitymodel,jump-diusionmodel,etc).Wewillrefertothesemethodsasprocess-specicmethods.Inordertojudgetheadvantagesoftheproposedalgorithmagainsttheprocess-specicmethods,weshouldcomparethemstepbystep. 120

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Ouralgorithmdoesnotrelyonsomespecicmodelanddoesnothaveerrorsrelatedtothechoiceofthespecicprocess.Also,wehaverealisticassumptions,suchasdiscretetrading,non-self-nancinghedgingstrategy,andpossibilitytointroducetransactioncosts(thisfeatureisnotdirectlypresentedinthepaper). Calibrationofprocess-specicmethodsusuallyrequireasmallamountofmarketdata.Ouralgorithmcompeteswellinthisrespect.Weimposeconstraintsreducingfeasiblesetofhedgingstrategies,whichallowspricingwithverysmallnumberofsamplepaths. Themajoradvantageofouralgorithmisthattheerrorsofback-testinginourcasecanbemuchlowerthantheerrorsofprocess-specicmethods.Thereasonbeing,theminimizationoftheback-testingerroronhistoricalpathsistheobjectiveinouralgorithm.Minimizationofthesquarederroronhistoricalpathsensuresthattheneedofadditionalnancingtopracticallyhedgetheoptionisthelowestpossible.Noneoftheprocess-specicmethodspossessthisproperty. 121

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Thispaperistherstintheseriesofpapersdevotedtoimplementationofthedevelopedalgorithmtovarioustypesofoptions.OurtargetispricingAmerican-styleandexoticoptionsandtreatmentactualmarketconditionssuchastransactioncosts,slippageofhedgingpositions,hedgingoptionswithmultipleinstrumentsandotherissues.Inthispaperweestablishedbasicsofthemethod;thesubsequentpaperswillconcentrateonmorecomplexcases. 122

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Impliedvolatilityvs.strike:CalloptionsonS&P500indexpricedusing100samplepaths.BasedonpricesincolumnsCalc.Vol(%)andAct.Vol(%)ofTable 3-3 .CalculatedVol(%)=impliedvolatilityofcalculatedoptionsprices(100sample-paths),ActualVol(%)=impliedvolatilityofmarketoptionsprices,strikepriceisshiftedleftbythevalueoftheloweststrike. Figure3-2. Impliedvolatilityvs.strike:PutoptionsonS&P500indexpricedusing100samplepaths.BasedonpricesincolumnsCalc.Vol(%)andAct.Vol(%)ofTable 3-3 .CalculatedVol(%)=impliedvolatilityofcalculatedoptionsprices(100sample-paths),ActualVol(%)=impliedvolatilityofmarketoptionsprices,strikepriceisshiftedleftbythevalueoftheloweststrike. 123

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Impliedvolatilityvs.strike:CalloptionsinBlack-Scholessettingpricedusing200samplepaths.BasedonpricesincolumnsCalc.Vol(%)andB-S.Vol(%)ofTable 3-1 .CalculatedVol(%)=impliedvolatilityofcalculatedoptionsprices(200sample-paths),ActualVol(%)=atvolatilityimpliedbyBlack-Scholesformula,strikepriceisshiftedleftbythevalueoftheloweststrike. Figure3-4. Impliedvolatilityvs.strike:PutoptionsinBlack-Scholessettingpricedusing200samplepaths.BasedonpricesincolumnsCalc.Vol(%)andB-S.Vol(%)ofTable 3-1 .CalculatedVol(%)=impliedvolatilityofcalculatedoptionsprices(200sample-paths),ActualVol(%)=atvolatilityimpliedbyBlack-Scholesformula,strikepriceisshiftedleftbythevalueoftheloweststrike. 124

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Black-Scholescalloption:distributionofthetotalexternalnancingonsamplepaths.Initialprice=$62,strike=$62timetoexpiration=70,risk-freerate=10%,volatility=20%.Stockpriceismodelledwith200Monte-Carlosamplepaths. Figure3-6. Black-Scholescalloption:distributionofdiscountedinows/outowsatre-balancingpoints.Initialprice=$62,strike=$62timetoexpiration=70,risk-freerate=10%,volatility=20%.Stockpriceismodelledwith200Monte-Carlosamplepaths. 125

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SPXcalloption:distributionofthetotalexternalnancingonsamplepaths.Initialprice=$1183:77,strikeprice=$1190timetoexpiration=49days,risk-freerate=2:3%.Stockpriceismodelledwith100samplepaths. Figure3-8. SPXcalloption:distributionofdiscountedinows/outowsatre-balancingpoints.Initialprice=$1183:77,strikeprice=$1190timetoexpiration=49days,risk-freerate=2:3%.Stockpriceismodelledwith100samplepaths. 126

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PricesofoptionsonthestockfollowingthegeometricBrownianmotion:calculatedversusBlack-Scholesprices. StrikeCalc.B-SErr(%)Calc.Vol.(%)B-S.Vol.(%)Vol.Err(%) Calloptions1.1450.00370.0038-3.7819.6320.00-1.861.1130.00750.00741.3519.9120.00-0.461.0810.01340.01330.6519.8720.00-0.651.0480.02260.0227-0.0419.7920.00-1.041.0160.03640.03610.8019.9420.00-0.281.0000.04460.04450.1919.8220.00-0.920.9680.06510.06480.4719.9420.00-0.310.9350.08910.0892-0.0819.5920.00-2.070.9030.11660.1168-0.1119.2920.00-3.560.8710.14640.1465-0.0718.7120.00-6.44Putoptions1.1450.12740.1276-0.1619.7320.00-1.361.1130.09950.09940.0420.0320.000.171.0810.07380.07380.0520.0220.000.121.0480.05140.0514-0.1019.9720.00-0.161.0160.03340.03320.7120.1420.000.681.0000.02580.02580.1520.0220.000.110.9680.01470.01441.8220.1920.000.930.9350.00700.0071-1.6019.8920.00-0.560.9030.00290.0031-5.7719.7120.00-1.450.8710.00100.0011-12.8819.5220.00-2.41 Note:Initialprice=$62,timetoexpiration=69days,risk-freerate=10%,volatility=20%,200samplepathsgeneratedbyMonte-Carlosimulation. Strike($)=optionstrikeprice,Calc.=obtainedoptionprice(relative),BS=Black-Scholesoptionprice(relative),Err=(FoundBS)=BS,Calc.Vol.=obtainedoptionpriceinvolatilityform,BS.Vol.(%)=Black-Scholesvolatility,Vol.Err(%)=(Calc:Vol:BS.Vol.)=BS.Vol.

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TheS&P500optionsdataset. StrikeBidAskPriceRel.Pr StrikeBidAskPriceRel.Pr CalloptionsPutoptions1500N/A0.5N/AN/A 1500311.3313.3312.30.263813250.30.50.40.0003 1300112.7114.7113.70.096013000.450.80.6250.0005 127588.890.889.80.075912751.151.651.40.0012 122546.948.947.90.040512503.74.23.950.0033 121036.938.937.90.032012258.69.69.10.0077 120031.033.032.00.0270121013.214.814.00.0118 119026.128.127.10.0229120017.518.918.20.0154 117519.821.420.60.0174119022.124.123.10.0195 115012.514.013.250.0112117530.832.831.80.0269 11258.09.08.50.0072115048.050.049.00.0414 11005.15.95.50.0046112568.369.568.90.0582 10753.34.13.70.0031110090.292.291.20.0770 10502.23.02.60.0022500682.1684.1683.10.5771 10251.552.051.80.0015 Note:Strike($)=optionstrikeprice,Bid($)=optionbidprice,Ask($)=optionaskprice,Price($)=optionprice(averageofbidandaskprices),Rel.Pr=relativeoptionprice 128

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PricingoptionsonS&P500index:100paths StrikeCalc.ActualErr(%)Calc.Vol.(%)Act.Vol.(%)Vol.Err(%) Calloptions1.1190.00020.0003-40.0013.1714.14-6.821.0980.00050.0005-5.2812.8012.92-0.901.0770.00130.001211.5712.7012.402.421.0560.00350.00335.7013.0312.801.781.0350.00790.00773.1513.3813.181.521.0220.01170.0118-0.7513.4313.49-0.481.0140.01560.01541.3213.9113.771.031.0050.01950.01950.0114.0714.060.010.9930.02690.02690.1814.6314.600.230.9710.04160.04140.5015.5715.401.090.9500.05890.05821.1216.8116.134.250.9290.07750.07700.6218.0417.353.940.4220.57890.57710.3369.39N/AN/APutoptions1.2670.26330.2638-0.2022.5029.02-22.441.0980.09560.0960-0.4713.8815.14-8.351.0770.07560.0759-0.3613.7114.18-3.321.0350.04060.04050.3314.2214.110.771.0220.03190.0320-0.2514.2914.35-0.401.0140.02740.02701.2614.7514.511.621.0050.02290.0229-0.0114.8914.90-0.010.9930.01760.01741.3815.4715.301.100.9710.01110.0112-0.5216.4316.47-0.280.9500.00700.0072-1.9517.5817.72-0.790.9290.00450.0046-3.4218.8419.05-1.090.9080.00280.0031-10.0020.0220.57-2.680.8870.00150.0022-32.2720.4622.24-7.990.8660.00110.0015-26.0022.4623.78-5.54 Note:Initialprice=$1183:77,timetoexpiration=49days,risk-freerate=2:3%.Stockpriceismodelledwith100samplepaths.Griddimensions:P=15,N=49. Strike=optionstrikeprice(relative),Calc.=calculatedoptionprice(relative),Actual=actualoptionprice(relative),Err=(Calc:Actual)=Actual,Calc.Vol.=calculatedoptionpriceinvolatilityform,Act.Vol.(%)=actualoptionpriceinvolatilityterms,Vol.Err(%)=(Calc:Vol:Act:Vol:)=Act:Vol:

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PricingoptionsonS&P500index:20paths StrikeCalc.ActualErr(%)Calc.Vol.(%)Act.Vol.(%)Vol.Err(%) Calloptions1.1190.00050.000345.0014.9514.145.781.0980.00100.000588.8014.4812.9212.091.0770.00200.001266.8613.9512.4012.501.0560.00470.003341.8014.3912.8012.381.0350.00920.007719.8414.4313.189.421.0220.01320.011811.4114.4713.497.261.0140.01600.01544.0314.2013.773.131.0050.01950.01950.0014.0614.060.000.9930.02640.0269-1.6614.2814.60-2.150.9710.03930.0414-5.0113.6715.40-11.230.9500.05480.0582-5.7612.0116.13-25.520.9290.07370.0770-4.358.3917.35-51.650.4220.57900.57710.34N/AN/AN/APutoptions1.2670.26330.2638-0.1923.4529.02-19.161.0980.09590.0960-0.1314.8215.14-2.111.0770.07620.07590.4014.6714.183.451.0350.04150.04052.4914.9214.115.721.0220.03320.03203.6915.2014.355.931.0140.02780.02702.7415.0314.513.541.0050.02290.02290.0114.9014.900.010.9930.01680.0174-3.3114.9015.30-2.630.9710.00890.0112-20.7214.5816.47-11.480.9500.00300.0072-58.7312.9917.72-26.730.9290.00000.0046-100.004.3819.05-77.000.9080.00000.0031-100.006.0720.57-70.500.8870.00000.0022-100.007.6822.24-65.480.8660.00000.0015-100.008.9823.78-62.21 Note:Initialprice=$1183:77,timetoexpiration=49days,risk-freerate=2:3%.Stockpriceismodelledwith20samplepaths.Griddimensions:P=15,N=49. Strike=optionstrikeprice(relative),Calc.=calculatedoptionprice(relative),Actual=actualoptionprice(relative),Err=(Calc:Actual)=Actual,Calc.Vol.=calculatedoptionpriceinvolatilityform,Act.Vol.(%)=actualoptionpriceinvolatilityterms,Vol.Err(%)=(Calc:Vol:Act:Vol:)=Act:Vol:

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SummaryofcashowdistributionsforobitainedhedgingstrategiespresentedonFigures 3-5 3-6 3-7 ,and 3-8 TotalnancingRe-bal.cashowTotalnancingRe-bal.cashow Black-ScholesCallSPXCallmean0.00.00.00.0st.dev.0.62740.044916.15491.2730median0.0770-0.00080.2695-0.0314 Note:Totalnancing($)=thesumofdiscountedinows/outowsofmoneyonapath;Re-bal.cashow($)=discountedinow/outowofmoneyonre-balancingpoints. Black-ScholesCall:Initialprice=$62,strike=$62,timetoexpiration=70,risk-freerate=10%,volatility=20%.Stockpriceismodelledwith200Monte-Carlosamplepaths. SPXCall:Initialprice=$1183:77,strikeprice=$1190,timetoexpiration=49days,risk-freerate=2:3%.Stockpriceismodelledwith100samplepaths. Table3-6. Calculationtimesofthepricingalgorithm. #ofpathsPNBuildingtime(sec)CPLEXtime(sec)Totaltime(sec) 2020490.88.29.010025491.612.614.220025705.531.737.2 Note:CalculationsaredoneusingCPLEX9.0onPentium4,1.7GHz,1GBRAM. #ofpaths=numberofsample-paths,P=verticalsizeofthegrid,N=horizontalsizeofthegrid,Buildingtime=timeofbuildingthemodel(preprocessingtime),CPLEXtime=timeofsolvingoptimizationproblem,Totaltime=totaltimeofpricingoneoption. Table3-7. Numericalvaluesofinexionpointsofthestockpositionasafunctionofthestockpriceforsomeoptions. Expir.(days)Strike($)Inexion($)Error(%) 06260.1263.02356261.0561.52696261.9750.0405452.3683.02355453.1781.52695453.9740.0507168.8553.02357169.9191.52697170.9670.05 Note:Expir.(days)=timetoexpiration,Strike($)=strikepriceoftheoption,Inexion($)=inexionpoint,Error(%)=(Strike-Inexion)/Strike. 131

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ValeriyRyabchenkowasbornin1981,inChernigov,Ukraine.In1998,hecompletedhishighschooleducationatHighSchool#12inChernigov.Hereceivedhisbachelor'sdegreeinappliedmathematicsfromKyivNationalTarasShevchenkoUniversityinKyiv,Ukraine,in2002.In2004,hereceivedhismaster'sdegreeincomputersciencefromKyivNationalTarasShevchenkoUniversityinKyiv,Ukraine.InAugust2004,hebeganhisdoctoralstudiesintheIndustrialandSystemsEngineeringDepartmentattheUniversityofFlorida.HenishedhisPh.D.inindustrialandsystemsengineeringinAugust2008. 136