Option valuation, optimization and excursions of commodity indices [Elektronische Ressource] / Melanie Hollstein
155 Pages
English

Option valuation, optimization and excursions of commodity indices [Elektronische Ressource] / Melanie Hollstein

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Published 01 January 2010
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Option Valuation, Optimization and
Excursions of Commodity Indices
Melanie Hollstein
Vom Fachbereich Mathematik
der Technischen Universit¨at Kaiserslautern
zur Verleihung des akademischen Grades
Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr.rer.nat)
genehmigte Dissertation
1. Gutachter: Prof. Dr. Ralf Korn
2. Gutachter: Prof. Dr. Rudi Zagst
Datum der Disputation: 25. August 2010
D 386To my sonACKNOWLEDGEMENTS
First and foremost, I would like to thank my advisor, Prof.Dr.Ralf Korn, for his support
andguidancethroughoutmyPhDstudiesandindustrialprojects. Iamtrulygratefulforthe
opportunity to work at the Fraunhofer Institute for Industrial Mathematics ITWM and the
Technical University Kaiserslautern. For all this and his constant encouragement, I am very
thankful.
I would also like to extend my gratitude to my colleagues for the wonderful time especially,
Prof.Dr.MarleneMu¨ller, Dr.PeterRuckdeschel, Dr.GeraldKroisandt, Dr.KalinaNatcheva
AcarandDr.JohandeKock,whohaveallinfluencedmeverymuch. Ihavebenefittedtremen-
douslyfromtheinspiringworkingatmosphere,forwhomIwouldliketothankDr.J¨orgWenzel,
Dr.Ulrich N¨ogel, Dr.Stefan Lorenz, Dr.Christina Erlwein, Tilman Sayer, Sascha Desmettre,
Dr.FrankSeifriedandNicoleTschauder.
Ifeelveryfortunatetohavebeenblessedwithalovingfamily. Myparentsdailylove,encour-
¨agementandsupportisbeyondwords. Lastbutnotleast,IwouldliketothankOnderGu¨ng¨or
forhisinspiration,loveandenrichmentinmylife. Finally,Ioweeverythingtomyson,Leofor
hisloveandpatienceforwhichIdedicatethisthesistohim.
SpecialthanksgoestotheRaiffeisenCentrobankAGfortheirkindpermissionofinsertingtheir
pictureonthetitlepage.Contents
Preface 1
1. Commodity Indices 5
1.1. IntroductiontoCommodityInvestmentandModeling . . . . . . . . . . . . . . . 5
1.2. DescriptiveAnalysisofaCommodityIndex . . . . . . . . . . . . . . . . . . . . . 9
2. Heston Model for Commodity Indices 23
2.1. TheModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2. ClosedformsolutionforCommodityFutures . . . . . . . . . . . . . . . . . . . . 28
2.3. EuropeanOptionsonFutures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4. ImplementationandVerificationwithMonteCarloSimulation . . . . . . . . . . 45
3. Construction of Optimal Commodity Indices and Portfolios 61
3.1. OptimalRollingofCommodityFutures . . . . . . . . . . . . . . . . . . . . . . . 61
3.2. MarkowitzOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3. MinimizationofConditionalValue-at-RiskforGeneralLossDistributions . . . . 77
4. Excursions and Statistical Arbitrage 103
4.1. Excursions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2. StatisticalArbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A. Put-Call Parity for European Options on Futures 135
B. Hesse Matrix 137
1Preface
Thisthesisdealswiththesolutionofspecialproblemsarisinginfinancialengineeringorfinancial
mathematics. Themainfocusliesoncommodityindices. Commodityindicesconsistoffutures
orspotvaluesofenergy,livestock,grains,industrymetals,preciousmetalsandsofts. Afutures
contractisanagreementtobuyorsellanassetatacertainfuturetimeforacertainprice.Today,
commodityindicesrepresenttheeasiestwayofgettingexposuretocommodities. Commodity
indexoptionsaretiedtomajorcommodityindices. Theissuersofsuchnotesareinvestment
banks,financialinstitutionsorindividualcommodityproducersraisingcapitaltoextendtheir
operationswhileprovidinginvestorswithexposuretothecommoditysector.
Wefirstdedicateourselveswiththemodelingofcommodityindices. Chapter 1addressesthe
importantissueforthefinancialengineeringpracticeofdevelopingwell-suitedmodelsforcertain
assets(here: commodityindices). DescriptiveanalysisoftheDowJones-UBScommodityindex
comparedtotheStandard&Poor500stockindexprovidesuswithfirstinsightsofsomefeatures
ofthecorrespondingdistributions. Statisticaltestsofnormalityandmeanreversionthenhelps
usinsettingupamodelforcommodityindices. Additionally,chapter1encompassesathorough
introductiontocommodityinvestment,historyofcommoditiestradingandthemostimportant
derivatives, namelyfuturesandEuropeanoptionsonfutures. Theimportanceofcommodity
indicesininvestmenttodayisoutlined,too.
Chapter 2proposesamodelforcommodityindicesandderivesfairpricesforthemostimpor-
tantderivativesinthecommoditymarkets. ItisaHestonmodel[Hes93]supplementedwitha
stochasticconvenienceyield. TheHestonmodelbelongstothemodelclassofstochasticvolatil-
itymodelsandiscurrentlywidelyusedinstockmarkets. Fortheapplicationinthecommodity
marketsthestochasticconvenienceyieldisincludedinthedriftoftheinstantaneousspotreturn
process. Motivatedbytheresultsofchapter1itseemsreasonabletomodeltheconvenienceyield
byameanrevertingOrnstein-Uhlenbeckprocess. Sincetradingdesksonlyapplyandconsider
modelswithclosedformsolutionsforoptionsIderivesuchformulasforcommodityfuturesby
solvingthecorrespondingpartialdifferentialequation. Additionally,semi-closedformformulas
for European options on futures are determined. The Cauchy problem with respect to these
optionsismorechallengingthanthefirstone. Asolutioncanbeprovidedbyapplyingthesame
1methodologyasinHeston[Hes93]andBakshiandMadan[BM00].
Besidestheoptimizationoftherollingprocedureforcommodityfutureswededicateourselves
in chapter 3withtheoptimizationoftheweightingsofthecommodityfuturesthatmakeup
theindex. Tothisend,IapplytheMarkowitzapproachormean-varianceoptimization. The
mean-varianceoptimizationpenalizesup-sideanddown-sideriskequally,whereasmostinvestors
donotmindup-siderisk. Toovercomethis, Iconsiderinthenextstepotherriskmeasures,
namelyValue-at-RiskandConditionalValue-at-Risk. Themainpartofchapter3isdevotedto
thepresentationofanapproachofRockafellarandUryasev[RU00]and[RU02]tooptimizinga
portfoliosoastoreducetheriskofhighlosses. TheConditionalValue-at-Riskisgeneralizedto
discontinuous cumulativedistributionfunctionsoftheloss. Forcontinuouslossdistributions,
theConditionalValue-at-Riskatagivenconfidencelevelisdefinedastheexpectedlossexceeding
theValue-at-Risk. Lossdistributionsassociatedwithfinitesamplingorscenariomodelingare,
however,discontinuous. Variousriskmeasuresinvolvingdiscontinuouslossdistributionsshall
beintroducedandcompared. Theydependonadecisionvariable xandtheconfidencelevel
α. Mycontributiontothistopicistobridgeagapintheproofofthecrucialtheoreminthe
articleofRockafellarandUryasev[RU02]. Furthermore,Ipresentanalternativewayofproving
somepartsofthemaintheorem. Ithenapplythetheoreticalresultstothefieldofportfolio
optimizationwithcommodityindices.
Furthermore, I uncover graphically the behavior of these risk measures. For this purpose, I
considertheriskmeasuresasafunctionoftheconfidencelevel α. Basedonaspecialdiscrete
lossdistribution,thegraphsdemonstratethedifferentpropertiesoftheseriskmeasures. One
recognizes graphically that the definition of the Conditional Value-at-Risk as given in Rock-
afellarandUryasev[RU02]isthemostreasonablegeneralizationtodistributionswithpossible
discontinuities.
Thegoalofthefirstsectionof chapter4istoapplythemathematicalconceptofexcursionsfor
thecreationofoptimalhighlyautomatedoralgorithmictradingstrategies. Algorithmictrading
iswidelyusedbypensionfunds,mutualfunds,institutionaltradersandhedgefunds. Theidea
istoconsiderthegainofthestrategyandtheexcursiontimeittakestorealizethegain. Inthis
sectionIcalculateformulasfortheOrnstein-Uhlenbeckprocess. Ishowthatthecorresponding
formulascanbecalculatedquitefastsincetheonlyfunctionappearingintheformulasistheso
calledimaginaryerrorfunction. Thisfunctionisalreadyimplementedinmanyprograms,such
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