68Real Options Valuation Tools in Corporate Financ

RealOptionsValuationToolsinCorporateFinance4.1IntroductionInChapter4themostcommonvaluationmethodsforrealoptionsarepre­sentedintheory.ThesetoolsareimplementedinthecomputersimulationprogramandwillbeusedinChapter5topricevariousrealoptionsbyap­plyinganon-constantinterestratewhichismodelledviathetermstructuremodelsintroducedinthepreviouschapter.Chapter4isorganizedinthreemainsections:•Section4.2:NumericalMethodsforRealOptionsPricingwithConstantInterestRates•Section4.3:Schwartz-MoonModel•Section4.4:RealOptionsPricingwithStochasticInterestRatesThefinalsection(4.5)contains,asusual,thesummaryofthischapter.Theveryfirstmethodusedtopricearealoptionisthedecision-treeanalysis(DTA)whichwasalreadypresentedandanalyzedinSection2.6.1.ButDTAexhibitstwomajordrawbacks.First,thedecisiontreestructurebecomescom­plicatedeveninsimplerealworldsituations.Second,theDTAcalculationsuserealprobabilitiesandarisk-adjustedinterestrate.Thisinterestrateisconstantanddoesnotchangeeveniftherearedecisionsinthedecisiontreethatchangetheriskstructureoftheproject.Moreover,inusuallycomplexinvestmentsituations,DTAturnsintodecision-bushanalysisthatcannotbehandledappropriately.Thesedrawbacksareeliminatedinthecontingent-claimsanalysis(CCA)byintroducingrisk-adjustedprobabilities,amethodalreadypresentedin2.6.2inthecontextoftheCox-Ross-Rubinsteinbinomialtreemethod.Thismethodwillbeusedatgreatlengthand,therefore,willbereviewedbrieflyin4.2.1intheframeworkofaconstantinterestrate.Anotherlatticemethodthatwillbe1324RealOptionsValuationToolsinCorporateFinanceexplainedindetailisalog-transformedbinomiallatticemethodintroducedbyTrigeorgis^in1991.BoththeCox-Ross-RubinsteinmethodandtheTri-georgislog-transformedbinomialtreemethodarepresentedinSection4.2.1,NumericalMethodsforRealOptionsPricingwithConstantInterestRates.Section4.2.2,FiniteDifferenceMethods^dealswithvariousfinitedifferencemethods.Realoptionsthatcanbedescribedmathematicallyviaapartialdifferentialequationcanbepricedbyusinganapproximationofthispartialdifferentialequationwithfinite(incontrasttoinfinitesimal)differencesthatreplacethepartialderivativesintheequation.Althoughfinitedifferencescanbeapplieddirectlytothepartialdifferentialequationwiththepriceofthetwinsecurityortheprojectvalueastheunderlying,suchafinitedifferencemethodexhibitspoornumericalproperties^.However,ifalog-transformationoftheunderlyingisusedandthepartialdifferentialequationisbuilt,theresultingfinitedifferencemethodexhibitsadvantageousnumericalproperties.Therefore,onlylog-transformedfinitedifferencemethodswillbepresentedhere.Theyarealsoimplementedinthecomputersimulationprogram.Dependingonthetypeofapproximation,distinctionscanbemadebe­tweenvarioustypesoffinitedifferencemethods.Thisthesispresentsthelog-transformedexplicitandimplicitfinitedifferencemethods,twobasickindsoffinitedifferencemethods.Eachmethodwillbeexplainedwithrespecttoitsmathematicaltheory,numericalproperties,andpracticallimitations.OthermethodsliketheCrank-Nicholsonalgorithm,ahybridbetweentheexplicitandtheimplicitmethod,willnotbeexplainedhere^.Anothermethodthatbuildsonthelog-transformedexplicitfinitedifferencemethodandalsoin­cludesatime-dependentrisk-freeratemodelledbyusingashort-rateprocesswaspresentedinHull&White[64].Thismethodwillnotbepresentedheresincethisthesisfocusesonmodellingthecompleteyieldcurveinsimulationsandhistoricalbacktestingandnottheshort-rateprocessonly.Inaddition,themodificationoftheCox-Ross-RubinsteinbinomialtreemethodandtheTri-georgislog-transformedbinomialtreemethodwhichisrealizedinthisthesisbyincorporatingastochasticrisk-freerateislesscomplicated(and,there­fore,morelikelytobeappliedinpractice)thanthemodificationofthelog-transformedexplicitfinitedifferencemethodcarriedoutbyHullandWhite.Section4.3presentstheSchwartz-Moonmodel,aspecificmodelforpricingrealoptionsinR&Dofnewdrugdevelopment.Thismodelusesvariousstoch­asticprocessestomodeltheunderlyingandthecostprocesses.Themodelhasadeterministiccounterpartthatappliesaconstantinterestrateandthatis^SeeTrigeorgis[132],pages320-328.^SeeTrigeorgis[132],page314.^FormoreinformationontheCrank-NicholsonalgorithmseeTrigeorgis[132],page319.4.1Introduction133usedbySchwartzandMoonasthebenchmarktotheirstochasticmodel.Tointroducetheideaofanon-constantinterestrate,thisdeterministiccounter­partwillbeusedinconnectionwithadeferredprojectstartforsystematicallychanginginterestratesinordertoevaluatetheeffectthatsuchamodificationhasontheNPVofaprojectandtherefore,onthemanager'sinvestmentde­cision.Section4.4,RealOptionsPricingwithStochasticInterestRates,presentsrealoptionsvaluationtoolsthataccommodateforanon-constantrisk-freeinter­estrate,andexplainsmodificationsofexistingvaluationmethodstoincludeanon-constantrisk-freerate.Inthebeginningofthissection,thepioneeringideaofIngersollandRosswillbeexplained(4.4.1).Theywerethefirsttoshowtheexistenceofarealoptioneveninthecaseofdeterministicfuturecashflowsbutwithstochasticinterestrates.Theirpublicationin1992wastheveryfirsttodealwiththisideabyapplyingtheItocalculustoinvestigateanoptiontodeferonthepurchaseofaZe