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American Option Pricing under Stochastic Volatility

Permanent Link: http://ufdc.ufl.edu/UFE0022624/00001

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Title: American Option Pricing under Stochastic Volatility Empirical Evaluations
Physical Description: 1 online resource (74 p.)
Language: english
Creator: Goswami, Manisha
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: american, lsm, stochastic, volatility
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: Over the past few years, model complexity in quantitative finance has increased substantially in response to earlier models that did not capture critical events for risk management. However, it is still not clear that the increased complexity is matched by additional accuracy in the ultimate result. We studied the empirical performance of a stochastic volatility model that has gained wide acceptance among both practitioners and academics. Specifically its focus is on pricing and hedging of American options using Heston's model, which is compared against the benchmark constant volatility model of Black and Scholes. The stochastic volatility model is calibrated with data on S & P 100 Index European style options using two-step procedure based on indirect inference and non-linear least squares. This approach provides the advantage of incorporating information from both equity and options market. The calibrated model is then used to price American style options through a modification of the least-squares Monte-Carlo (LSM) algorithm of Longstaff and Schwartz to account for stochastic volatility. The performance of this model is then assessed by comparing it against the constant volatility model of Black and Scholes on the basis of pricing and hedging errors. Empirical results from the study support the need for stochastic models. However, the resulting pricing and hedging process procedures are time consuming. Thus an approximation method is also developed in the dissertation that considerably reduces the computation time without compromising the accuracy of result and retaining the randomness of volatility in the modeling of asset process. The approximate method to price American options makes use of the fact that accurate pricing of these options does not require exact determination of the early exercise boundary. Thus, the procedure mixes the two models of constant and stochastic volatility. The idea is to obtain early exercise boundary through constant volatility model using the approximation methods of AitSahlia and Lai or Ju and then utilize this boundary to price the options under stochastic volatility models. The data on S & P 100 Index American options is used to analyze the pricing performance of the mixing of the two models. The performance is studied with respect to percentage pricing error and absolute pricing errors for each money-ness maturity group.
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 Manisha Goswami.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: AitSahlia, Farid.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-06-30

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Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022624:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022624/00001

Material Information

Title: American Option Pricing under Stochastic Volatility Empirical Evaluations
Physical Description: 1 online resource (74 p.)
Language: english
Creator: Goswami, Manisha
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: american, lsm, stochastic, volatility
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: Over the past few years, model complexity in quantitative finance has increased substantially in response to earlier models that did not capture critical events for risk management. However, it is still not clear that the increased complexity is matched by additional accuracy in the ultimate result. We studied the empirical performance of a stochastic volatility model that has gained wide acceptance among both practitioners and academics. Specifically its focus is on pricing and hedging of American options using Heston's model, which is compared against the benchmark constant volatility model of Black and Scholes. The stochastic volatility model is calibrated with data on S & P 100 Index European style options using two-step procedure based on indirect inference and non-linear least squares. This approach provides the advantage of incorporating information from both equity and options market. The calibrated model is then used to price American style options through a modification of the least-squares Monte-Carlo (LSM) algorithm of Longstaff and Schwartz to account for stochastic volatility. The performance of this model is then assessed by comparing it against the constant volatility model of Black and Scholes on the basis of pricing and hedging errors. Empirical results from the study support the need for stochastic models. However, the resulting pricing and hedging process procedures are time consuming. Thus an approximation method is also developed in the dissertation that considerably reduces the computation time without compromising the accuracy of result and retaining the randomness of volatility in the modeling of asset process. The approximate method to price American options makes use of the fact that accurate pricing of these options does not require exact determination of the early exercise boundary. Thus, the procedure mixes the two models of constant and stochastic volatility. The idea is to obtain early exercise boundary through constant volatility model using the approximation methods of AitSahlia and Lai or Ju and then utilize this boundary to price the options under stochastic volatility models. The data on S & P 100 Index American options is used to analyze the pricing performance of the mixing of the two models. The performance is studied with respect to percentage pricing error and absolute pricing errors for each money-ness maturity group.
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 Manisha Goswami.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: AitSahlia, Farid.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022624:00001


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Iwouldliketoacknowledgetheadviceandguidanceofmythesisadvisor,Dr.FaridAitSahlia.Ialsothankthemembersofmythesiscommittee,Dr.StanislavUryasev,Dr.HaniDossandDr.LiqingYanfortheirguidanceandencouragement.IthankmygraduatecolleaguesintheIndustrialandSystemsEngineeringDepartment,myparentsandparents-in-lawfortheirsupportandmyhusbandforhispatienceandunderstanding. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 11 2PERFORMANCECOMPARISONOFCONSTANTVOLATILITYMODELANDSTOCHASTICVOLATILITYMODEL ................... 18 2.1Introduction ................................... 18 2.2Heston'sStochasticVolatilityModel ..................... 21 2.2.1ClosedFormSolution .......................... 22 2.2.2Risk-NeutralPricing .......................... 23 2.3DataDescription ................................ 23 2.3.1DataforParameterEstimation ..................... 23 2.3.2DataforOut-of-SampleAnalysis .................... 24 2.4CalibrationoftheHestonModel ........................ 24 2.4.1StructuralParameterEstimation .................... 26 2.4.2OptionPricingCalibration ....................... 29 2.4.3DiscussionofEstimatedParameters .................. 29 2.5PricingAmericanOptions ........................... 30 2.5.1TheLSMFramework .......................... 30 2.5.2Longsta-SchwartzAlgorithm ..................... 32 2.6EmpiricalEvaluation:PricingS&P100PutOptions ............ 33 2.6.1OptionsClassication .......................... 34 2.6.2MeanPercentagePricingError ..................... 37 2.6.3AbsolutePricingErrors ......................... 37 2.6.4ResultsAnalysis ............................. 42 2.7HedgingPerformance .............................. 44 2.8Conclusions ................................... 46 3CONSTANTVOLATILITYAPPROXIMATIONSFORSTOCHASTICVOLATILITYMODELS ....................................... 48 3.1Introduction ................................... 48 3.2EarlyExerciseBoundaryApproximationviaConstantVolatility ...... 50 3.2.1LinearSplineApproximation ...................... 51 3.2.2BoundarySurfaceComparison ..................... 54 3.3AmericanOptionPricingApproximationviaConstantVolatility ...... 57 5

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.............. 59 3.4.1Hestons'sModelParameters ...................... 59 3.4.2DataSetforAnalysis .......................... 59 3.5ClassicationofOptionsandMeasuresofPerformance ........... 60 3.5.1OptionsClassication .......................... 60 3.5.2MeasuresofPerformance ........................ 60 3.6EmpiricalResults ................................ 60 4CONCLUSION .................................... 67 REFERENCES ....................................... 69 BIOGRAPHICALSKETCH ................................ 74 6

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Table page 2-1TheGARCHestimatesofauxiliaryparameters .................. 29 2-2Nonlinearleast-squaresestimatesofoptionpricingparameters .......... 29 2-3Out-of-samplepercentagepricingerrorsforspotvolatilityasinitialvolatility: 38 2-4Longtermaverageasinitialvolatilityforout-of-samplepercentagepricingerrors 39 2-5Spotvolatilityasinitialvolatilityforout-of-sampleabsolutepricingerrors ... 40 2-6Absolutepricingerrorsforlongtermaverageasinitialvolatility ......... 41 2-7Absolutehedgeerrors ................................ 46 3-1ParameterestimatesofHeston'smodel ....................... 59 3-2Relativepricingerrorsforinitialandconstantvolatilitysetatspotvolatility .. 63 3-3Longtermaverageasinitialandconstantvolatilityforrelativepricingerrors .. 64 3-4Spotvolatilityasinitialandconstantvolatilityapproximationsforabsoluterelativepricingerrors ..................................... 65 3-5Absolutepricingerrorsforconstantvolatilityapproximationsetatlong-termaverage ........................................ 66 7

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Figure page 2-1Rawdatasetforoneday ............................... 25 2-2Insampletofthedata ............................... 31 2-3Spotvolatilityasinitialvolatility:out-of-samplepricingperformanceforCVandSVmodelsascomparedtomarketprice .................... 35 2-4Longtermaverageasinitialvolatility:out-of-samplepricingperformanceforCVandSVmodelsascomparedtomarketprice .................. 36 3-1Spotvolatilityasconstantvolatilityforevolutionofboundarysurfaceplotsfromone-piecetothree-pieceapproximationsascomparedtotheactualboundary .. 55 3-2Evolutionofboundarysurfaceplotsfromone-piecetothree-pieceapproximationsascomparedtotheactualboundarywithlongtermaverageasconstantvolatility 56 8

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Overthepastfewyears,modelcomplexityinquantitativenancehasincreasedsubstantiallyinresponsetoearliermodelsthatdidnotcapturecriticaleventsforriskmanagement.However,itisstillnotclearthattheincreasedcomplexityismatchedbyadditionalaccuracyintheultimateresult. Westudiedtheempiricalperformanceofastochasticvolatilitymodelthathasgainedwideacceptanceamongbothpractitionersandacademics.SpecicallyitsfocusisonpricingandhedgingofAmericanoptionsusingHeston'smodel,whichiscomparedagainstthebenchmarkconstantvolatilitymodelofBlackandScholes.ThestochasticvolatilitymodeliscalibratedwithdataonS&P100IndexEuropeanstyleoptionsusingtwo-stepprocedurebasedonindirectinferenceandnon-linearleastsquares.Thisapproachprovidestheadvantageofincorporatinginformationfrombothequityandoptionsmarket. ThecalibratedmodelisthenusedtopriceAmericanstyleoptionsthroughamodicationoftheleast-squaresMonte-Carlo(LSM)algorithmofLongstaandSchwartztoaccountforstochasticvolatility.TheperformanceofthismodelisthenassessedbycomparingitagainsttheconstantvolatilitymodelofBlackandScholesonthebasisofpricingandhedgingerrors. Empiricalresultsfromthestudysupporttheneedforstochasticmodels.However,theresultingpricingandhedgingprocessproceduresaretimeconsuming.Thusanapproximationmethodisalsodevelopedinthedissertationthatconsiderablyreduces 9

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TheapproximatemethodtopriceAmericanoptionsmakesuseofthefactthataccuratepricingoftheseoptionsdoesnotrequireexactdeterminationoftheearlyexerciseboundary.Thus,theproceduremixesthetwomodelsofconstantandstochasticvolatility.TheideaistoobtainearlyexerciseboundarythroughconstantvolatilitymodelusingtheapproximationmethodsofAitSahliaandLaiorJuandthenutilizethisboundarytopricetheoptionsunderstochasticvolatilitymodels.ThedataonS&P100IndexAmericanoptionsisusedtoanalyzethepricingperformanceofthemixingofthetwomodels.Theperformanceisstudiedwithrespecttopercentagepricingerrorandabsolutepricingerrorsforeachmoney-nessmaturitygroup. 10

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Inmoderntimes,amajorissuefornancialinstitutionsisecientandrealisticcomputationsofpricesandhedgesforderivativeproducts,whichderivetheirvaluefromsomeotherasset.Thisinvolvesthedevelopmentofincreasinglycomplexnancialmodelsrequiringtheuseofadvancedstochasticandnumericalanalysistechniques.Attheheartofthismodern-daynancialmathematicsisthemodeling,pricingandhedgingofoptioncontracts,whicharecontractsthatgivethebuyertheright,butnottheobligation,tobuy(Call)orsell(Put)anunderlyingassetataspecicpriceonorbeforeacertaindate.Ifthisso-calledexerciserightisgrantedforthisdate,thentheoptionislabeledEuropean;otherwise,itisAmerican.Theimportanceofoptionpricingstemsfromthefactthatoptionsplayapivotalroleinthedailyriskmanagementpracticesandtradingstrategiesofnancialinstitutions.Therefore,theabilitytoimplementmathematicalmodelsthatsucceedincapturingthebehaviorofmarketoptionpricesmoreaccuratelyandrealisticallyacrossvariousunderlyingmarketvariablesisofsignicantimportancetothenancialengineeringcommunity. Sinceitsintroduction,manymodicationsandextensionstothe BlackandScholes ( 1973 )modelhavebeensuggestedandtestedinordertoovercomeitsdrawbacks.Onesuchmodicationistherelaxationofconstantvolatilityassumption.Intheparticularcaseofstochasticvolatilitymodels,whichhavereceivedsignicantattention,recentempiricalstudieshavefocusedprimarilyonEuropeanoptions BakshiG.andZ. ( 1997 ).Inthisdissertation,theseempiricalevaluationsareextendedtoAmericanoptions,astheiradditionalopportunityforearlyexercisemayincorporatestochasticvolatilityinthepricingdierently.Specically,thisworkcomparestheempiricalpricingperformanceofthecommonlyadoptedstochasticvolatilitymodelof Heston ( 1993 )againstthetraditionalconstantvolatilitybenchmarkof BlackandScholes ( 1973 ).Forthestudy,dataonS&P100optionsareused.Resultsfromthisstudyconrmtheusefulnessofstochasticvolatility 11

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AitSahliaandLai ( 2001 )).Asaresult,weshowinthepresentworkthatapproximationsbasedontheconstantvolatilitymodelcanbeusedtopriceaccuratelyandecientlyAmericanoptionsunderstochasticvolatilityassumptions. Modernoptionpricingtechniques,withrootsinstochasticcalculus,areoftenconsideredamongthemostmathematicallycomplexofallappliedareasofnance.Therstattemptstoexplainthehedgingandspeculationaspectsofoptionscanbetracedbackto1877,withthebook"TheTheoryofOptionsinStocksandShares"writtenby Castelli ( 1877 ),however,Castelli'sbooklackedanysubstantialtheoreticalbasis.Louis Bachelier ( 2006 ),inhis1900dissertation,ThoriedelaSpculationproposedmostofthestandardconceptsinmathematicalnance:randomwalkofnancialmarketprices,Brownianmotionandmartingales.TherehavebeenmanypioneeringcontributionssincetheninnancebythelikesofPaulSamuelsonwhoalsorediscoveredBachelier'swork,RichardKruizenga,astudentofPaulSamuelson,RobertMerton,A.JamesBoness.InfactitwastheworkofBonessthatservedastheprecursorforFischerBlackandMyronScholeslandmarkoptionpricingmodelof1973whichprovidedtheclosedformsolutionforpricingEuropeanstyleoptions( Bernstein ( 1992 )). However,pricingandhedgingofAmericanstyleoptionsstillremainsachallengingproblem.Aclosed-formformulafortheseoptionshasnotyetbeenfoundanditisnotlikelytobefoundsoon.Therefore,eortsforpricingtheseoptionshaveconcentratedonapproximatemethodseversince. McKean ( 1965 )and Merton ( 1973 )showedthatthepricingofAmericanoptionisafree-boundaryproblem.Astheseoptionscanbeexercisedanytimeuptomaturity,apartofthevaluationproblemconsistsinidentifyingtheearlyexerciseboundary.Someearlierapproximatemethodsconcentratedonnumericalmethodssuchasnite-dierencesdueto BrennanandSchwartz ( 1977 )andbinomialtree 12

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CoxandRubinstein ( 1979 ).Thesemethodsarestillwidelyusedforcomplexoptionssincetheyareeasytoimplementandarequiteexible.However,theyareverytimeconsumingandsoresearcheortshaveconcentratedonndingfastermethodswithoutloosingaccuracy.Therehavealsobeenanalyticalapproaches,suchasthoseof MacMillan ( 1986 )and Barone-AdesiandWhaley ( 1987 ).Thesearefastbutarenotveryaccurateespeciallyforlong-maturityoptions. AnothermethodreliesondevelopinglowerandupperboundsforAmericanoptions.Itrstobtainslowerandupperboundsfortheoptions,followedbyaninterpolationschemetopriceAmericanoptions( Johnson ( 1983 )and BroadieandDetemple ( 1996 )).Thismethodcanbequitefastbutrequiresaccuratecomputationofoptionvaluesforalargesetofdatatodeterminesomerequiredregressioncoecients.Asforanalyticalapproximationmethods,theyarealsonotconvergent. AfourthapproachistheintegralrepresentationformulaforpricingAmericanoptions.Itcanbetracedbackto GeskeandJohnson ( 1984 ).Inthisapproach,theAmericanoptionpriceisrepresentedasthesumofitsEuropeancounterpartplusanearlyexercisepremium,whichisrepresentedasanintegral.Itisattributedto Kim ( 1990 ), Jacka ( 1991 )and CarrandMyneni ( 1992 ).Therepresentationformularequiresthedeterminationoftheearlyexerciseboundarywhichcharacterizestheoptimalexercisestrategyasthersttimetheunderlyingstochasticprocesshitsthisboundary. Ju ( 1998 )and AitSahliaandLai ( 2001 )haveshownthattheintegralintheearlyexercisepremiumdoesnotdependcriticallyontheexactdeterminationoftheboundary.Theirmethodsarenotonlycomputationallyfastbutalsoarequiteaccurate. Anotherapproachthatdoesnotrequiretheidenticationoftheearlyexerciseboundaryisthevariationalinequalityapproachdevelopedin JailletandLapeyre ( 1990 ).Itreformulatestheproblemintermsofavariationalinequalitywhichdoesnotdependontheboundary. 13

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Carrire ( 1996 )and LongstaandSchwartz ( 2001 ),whoincontrasttoearlyattemptswiththisapproachsuchas Tilley ( 1993 )and BroadieandGlasserman ( 1997 ),devisedecienttechniquesbasedonleast-squarestoestimatethehard-to-obtainvalueofcontinuation.AmajorappealofMonteCarlobasedapproachesistheirrelativeinsensitivitytoincreasesinthestatedimensionoftheproblem,contrarytoalltheotheralternatives,whichquicklybecomeinecientassoonasthestatespaceisthreeorfour-dimensional. AlmostallthemethodsdetailedabovetopriceAmericanoptionshavebeenimplementedinthestandardBlack-Scholesmodelwiththeassumptionofconstantvolatility.Thephenomenonofimpliedvolatilitysmilehasshownthatvolatilityisnotconstant. BollerslevandK.F. ( 1992 )alsoprovidedevidenceregardingthepersistentchangesinthevolatilityofassetreturns.However,incorporatingvolatilityasastochasticcomponentismadechallengingbythefactthatitisahiddenprocess:itcannotbedirectlyobservedbutisinherentinthepriceprocess.Severaloptionpricingmodelsforstochasticvolatilityhavebeenproposed.Thediscretetimemodelsaredominatedbyvariantsoftheautoregressiveconditionalheteroscedasticitymodel(ARCH)introducedby Engle ( 1982 ),withthemostwidelyusedformulationbeingthegeneralautoregressiveconditionalheteroscedasticitymodel,GARCHintroducedby Bollerslev ( 1986 ).ThecontinuousmodelsarerepresentedbystochasticdierentialequationswiththemostpopularbeingtheOrnstein-Uhlenbeckprocessforthelogofinstantaneousconditionalvolatility:d(ln)=(ln)dt+vdW

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CoxandRoss ( 1985 )squarerootstochasticprocessisalsowidelyused:d(2)=(2)dt+vp HullandWhite ( 1987 ),themeanrevertingmodelsof Scott ( 1987 ), MelinoandTurnbull ( 1990 ), MelinoandTurnbull ( 1995 ), SteinandStein ( 1991 ),and Heston ( 1993 ).Thelatteristhemostpopularsinceitallowsacorrelationbetweentheassetpriceandvolatilityprocess,canensurethatthevolatilityprocessremainspositiveandhasananalyticsolutiontopriceEuropeanoptions.ThusHeston'smodelisournaturalchoicetoaccountforstochasticvolatility. Oncethestochasticvolatilitymodelischosen,itrequirescalibration.Discrete-timemodelsareeasiertoestimateasstatisticalinferencetechniquescanbeappliedtothesemodels,however,noarbitrageconditionsarerepresentedbestbythecontinuous-timemodelsduetotheframeworkofstochasticdierentialequations.Oneofthebasicissuesincalibratingthesemodelsiswhichdatatouse:thedataontheunderlyingassetpricesordatafromtheoptionsmarket?Therehavebeenseveraltechniquesappliedtocalibratethecontinuousmodelsusingthecross-sectionaldatafromtheoptionsmarketortimeseriesdatafromtheassetmarketorboth.Thenon-linearleastsquareapproachappliedby BakshiG.andZ. ( 1997 )usescross-sectionaldata,theindirectinferencemethodof EngleandLee ( 1996 )usestimeseriesdataandthetwostepestimationprocedurecombiningthesetwomethodsusedby ZhangandShu ( 2003 )makesuseofdatafrombothmarkets.Usingdatafromboththemarketsisessentialforestimatingtheriskpremiumonvolatility.TherehasalsobeentheBayesianapproachinMarkovChainMonteCarlomethods,suchas JacquierandRossi ( 1994 )and ChibandShephard ( 1998 )tocalibrate 15

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ThisdissertationisanattempttostudyempiricallytheeectsofpricingandhedgingAmericanoptionsunderstochasticvolatility.ThestochasticvolatilitymodelisttedtothemarketdataandthenAmericanoptionsarepricedusingthecalibratedmodel.Thepricingandhedgingperformanceofstochasticvolatilitymodeliscomparedwiththatofconstantvolatilitymodel.Asanimprovementofsimplicityandtimeoverpurestochasticvolatilitymodels,theperformanceofmixingtheconstantandstochasticvolatilitymodelisalsostudied.Therestofthedissertationisorganizedasfollows. Chapter2revisitsHeston'sstochasticvolatilitymodelanditsclosedformsolutionforpricingEuropeanoptions.Datadescriptionandlteringareexplained.Heston'smodeliscalibratedusingatwo-stepstatisticaltechniquebasedonindirectinferenceandnon-linearleastsquareestimation.Theresultsobtainedfromthetwoproceduresarethendiscussed. Afterthemodeliscalibrated,theMonteCarloalgorithmofCarrire/LongstaSchwartzisadaptedtopriceAmericanoptionsunderstochasticvolatility.ThepricingandhedgingperformanceofstochasticvolatilityandconstantvolatilitymodelsarecomparedonthebasisofS&P100indexdata. TheresultsfromChapter2arepromisingforstochasticvolatilitymodels.TheyshowconsiderableimprovementinpricingandhedgingofAmericanoptionsoverconstantvolatilitymodels.However,incorporatingvolatilityasastochasticcomponentincreasesthecomplexityoftheproblemandisalsoatime-consumingprocess.ThisinspiresthestudyinChapter3whereitisproposedtocombinethetwomodels.SincethepricingofAmericanoptionsrequirestheknowledgeoftheearlyexerciseboundary,theaccuracyofwhichisnotcriticalforthepricingformula,itisapproximatedusingconstantvolatilitymodel.Eitheroftheapproximatemethodsdueto AitSahliaandLai ( 2001 )or Ju ( 1998 )canbeemployedtoobtaintheconstantboundary. 16

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Themixingofconstantvolatilityandstochasticvolatilitymodelgivesrisetoanotherimportantissuei.e.constantvolatilityestimation.Inchapter3,wealsotestempiricallytheeectofusingspot,realized,andlong-termmeanp Chapter4concludesthecurrentworkandgivesfutureresearchdirections. 17

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BlackandScholes ( 1973 )formulaforEuropeanoptionsiswithoutdoubtthemostsuccessfulpricingformula.However,ithasknownbiasesforwhichremedieshavebeenattemptedinanumberofdierentdirections.Theycanbetracedbackto Black ( 1976 )whoobservedthattime-seriesofequityreturnsdisplayedfattertailsthanimpliedbythenormaldistributionunderthegeometricBrownianmodelof BlackandScholes ( 1973 ).Additionally,thenon-constantimpliedvolatilityacrosstimeandstrikeprices,togetherwiththephenomenonofvolatilityclustering(sustainedperiodsofhigh-variabilityalternatingwithsustainedperiodsoflow-variability)asobservedinmanystudieshasprovidedenoughevidencetoseekalternativeoptionpricingmodels( Rossi ( 1996 ).Indeed,severalsuchmodelshavebeenproposedparticularlyinthelasttwodecades.Theyincludefeaturessuchasstochasticvolatility,stochasticinterestrates,andstochasticjumpsthatareconsideredinvariousspecicationsseparatelyorincombinations.Asamplingoftheliteratureincludesthestochasticinterestratemodelof AminandJarrow ( 1992 );thestochasticvolatilitymodelsof HullandWhite ( 1987 ), Scott ( 1987 ), MelinoandTurnbull ( 1990 ), MelinoandTurnbull ( 1995 ), SteinandStein ( 1991 ), Heston ( 1993 );thestochastic-volatilityandstochasticinterestratemodelsof BaileyandStulz ( 1989 ), AminandNg. ( 1993 ),and BakshiandChen. ( 1997a ), BakshiandChen. ( 1997b );thestochastic-volatilityjump-diusionmodelsof Bates ( 1996 )and Scott ( 1997 ).Thislistisfarfrombeingexhaustivebutalreadypointstothedierentorientationsoftheeortstoimproveupontheoriginalmodelof BlackandScholes ( 1973 ). Asthesemodelsbecomeincreasinglycomplex,requiringadditionalparameterstobeestimatedandmoresophisticatednumericaltechniquesfortheirpricingandhedging,itisunclearwhetherthiscomplexitycostismatchedbyadditionalaccuracyinpricing.InacomprehensivestudybasedonS&P500data, BakshiG.andZ. ( 1997 )compare 18

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BakshiG.andZ. ( 1997 )usecross-sectionalinformationcontainedinoptionpriceswithdierentmaturitiesandstrikepricesresultinginimpliedvolatilitiesinordertoinferestimatesforthestructuralparametersofthestochasticvolatilitymodel.Ontheotherhand,itappearsthattheseimpliedstructuralparametersdeviatesubstantiallyfromtheirtime-seriescounterparts BakshiG.andZ. ( 1997 )useimpliedvolatilitiestoestimatethecorrelationcoecientoftheassetreturninnovationwiththatofitsstochasticvolatilityas0.76whereastheirestimatebasedontheunderlyingassettime-seriesis0.23.)Asanalternative, ZhangandShu ( 2003 )haveproposedtouseatwo-stepproceduretoestimaterst,viatheindirectinferencemethodof GourierouxandRenault ( 1993 ),structuralparametersfortheunderlyingasset,followedbyasecondsetofadditionalparametersneededforoptionpricingviaamarketpricecalibrationbasedonleast-squares.Anexampleforthelatteristheempiricallydeterminedvolatilityriskpremium,whichisrequiredsincethepricingmodelisincomplete(i.e.,non-uniquenessofrisk-neutralequivalentmeasure). ZhangandShu ( 2003 )applythistwo-stepapproachintheirstudycomparingthepricingaccuracyofthe Heston ( 1993 )stochasticvolatilitymodelagainsttheBlack-Scholesconstantvolatilitymodel.TheyuseS&P500datatoshowthattheHestonmodelsignicantlyoutperformstheBlack-Scholesmodelinalmostallmoneynessmaturityclasses.AstheirstudyislimitedtoEuropeanoptions,itremainstobeseenhowtheygeneralizetoAmerican-styleoptions,whichhavetheadditionalearly-exercisefeaturethatmayamplifythestochasticvolatilitycharacteristicoftheunderlyingasset.Theseoptionsarenotoriouslydiculttopriceincomparison.Forexample, Heston ( 1993 )derivesFourier-basedexpressionsforEuropean 19

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Carrire ( 1996 )and LongstaandSchwartz ( 2001 ),whichapproximatesthecontinuationvalueintherelatedoptimalstoppingproblembyaregressionforwhichdataaregeneratedviasimulation.ItshouldbenotedthatalternativemethodsbasedonnumericalresolutionofpartialdierentialequationshavebeendevelopedtopriceAmericanoptionsunderstochasticvolatilityconsideration( IkonenandToivanen ( 2007 )).However,wechosetheLSMapproachasitpresentsmoreexibilityforourapproach,particularlyinitsanticipatedextensionstohigherdimensionsandpath-dependentoptionpayos,whichalsoexplainspartlyitswideadoption. Theremainderofthechapterisorganizedasfollows:Section2reviewsthestochasticvolatilitymodelof Heston ( 1993 )thatweadoptforoptionpricing;Section3describesthedataandtheirsourcesforthisstudy;Section4concernsthetwo-stepstatisticaltechniquethatwefollowforparameterestimationoftheresultingbivariatediusionprocess;inSection5weadapttheMonteCarloalgorithmofLongstaSchwartztocomputepricesandhedgingparametersforAmericanoptionsunderstochasticvolatility;Section6containsnumericalresultswithrespecttoconsideringtwodierentestimatesofvolatilitynamelyspotvolatilityandlongtermmeanp 20

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( 1993 )modelassumesboththeunderlyingassetanditsvolatilitytobestochasticprocessesdenedbythefollowingstochasticdierentialequations: where,undertheoriginalmeasureP,andareagaintheunderlyingmeanreturnanddividendrates,respectively,oftheasset;p Feller ( 1951 )). WhereastheBlack-Scholesmodelconsidersonlyoneobservableprocess,fStg,whichcanbeusedtoestimateitssoleunknownparameterrequiredforpricing,namelytheconstant,theHestonmodelintroducesanadditionalprocessfvtgthatisnotobservableandforwhichthreeparameters,,,,mustbeestimated.Thisvariationincreasesimmediatelythecomplexityintheestimationprocedure.ThoughanestimateofcanbeobtainedonthebasisofpastobservationsofS,thecommunityofnancialpractitionershasnowgrownaccustomedtoestimatingtheconstantthroughtheso-calledimpliedvolatilityapproach,whichusesoptiondataaswell.ItsetstheBlack-Scholesfunctionevaluatedforanat-the-moneyoptionequaltothecorrespondingobservedprice,C(K;T)=Cobs,fromwhichisinverted,thusnowbecomingafunctionofTandK.ThoughthisapproachmayseemunfoundedsincetheBlack-Scholesformularesultsfromtheassumptionofconstant,itisappealingwhenconsideringtheresultingestimateasbeingaforward-lookingestimateof. 21

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Heston ( 1993 ),thismarketpriceofvolatilityriskisassumedtobeproportionaltotheinstantaneousvolatility;i.e.:(St;vt;t)=vt; wherePj(St;vt;T;K);j=1;2maybeinterpretedasadjustedorriskneutralizedprobabilityfunctionsandareobtainedbyinvertingtheircorrespondingcharacteristicfunctions(?):Pj(x;v;T;K)=1 2+1 forj=1;2,withthecharacteristicfunctionfjdenedby:fj(x;v;T;)=exp[Cj(Tt;)+Dj(Tt;)v+ix]Cj(;)=ri+a 2(bji+dj)2ln1gjedj

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2;u2=1 2;a=;b1=+;b2=+ 2 and 2 become: where=+,==(+). Themodelisanalyzedintermsofthisrisk-neutralizedprocessbecausethisprocessdeterminesthepricesexclusivelyasintheBlack-Scholesmodel.Thus,wecouldestimate(;;)fromthetrueprocessof 2 and 2 andthenapplytheaboveprocesstoobtainrisk-neutralprices. 2.3.1DataforParameterEstimation 23

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8thareomitted;and(vi)forestimationpurposes,weuseEuropeancalloptionsonlyfromtheperiodJanuary1,2002toDecember31,2005.Oncetheselterrulesareapplied,weareleftwith33;860observationsfor1,007days(anaverageof34optionsperday). Figure 2-1 showsonedayofrawdataonoptionsonS&P100Index.Thesnapshotofdatabaseshowstheattributesforwhichthedataisavailableforanoption.Forexamplebestbidprice,bestaskprice(foranalysis,thepriceoftheoptionistakenastheaverageofbestbidandbestask),openinterest,lasttradeddate,strikeprice,maturitydate,closingpriceoftheindexetc. 24

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Rawdatasetforoneday

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ZhangandShu ( 2003 )usethesametreatmenttoestimatethestochasticvolatilityprocessforS&P500indexoptions,whichhaveEuropeanexercisestyle.(1997)usetheleast-squaresmethodoncross-sectionaloptiondatatoestimatetheparametersoftheHestonModel.Asnotedinourintroduction,weopttofollowinsteadtheindirectinferencemethodof GourierouxandRenault ( 1993 )toestimatethestructuralparameters(;;;).Ourstudyof5-minutereturnsontheS&P100indexdoesnotshowthattheiraverageissignicantlydierentfromzero,wethereforesettobezero. GourierouxandRenault ( 1993 ))involvesdeninganauxiliaryparameterthatisrstestimatedthroughaneconometricapproachonthebasisofactualdataandthenusedtoestimateoriginal(structural)parametersofinterestthroughmomentmatching.Thisapproachisparticularlyusefulwhenlikelihoodfunctions,oranyotherestimationcriterion,fortheoriginalmodelarediculttoevaluate,incontrasttotheauxiliarymodel.Thelatterneednotbecorrectlyspecied.Ifitis,thenindirectinferenceisequivalenttomaximumlikelihood. Inourpresentcaseofstochasticvolatility,thesimulationisgeneratedviaanEulerdiscretizationof( 2 ): wherertisthereturnovertheperiod(t;t],withbeingthetimediscretizationincrement,and1tand2tareuncorrelatedstandardnormalrandomvariables.Thisdatageneratingprocess(DGP)isfullyspeciedwhentheparameter=(;;)isxed.On 26

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wheretN(0;ht),withtheconditionalvariancehtVar(rtjFt1)basedontheinformationsetFt1uptotimet1.Thischoiceispredicatedontheresultsof Nelson ( 1990 )whoarguesthatwhenthetimeincrementof( 2 2 )isarbitrarilysmall,therstandsecondmomentsofvolatilityinGARCH(1,1)matchthoseofthediusion( 2 ). Theparameter=(!;;)oftheGARCHmodelisrstestimatedonthebasisofobservedreturns,yieldingtheestimate(0)=!(0);(0);(0).Anothersetofdata,nowgeneratedviatheDGP( 2 2 ),willbeusedtogetanotherestimate(1)of.AstheDGP( 2 2 )requirestheknowledgeof=(;;),theideaofindirectinferenceistogetaninitialestimate(0)oftoget(1),whichinturnwillleadtotherightestimate(1)ofviaanoptimalitycriterion.Forthispurpose,wefollow EngleandLee ( 1996 )whosuggestanintermediateestimate(0)throughthematchingofthersttwomomentsof( 2 2 )with( 2 2 ),resultingin: whereistheconditionalkurtosisofthevolatilityshocksestimatedfromtheactualdata. With=(0)nowxed,asampleofNobservationsisgeneratedviatheDGP( 2 2 )inordertotaGARCH(1,1)modeloftheform( 2 2 ),yieldinganestimate(1),whichclearlydependson(0).WithRtrt((0))denotingthesimulatedreturnsandltlnhtR2t 27

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EngleandLee ( 1996 )determinethattheappropriatemetrictouseforthispurposeisdenedbythematrix=1 wherethefunctionmisasin( 2 ),withnolongerxedat(0).Asaresult,thesimulatedobservationsrt(),whicharegeneratedthroughnormaldistributionsintheDGP( 2 2 ),willappearexplicitlyintermsof=(;;).Weshouldalsonotethatthematrixispositivedenitebywayof1( NeweyandWest ( 1987 ).) Table 2-1 reportstheestimatedparametersfromGARCHtting.Theestimatesontherstrowarebasedonmarketdata.Thoseonthesecondarebasedonsimulatedindexreturnswiththestartingstructuralparameterssetat(0)=(0:0044;6:93e5;0:0653)assetin( 2 ).Thesereturnsareat5-minuteintervals,withthestartingindexlevelandvolatilitysettothosereportedfortheindexlevelandimpliedvolatilityonJanuary1,2002,thatis588:98and0:187377,respectively.Everyday,thereareapproximately775-minutetimeintervals,resultinginassimulationlengthofsimulating1,007days.Inotherwords,wegeneratesamplepathsfor77;539timesteps,with=0:012987013.Thentheminimizationproblemin( 2 )becomesminf124:893245:11+0:08+0:781:05210:1+0:000252+21081:6422+0:0000152+0:001222g 28

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whereptj(t;t;vt)areobtainedviaHeston'sformulaforEuropeanoptionprice( 2 ). LetJbethenumberofoptionsondayt.Theestimate(^t;^vt;^t)fordaytminimizesthesumofsquaresoferrorsPJj=1p2tj;,whichweperformedthroughthenon-linearleast-squaresroutineinMATLAB. Thesecalculationsaredoneforeachday,andforatotalnumberTofdaysinthesampleset,withthenalestimatesarereportedastheiraverages:^=1 2-2 )reportstheaveragesofthesevalues. Table2-1. TheGARCHestimatesofauxiliaryparameters 0:5457 Nonlinearleast-squaresestimatesofoptionpricingparameters Mean Median Std.Dev. Minimum Maximum 0:4562 0:999 ^v 1:5e8 2:2e14 ^ 10

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Figure 2-2 showstheinsampletoftheconstantvolatilityandstochasticvolatilitymodelstothedata.Thegureshowsthatstochasticvolatilitymodelisthebettertthantheconstantvolatilitymodel.Thisisanexpectedresultsincewearetryingtotatightermodelbyaddingmoreparametersandincorporatingmoreinformationfromtheavailabledatainourmodelwhichaccountsforthebettertofthestochasticvolatilitymodel. 2.5.1TheLSMFramework ( 1996 )and LongstaandSchwartz ( 2001 )proposedasimulation-basedtechniquewithregressionttingtopriceAmericanoptionswithconstantvolatility.InourstudyweuseittopriceAmericanoptionsformodelswithbothconstantandstochasticvolatilities. Asisstandard,weassumeacompleteprobabilityspace(;F;P)andanitetimehorizon[0;T],whereisthesetofallpossiblerealizationsofthestochasticprocesses,(St;vt)over[0;T].Agenericelementofwillbedenoted!i.FandParetheassociatedprobabilityltrationandoriginal/physicalmeasure,respectively. OnewaytoevaluatethepriceofanAmericanoptionistorstdetermineitsoptimalexercisedate.Sincethelatterisrandom(dependingonthepathfollowedbytheunderlyingasset),thecorrespondingpayowillberandom.Wearethereforeinterestedinobtainingtheoptimalexercisestrategy,whichmaximizestheexpectedvalueofthispayo,intheinterval[0;T].Assumethattheoptioncanonlybeexercisedatdiscretetimepoints 30

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Insampletofthedata

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wherer(wi;t)istherisklessinterestrateandtheexpectationisconditionalontheinformationavailableuptotimetm.Withthissetup,theproblemreducestoevaluatingtheconditionalexpectedpayoF(wi;tm)ateverytimesteptm,foreverypath!i,andcomparingitwithimmediatepayo.Theoptionwillthenbeexercisedimmediatelyifthelatterishigher. Carrire ( 1996 )and LongstaandSchwartz ( 2001 )proposedleast-squaresapproachtoapproximatetheaboveconditionalexpectationattm,m=1;2;:::;M1.Movingbackwardsintime,theyassumedthatattimetM1,theunknownfunctionalformofF(wi;tM1)inequation( 2 )canberepresentedasalinearcombinationofacountablesetofFtM1-measurablefunctions.ForthepurposeofthisstudywechoosethebasisfunctionsasthesetofweightedLaguerrepolynomials:Ln(X)=expX

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LongstaandSchwartz ( 2001 ),weusetherstthreeLaguerrepolynomials,(L0(X);L1(X);L2(X)).ThenF(wi;tM1)canbeapproximatedas: whereAjarethecoecientsoftheregressionequation.Asin LongstaandSchwartz ( 2001 ),weapproximatethevalueofF(wi;tM1)byregressingthediscountedpayos,C(!i;t:tM1;T)ontothebasisfunctionsforthepathswheretheoptionisinthemoney. 2 and 2 ( Glasserman ( 2003 )).Foraxedtimestepsizeh(weuseh=1=365)andletting1=and2=p 2r2sih2+si 42si(w1w2+)vi+1=h+(1h)vi+p 2()2(vi)h2+ 421(w21h)+1 422(w22h)+1 212w1w2; Forthepurposeofourstudy,weinitializetheassetpricetobetheindexvalueonthecurrentdayandvolatilitytobethespotvolatilityestimatethatweobtainedfromcalibrationabove.Thensuccessivevaluesof(si;vi)aregeneratedwitheachsamplingof 33

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LongstaandSchwartz ( 2001 ),followedbythesampleaverageofdiscountedcashowstocomputetheoptionprice. Figures 2-3 and 2-4 showtheoutofsamplepricingperformanceofconstantvolatilityandstochasticvolatilitymodelsascomparedtothemarketpricewheninitialvolatilityissettospotvolatilityandlongtermaveragerespectivelyforonedayintheoutofsampledataset.Theplotsshowthatstochasticvolatilitymodelperformsbetterthantheconstantvolatilitymodelandthemodeldoesparticularlywellforinthemoneyandatthemoneyoptions.Fordeepinthemoneyoptionsthestochasticvolatilitymodelattainsbetteraccuracythentheconstantvolatilitymodel,howeverintheplotthedierenceisnotvisiblesincethepricingdierenceoccursintheseconddecimalplace. Theplotsalsoshowthatthepricingperformanceisbetterforboththemodelswhentheinitialvolatilityissettolongtermaverageratherthanbeingspotvolatility.Sincehighervarianceraisestheoptionpriceandthespotvolatilityestimatesforoursampledatasetaremostlygreaterthanthelongtermaverage,thustheobservationoflongtermaverageperformingbetter. s1.Inouranalysis,short-termoptionsarethoseforwhich(T<45);mid-termoptionshave(45T<90)andthosewith(T90)arelabelledlong-termOptions.Withrespecttomoneyness,weclassifyoptionswithx>0:05asdeep-in-the-money(DITM),thosewithx2(0:02;0:05)asin-the-money(ITM)options,thosewithx2(0:02;0:02)asat-the-money(ATM)options,thosewithx2(0:05;0:02)asout-of-the-money(OTM)options,andthosewithx>0:05asdeep-out-of-the-money(DOTM)options. 34

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Spotvolatilityasinitialvolatility:out-of-samplepricingperformanceforCVandSVmodelsascomparedtomarketprice

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Longtermaverageasinitialvolatility:out-of-samplepricingperformanceforCVandSVmodelsascomparedtomarketprice

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Theresultsintables 2-3 and 2-4 areobtainedbycomputingtheaverageofpercentagepricingerrorsforeachmoneyness-maturitygroup.Sincewehadthechoiceofconsideringtheinitialvolatilitytobespotvolatilityorthelongtermmean,theresultsintable 2-3 areobtainedbypricingoptionsconsideringinitialvolatilitytobespotvolatilityandintable 2-4 areobtainedwheninitialvolatilityistakenasp 2-5 and 2-6 areobtainedbycomputingtheaverageofabsolutepricingerrorsforeachmoneyness-maturitygroupandforoptionpricesintable 2-5 initialvolatilitywastakentobespotvolatilityandintable 2-4 tobep 37

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Out-of-samplepercentagepricingerrorsforspotvolatilityasinitialvolatility: Maturity Money-nessModelShort-TermMid-TermLong-TermOverall DITMCV0:0080:0540:0640:042(0:002)(0:004)(0:005)(0:004)SV0:00250:0190:0190:013(0:0012)(0:0017)(0:003)(0:002)

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Longtermaverageasinitialvolatilityforout-of-samplepercentagepricingerrors Maturity Money-nessModelShort-TermMid-TermLong-TermOverall DITMCV0:00660:01740:0230:0113(0:0007)(0:002)(0:004)(0:002)SV0:00870:0000690:0030:0019(0:0007)(0:001)(0:002)(0:001)

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Spotvolatilityasinitialvolatilityforout-of-sampleabsolutepricingerrors Maturity Money-nessModelShort-TermMid-TermLong-TermOverall DITMCV0:0240:0580:0660:0493(0:002)(0:004)(0:005)(0:004)SV0:0190:0280:0290:0258(0:001)(0:002)(0:003)(0:002)

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Absolutepricingerrorsforlongtermaverageasinitialvolatility Maturity Money-nessModelShort-TermMid-TermLong-TermOverall DITMCV0:0140:0280:0360:0256(0:0005)(0:002)(0:004)(0:002)SV0:0150:0170:0190:017(0:0005)(0:0007)(0:0014)(0:0008)

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2-3 andTable 2-4 reportthemeanpercentagepricingerrorforconstantvolatility(CV)andstochasticvolatility(SV)models,withstandarddeviationsinparentheses.Theresultsareclassiedintoeachoptiongroupasdenedabove.Weobservedthatin-sampleperformanceofstochasticvolatilitymodelisbetterthantheconstantvolatilitymodel,onemayarguethatsincewehavemorecomplexmodelswithadditionalparameterstocapturemoreinformationthein-sampleperformancewouldbebetterandthisoverttingcanhaveadverseeectonout-of-samplepricingperformanceoftheSVmodel.However,itisevidentfromtheresultsobtainedforout-of-sampleanalysisthatstochasticvolatilityperformsbetterthantheCVmodel. LetusrstconsiderthepercentageandabsolutepricingerrorslistedinTable 2-3 and 2-5 correspondingtopricesobtainedwheninitialvolatilityissettospotvolatility.Consideringbothpricingerrors,weobservethattheSVmodelperformsmuchbetterthanCVmodelforallmoneyness-maturitygroups,exceptthedeepout-of-themoneyoptions,forallmaturities,whereCVmodelhaslesspricingerror.ThepricingperformanceisstrikinglyremarkableforshorttermdeepinthemoneyoptionswhereSVmodelgivesalmost0percentpricingerrorandevenmid-termandlong-termoptionsforthismoneynessgrouphaveverysmallpercentpricingerrors.Thisalsoshowsthatfordeepin-the-moneyoptions,theSVmodelhasnoparticularbiastowardsoverpricingorunderpricingofoptions.Theabsolutepricingerrorsshowthesamestoryforthismoneyness-maturitygroupandhasconsiderablylowererrorsthantheCVmodel.Forexample,theoverallperformanceofSVmodelfordeepinthemoneyoptionshasanabsoluteerrorof0.035whereasforCVmodelitisalmost0.05.Forinthemoney,atthemoneyandout-of-the-moneyoptions,theSVmodelperformsmuchbetterintermsofpercentandabsolutepricingerrorsboth.Thescenarioisslightlydierentfordeepout-of-the-moneyoptions.However,itcanalsobeobservedthatoverallthemoneyness-maturitygroupsdeepout-of-the-moneyhavethehighestpercentandabsolute 42

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LetusnowconsiderthepercentageandabsolutepricingerrorslistedinTable 2-4 and 2-6 correspondingtopricesobtainedwheninitialvolatilityissettolongtermaverage.Ifthetablesarecomparedwiththeprevioustwotables,itisevidentthatlongtermaveragegivesbetterresultsthansettingtheestimateasspotvolatility.Thiscanalsobeattributedtothefactthattheestimatedspotvolatilitiesaregreaterthanthelongtermaverageforalmostallsampledays. AgaininthiscasetheSVmodelperformsbetterthantheCVmodelinallthecasesexceptforout-of-the-moneyanddeepout-of-the-moneyoptions.Thiscanbeattributedtothefactourdatasetforestimationprocedurehadmoreinthemoneyoptions(alsoatthemoneyoptions)thanout-of-the-moneyoptions,whichresultedinpoorttingforout-of-the-money(ordeepout-of-the-money)options. Also,observethatfordeepout-of-the-money,shorttermoptionsthepercentpricingerrorisnegativeshowingabiastowardsunderpricing.However,thebiasismorepronouncedforCVmodelthenSVmodel,inwhichcasetheerrorisalmostzeroagainshowingalmostnobias.Fortherestoftheoptiongroupsthebiasistowardsoverpricingforboththemodels. Thuswecanconcludefromtheseobservationsthatstochasticvolatilityplaysanimportantroleinoptionpricing.However,thepricingbiascannotbecompletelyremoved,necessitatingtheneedforbetterpricingtechniques,representativedatainclusionsifavailableandincorporatingotherfactorssuchasstochasticinterestandrandomjumps. 43

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whereS(t;)andV(t;)arethedeltaandvegawhichareestimatedusingthebackwarddierence,i.e.,ifCt=f(S;K;r;tt;;v)istheoptionpricingformulathen;S(t;)=f(S;K;r;Tt;;v)f(SS;K;r;Tt;;v) S v Andtheresultingcashpositionisgivenby: Intuitively,equation( 2 )canbeinterpretedasfollows.Ifvolatilityisstochasticandcorrelatedwithassetreturns,thenthepositionintheassetisgovernednotonlybytheimpactofassetpricechangesbutalsobythevolatilitychanges,whicharecorrelatedwithassetpricechanges.FortheCVmodelweneedonlyconsiderthersttermontherighthandsideofequation( 2 )sincevolatilityisconstant. 44

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Thenreconstructtheselfnancedportfolio,repeatthehedgingerrorcalculationattimet+2t,andsoon.WerecordthehedgingerrorsH(t+jt),forj=1;:::;Mandthencomputetheaverageabsolutehedgingerrorasafunctionofrebalancingfrequencyt:H(t)=1 ToobtainthehedgingresultsinTable 2-7 ,weusetheindex,impliedvolatility,andinterestratesvaluesavailablefromdayt1.Thenondaytusethesevaluesandthecurrentday'sindexandinterestvaluestoconstructthedesiredhedgeasgiveninequation( 2 )(creatingahedgebothusingCVandSVmodel).Finally,sincewearerebalancingthehedgedaily,ondayt+1,calculatethehedgingerror.ThesestepsarerepeatedforeachoptionandeverytradingdayinthemonthofJanuary2006.Theaverageabsolutehedgingerrorsforeachmoneyness-maturitygroupisreportedinTable 2-7 BasedontheresultsinTable 2-7 ,theSVmodelisthebetterofthetwomodelsforallthegroupsexcepttheshort-termDeep-In-the-Moneygroup.Buteveninthiscase,SVmodelisessentiallydoingbettersinceintheCVmodeltheoptionisimmediatelyexercised;thusnohedgeiscreatedandsothemodelreports0hedgingerror.However,intheSVmodeltheoptioniscontinuedinmostofthecasestillmaturity,resultinginhigherprots. 45

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Absolutehedgeerrors Maturity Money-nessModelShort-TermMid-TermLong-TermOverall DITMCV0:2840:4920:4370:404SV0:3280:2380:1610:242 46

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Theanalysiswasalsodonewithrespecttohedgingperformanceofthetwomodels.TheresultsshowthattheSVmodelperformsbetterhedgingthentheCVmodelforalltheoptiongroups.Eveninthecaseofshorttermdeepin-the-moneyoptionswhereCVmodelreportslesserrorthantheSVmodel,theSVmodelisinherentlydoingbetterbecauseithasmoreprotableexercisestrategythantheCVmodel. ThusourstudyinthischapterhasshownthataccountingforstochasticvolatilityresultsinimprovementupontheconstantvolatilitymodelforpricingAmerican-styleoptions.Itnotonlyprovidesabetterttothedata,whichisstatisticallyobviousbutalsohasbetterout-of-samplepricingandhedgingperformance.Thechallengeinusingstochasticvolatilitymodelistheestimationprocedureofitsparameters.Withproperestimationofparameters,themodelcanbepromisingandhelpfulinpricingandhedgingofAmericanoptionswhichinthemselvesprovideachallengeinpricingsincethereisnoclosed-formsolutionforthem.Thisstudycanbefurtherextendedinanumberofdierentways:(i)byincludingmoreparametersinthemodeltoaccountforfeaturessuchasjumpsandstochasticinterestrates;(ii)byconductingtheempiricalassessmentusingotherunderlyingassetsandoptions;and(iii)byadditionalcomparisonswithalternativestochasticvolatilitymodelssuchasthoseof HullandWhite ( 1987 ), Scott ( 1987 ),and SteinandStein ( 1991 ). 47

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AitSahliaandLai ( 2001 )whichisaimprovementof Ju ( 1998 ),todeterminetheconstantvolatilityexerciseboundary.WethenuseS&P100datatoevaluateempiricallytheresultingpricingapproximationforAmericanoptionsunderstochasticvolatility. Overtheyears,severalanalyticalapproximationsforconstantvolatilitymodelshavebeendevelopedtoimprovecomputationaleciency.Theaccuracyofthesemethodsimprovesasmoreandmoretermsareincludedintheapproximationformula.Butincludingmoretermsdefeatsthepurposeofreducingcomputationaltime.Therstattemptsattheapproximationmethodscanbeattributedto GeskeandJohnson ( 1984 )whousedRichardsonextrapolationwiththreeorfourpointsforapproximatingtheirAmericanoptionformulawhichinvolvesaninniteseriesofmultivariatecumulativenormalfunctions.However,thisrequirescomputationsofmulti-dimensionintegralswhichmakesitdiculttoincludemoretermsintheapproximation.Toavoidthecomputationsofmulti-dimensionintegrals BunchandJohnson ( 1992 )implementedamodiedtwo 48

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Kim ( 1990 ), Jacka ( 1991 )and CarrandMyneni ( 1992 )independentlydemonstratedthattheAmericanputoptionvaluecanbecharacterizedasthevalueofcorrespondingEuropeanputoptionandanearlyexercisepremiumdeterminedbythecriticalstockprice,wheretheearlyexercisepremiumisanintegralinvolvingtheunknownoptimalexerciseboundary. HuangandYu ( 1996 )developedafour-pointRichardsonextrapolationschemetoavoidcomputingtoomanyearlyexercisepoints.Theirschemeonlyneedssixpointsforanapproximationtotheearlyexerciseboundary.Thismethodisveryfastbutnotaccurate,especiallyformoderateandlongmaturityoptions. Ju ( 1998 )suggestedanimprovementtothismethodbyapproximatingboundaryasamulti-pieceexponentialfunctionandthenuseiterationtogetoptionvalue.Hismethodmakesuseoftheobservationthattheearlyexercisepremiumdoesnotcriticallydependontheexactvaluesoftheearlyexerciseboundary. AitSahliaandLai ( 1999 )developedanimprovementofJu'sapproximationboundarybychangingvariablesunderwhichtheexponentialfunctionsaretransformedtoalinearforms.Theyshowthattheboundaryiswellapproximatedbyalinearsplinehavingafewknotsanddemonstratedthroughacanonicalreductionforawiderangeofmaturities,interestrates,dividendratesvolatilitiesandstrikeprices,thattheearlyexerciseboundaryisverywellapproximatedbyapiecewiselinearboundarywhichusesasmallnumberofpieces. WeuseAitSahliaandLai'sapproximationmethodofobtainingtheearlyexerciseboundaryandthensimulateMonte-Carlopathsusingasecond-orderdiscretizationofHeston'smodelandstoppingthesimulation.Theoptionpriceisthenapproximatedastheaverageofthepayosofwhenthegeneratedpathsarestoppeduponhittingthisboundary. Theremainderofthechapterisorganizedasfollows:section2revisitsthemethodof AitSahliaandLai ( 1999 )and AitSahliaandLai ( 2001 )ofapproximatingtheearlyexerciseboundaryforconstantvolatilitymodel.Section3developsthemethodofapplyingthe 49

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Kim ( 1990 ), Jacka ( 1991 )and CarrandMyneni ( 1992 )canberepresentedasfollows: where:d1(x;y;t)=log(x=y)+(r+2=2)t p 3 thatBtsolvesthefollowingintegralequation;KBt=PE(Bt;K;Tt)+

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Ju ( 1998 )proposedapproximatingtheboundaryasamulti-pieceexponentialfunctionwhichsimpliestheestimationoftheboundary. AitSahliaandLai ( 2001 )furtherimprovedtheprocessbyintroducingthefollowingchangeofvariableswhichtransformstheexponentialboundaryBttoz(s)withBt=Ke(z(s)+(1 2)s):s=2(tT);z=log(S=K)(1 2)s; where=r=2;==r.Thissimpliestheintegralequationin 3 to:1ez(s)+(1 2)s=eshN(z(s)=p s)ez(s)1 2sN(z(s)=p sp s)i+ 2s)+z(s)Nz(u)z(s) Equation 3 requirestheevaluationofintegralsinitslasttwoterms,whichcanbedoneusingpiecewiselinearapproximationtoz(:).Becauseofthelinearapproximationthelasttwointegraltermshaveclosedformexpressionsgivenby:esZ0seuNz(u)z 2s)+zNz(u)z 2)s(1es)Zs0etNzz(s+t) 2)sZs0etNzz(s+t)+t

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3 .Tosolvefortheboundarypointsandthelinearsplineswefollowtheprocedureasillustratedin AitSahliaandLai ( 2001 ).Theintegralin 3 isevaluatedrecursivelyforthetimepoints0>s1:s2>:::::::>smwheresm=2T,initializingtherecursionats0=0withz(0)=0if01andz(0)=lnotherwise,withz(j)=z(sj).In AitSahliaandLai ( 1999 ),AitSahliaandLaishowedthattheearlyexerciseboundaryz(s)isboundedabovebyzu(s)andbelowbyzl(s),where;zu(s)=[(1)0:5]sln()+;zl(s)=[(1)0:5]sln(=(1));=[(1)0:5]([(1)0:5]2+2)1 2 3 withrespecttothelinearapproximation; 52

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2)sk=eskhN(z=p sk)ez1 2skN(zp skp sk)i+1eskez+(1 2)sk(1esk)+ek1N(b(z)1=2k1)1 2b(z) 2)sk"~b(z) ~a(z)[N(~a(z)1=2k1)0:5]# 2esk1N(~b(z)1=2k1)+k1Xi=1~Ai(z) Ai(z)and~Ai(z)arerespectivelythevaluesoftwointegralsin 3 obtainedovertheinterval[i;i1]( AitSahliaandLai ( 2001 )). Thisprocedureisfollowedtoobtainboundaryvaluesforalltimepointsandtodeterminetheparametersoflinearsplinesforallthepieces.Inourstudyweanalyzetheeectofusingonepiece,twopieceandthreepiecelinearapproximationsonthepricingaccuracyandalsocomparetheboundaryplotsofthelinearapproximationswiththatoftheactualboundary. AitSahliaandLai ( 2001 )and AitSahliaandLai ( 2004 )demonstratedinthesepapersthatthismethodisecientandaccurateconrmingthatexactvaluesofearlyexerciseboundaryarenotneededtodeterminetheaccurateoptionvalues.Wemakeuseofthisfacttoapproximatetheearlyexerciseboundaryforthestochasticvolatilitymodelviatheecientconstantvolatilityapproachof AitSahliaandLai ( 2001 ).Inpractice,thisboundarywouldnormallybeusedinthepricedecompositionformulafor 53

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3-1 and 3-2 showthesurfaceplotsforboundarieswithone-piece,two-pieceandthree-pieceapproximationsascomparedwiththeactualboundary.Figure 3-1 plotshavebeenobtainedbysettingtheconstantvolatilityatspotvolatilityandgure 3-2 arewithconstantvolatilitysetatlongtermaverage.Theboundarysurfaceplotsforconstantvolatilitycasehavebeenobtainedbyreplicatingtheboundaryoverallthevolatilitygroups. Theevolutionofthegraphsfromone-piecetothree-pieceshowthatasweincreasethenumberofpiecesintheapproximation,theboundarysurfaceconvergestotheactualboundary.The1-pieceand2-pieceboundariesforconstantvolatilitysetatlongtermaverageareslightlyhigherthanwithconstantvolatilitysetatspotvolatilitywhereasfor3-pieceitisvice-versa.Ascomparedtotheactualboundary,theapproximateboundariesarehigherbutasweincreasethenumberofpiecesintheapproximationsitcanbeseenthattheyconvergetotheactualboundary. 54

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[Piece2] [Piece3] [Actual] Figure3-1. Spotvolatilityasconstantvolatilityforevolutionofboundarysurfaceplotsfromone-piecetothree-pieceapproximationsascomparedtotheactualboundary Theplotspiece1,piece2andpiece3areobtainedusingone,twoandthreepiecelinearapproximationswithconstantvolatilitysetatspotvolatilityandreplicatedoverallthevolatilitygroups.TheactualboundaryisobtainedusingLSMmethod 55

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[Piece2] [Piece3] [Actual] Figure3-2. Evolutionofboundarysurfaceplotsfromone-piecetothree-pieceapproximationsascomparedtotheactualboundarywithlongtermaverageasconstantvolatility Theplotspiece1,piece2andpiece3areobtainedusingone,twoandthreepiecelinearapproximationswithconstantvolatilitysetatlongtermaverageandreplicatedoverallthevolatilitygroups.TheactualboundaryisobtainedusingLSMmethod 56

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Heston ( 1993 )denedbythefollowingstochasticdierentialequations: where,undertheoriginalmeasureP,andaretheunderlyingmeanreturnanddividendrates,respectively,oftheasset;p Undertherisk-neutralizedmeasurethemodelin 3 becomes: where=+,==(+). InthepreviouschapterLongstaSchwartzalgorithmwasusedtopriceAmericanoptionsunderconstantandstochasticvolatility.Itwasfoundthatstochasticvolatilitymodelimprovesthepricingandhedgingperformanceovertheconstantvolatilitymodel,however,itisatimeconsumingprocess.ItisknownthattopriceAmericanoptionsrequirestheknowledgeofearlyexerciseboundaryandtheboundaryevaluationconsumesmostofthetimeinthecalculationofoptionprice.Thus,todecreasethetimeofcomputationandcomplexityofthemodelobtaintheboundaryusingconstantvolatilitymodelandthenapplythisboundarytothestochasticvolatilitymodel. SecondorderdiscretizationofHeston'srisk-neutralstochasticvolatilitymodel( 3 ,givenbelow,isusedtogenerate100,000samplepaths(50;000plus,50;000antithetic), 57

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2r2sih2+si 42si(w1w2+)vi+1=h+(1h)vi+p 2()2(vi)h2+( 421(w21h)+1 422(w22h)+1 212w1w2 Thenumberoftimeperiodsforgenerationofpathsdependsuponthematurityoftheoption.Thus,iftheoptionmaturesin50days,thenthepathsaregeneratedfor50timesteps.Also,foreverypath;ateverytimesteptstartingfromtherst,thegeneratedstockpriceStiscomparedwiththecorrespondingboundaryvaluebtobtainedusingtheapproximatemethod,ifStbtthenthefurthergenerationofStforthatparticularpathisstoppedsincewefoundthetimestepatwhichtheoptionwouldbeexercisedandthepayoatthispointwouldbe(KSt)+,forAmericanput.Thisprocedureisfollowedforallthepaths.Thusattheendofthesimulationruns,thepayosforeachpathareobtainedandtheAmericanputoptionpriceisthentheaverageofpayosforallthepathsi.e.thetime-0Americanputoptionpriceis: 58

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2 usingthemethodsofIndirectInferenceandNon-LinearLeastSquaresestimation.Sincethedatasetfortheanalysisinthischapteristhesame,weusethesamesetofestimatesfortheHestonModelinthischapteralso.Forconveniencethetableenlistingtheestimatesofparametersisreproducedbelow: Table3-1. ParameterestimatesofHeston'smodel ParameterEstimate 59

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s1.Dierentoptiongroupsaredenedbelow.Alsodenedarethetoolsthatwouldbeusedformeasuringtheperformanceofeachmodel. 3-2 and 3-3 areobtainedbycomputingtheaverageofpercentagepricingerrorsforeachmoneyness-maturitygroupwhenconstantandinitialvolatilityaresettospotvolatilityandlongtermaverage,respectively.The 60

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3-4 and 3-5 areobtainedbycomputingtheaverageofabsolutepricingerrorsforeachmoneyness-maturitygroupwhenconstantandinitialvolatilityaresettospotvolatilityandlongtermaverage,respectively.Pure-SVmodelrepresentstheresultsfromStochasticVolatilitymodelobtainedinthepreviouschapter.1-CV,2-CVand3-CVaretheresultsobtainedfrommixingthetwomodels,whereearlyexerciseboundaryisestimatedbyone-piece,two-pieceandthree-piecelinearfunctionsandthenusedinthesimulationoftheSVmodel. Letusrstconsiderthepercentageandabsolutepricingerrorslistedintables 3-2 and 3-4 whichcorrespondtopricesobtainedwheninitialvolatilityissettospotvolatility.Consideringbothpricingerrors,weobservethatourapproximationapproachperformsbetterascomparedtopureSVmodelforallmoneyness-maturitygroups,exceptthedeep-in-the-moneyoptions,whereSVmodelhaslesspricingerror.However,theerrorsfromapproximationarewithin2%dierence.Foralltheotheroptiongroups,theapproximationmethodisperformingbetterandwithinthese,one-pieceapproximationisoutperformingtheothertwoapproximationsby.TherelativepricingerrorplotalsoshowthatfordeepinthemoneyoptionsallthemodelshavenoparticularbiastowardsoverpricingorunderpricingofoptionsandtheabsolutepricingerrorsforthismoneynessgrouphasconsiderablylowererrorsthantheCVmodel.Forexample,theoverallperformanceofSVmodelfordeepinthemoneyoptionshasanabsoluteerrorof0.04andboth2-pieceand3-pieceapproximationhaveanerrorof0.046.Forin-the-money,at-the-money,out-of-the-moneyanddeep-out-of-the-moneyoptions,the1-pieceapproximationperformsfarbetterthananyothermodelandthe2-pieceand3-pieceapproximationsarestillbetterthanthepureSVmodelintermsofpercentandabsolutepricingerrorsboth.Althoughfortheseoptiongroupsallthemodelsshowabiastowardsoverpricing. LetusnowconsiderthepercentageandabsolutepricingerrorslistedinTable 3-3 and 3-5 correspondingtopricesobtainedwheninitialvolatilityissettolongtermaverage. 61

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Thuswecanconcludefromtheseobservationsthatapproximatingtheboundaryusingconstantvolatilityandapplyingittostochasticvolatilitymodelisanecientmethod.ThepricingaccuracyfortheseapproximationsiseitherhigherorequivalenttopureSVmodel. 62

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Relativepricingerrorsforinitialandconstantvolatilitysetatspotvolatility Maturity Money-nessModelShort-TermMid-TermLong-TermOverall DITMPure-SV0:00250:01860:01920:01351-CV0:00160:01160:00690:00562-CV0:00170:01120:00590:00513-CV0:00090:01250:00770:0064

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Longtermaverageasinitialandconstantvolatilityforrelativepricingerrors Maturity Money-nessModelShort-TermMid-TermLong-TermOverall DITMPure-SV0:00870:0000690:00290:00191-CV0:0150:00610:00960:01012-CV0:0140:00410:00530:00773-CV0:0130:00290:00420:0068

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Spotvolatilityasinitialandconstantvolatilityapproximationsforabsoluterelativepricingerrors Maturity Money-nessModelShort-TermMid-TermLong-TermOverall DITMPure-SV0:01890:02840:02990:02581-CV0:01970:02940:03410:02772-CV0:01890:02740:03100:02583-CV0:01900:02800:03120:0261

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Absolutepricingerrorsforconstantvolatilityapproximationsetatlong-termaverage Maturity Money-nessModelShort-TermMid-TermLong-TermOverall DITMPure-SV0:01510:01660:01870:01681-CV0:01650:02090:02770:02172-CV0:01560:01800:02170:01843-CV0:01540:01840:02230:0187

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InthisworkweperformedanempiricalstudytosupporttheimportanceofhavingvolatilityasastochasticprocessintheoptionpricingsetupoverthestandardBlack-Scholesmodel.ThisistherstattempttotestthepricingandhedgingperformanceofstochasticvolatilityagainstconstantvolatilitymodelforAmericanstyleoptions.ForourstudyweusedHeston'smean-revertingstochasticvolatilitymodel,whichwecalibratedwithreal-worlddataonS&P100Indexoptions,andthenusedthecalibratedmodeltoperformout-of-samplepricingandhedginganalysisofAmericanoptionsforthetwomodels.Ourpricingerroranalysisshowsthatthestochasticvolatilitymodelperformssignicantlybetterthantheconstantvolatilitymodelforallthemoney-nessmaturitygroupsofoptionsexceptdeep-out-of-the-money.Ithaslesspricingerrorandgivesverygoodresultsfordeep-in-the-moneyoptionsandin-the-moneyoptions.ThusHeston'sstochasticvolatilitymodelcapturesmarketinformationbetterthanitsconstantcounterpart.inadditiontoimprovementinpricingaccuracy,themodelalsohasabetterhedgingperformanceanddoesparticularlywellfordeepinthemoneyandinthemoneyoptions.Theanalysiswasalsodonewithrespecttodierentestimatesofvolatility(initialvolatilityincaseofstochasticvolatilityandvolatilityforconstantvolatilitymodel),inparticular,estimatesofspotvolatilityandlongtermmeanp Resultsfromthestochasticvolatilitymodelalsoinspiredthestudyofmixingthetwomodels,constantandstochastic.SinceforpricingAmericanoptions,itisknownthatanexactboundaryisnotneeded,westudiedtheeectofobtainingtheapproximateboundaryusingconstantvolatilitymodel,whichisthenusedforthestochasticvolatilitysetup.Theanalysiswasperformedforthesamedatasettocomparetheresultsagainstthepurestochasticvolatilitymodel.Theresultsareveryencouraging.Theapproximatemethodofobtainingboundaryisveryfastandalsoperformsbetterthanthepure 67

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ThisworkisasmallsteptowardsestablishingtheimportanceofhavingvolatilityasastochasticcomponentinthepricingofAmericanoptions.Asthisincreasesthemodelcomplexity,whichinturnincreasesthecomputationtime,amethodofmixingstochasticvolatilityandconstantvolatilitywasempiricallyevaluatedasbeingsuperior.Itsignicantlyreducesthecomputationtimewhileimprovingaccuracy. 68

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ManishaGoswamiwasborninNewDelhi,India.Sheearnedherbachelor'sdegreeinStatisticsandmaster'sdegreeinoperationalresearchfromUniversityofDelhi,India.Sheworkedforfouryearwiththestandards-makingbodyofIndia,BureauofIndianStandards.In2004shebeganherdoctoralstudiesintheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFlorida.ShenishedherPh.D.withaconcentrationinquantitativenanceinDecember2008. 74