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American Option Pricing under Stochastic Volatility

Permanent Link: http://ufdc.ufl.edu/UFE0022641/00001

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Title: American Option Pricing under Stochastic Volatility Efficient Numerical Approaches
Physical Description: 1 online resource (74 p.)
Language: english
Creator: Guha, Suchandan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: american, engineering, financial, lsm, option, pricing, 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

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Abstract: We developed two new numerical techniques to price American options when the underlying follows a bivariate process. The first technique exploits the semi-martingale representation of an American option price together with a coarse approximation of its early exercise surface that is based on an efficient implementation of the least-squares Monte Carlo method. The second technique exploits recent results in the efficient pricing of American options under constant volatility. Extensive numerical evaluations show these methods yield very accurate prices in a computationally efficient manner with the latter significantly faster than the former. However, the flexibility of the first method allows for its extension to a much larger class of optimal stopping problems than addressed in this paper.
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 Suchandan Guha.
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 2009-02-28

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022641:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022641/00001

Material Information

Title: American Option Pricing under Stochastic Volatility Efficient Numerical Approaches
Physical Description: 1 online resource (74 p.)
Language: english
Creator: Guha, Suchandan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: american, engineering, financial, lsm, option, pricing, 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: We developed two new numerical techniques to price American options when the underlying follows a bivariate process. The first technique exploits the semi-martingale representation of an American option price together with a coarse approximation of its early exercise surface that is based on an efficient implementation of the least-squares Monte Carlo method. The second technique exploits recent results in the efficient pricing of American options under constant volatility. Extensive numerical evaluations show these methods yield very accurate prices in a computationally efficient manner with the latter significantly faster than the former. However, the flexibility of the first method allows for its extension to a much larger class of optimal stopping problems than addressed in this paper.
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 Suchandan Guha.
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 2009-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022641:00001


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MysincerethanksrstofalltomyadvisorDr.FaridAitSahlia,whoguidedanddirectedmethroughmyresearch.Hehasbeenaninspirationforme.Iwouldalsoliketothankmycommitteemembers(Dr.StanislavUryasev,Dr.HaniDossandDr.LiqingYan)fortheircooperationandvaluableinsights.IalsoextendmygratitudetoManishaGoswami,afellowstudent,whoIhavebeenworkingwithforthelastfewyears.Ithasbeenagreatpleasureworkingwithher.MydeepestthanksgotoMatajiShriNirmalaDeviforherblessingsthroughoutmylife.Thiseffortwouldnothavebeenpossiblewithouttheconstantsupportofmyfamily,especiallymygrandparents,myparents,mysisterandmybrother-in-law.IamdeeplygratefultomywifeNandiniwhostoodbymeandhasbeenaconstantmoralandemotionalsupportforthelastfewyears.Thisacknowledgementwouldremainincompletewithoutthementionofallmyfriendsinvariouspartsoftheworld. 4

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page ACKNOWLEDGMENTS .................................... 4 LISTOFTABLES ....................................... 7 LISTOFFIGURES ....................................... 9 ABSTRACT ........................................... 10 CHAPTER 1INTRODUCTION .................................... 11 1.1Introduction ..................................... 11 1.2AmericanOptionPricingFormulations ....................... 12 1.2.1TheFree-BoundaryApproach ........................ 12 1.2.2VariationalInequalities ........................... 13 1.2.3IntegralRepresentationApproach ...................... 14 1.3NumericalMethodsforPricingAmericanOptions ................. 15 1.3.1FiniteDifferenceMethods ......................... 16 1.3.2LatticeMethods ............................... 16 1.3.3MonteCarloMethods ............................ 17 1.3.4AnalyticalApproximations ......................... 18 1.4ResearchMotivation ................................ 18 2TESTINGTHEPROPOSEDMETHODONACONSTANTVOLATILITYMODEL 20 2.1Introduction ..................................... 20 2.2Method ....................................... 20 2.2.1StockPriceSimulation ........................... 20 2.2.2BoundarywithConstantVolatility ..................... 22 2.2.3UsingtheBoundaryintheDecompositionFormula ............ 23 2.3NumericalImplementationandResults ....................... 23 2.4Conclusion ..................................... 25 3USINGTHEPROPOSEDMETHODONASTOCHASTICVOLATILITYMODEL 26 3.1Introduction ..................................... 26 3.2HestonPricingModel ................................ 26 3.3BoundaryEvaluation ................................ 30 3.4NumericalImplementation ............................. 33 3.5Conclusion ..................................... 36 5

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............................. 42 4.1Introduction ..................................... 42 4.2Method ....................................... 42 4.2.1BoundarywithConstantVolatility ..................... 42 4.2.2UsingtheDecompositionFormulatoObtaintheOptionPrice ....... 45 4.3NumericalImplementation ............................. 46 4.3.1Scenario1 .................................. 47 4.3.2Scenario2 .................................. 49 4.3.3Scenario3 .................................. 51 4.4Conclusion ..................................... 52 5CONCLUSIONANDFUTUREWORK ......................... 54 5.1Conclusion ..................................... 54 5.2FutureWork ..................................... 55 APPENDIX ASTOCHASTICVOLATILITYAPPROXIMATIONSFORASVMODEL ....... 57 BCONSTANTVOLATILITYAPPROXIMATIONSFORASVMODEL ........ 61 REFERENCES ......................................... 71 BIOGRAPHICALSKETCH .................................. 74 6

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Table page 2-1=0:375,=0:3,Strike=100:AmericanPutOption ................. 24 2-2=1:220,=0:4,Strike=100:AmericanPutOption ................. 24 2-3=0:305:SummaryAnalysis .............................. 25 2-4=1:220:SummaryAnalysis .............................. 25 3-1=0:0225:SummaryAnalysis .............................. 41 3-2=0:09:SummaryAnalysis ............................... 41 3-3=0:2:SummaryAnalysis ............................... 41 4-1=p ..................... 47 4-2=;=0:0225:SummaryAnalysis .......................... 48 4-3=+p ..................... 48 4-4=0:0225:ComputationalTime ............................. 49 4-5=0:0225:StochasticVolatilityResults ......................... 49 4-6=p ...................... 49 4-7=;=0:09:SummaryAnalysis ........................... 50 4-8=+p ...................... 50 4-9=0:09:Computationaltime .............................. 51 4-10=0:09:StochasticVolatilityResults .......................... 51 4-11=p ....................... 51 4-12=;=0:2:SummaryAnalysis ............................ 52 4-13=+p ....................... 52 4-14=0:2:Computationaltime ............................... 53 4-15=0:2:StochasticVolatilityResults ........................... 53 A-1=0:0225:Americancalloptionprices ......................... 58 A-2=0:09:Americancalloptionprices .......................... 59 A-3=0:2:Americancalloptionprices ........................... 60 7

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.............. 62 B-2=p ................... 63 B-3=p .............. 64 B-4=p ................ 65 B-5=p ..................... 66 B-6=p ................ 67 B-7=p ................ 68 B-8=p ..................... 69 B-9=p ................ 70 8

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Figure page 3-11000and10,000SamplePaths:VariationofAmericanCallOptionBoundary ..... 37 3-2100,000and1,000,000SamplePaths:VariationofAmericanCallOptionBoundary .. 38 3-325and20TimeSteps:VariationofAmericanCallOptionBoundary .......... 39 3-410and5TimeSteps:VariationofAmericanCallOptionBoundary ........... 40 3-5AmericanCallOptionBoundarywithT=0.25,=0:04,=0:1,r=0:03,q=0:05 9

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WedevelopedtwonewnumericaltechniquestopriceAmericanoptionswhentheunderlyingfollowsabivariateprocess.Thersttechniqueexploitsthesemi-martingalerepresentationofanAmericanoptionpricetogetherwithacoarseapproximationofitsearlyexercisesurfacethatisbasedonanefcientimplementationoftheleast-squaresMonteCarlomethod.ThesecondtechniqueexploitsrecentresultsintheefcientpricingofAmericanoptionsunderconstantvolatility.Extensivenumericalevaluationsshowthesemethodsyieldveryaccuratepricesinacomputationallyefcientmannerwiththelattersignicantlyfasterthantheformer.However,theexibilityoftherstmethodallowsforitsextensiontoamuchlargerclassofoptimalstoppingproblemsthanaddressedinthispaper. 10

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BabaandGallardo ( 2008 )fromtheBankofInternationalSettlementsreportthatthenotionalamountsofallcategoriesofover-the-countercontractsreached$596trillionattheendofDecember2008,followinga24%increaseinthersthalfoftheyearwhereasthenotionalamountsofoutstandingcreditdefaultswaps(CDSs)was$58trillion. Amongdifferenttypesofderivatives,optionsarecontractswhichgiveaholdertherighttobuy(foracalloption)orsell(foraputoption)anassetatsomefuturedatebyaspeciedexpirationdate(maturity).OptionsthatgrantthisexercisefortheexpirationdateonlyaretermedEuropean.ThosethatallowtheusertoexercisethisrightanytimeuptothematuritydateareknownasAmericanoptions.PricingAmericanoptionsischallengingbecauseofthisexibilityofexerciseavailabletotheuser.Variouspricingtechniqueshavebeenproposedbyresearchersovertheyears.OurworkinvolvespricingAmericanoptionsunderstochasticvolatility.Insolvingthisproblemwearefacedwithtwomajorissues.Therstoneisduetoexibilityofexercise,thesecondissueisbecauseofstochasticvolatility.ThecurrentliteratureisrichwithmethodsforpricingAmericanoptions,evenwithstochasticvolatility.Howevertheproblemwiththeseexistingapproachesisthatmostofthemareverydemandingintermsofcomputationaltime.ThusoneofourgoalsofresearchwastodesignefcientalgorithmswhichcanpriceAmerican 11

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Samuelson ( 1965 )inamodelthatwasaprecursortothewidelyadopted BlackandScholes ( 1973 )framework.Inthisclassicalcontext,thepriceStoftheunderlyingasset(labeledstock)isassumedtofollowageometricBrownianmotion,thatissatisfying whereristheprevailingrisklessrateinthemarket,thedividendrateoftheassetanditsvolatility.ThepriceVt=V(St;t)ofanoptionwithexerciseprice(strike)Konthisassetisthenthevaluefunctionofthefollowingoptimalstoppingproblem: whereT[t;T]isthesetofoptimalstoppingtimesin[t;T]andg(S)isthepayoffuponexercise,withg(S)=(SK)+andg(S)=(KS)+foracallandaput,respectively.(Foramoredetailedreview,referto BroadieandDetemple ( 2004 ).) Thereareanumberofdifferentapproachestosolvethisoptimalstoppingproblem.Amongthemostcommonandsuccessfularethefree-boundary,variationalinequalities,andintegralrepresentationapproaches,whicharereviewedbelow. KaratzasandShreve ( 1988 )),theequationcanbederivedas @t+@V @SSt(r)+1 2@2V @S2S2t2dt+@V @SStdWt(1) 12

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dS(S)for(S;t)=(B(t);t)(1) Therstconditionstatesthatthevalueoftheoptionatmaturity(T)isequaltothepayoffobtainedatthattime.Thesecondoneensuresthatthevalueisequaltothepayoffwhenthestockpriceequals0(toensureno-arbitrage).Thethirdconditionexpressesthevalueasthepayoffuponexercise,i.e.uponhittingthestopping(a.k.a.exercise)boundaryfB(t)g.ThefourthconditionensurestheoptimalityoftheexerciseboundaryfB(t)gatanytimet. Tracesofthismethodcanbefoundin Samuelson ( 1965 ), McKean ( 1965 ), Taylor ( 1967 )and Merton ( 1973 ).fB(t)gistheboundarythatseparatesthecontinuationandexerciseregionsCandE,respectively.ThecontinuationregionCisdenedasthesetofstockpricesattimetforwhichitisoptimalnottoexercisetheoption.ThusC=f(S;t)2<+[0;T]:V(S;t)>g(S)gandE=f(S;t)2<+[0;T]:V(S;t)=g(S)g BrennanandSchwartz ( 1977 )andfurtherdevelopedby Jailletetal. ( 1990 ).Afterundergoingthefollowingchangeinvariablesonequation 1 ,x=ln(S=K);=1 22(Tt);u(x;)=ex+V(x;)

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22)=2;=2+2r=2,thepricingproblemisrsttransformedintoalinearcomplementarityform.Afterafewstepsofintegrationandrelaxingsomeconditions,thevariationalinequalityformisobtained,whereintheproblemistondu2Vsuchthatthefollowingequation @((x;)u(x;))dx+Z1@u @x((x;)u(x;))dx(1) holdsforallt2[0;T],foralltestfunctions2V,whereVisthesetoftestfunctions(x;)thatarecontinuous,continuouslydifferentiablein,differentiablealmosteverywhereinxandsatisfythefollowingconditions 22Tlimx!1exu(x;)=g(1)for20;1 22T(x;t)^g(x;)(1) where^g(x;)=ex+u(x;)g(Kex) AitSahliaandCarr ( 1997 )forareview). Analternativeapproach,basedonfundamentalprobabilisticconsiderationsisthatofintegralrepresentation(alsocalledthepricedecompositionformula)wherethepriceofanAmericanoptionisexpressedasthesumofthecorrespondingEuropeanoptionpriceandanintegraltermrepresentingthevalueofearlyexercise.Forexample,thepriceC(St;t;K)ofanAmericancall 14

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whered(X;K;t)=1 22tc(St;t;K)isthepriceofthecorrespondingEuropeanoptionandN(x)isthecumulativestandardnormaldistributionfunction.Thisdecompositionisattributedtovariousauthorssuchas Kim ( 1990 ), Jacka ( 1991 )and Carr,Jarrow,andMyeni ( 1992 ),whoderiveitthroughdifferenttechniques(seealso KaratzasandShreve ( 1998 )). AstheboundaryB()isunknowninthisequation,weusetheknowledgethattheoptionisexercisedwhenSt=BtandthusC(B(t);t;K)=B(t)Ktogetanintegralequationforthisboundary: 1. FiniteDifferenceMethods 2. LatticeMethods 3. MonteCarloMethods 4. AnalyticalApproximations 15

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13 )alongwiththeboundaryconditionsgiveninequation 1 numerically.Theadvantagesassociatedwiththenitedifferencemethodistheavailabilityofawiderangeofnumericaltechniquesintheexistingliteratureandtheexibilityofthemethodregardingtosolvingforthepriceformorecomplexprocesses.Themethod,rstproposedby BrennanandSchwartz ( 1977 )isoftenusedtocalculatebenchmarkvaluesfordifferentmodels. Randalletal. ( 1997 )havedevelopedahigherlevellanguageforautomaticcodegenerationfornitedifferencemethodsmakingitmoreeasiertousethistool. Parkinson ( 1977 )and Cox,Ross,andRubinstein ( 1979 ),thebasicideabehindlatticemethodsistodiscretizethestatespaceintoagridandcalculatevaluesateachgridpointthroughdynamicprogrammingtechniques.Thebinomialoptionpricingmodel,proposedby Coxetal. ( 1979 ),usesadiscretetimeframeworktocalculatevaluesofthestatevariablethroughabinomialtreeforagivennumberoftimesteps.Eachnodeinthetreerepresentsapossiblepriceatthatpointoftime.ThusifthestockpriceattimetisSttherearetwopossiblemovementsforthenexttimestep:upordownwiththepossiblepricesgivenasSt+1(up)=St:uandSt+1(down)=St:d 16

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wheretheprobabilityofupmovementpisgivenasp=e(r)td ud Foratrinomiallatticethereare3movementspossibleforthenexttimestepinsteadof2inbinomialcase.Thetrinomiallatticegenerallyproducesresultswithgreateraccuracy,althoughatthecostofgreatercomputationaltime.(c.f. JarrowandRudd ( 1983 ), Amin ( 1991 )and Boyle ( 1986 )) Bothlatticeandnite-differencetechniquesareknowntobecomputationallyextensivewithlatticemethodstakingO(mn)timeforcomputation,mwhereisthenumberoftimestepswhilenisthenumberofassetpricelevels.Forvariousmodelswehavem=O(n)thusmakingthetotalcomputationtimeO(n2). 1 ,theAmericanoptionpriceisactuallytheexpectedvalueofthediscountedpayoffoccuringattheoptimalstoppingtime. Boyleetal. ( 1997 )and Glasserman ( 2004 )surveythevariousMonteCarlomethodsusedinthepricingofoptionsandothernancialderivatives.VariousadvantagesofusingMonteCarlotechniquesinthisregardareasfollows: 17

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LongstaffandSchwartz ( 2001 ).Intheirapproach,thevaluetocontinue,whichisthemostdifculttoobtain,isapproxi-matedbyleast-squaresregression.Infactthisconcepthasalsoappearedinasomewhatdifferentmannerin Carriere ( 1996 ).Furthervariationsonthissamethemewerealsoderivedin TsitsiklisandVanRoy ( 1999 2001 )whereasthesomewhatrelatedstochasticmeshconceptisdevelopedin BroadieandGlasserman ( 2004 )toaddresstheissueofexponentialordercomputationtime. GeskeandJohnson ( 1984 )evaluatedAmericanoptionsascompoundEuropeanoptionsandusedtheRichardsonextrapolationforthepurposeofnumericalapproximation. AitSahliaandLai ( 2001 ), Ju ( 1998 )and Ingersoll ( 1998 )reviewvariousapproximationmethodsfortheearlyexercisepremium.Asknowledgeoftheboundaryisnecessaryforthecalculationofearlyexercisepremium, Huang,Subrahmanyam,andYu ( 1996 )proposedtoapproximateitasapiecewiseconstantfunctionandthenusedathree-pointRichardsonextrapolationschemetoobtainthepriceoftheoption. Ju ( 1998 )didthesameanalysisusingpiecewiseexponentialfunctionapproximationfortheboundaryyieldingbetteraccuracythan Huang,Subrahmanyam,andYu ( 1996 )and GeskeandJohnson ( 1984 )andbettercomputationalefciency. AitSahliaandLai ( 2001 )improveupon Ju ( 1998 )bysolvingaone-dimensionalintegralequationtodeterminetheexerciseboundary,insteadofJu'stwodimensionalequation,whichcanbepronetonumericalstabilityissues. AitSahliaandLai ( 1999 ),asalludedtoin Huangetal. ( 1996 )and Ju ( 1998 )alsoestablishthatanaccurateevaluationoftheboundaryisnotessentialforcalculatingthepriceoftheoption. 18

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ChiarellaandZiogas ( 2005 ))withacoarseimplementationoftheleast-squaresMonteCarlodescribedabove.Thelattertechniqueisusedtodeterminequicklyanapproximationoftheoptimalstoppingsurfacetobeusedinthepricedecompositionformula.Forthispurpose,asignicantlysmallnumberofsamplepathsaregenerated:afewthousands,comparedtothetensorhundredsofthousandsrequiredforaccurateMonteCarlopricing.Additionally,onlyaverylimited(aroundve)exercisedatesareallowed,whichisanorderofmagnitudelessthantheoriginalformulation.Insodoingacoarseapproximationoftheexercisesurfaceisgeneratedandthenusedintheintegralrepresentationformula.Thisapproachisinthespiritof AitSahliaandLai ( 1999 )whodemonstratedthatoneneednotknowtheexerciseboundaryaccuratelytopriceAmericanoptionswhenvolatilityisconstant.Thisnewapproachisrsttestedinchapter 2 onaspecialcaseofstochasticvolatility,namelythatofconstantvolatility.Chapter 3 thencontainsafull-edgedtestonagenuinestochasticvolatilitymodel. Thesecondapproachistoapproximatethestoppingsurfaceviatheintegralequationforacorrespondingconstantvolatilityforwhichtheefcientmethodof AitSahliaandLai ( 2001 )isapplied.Chapter 4 containsextensivenumericalresultsindicatingthatitisveryaccurate.Itisalsosignicantlyfasterthantherst.However,thelattercanbeextendedmorereadilytohigherdimensionsthankstoits(partial)relianceonMonteCarlosimulation. 19

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Carriere ( 1996 )and LongstaffandSchwartz ( 2001 ).Theirbasicbreakthroughistoapproximatethecontinuationvalueintheassociatedoptimalstoppingproblembyanefcientregressioncalculation.Thecomparisonofthisvaluewiththestoppingpayoffateverystateandtimeisattheheartofthebackwarddynamicprogrammingalgorithmusedtosolvethisproblem.Inthischapter,theapproachisfollowedtotheextentthatitusessignicantlylesssamplepathsandoveragreatlyreducednumberofexercisedates.Thisisbecauseofourgoalofcoarselyapproximatingtheearlyexerciseboundaryonly,whileforegoinganyinterestincomputingtheoptionthroughthismethodasitwillthroughthedecompositionformula,whichrequiresknowledgeofsaidboundary. Thischapterisorganizedasfollows.Section 2.2.1 simulatesthestockpricewhilesection 2.2.2 outlinesthestepsfortheevaluationoftheearlyexerciseboundary.Finallysection 2.2.3 calculatesthepriceoftheoptionusingthedecompositionformulausingtheboundaryobtainedinsection 2.2.2 .Theresultsareshownandanalyzedinsection 2.3 2.2.1StockPriceSimulation BlackandScholes ( 1973 )model,thebehaviorofthestockpriceStattimetisgivenas 20

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ForthischapterwefocusonAmericanputsunderthegivenconstantvolatilitymodel(calloptionscanbestudiedsimilarly).ThepriceofanAmericanoptionattimetisgivenastheoptimumvalueinthefollowingoptimumstoppingproblem whereTisthetimetheoptionmatures.Thepayofffunctionf(S)withastrikepriceofKisgivenas ForpricingtheAmericanputoptionweusethedecompositionformulawhereintheAmericanoptionpriceisexpressedastheEuropeanoptionpriceplusanearlyexercisepremium. Kim ( 1990 ), Jacka ( 1991 ), Carretal. ( 1992 )havederivedtheAmericanputpriceU(S;t;K)as: 2 isgivenas whered1(x;y;)=lnx y+(rq+1 22) p

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2 ,weneedtoknowthevalueoffBtg,theearlyexerciseboundary.Asstatedinchapter 1 ,wewanttocalculatethisboundaryefcientlyusingsimulation.Thusinordertoobtainthepriceoftheoptionweneedtodothefollowinginthegivenorder: 1. Evaluatetheearlyexerciseboundaryusingarangeofnumberoftimestepsandsamplepaths(doneinsection 2.2.2 ). 2. Usetheobtainedboundaryinthedecompositionformulatoobtainthepriceoftheoption(doneinsection 2.2.3 ). 2 )canbediscretizedas 22t+W1(t)(2) Thetimedomain[0;T]isdividedintonequidistanttimestepssuchthatti=iT n8i=0;1;2;:::;n 2 of 2 Next,theearlyexerciseboundaryisobtainedusingtheLSMalgorithm.Thisisdonestartingfromthepointofmaturityandgoingbackwardsusingthefollowingsteps: 1. Attimet=T;B(t)issetas(c.f. Kim ( 1990 ))B(T)=(Kifrq(r=q)Kifr>q Ateachtimestep,thepayoffforeachsamplepathiscalculated.Lookingatthepayoffsatthesubsequentpaths,thepayofffromcontinuationiscalculatedforeachin-the-moneypath(c.f. LongstaffandSchwartz ( 2001 )).Thisarrayofpayoffsfromcontinuationisregressedagainstthecorrespondingstockpricestoobtainthefunctionfortheexpectedfuturepayoff. 3. Usingthevaluesofthestockpricesateachpath,theexpectedfuturepayoffiscalculatedforeachpathwhichiscomparedtothecurrentpayofftodeterminewhetherexercisehappensornot( LongstaffandSchwartz 2001 ). 22

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Theboundaryvalueattimetisobtainedasthemaximum(forPut)orminimum(forCall)valueamongthestockpricesfordifferentsamplepaths.b(t)=(maxfS(t)jS(t)2EtgforPutminfS(t)jS(t)2EtgforCall whereEtisthesetofstockpricesattimetresultinginexercise. Afterthiscalculation,weobtaina2dimensionalboundaryrepresentedasfB(t0);B(t1);:::;B(tn)g. 2 ).AlthoughUEiseasilycalculated,weneedtoreverttonumericalapproximationtechniquesinordertocalculatetheintegralinequation 2 .Theintegralinthiscaseisaboundedoneandisthuscalculatedusingasimpletrapezoidalrule.WealsotriedtheSimpson'srule,howeverwithoutmuchextrabenetsandthereforedecidedtoremainwiththesimplertrapezoidalrule. AitSahliaandLai ( 1999 )forAmericanputoptionprices.Thenormalizedpricesweremultipliedbythestrikevaluetogettherealprice.Theparametervaluesfortheseprices,reportedintheircanonicalformsareconvertedtothenormalformtoretrieve,randT.Fixingataparticularvalue,wegetthevaluesofrandTfromthereportedvaluesofandsasfollows:r=2T=s 2 2.2 ,pricesarecalculatedforAmericanputoptionsusingtheobtainedvaluesfortheparameters.Tables 2-1 and 2-2 showthepricesoftheAmericanputoptionsfor=0:305and1:220respectively,calculatedusingdifferentmethods.ThecolumnsBMandSAserveasourbenchmark.BM(Bernoullimethod)givesthepriceobtainedby AitSahliaandLai ( 1999 )usingaconvergentdynamicprogrammingalgorithm,whereasSA(splineapproximations)givesthepricecalculatedusingtheirinterpolationsplines.Thecolumnstotheirrightshowthepricesobtainedusingtheproposedalgorithm.AcolumnshowingMKN 23

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2 Table2-1. AmericanPutOptionpricesforStrike=100,=0:375,=0:3 0.1488020.03019.94419.94419.94020.03920.04520.02720.02920.0359010.78510.80010.83210.78610.84910.84910.85910.82110.8211004.3854.3934.4104.3924.3954.3924.4044.3874.3891200.2780.2790.2810.2800.2780.2790.2800.2780.278 0.2978020.15220.19220.27820.20320.19920.19220.31020.19920.2379011.87411.90712.09211.92111.89511.90711.98111.88811.9251006.0756.0886.2326.1076.0876.0926.1186.0816.0941201.0551.0571.0821.0651.0591.0581.0631.0581.057 0.4448020.47320.49520.46320.59420.69420.61920.69420.61220.5909012.79912.82112.86012.87012.89712.88412.92712.84512.8581007.3107.3277.3917.3627.3607.3637.3797.3377.3421201.8731.8801.9141.8981.8881.8891.8941.8841.882 Columns1and2givethevalueofmaturityandspotprice.ColumnBMgivesthebenchmarkpriceobtainedusingtheBernoullimethodwhilecolumnSAgivestheoneobtainedusingsplineapproximationmethodin AitSahliaandLai ( 2001 ).Thecolumnstotherightofthebenchmarkgivethepricesobtainedbyusingtheproposedmethod.MKNdenotespriceobtainedusingM1000samplepathsandNtimesteps AmericanPutOptionpricesforStrike=100,=1:220,=0:4 0.0218020.00220.00019.95119.96019.98019.98019.96019.98019.9929010.00010.0179.8459.9729.9379.9589.9529.9419.9271002.1382.1512.1222.1232.1142.1122.1212.1122.1061200.0010.0010.0010.0010.0010.0010.0010.0010.001 0.0838020.00220.00219.85019.89719.94319.96119.89519.94719.9699010.35410.37710.43310.52310.40810.48710.50210.42810.3561003.9423.9924.0114.0083.9713.9744.0073.9643.9371200.1900.2260.2280.2270.2230.2230.2270.2230.221 0.1468020.00220.07519.90420.00920.00720.05219.98020.01720.0069010.84210.92411.05611.20511.00711.07211.15011.01210.8931004.9795.0285.0805.1325.0495.0435.1165.0384.9791200.6510.6600.6700.6730.6600.6590.6710.6590.650 Columns1and2givethevalueofmaturityandspotprice.ColumnBMgivesthebenchmarkpriceobtainedusingtheBernoullimethodwhilecolumnSAgivestheoneobtainedusingsplineapproximationmethodin AitSahliaandLai ( 2001 ).Thecolumnsontherightofthebenchmarkgivethepricesobtainedbyusingtheproposedmethod.MKNdenotespriceobtainedusingM1000samplepathsandNtimesteps

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2-3 and 2-4 dotheanalysisfortables 2-1 and 2-2 respectively.Asexpected,itisobservedthatthesevaluesimproveasweincreasethenumberoftimestepsandsamplepaths. Table2-3. SummaryAnalysisfor=0:305 MaximumDeviationfromBM0.2180.1210.2210.1460.2210.1390.117 AverageDeviationfromBM0.0730.0420.0460.0400.0710.0270.035 MaximumDeviationfromSA0.1850.0990.1990.1240.1990.1170.095 AverageDeviationfromSA0.0550.0230.0400.0320.0630.0250.028 Foreachcolumn,MKNdenotespriceobtainedusingM1000samplepathsandNtimesteps SummaryAnalysisfor=1:220 MaximumDeviationfromBM0.2140.3630.1650.2300.3080.1700.073 AverageDeviationfromBM0.0830.0840.0440.0570.0790.0450.020 MaximumDeviationfromSA0.1720.2810.0830.1480.2260.0880.090 AverageDeviationfromSA0.0700.0700.0350.0400.0670.0360.035 Foreachcolumn,MKNdenotespriceobtainedusingM1000samplepathsandNtimesteps 25

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Carriere ( 1996 )and LongstaffandSchwartz ( 2001 ).Extensivenumericalresultsshowthatourapproachyieldsveryaccuratepricesinacomputationallyef-cientmanner.Inaddition,theexibilityofthemethodallowsforitsextensiontoamuchlargerclassofoptimalstoppingproblemsthanaddressedinthischapter. Thischapterisorganizedasfollows.Inthenextsection,thestochasticvolatilitymodelof Heston ( 1993 )isreviewed.Section3developsourapproximationapproachtopriceAmer-icanoptionsunderthismodel.Section4containsasystematicnumericalevaluationofthisapproximationandSection5concludes. whererandqdenotetherisk-freerateandthedividendyieldrespectively,withW1andW2twoindependentstandardBrownianmotionsdenedonacommonunderlyingcompletelteredprobabilityspace(;(Ft)t;P),wherePistherisk-neutralmeasure.ThevolatilityV(t)

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Forillustrativepurposes,weconsideranAmericancalloptionwithstrikepriceK.LetCA(S;v;)denotethepriceofthisoptionwhentheunderlyinghaspriceSandspotvolatilityv,withunitsoftimelefttoexpiry.Usingstandardarbitragearguments,thispriceCAcanbeshowntosatisfythefollowingpartialdifferentialequation@CA ChiarellaandZiogas ( 2005 )usethemethodin Jamshidian ( 1992 )toconverttheabovehomogeneousPDEdenedintheregionDtoaninhomogeneousoneinanunrestricteddomain: 27

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2;x=0;0;x<0 ToobtainCAthroughequation( 3 ),onestillneedstheknowledgeoftheoptimalstopping(exercise)boundaryb(v;t).Intheclassicalcontextofconstantvolatilityfortheunderlyingassetreturn, AitSahliaandLai ( 1999 )haveshownthatthisboundaryiswell-approximatedbylinearsplineswithveryfewknots,typically3or4.Whenthevolatilityoftheunderlyingassetitselffollowsastochasticprocessasin( 3 )above, Broadieetal. ( 2000 )haveproducedempiricalevidencetosuggestthatthecorrespondingoptimalstoppingsurfacecanbewell-approximatedinalog-linearfashionnearthelong-termvariancelevel;i.e.:lnb(v;)b0()+vb1();near, thusreducingthedeterminationofb(v;)tothatofb0()andb1().Underthisassumption, ChiarellaandZiogas ( 2005 )thenexpressthesolutionforthePDE( 3 )asthefollowing 28

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where 2+1 Z10Refj(S;v;T;;w)eilnb forj=1;2andf1(S;v;;;w)=elnSe(rq)()f2(S;v;T;;w)f2(x;v;;;)=expg0(;;)+g1(;;)x+g2(;;)v; 2(i+D2)()2ln1G2()eD2() Carriere ( 1996 )and LongstaffandSchwartz ( 2001 ).ThisexiblemethodusesacombinationofMonte-Carlosimulationwithleast-squaresregressiontoevaluateAmericanoptionprices.Inouradaptationofthisapproach,weshallestimateb0()andb1()overanitesubsetofdiscrete 29

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AitSahliaandLai ( 1999 ).)Inaddition, Glasserman ( 2004 )alsoshowsthatsimulation-basedvaluationsofAmericanoptionpricesdonotcriticallydependonanaccurateevaluationoftheoptimalexercisestrategy. 3 )-( 3 ).AnaturalchoiceistheEulerscheme:^Si+1=^Si+^Sit+q However,wefollow Glasserman ( 2004 )whosuggeststhatthesecond-orderschemeof Milstein ( 1978 )and Talay ( 1982 )givenbelowhasabetterconvergence(lessbias)foroption 30

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2r2^Sih2+r+ 16^Si 2^Si^Vi+ 4p 22(^Vi)h2+ 161 42(W21h)+1 4(12)(W22h)+1 2p 1. Ataparticulartimesteptk,werstcalculatethecontinuationcashowfC(!;tj;tk;T):k+1jNgforeachsamplepathasin LongstaffandSchwartz ( 2001 ),whoalsoshowthatthevalueofcontinuationF(!;tk)canbeexpressedas 31

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whereKisthestrikepricefortheoptionandisthelongrunmeanofthevolatilityasgiveninHeston'smodelandL0(X)=expX 3 ).Thenforeachsamplepathinthattimestep,theexpectedcashowfromcontinuationiscalculatedandcomparedwiththecurrentpayoff.Theoptionisexercisedforaparticularsamplepathifthecurrentpayoffisgreaterthantheexpectedpayofffromcontinuation.ThisisdoneforalltheMstockprices(oneforeachsamplepath)atthistimestep.LetnedenotethenumberofsamplepathsoutofthetotalNSwhereexercisehappens.Letthesenepointsbedenotedas(si;vi)i=1;:::;ne. 2. TheobtainedexercisepointsarethendividedintoclassesC1;C2;:::;Cnvsothat(si;vi)2Cjifvi2(Vj;Vj+1)whereVj(j=1;:::;nv)areequidistantvaluesofvolatility,suchthatV0=vMinandVnv=vMax,Vj=V0+jVnvV0 3 )forHeston'sprocess.Thevaluesnv,vMin,andvMaxclearlydependonthestochasticvolatilitymodelandcanbesetinadvanceinanumberofdifferentways.Inourparticularcase,werannumericalsimulationsofthevolatilityprocessaloneaheadofthepricingcalculationsanddeterminedthatvMincouldbesettoitsnaturalvalueof0,asitisavariance,andvMaxwassetto:70,whichisconservativevalueasobservedvariancevaluesareoverwhelminglyless:06.Numericalexperimentswereperformedasdryrunsforthene-tuningofnv,setto10,asitwasthevaluealittlehigherthanobserved,wherepricingqualitystartedtodeteriorate. 3. Theboundaryvaluesforthistimesteptkcorrespondingtovolatilitiesfv1;v2;:::;vnvg(vj=Vj+Vj+1 32

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4. Empiricalevidencein Broadieetal. ( 2000 )pointtotheexistenceofalinearrelationshipbetweenlnb(v;t)andv,arelationshipalsousedby TzavalisandWang ( 2003 ): Thusforourtimediscretizedversion,weneedtoevaluatethevaluesofb0(tk)andb1(tk)intheequation Inordertocalculateb0(tk)andb1(tk),theobtainedboundaryvaluesfb1(tk);b2(tk);:::;bnv(tk)gfordifferentclassesC1;C2;:::;Cnvareregressedagainstthemidpointsofthecorrespondingclassesfv1;v2;:::;vnvg. 3 ).ThevaluesoftheparametersofHeston'smodelareassumedtobeknown.Asisseenin( 3 ),inordertoevaluatethepriceoftheoption,weneedtocomputethevaluesofsomeintegrals.Theouterintegralsin( 3 )fortheearlyexercisepremiumaretimeintegralsandarecomputedusingSimpson'srule.Thenumberofpointsfortheintegrationissameasthenumberoftimestepsoverwhichtheboundaryiscalculated.However,withintheseintegralsandforthecalculationsoftheEuropeanoption,additionalnumericalintegrationisrequired,overunboundedintervals.WeproceedwiththeuseofGauss-Laguerrequadrature,whichapproximatesanintegraloftheformR10f(x)dxusingoptimallychosennpointsxiwithweightswi,resultingintheapproximationZ10f(x)dxi=nXi=1f(xi)wi:ThesepointsxiaretherootsoforthogonalLaguerrepolynomialsLn(x)=ex WilliamH.Press ( 1992 ).)ForourcodeweusedreadilyavailableroutinesfromtheQuantLiblibrary(http://quantlib.org). Tocomputethebenchmarkvaluesagainstwhichtocomparethosegeneratedthroughtheproposednumericalapproach,wefollowthestepsbelow: 33

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ApplyLSMalgorithmtocalculatetheearlyexercisesurface. 2. Stopsimulatednerpathsatthisboundarytoobtainoptionprice. Instep1,werstsimulatethestockpricepathsfollowingthesecond-orderschemedescribedinsection3.LSMisthenappliedtothesesimulatedpathstocalculatetheboundaryusingtheproceduregivenin3.1.Thecalculatedvaluesoftheboundaryarestoredforthenextstep,whichgeneratesafreshsetofsamplepathsandstopsthemattheboundarytogetthepriceforeachpath.Inordertocheckifaparticularpathcanbestoppedataparticulartime,thestockpriceischeckedwiththeboundarypriceforthecorrespondingvolatilityclass.Theaverageoverallsamplepathsgivesthepriceoftheoption.Sincethisstepdoesnotrequireanycomplexcalculations(regressionsinLSM),itisnotmuchcomputationallyintensive.Asaresult,wecanincreasethenumberofsamplepathsconsiderablywithoutaffectingthecomputationaltimealotandtherebyobtainingapricewithamuchnarrowercondenceintervalthantheoriginalLSM.Thenumberofpotentialdateshoweverhastobethesameasinstep1above.Thebenchmarkboundaryisgeneratedusing1,000,000samplepathsandtimeincrementsof0.01yeareach.Pricesaresubsequentlycalculatedbygeneratingafreshsetof10,000,000samplepathsandstoppingthemattheboundaryobtainedinthepreviousstep. Inournumericalexperimentation,theparametervaluesaresetasT=0:25;=0:0;=1:0;=0:09;=0:1,aswellasvariouscombinationsofsamplesizesandnumbersoftimesteps,areconsidered.Inaddition,differentmaturities2f0:2;0:4;0:6;0:8;1:0gandspotpricesareused.Foreachvalueof,thespotpricess0aretakenfromthesetf90;95;100;105;110g. Inordertostudytheeffectofchangingthenumberofsamplepaths,weevaluatetheboundarysurfaceusing10K,50K,100Kand1000Knumberofsamplepaths,keepingthenumberoftimestepsconstantat25.Asshowningures 3-1 and 3-2 ,startingwith1,000samplepaths,theapproximatedboundarysurfacebecomesprogressivelybetter.Infact,thereislittlenoticeabledifferencebetweentheonegeneratedwith100,000samplepathsandthatwhichisgeneratedwith1million. 34

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3-3 and 3-4 ,whichshowsthatpiecewiselinearapproximationswithonlyafewtimestepsandsamplepathsarewarranted. Figure 3-5 showstheboundaryobtainedby ChiarellaandZiogas ( 2005 )bysolvingtheintegralequationsystemfortheboundaryusingnumericalintegrationtechniquesforVolterraintegralequations.Asitisapparentfromthegures,theboundaryinFigure 3-5 iswellrepresentedbytheboundaryinFigure 3-4B .Next,wecomputeoptionpricesaccordingtoourmethod.Table A-1 showscorrespondingresultscomparedagainstbenchmarkvalues.Sincethelatteraregeneratedviasimulation,thecorrespondingcolumnlabelledBMcontainsbothaveragesand95%condenceintervals.ThecolumnheadingMkNindicatesthatpricesarecalculatedonthebasisofanapproximateboundaryobtainedwithNtimestepsandM1;000samplepaths. Toassesstheaccuracyofourapproach,foreachcolumnwedeterminewhetherapricefallsinthe95%benchmarkcondenceinterval(CI).IfitmissestheCI,werecordtheamountbywhichitmissesandanaverageistakenforthecorrespondingcolumn.Tables A-1 A-2 A-3 givethepricesobtainedforvaluesofof0:0225,0:09,and0:2,respectively,withalltheotherparametersremainingthesameasabove.Tables 3-1 3-2 3-3 arefortheresultinganalysis.Itcanbeobservedthattheproposedmethodworksgenerallywell,andevenbetterforlowstandardvolatilityvalues(i.e.whenthelongrunmeanofthevolatilityislow).Thustheresultsfor=0:0225arebetterthanthatwith=0:09,whichareinturnbetterthattheonewith=0:2.However=0:09correspondstoavolatilityof30%whichbyitselfisfairlycommonmarketvolatilityvalue.Foreachvalueof,weobservethatthepriceestimateimprovesasthenumberofsamplepathsandthenumberoftimestepsareincreased. Alsoreportedintables 3-1 3-2 3-3 arethecomputationtimeforeachcolumn.Itisclearthatareductionincomputationtimecomesatthecostofincreasingvalueoftheerror.However 35

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36

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BBoundarycalculatedusing10,000samplepaths ApproximateBoundaryforAmericancalloptionwithT=0.25,=0:04,=0:1,r=0:03,q=0:05,evaluatedusing25timesteps 37

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BBoundarycalculatedusing1000,000samplepaths ApproximateBoundaryforAmericancalloptionwithT=0.25,=0:04,=0:1,r=0:03,q=0:05,evaluatedusing25timesteps 38

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BBoundarycalculatedusing20timesteps ApproximateBoundaryforAmericancalloptionwithT=0.25,=0:04,=0:1,r=0:03,q=0:05,evaluatedusing100,000samplepaths 39

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BBoundarycalculatedusing5timesteps ApproximateBoundaryforAmericancalloptionwithT=0.25,=0:04,=0:1,r=0:03,q=0:05,evaluatedusing100,000samplepaths 40

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Boundaryobtainedby ChiarellaandZiogas ( 2005 )withT=0.25,=0:04,=0:1,r=0:03,q=0:05bysolvingtheintegralequationsystemfortheboundaryusingthenumericalapproximationtechniquesforVolterraequations Table3-1. SummaryAnalysisfor=0:0225 MaxdistancefromCI0.4390.1690.0290.0280.0760.0260.030 AveragedistancefromCI0.0840.0580.0080.0100.0400.0110.002 %inCI0%4%16%12%0%12%68% ComputationTime(sec)0.080.320.562.983.046.616.8 Foreachcolumn,MkNdenotespriceobtainedusingM1000samplepathsandNtimesteps SummaryAnalysisfor=0:09 MaxdistancefromCI0.7840.4320.0990.0110.0350.0150.002 AveragedistancefromCI0.1740.1150.0120.0040.0190.0070.000 %inCI0%0%24%32%8%16%76% ComputationTime(sec)0.080.320.562.983.046.616.8 Foreachcolumn,MkNdenotespriceobtainedusingM1000samplepathsandNtimesteps SummaryAnalysisfor=0:2 MaxdistancefromCI1.0231.0320.7380.4140.4460.4370.462 AveragedistancefromCI0.3010.3140.1560.0920.1030.1010.100 %inCI4%0%10%12%16%20%24% ComputationTime(sec)0.080.320.572.973.086.6316.77 Foreachcolumn,MkNdenotespriceobtainedusingM1000samplepathsandNtimesteps

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A-1 A-2 and A-3 showthatpricesofoptionsobtainedbyrstcalculatingtheboundaryusingdifferent(lesser)numberofsamplepathsanddifferent(lesser)numberoftimestepsandthenusingtheboundaryinthedecompositionformulawerefairlyclosetothechosenbenchmarkfortheprices.Thedifferentapproximateformsoftheboundarythatwecalculatedasaresultareshowningures 3-1 3-2 3-3 and 3-4 Lookingattheseguresandusingourresultthatweneedtohaveonlyaroughestimateoftheboundary,ourintuitiontookusonestepfurtherbyconsideringanapproximationfortheboundarybasedontheconstantvolatilitymodel.Oncethisboundaryisapproximated,itisthenusedinthedecompositionformulaforstochasticvolatilitytoobtainthepriceoftheoption. 2 and 3 herealso,thepriceiscalculatedintwobroadstepsasfollows: 1. Evaluatetheearlyexerciseboundaryapproximatelyusingaconstantvolatilitymodel. 2. Calculatethepriceoftheoptionforthestochasticvolatilitymodelusingtheaboveboundaryinthedecompositionformula 3 ,theboundarywascalculatedusingtheLSMmethodfrom LongstaffandSchwartz ( 2001 )andinchapter 2 ,thesameideawasbeingtestedwithconstantvolatility. 42

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Kim ( 1990 ), Jacka ( 1991 )and Carretal. ( 1992 ).Thisboundarywasapproximatedby Huang,Subrahmanyam,andYu ( 1996 )aspiecewiseconstantfunctionsandwascalculatedusingn=1;2&3pieces. Ju ( 1998 )didthesameanalysisusingpiecewiseexponentialfunctionapproximationfortheboundaryinsteadofthepiecewiseconstantapproximation.Anumberofdifferentcalculationsforboundariescarriedoutby AitSahliaandLai ( 1999 )showedthatapiecewiseexponentialboundarycanverywellapproximatetherealboundary. AitSahliaandLai ( 2001 )solvetheintegralequationfortheboundarybyrstundergoingachangeinvariablesandthensolveitusinganumericallystablerootndingalgorithmastheirapproachisone-dimensionalincontrasttothetwo-dimensionalapproachof Ju ( 1998 ).Inaddition,theirboundaryapproximationistheonlyonethatiscontinuous,whichisinconformitywithitstheoreticalcharacterization.Weusethesamemethodasin AitSahliaandLai ( 2001 )toapproximatetheboundarywithafewtimesteps. LetS;Krepresentthestockpriceandthestrikeprice,respectively.Wedenotebyr;;Ttheparametersforrisk-freerate,volatilityandmaturityrespectively.Applyingthefollowingchangeofvariables(asin AitSahliaandLai ( 1999 ))s=2(tT);z=log(S=K)(1 2)s 1 weobtainthefollowingintegralformulafortheboundary(z(s))inthecanonicalform 2)s=ese 2sN s+p sN sesZ0seu1 2s+zNz 43

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AitSahliaandLai ( 2001 ),theboundaryz()issolvedrecursivelystartingfrom Astheapproximatingboundaryispiece-wiselinear,eachinterceptisdeterminedbythepreviouspieceandthusonlythecorrespondingslopeneedstobedeterminedasrootofanon-linearequationasexplainednext. Withzj=z(sj)andj=sjsm,oncez0;:::;zm1aredetermined,zmcanbedeterminedbysolvingthefollowingequationforz ez+(1 2)sm=esmez1 2smNz sm+p smNz sm+em1Nb(z)1=2m11 2b(z) 2m1Xi=1Ai(z)+ez+(1 2)sm"~b(z) ~a(z)N~a(z)1=2m11 21 2em1N~b(z)1=2m1+m1Xi=1~Ai(z)#(4) whereAi(z)and~Ai(z)aregivenbytheRHSofequationnumbers9and10of AitSahliaandLai ( 2001 )andb(z)=zzm1 4 weusethebisectionmethod,forwhichweusethelowerandupperboundsforthecalloptionboundary( AitSahliaandLai ( 1999 ))asstartingpoints 2s 44

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2+n(1)1 22+2o1=2 2)s(4) wheres=2(tT). Afterthiscalculation,weobtaina2dimensionalboundaryrepresentedasfb(t0);b(t1);:::;b(tk)g.Inordertostretchittoa3-dimensionalone,thevaluesarejustre-peatedoverthevolatilityaxis.Thusifthefv1;v2;:::;vnvgarepointsalongthevolatilityaxis,theresulting3-dimensionalboundaryisformedasb(ti;vj)=b(ti)8i;j 3 ,weknowthatempiricalevidencein Broadieetal. ( 2000 )pointstotheexistenceofalinearrelationshipbetweenlnb(v;t)andv,arelationshipalsousedby TzavalisandWang ( 2003 ): Thus,unlikelastchapter,wedonothavetoperformaregressiontoevaluateb0(t)andb1(t).Instead,thesevaluesareeasilycalculatedfromtheobtainedboundaryb(t)as:b0(ti)=logb(ti);b1(ti)=08i=1;2;:::;nt 4.2.1 ,we 45

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where 2+1 Z10Refj(S;v;T;;w)eilnb forj=1;2andf1(S;v;;;w)=elnSe(rq)()f2(S;v;T;;w)f2(x;v;;;)=expg0(;;)+g1(;;)x+g2(;;)v 2(i+D2)()2ln1G2()eD2() 3 .WeuseSimpson'sruletocalculatethetimeintegralandGaussianquadraturetocalculatetheotherintegrals. 3 ,herealsowedividedourproblemintothreedifferentscenariosdependingonthevalueofthelongrunmeanofvolatility. 46

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Scenario1:Lowvalueof(=0:0225orp Scenario2:Midvalueof(=0:09orp Scenario3:Highvalueof(=0:2orp 3 ),itisreasonabletocalculatetheearlyexerciseboundarybyassumingtobeequaltooneofthethreedifferentvaluesgivenby2=p 3 .Foreachcolumnwecalculatethedistancefromthebenchmark95%condenceinterval(CI)foreachentityandnallythemaximumandaveragedistancefromtheCIfortheentirecolumn.Eachofthethreeanalysistables(oneforeachofthethreevaluesof)has4sub-tables.Therstthree(a,bandc)correspondtothethreedifferentvaluesof(intheconstantvolatilitymodel)usedforcalculationoftheboundary.Thelastsub-table(d)iswhereweapplythemethodfromthepreviouschapteronafull-edgedstochasticvolatilitymodel.Theresultsareobtainedasfollows. SummaryAnalysisfor=p 35102550 MaxdistancefromCI1.1721.2041.1961.1881.185 AveragedistancefromCI0.3530.3520.3500.3490.348 %inCI0%0%0%0%0% Tables B-1 B-2 and B-3 givethepricesofAmericanputoptionswithalongrunmeanofvolatility=0:0225.Thevaluesofusedintheconstantvolatilitymodeltocalculatethe 47

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SummaryAnalysisfor=;=0:0225 35102550 MaxdistancefromCI0.4490.4390.4350.4350.434 AveragedistancefromCI0.1610.1520.1470.1460.146 %inCI0%0%4%4%4% Table4-3. SummaryAnalysisfor=+p 35102550 MaxdistancefromCI0.2360.2180.2110.2090.209 AveragedistancefromCI0.1020.0850.0770.0740.074 %inCI0%0%0%4%8% boundaryinthesetablesaregivenrespectivelyas2=8>>>>>><>>>>>>:p B-1 B-2 B-3 Tables 4-1 4-2 and 4-3 summarizethesethreetablesbycalculatingthemaximumandaveragedistancesfromthe95%CIoftheBenchmark.Table 4-1 showsthatpricesintable B-1 areasmuchas117awayfromthe95%CI.Intable 4-2 themaximumdeviationfromtheCIis45.Mostoftheentriesinthistableshowbetterresultsthantheprevioustable.Fromtable 4-3 ,itcanbenotedthatthepricesareclosesttothebenchmarkfortable B-3 ,bothintermsofthemaximumdeviationaswellastheaveragedeviation.Inthiscase,eventhefastestapproximationwith3timestepsproducesamaximumdeviationof24fromthebenchmark95%CI. However,wenoteinterestingresultswhentables 4-1 4-2 and 4-3 arecomparedto 4-5 .Althoughtable 4-5 isbetterthan 4-1 and 4-2 forallvalues,suchisnotthecasewhilecomparingitwithtable 4-3 .Table 4-3 hasbettervaluesthan 4-5 when3or5timestepsareusedtocalculatetheboundary.Thus,itisobservedthatwithasuitablychosenvaluefor,wegetbetterresultsbycalculatingtheboundaryusingaconstantvolatilitymodel,whenweareusingalessernumberoftimesteps.Forhighernumberoftimesteps,stochasticvolatilitygivesabetterresult.However 48

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Computationaltime(inseconds)for=0:0225 ComputationTime0.0150.016240.026840.103720.34872 Table4-5. StochasticVolatilityResultsfor=0:0225 MaxdistancefromCI0.4390.1690.0290.0280.0760.0260.030 AveragedistancefromCI0.0840.0580.0080.0100.0400.0110.002 %inCI0%4%16%12%0%12%68% ComputationTime0.080.320.562.983.046.616.8 ithastobenotedthattheresultsobtainedusingconstantvolatilityarealsoaccurateenough(averagedifferenceof7-8)andtakeapproximatelyhalftheamountofcomputationaltime. SummaryAnalysisfor=p 35102550 MaxdistancefromCI0.1890.1730.1670.1640.163 AveragedistancefromCI0.0880.0760.0710.0700.069 %inCI0%0%0%0%0% Tables B-4 B-5 and B-6 givethepricesofAmericanputoptionswithalongrunmeanofvolatility=0:09.Thevaluesofusedintheconstantvolatilitymodeltocalculatetheboundaryinthesetablesaregivenas2=8>>>>>><>>>>>>:p B-4 B-5 B-6 Inthiscase,onedifferencewiththepreviousanalysisforscenario1isthatherewehaveasituationwherethespotvolatility(=0:04)isdifferentfromthelongrunmeanofvolatility(=0:09).Tables 4-6 4-7 and 4-8 analyzethesethreetablesbycalculatingthemaximumandaveragedistancesfromthe95%CIoftheBenchmark.Table 4-6 showsthatpricesintable B-4 areasmuchas18awayfromthe95%CI.Intable 4-7 themaximumdeviationfromtheCIis 49

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SummaryAnalysisfor=;=0:09 35102550 MaxdistancefromCI0.0810.0650.0610.0600.060 AveragedistancefromCI0.0530.0330.0240.0210.021 %inCI0%0%0%0%0% Table4-8. SummaryAnalysisfor=+p 35102550 MaxdistancefromCI0.0700.0510.0600.0620.062 AveragedistancefromCI0.0410.0190.0130.0120.012 %inCI4%16%16%20%20% 8.Alltheentriesinthistableshowbetterresultsthantheprevioustable.Fromtable 4-8 ,itcanbenotedthatthepricesareclosesttothebenchmarkfortable B-6 ,bothintermsofthemaximumdeviationaswellastheaveragedeviation.Inthiscaseeventhefastestapproximationwith3timestepsproducesamaximumdeviationof7fromthebenchmark95%CI. Nextwecomparetables 4-6 4-7 and 4-8 to 4-10 .Herewendthattables 4-6 and 4-7 arebetterthantable 4-10 forboundariescalculatedwith3or5timestepswhilethereverseistrueforothertimesteps.However,intermsofaveragedeviationfromtheCI,bothtables 4-6 and 4-7 seemtodoverywell.Similarly,table 4-8 alsohasbettervaluesthan 4-10 forthe3,5or10timesteps.Thussimilartosection 4.3.1 ,itisobservedthatwithasuitablychosenvaluefor,wegetbetterresultsbycalculatingtheboundaryusingaconstantvolatilitymodel,whenweareusingafewertimesteps(3,5or10).Forahighernumberoftimesteps,stochasticvolatilitygivesabetterresult.However,thistimetheresultsobtainedusingconstantvolatilityarecloserthaninsection 4.3.1 (differenceof2-4)andagaintakelessthanhalfthecomputationaltime. 50

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Computationaltime(inseconds)for=0:09 ComputationTime0.013120.013720.028720.101840.35124 Table4-10. StochasticVolatilityfor=0:09 MaxdistancefromCI0.7840.4320.0990.0110.0350.0150.002 AveragedistancefromCI0.1740.1150.0120.0040.0190.0070.000 %inCI0%0%24%32%8%16%76% ComputationTime0.080.320.562.983.046.616.8 SummaryAnalysisfor=p 35102550 MaxdistancefromCI0.3570.3560.3500.3460.345 AveragedistancefromCI0.0970.0850.0790.0760.076 %inCI4%8%16%16%16% Tables B-7 B-8 and B-9 givethepricesofAmericanputoptionswithalongrunmeanofvolatility=0:2.Thevaluesofusedintheconstantvolatilitymodeltocalculatetheboundaryinthesetablesaregivenas2=8>>>>>><>>>>>>:p B-7 B-8 B-9 Tables 4-11 4-12 and 4-13 analyzethesethreetablesbycalculatingthemaximumandaveragedistancesfromthe95%CIoftheBenchmark.Table 4-11 showsthatpricesintable B-7 areasmuchas36awayfromthe95%CI.Intable 4-12 themaximumdeviationfromtheCIis25.Againallentriesinthistableshowbetterresultsthantheprevioustable.Fromtable 4-13 ,itcanbenotedthatthepricesareclosesttothebenchmarkfortable B-9 ,bothintermsofthemaximumdeviationaswellastheaveragedeviation.Inthiscaseeventhefastestapproximation 51

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SummaryAnalysisfor=;=0:2 35102550 MaxdistancefromCI0.2570.2790.2600.2510.249 AveragedistancefromCI0.0710.0530.0460.0440.043 %inCI8%32%36%40%40% Table4-13. SummaryAnalysisfor=+p 35102550 MaxdistancefromCI0.2200.1860.1780.1750.175 AveragedistancefromCI0.0530.0330.0260.0250.024 %inCI28%44%56%60%64% with3timestepsproducesamaximumdeviationof22fromthebenchmark95%CIwhiletheaveragedeviationisjust5. Howeverwhencomparedwithtable 4-15 itisobservedthatallthetables 4-11 4-12 and 4-13 dobetterthanstochasticvolatilitytherebyshowingthatwegetabettervaluewhentheconstantvolatilityboundaryisusedforcalculatingpricesofoptionswithahigherlongrunmeanofvolatility. 3 ,wherewehadfoundthatcalculatingthepriceofAmericanoptionsusingroughapproximationsoftheearlyexerciseboundaryboregoodresults.Asubstantialamountoftimewassavedatthecostofalittlelossinaccuracy.Thisapproximationoftheearlyexerciseboundarywastakenastepfurtherinthischapterbyusingaconstantvolatilitymodeltocalculatetheboundary.Therstproblemthatcameupwasthattheboundaryobtainedwasonlyatwo-dimensionaloneratherthanthethree-dimensionaloneneededforthedecompositionformula( ChiarellaandZiogas ( 2005 ))giveninequation 4 Thisproblemwassolvedbyreplicatingtheboundaryalongthemissingvolatilityaxis,sothatitdoesnotvaryalongthisaxis.Oncetheboundaryisthusobtained,itispluggedintothestochasticvolatilitydecompositionformulatocomputethepriceoftheAmericanoption. 52

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Computationaltime(inseconds)for=0:2 ComputationTime0.01060.014840.029360.101240.40684 Table4-15. StochasticVolatilityResultsfor=0:2 MaxdistancefromCI1.0231.0320.7380.4140.4460.4370.462 AveragedistancefromCI0.3010.3140.1560.0920.1030.1010.100 %inCI4%0%10%12%16%20%24% ComputationTime0.080.320.572.973.086.6316.77 Theperformanceofthistechniquewasassessedbycomparingtheresultingpricesagainstbenchmarkvaluesbasedentirelyonthestochasticvolatilitymodel.Thismethoddidparticularlywellevenforsmallnumbersoftimesteps,whichisthemostimportant.Thisisespeciallycrucialastheresultingcomputationaltimereductionwasontheorderof70%to80%,thusmeetingourstatedobjective. 53

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Kim ( 1990 ), Jacka ( 1991 )and Carr,Jarrow,andMyeni ( 1992 ),whiletheoneforstochasticvolatilityisgivenby ChiarellaandZiogas ( 2005 ).However,inordertocalculatetheearlyexercisepremiuminthedecompositionformula,thecorrespondingearlyexerciseboundaryfortheoptionneedstobeknown.Thisboundarycanbecalculatedfromusingthedecompositionformulatoformanintegralrepresentationfortheboundary.Variousmethodshavebeensuggestedforsolvingthisintegralequation,includingthoseof Huang,Subrahmanyam,andYu ( 1996 ), Ju ( 1998 )and AitSahliaandLai ( 2001 ).Whilesolvingfortheboundary, Ju ( 1998 )and AitSahliaandLai ( 1999 )alsoobservethefactthatacoarselyapproximatedboundaryisenoughtogetanaccuratepriceoftheoptionwhenvolatilityisconstant.ItisthisfeaturethatguidedourworktoreducethecomputationaltimeassociatedwithpricingofAmericanoptionsunderstochasticvolatility.Thusourproposedapproachistorstcalculatetheearlyexerciseboundaryinaapproximatefashion(usingasmallnumberoftimestepsand,whencomputedviaMonteCarlo,asamplesize)andthenapplyittothedecompositionformulatoobtainthepriceoftheoption. Inordertotestthemodelonexistingresultswerstappliedittoaconstantvolatilitymodelinchapter 2 .Theconstantvolatilityboundaryinthiscasewasevaluatedusingcoarselytheleast-squaresMonteCarlo(LSM)methodgivenby LongstaffandSchwartz ( 2001 )withonlyafewsamplepathsandtimesteps.Theresultsobtainedbyusingthisboundaryintheconstant 54

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AitSahliaandLai ( 2001 ),thusvalidatingtheproposedmodelforconstantvolatility. Inchapter 3 theproposedmodelwasusedtopriceAmericanoptionsunderstochasticvolatility.Forthispurpose,theboundarywasrstcalculatedusingthecoarseLSMmethodwithafewmodicationsforstochasticvolatility.Theobtainedboundarywasthenappliedinthestochasticvolatilitydecompositionformulatoarriveattheoptionprice.ThebenchmarkpricewascalculatedusingthesameLSMalgorithm,butthistimewithaveryhighnumberofsamplepathsandtimesteps.Thepricesobtainedusingtheproposedmodelwerealsoveryclosetothebenchmarkforlowandmediumvaluesofthelongrunmeanofvolatility.Althoughtheerrorswerealittlehigherforalongrunvolatilitymeanofaround0.2(correspondingtovolatilitygreaterthan40%),theclosenessofthesepricestothe95%condenceintervalofthebenchmarkestablishedthevalidityofthemodelforstochasticvolatilityaswell. Chapter 4 tookadifferentapproachtotheboundaryapproximation.Itcalculatedthebound-aryfromaconstantvolatilitymodelandthenuseditinthestochasticvolatilitydecompositionformula.Theboundarycalculationwasperformedthroughthesplineapproximationmethodin AitSahliaandLai ( 2001 ).This2-dimensionalboundarywasthenconvertedintoa3-dimensionalformthroughreplicationalongthevolatilityaxis.Thepricesobtainedbypluggingthisboundaryintothestochasticvolatilitydecompositionformulawerecomparedtothesamebenchmarkasinchapter 3 .Itwasobservedthattheresultswerebetterthanthoseinchapter 3 forhighvolatilitycasesandworseforlowvolatility. Thus,overalltheproposedmodelwasfoundtoperformwellinreducingthecomputationaltimewithoutadverselyaffectingaccuracy.Inthisregard,itcanbenotedthatthecomputationaltimeformethodinchapter 4 werealllessthanasecond(usingacomputerwithWindowsXP,IntelPentiumMprocessor,1.60GHzand512MBofRAM). 4 ,weuseonlyoneboundaryobtainedforaparticularvalueoftheconstant 55

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Americancalloptionwithstrike=100,r=0:03,q=0:05,=0:1,=4:0,=0,=0:0225,V0=0:0225 1901.5451.6371.5891.5601.5581.5891.5571.550(1.5361.554)952.8783.0552.9702.9192.9182.9602.9162.903(2.8652.89)1004.8725.1234.9914.9024.9044.9584.9034.879(4.8554.888)1057.537.8757.6857.5407.5437.6127.5477.509(7.5117.55)11010.82811.29011.02010.80410.80410.89810.81410.775(10.80510.851) 0.8901.2211.2641.2631.2411.2421.2501.2451.232(1.2131.228)952.4952.5572.5562.5202.5192.5402.5242.506(2.4842.506)1004.464.5374.5454.4874.4844.5234.4894.466(4.4454.475)1057.1497.2517.2747.1797.1837.2377.1857.154(7.1317.168)11010.53710.66010.70410.54910.56910.63410.57110.531(10.51510.558) 0.6900.8710.9010.8950.8890.8900.8980.8870.883(0.8650.877)952.0342.0772.0672.0492.0532.0672.0502.040(2.0252.043)1003.9564.0284.0193.9833.9894.0143.9913.970(3.9433.969)1056.7266.8156.8166.7546.7616.8026.7676.737(6.7096.742)11010.2610.36910.38310.28210.29710.35310.30310.269(10.24110.28) 0.4900.5020.5150.5130.5070.5070.5140.5100.505(0.4980.506)951.4651.4881.4901.4781.4801.4901.4811.473(1.4581.472)1003.3383.3743.3743.3543.3583.3733.3553.344(3.3273.349)1056.2196.2866.2816.2486.2506.2796.2496.232(6.2056.234)1109.9910.06810.07110.01910.02210.06410.02610.000(9.97310.007) 0.2900.1390.1410.1400.1390.1390.1420.1400.140(0.1370.14)950.7460.7530.7530.7490.7490.7560.7500.747(0.7420.75)1002.4552.4782.4772.4662.4662.4772.4672.459(2.4482.463)1055.5795.6185.6165.5985.5985.6145.5975.586(5.5685.59)1109.7849.8419.8329.8099.8109.8319.8099.797(9.779.797)

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Americancalloptionwithstrike=100,r=0:03,q=0:05,=0:1,=4:0,=0,=0:09,V0=0:04 1905.4845.5995.5905.5315.4925.5235.4975.476(5.4775.491)957.5147.6527.6347.5757.5197.5587.5177.497(7.5067.523)1009.91110.09710.0809.9949.9119.9539.9159.894(9.9029.920)10512.67412.88812.90212.79612.66912.72512.68612.656(12.66412.685)11015.77816.03816.04215.95215.78215.86315.79215.765(15.76715.789) 0.8904.5054.5704.5734.5364.5184.5374.5194.504(4.4984.511)956.4456.5376.5446.4806.4566.4836.4566.440(6.4386.452)1008.8098.9118.9268.8578.8218.8618.8178.798(8.8008.817)10511.58311.72411.73711.65311.59511.64211.59311.569(11.57411.593)11014.74314.89014.95614.83314.74614.80014.75314.729(14.73214.753) 0.6903.3973.4333.4423.4103.4123.4293.4123.400(3.3923.402)955.2125.2635.2765.2285.2295.2515.2265.210(5.2065.218)1007.5197.5857.5987.5317.5357.5587.5327.512(7.5127.527)10510.30910.37410.41010.31910.32310.34410.32010.299(10.30010.317)11013.55113.63313.69213.56913.56513.58913.56013.538(13.54113.560) 0.4902.1422.1582.1612.1472.1502.1622.1482.143(2.1382.146)953.7433.7693.7713.7523.7543.7713.7543.745(3.7383.747)1005.9616.0026.0065.9725.9735.9925.9475.962(5.9555.967)1058.7928.8378.8538.8018.8078.8278.8088.792(8.7858.799)11012.18212.25012.25412.18712.20412.22712.20112.183(12.17412.190) 0.2900.7680.7760.7750.7720.7730.7770.7720.771(0.7670.770)951.9361.9511.9511.9441.9451.9521.9441.941(1.9331.939)1003.9773.9933.9983.9833.9853.9973.9853.978(3.9733.981)1056.9386.9646.9686.9486.9496.9646.9506.941(6.9336.943)11010.69310.72210.73310.70510.70910.72510.71010.700(10.68710.699)

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Americancalloptionwithstrike=100,r=0:03,q=0:05,=0:1,=4:0,=0,=0:2;V0=0:2 19011.130511.65511.57911.42011.30511.31111.30711.326(11.09411.167)9513.53614.18114.11513.92213.77413.78813.77013.794(13.49613.576)10016.19216.95016.90316.67016.47716.49316.48116.501(16.14916.235)10519.01919.94719.92519.65419.39619.41019.41819.436(18.97319.065)11022.076523.14823.15722.86322.53922.57122.56222.587(22.02822.125) 0.8909.74910.03910.0529.8919.8329.8249.8459.837(9.7159.783)9512.119512.45512.51212.31112.22412.21712.24212.234(12.08212.157)10014.725515.15115.24715.00314.88414.88814.90414.898(14.68514.766)10517.597518.09818.24017.95217.80017.80817.81817.816(17.55417.641)11020.68221.25021.46821.14520.95320.95720.97720.976(20.63620.728) 0.6908.07658.2238.2418.1278.1328.1408.1418.131(8.0478.106)9510.356510.53610.58110.42510.43310.44510.44410.432(10.32310.39)10012.94313.15813.23513.03313.04013.06113.05113.041(12.90612.98)10515.81316.07516.18615.93915.94115.96615.94815.943(15.77315.853)11018.982519.25019.40919.12219.11519.14119.12019.119(18.9419.025) 0.4907.13756.1166.1396.0876.0936.1066.1006.089(6.0696.13)958.23458.2868.3198.2418.2528.2688.2618.247(8.2068.263)10010.742510.83210.87810.76610.78410.80210.79410.778(10.71110.774)10513.62313.73713.79813.65013.67313.69013.68113.664(13.58813.658)11016.8216.97417.05216.86716.89116.91116.89716.882(16.78216.858) 0.2903.4513.5473.4693.4583.4603.4653.4633.458(3.4353.467)955.32955.4595.3565.3385.3425.3495.3455.339(5.315.349)1007.70957.8897.7597.7327.7387.7457.7427.735(7.6867.733)10510.61410.81810.66310.62310.63210.64010.63610.628(10.58710.641)11013.9514.19914.02413.96813.98113.99113.98513.977(13.9213.98)

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Americancalloptionwithstrike=100,r=0:03,q=0:05,=0:1,=4:0,=0,=0:0225,v0=0:0225,(boundarycalculatedusingconstantvolatility(SAmethod)modelwith=p TS0BM35102550 1901.5451.6571.6491.6461.6451.645(1.5361.554)952.8783.1243.1163.1143.1133.113(2.8652.89)1004.8725.3045.3015.2995.2995.299(4.8554.888)1057.538.2298.2538.2688.2718.271(7.5117.55)11010.82812.02312.05512.04712.03912.036(10.80510.851) 0.8901.2211.3011.2961.2941.2941.294(1.2131.228)952.4952.6662.6612.6592.6582.658(2.4842.506)1004.464.8004.7984.7974.7974.797(4.4454.475)1057.1497.7487.7737.7887.7927.793(7.1317.168)11010.53711.72611.69811.66411.64411.638(10.51510.558) 0.6900.8710.9240.9200.9180.9170.917(0.8650.877)952.0342.1482.1432.1402.1392.138(2.0252.043)1003.9564.2204.2174.2144.2134.212(3.9433.969)1056.7267.2167.2367.2477.2497.248(6.7096.742)11010.2611.28711.24311.21511.20211.196(10.24110.28) 0.4900.5020.5250.5220.5200.5190.519(0.4980.506)951.4651.5311.5271.5251.5241.524(1.4581.472)1003.3383.5043.5003.4983.4973.497(3.3273.349)1056.2196.5906.6016.6076.6086.608(6.2056.234)1109.9910.73710.72310.71810.71810.718(9.97310.007) 0.2900.1390.1420.1410.1410.1410.141(0.1370.14)950.7460.7650.7630.7620.7610.761(0.7420.75)1002.4552.5322.5302.5292.5292.528(2.4482.463)1055.5795.7895.8025.8115.8155.815(5.5685.59)1109.78410.23010.22910.22810.22810.228(9.779.797)

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Americancalloptionwithstrike=100,r=0:03,q=0:05,=0:1,=4:0,=0,=0:0225,v0=0:0225,(boundarycalculatedusingconstantvolatility(SAmethod)modelwith=p TS0BM35102550 1901.5451.6121.5971.5901.5881.588(1.5361.554)952.8783.0183.0002.9922.9892.988(2.8652.89)1004.8725.0815.0635.0555.0525.052(4.8554.888)1057.537.8497.8317.8257.8247.824(7.5117.55)11010.82811.30111.29011.28611.28611.285(10.80510.851) 0.8901.2211.2841.2711.2621.2581.258(1.2131.228)952.4952.6022.5842.5742.5702.569(2.4842.506)1004.464.6364.6194.6094.6064.605(4.4454.475)1057.1497.4457.4327.4267.4247.423(7.1317.168)11010.53710.99310.99310.99210.99210.992(10.51510.558) 0.6900.8710.9100.9020.8980.8960.896(0.8650.877)952.0342.1002.0892.0822.0802.080(2.0252.043)1003.9564.0934.0814.0744.0734.072(3.9433.969)1056.7266.9716.9616.9576.9566.956(6.7096.742)11010.2610.68010.67610.67610.67610.676(10.24110.28) 0.4900.5020.5200.5140.5120.5110.510(0.4980.506)951.4651.5091.5001.4951.4931.493(1.4581.472)1003.3383.4253.4153.4103.4083.408(3.3273.349)1056.2196.4076.3996.3966.3956.395(6.2056.234)1109.9910.34310.34410.34410.34410.343(9.97310.007) 0.2900.1390.1420.1410.1400.1400.140(0.1370.14)950.7460.7590.7550.7530.7530.752(0.7420.75)1002.4552.4972.4912.4892.4882.488(2.4482.463)1055.5795.6895.6865.6855.6855.685(5.5685.59)1109.78410.03710.04410.05010.05110.051(9.779.797)

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Americancalloptionwithstrike=100,r=0:03,q=0:05,=0:1,=4:0,=0,=0:0225,v0=0:0225,(boundarycalculatedusingconstantvolatility(SAmethod)modelwith=p TS0BM35102550 1901.5451.6081.5921.5781.5721.570(1.5361.554)952.8782.9982.9732.9552.9482.947(2.8652.89)1004.8725.0214.9924.9754.9684.967(4.8554.888)1057.537.7117.6847.6717.6677.667(7.5117.55)11010.82811.04911.02511.01811.01711.017(10.80510.851) 0.8901.2211.2721.2551.2481.2451.245(1.2131.228)952.4952.5802.5552.5432.5392.538(2.4842.506)1004.464.5904.5584.5444.5404.539(4.4454.475)1057.1497.3427.3137.3007.2967.296(7.1317.168)11010.53710.79410.77610.76810.76610.765(10.51510.558) 0.6900.8710.9140.9000.8930.8900.890(0.8650.877)952.0342.0972.0762.0662.0622.061(2.0252.043)1003.9564.0704.0444.0324.0274.026(3.9433.969)1056.7266.8956.8736.8636.8596.858(6.7096.742)11010.2610.50510.49410.49110.48910.489(10.24110.28) 0.4900.5020.5210.5130.5100.5080.508(0.4980.506)951.4651.5081.4931.4871.4841.484(1.4581.472)1003.3383.4113.3923.3833.3803.380(3.3273.349)1056.2196.3516.3356.3286.3266.326(6.2056.234)1109.9910.20510.20010.19910.19810.198(9.97310.007) 0.2900.1390.1430.1410.1400.1400.140(0.1370.14)950.7460.7600.7550.7520.7500.750(0.7420.75)1002.4552.4922.4832.4772.4762.475(2.4482.463)1055.5795.6555.6485.6465.6455.645(5.5685.59)1109.7849.9459.9469.9479.9489.948(9.779.797)

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Americancalloptionwithstrike=100,r=0:03,q=0:05,=0:1,=4:0,=0,=0:09,v0=0:04,(boundarycalculatedusingconstantvolatility(SAmethod)modelwith=p TS0BM35102550 1905.4845.5585.5495.5415.5385.537(5.4775.491)957.5147.6017.5927.5867.5847.584(7.5067.523)1009.91110.03210.01510.01110.01010.010(9.9029.920)10512.67412.83712.81412.80812.80712.807(12.66412.685)11015.77815.97815.96215.95615.95315.952(15.76715.789) 0.8904.5054.5704.5564.5514.5504.550(4.4984.511)956.4456.5316.5156.5106.5096.509(6.4386.452)1008.8098.9228.9048.8978.8968.896(8.8008.817)10511.58311.72811.71111.70311.70111.700(11.57411.593)11014.74314.91514.90414.90014.89714.896(14.73214.753) 0.6903.3973.4503.4363.4323.4303.430(3.3923.402)955.2125.2845.2675.2625.2605.260(5.2065.218)1007.5197.6157.5987.5927.5917.590(7.5127.527)10510.30910.43510.42010.41410.41310.412(10.30010.317)11013.55113.70713.69913.69513.69313.693(13.54113.560) 0.4902.1422.1752.1662.1602.1572.157(2.1382.146)953.7433.7953.7843.7773.7743.774(3.7383.747)1005.9616.0326.0226.0166.0146.014(5.9555.967)1058.7928.8898.8838.8798.8778.877(8.7858.799)11012.18212.32012.31512.31312.31312.313(12.17412.190) 0.2900.7680.7810.7770.7740.7730.773(0.7670.770)951.9361.9611.9541.9501.9491.949(1.9331.939)1003.9774.0154.0074.0034.0014.001(3.9733.981)1056.9387.0026.9976.9946.9936.993(6.9336.943)11010.69310.80010.79910.79910.79910.799(10.68710.699)

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Americancalloptionwithstrike=100,r=0:03,q=0:05,=0:1,=4:0,=0,=0:09,v0=0:04,(boundarycalculatedusingconstantvolatility(SAmethod)modelwith=p TS0BM35102550 1905.4845.5435.5195.5065.5025.501(5.4775.491)957.5147.5867.5477.5337.5307.529(7.5067.523)1009.91110.0019.9629.9429.9359.933(9.9029.920)10512.67412.73112.71712.70412.69812.696(12.66412.685)11015.77815.81815.80115.79715.79715.797(15.76715.789) 0.8904.5054.5674.5404.5304.5274.526(4.4984.511)956.4456.5306.4866.4746.4716.470(6.4386.452)1008.8098.8948.8628.8458.8398.838(8.8008.817)10511.58311.64311.63511.62211.61611.614(11.57411.593)11014.74314.79614.78014.77514.77314.773(14.73214.753) 0.6903.3973.4433.4303.4213.4183.417(3.3923.402)955.2125.2695.2475.2395.2375.236(5.2065.218)1007.5197.5947.5617.5537.5517.550(7.5127.527)10510.30910.39310.36410.35410.35110.350(10.30010.317)11013.55113.62113.61313.60413.60113.600(13.54113.560) 0.4902.1422.1732.1582.1522.1502.150(2.1382.146)953.7433.7933.7723.7643.7623.761(3.7383.747)1005.9616.0266.0045.9945.9915.991(5.9555.967)1058.7928.8678.8508.8418.8388.838(8.7858.799)11012.18212.26312.25512.25012.24912.248(12.17412.190) 0.2900.7680.7810.7770.7740.7720.772(0.7670.770)951.9361.9611.9531.9481.9461.945(1.9331.939)1003.9774.0114.0013.9953.9933.992(3.9733.981)1056.9386.9896.9796.9746.9736.973(6.9336.943)11010.69310.76610.76110.76010.75910.759(10.68710.699)

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Americancalloptionwithstrike=100,r=0:03,q=0:05,=0:1,=4:0,=0,=0:09,v0=0:04,(boundarycalculatedusingconstantvolatility(SAmethod)modelwith=p TS0BM35102550 1905.4845.5615.5135.4905.4825.480(5.4775.491)957.5147.5657.5337.5107.5007.498(7.5067.523)1009.9119.9549.9069.8899.8849.884(9.9029.920)10512.67412.70512.66612.64112.63112.629(12.66412.685)11015.77815.73815.71615.70715.70515.705(15.76715.789) 0.8904.5054.5814.5354.5184.5134.512(4.4984.511)956.4456.5046.4776.4586.4506.449(6.4386.452)1008.8098.8688.8198.8068.8028.802(8.8008.817)10511.58311.63511.58911.56911.56311.562(11.57411.593)11014.74314.73314.72514.71014.70314.702(14.73214.753) 0.6903.3973.4433.4243.4133.4103.409(3.3923.402)955.2125.2795.2435.2285.2235.223(5.2065.218)1007.5197.5837.5567.5377.5307.529(7.5127.527)10510.30910.35310.33210.31910.31510.314(10.30010.317)11013.55113.59113.55513.54613.54513.545(13.54113.560) 0.4902.1422.1742.1582.1502.1472.147(2.1382.146)953.7433.7953.7713.7603.7553.755(3.7383.747)1005.9616.0215.9985.9845.9795.978(5.9555.967)1058.7928.8508.8318.8208.8168.815(8.7858.799)11012.18212.24012.21912.21312.21212.211(12.17412.190) 0.2900.7680.7810.7750.7720.7710.771(0.7670.770)951.9361.9611.9511.9461.9441.943(1.9331.939)1003.9774.0123.9983.9903.9883.987(3.9733.981)1056.9386.9876.9726.9646.9626.962(6.9336.943)11010.69310.75310.74310.73910.73810.738(10.68710.699)

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Americancalloptionwithstrike=100,r=0:03,q=0:05,=0:1,=4:0,=0,=0:2,v0=0:2,(boundarycalculatedusingconstantvolatility(SAmethod)modelwith=p TS0BM35102550 19011.130511.30111.31111.29611.28811.286(11.09411.167)9513.53613.78313.75313.74313.74013.740(13.49613.576)10016.19216.44016.45416.44216.43516.433(16.14916.235)10519.01919.39319.34919.34519.34519.345(18.97319.065)11022.076522.48222.48122.47522.47122.470(22.02822.125) 0.8909.7499.8599.8419.8309.8269.825(9.7159.783)9512.119512.25112.21512.21112.21112.212(12.08212.157)10014.725514.90214.88214.87014.86614.865(14.68514.766)10517.597517.79017.77017.76717.76617.766(17.55417.641)11020.68220.92820.91820.90620.90220.901(20.63620.728) 0.6908.07658.1668.1398.1338.1328.132(8.0478.106)9510.356510.46410.44510.43510.43110.430(10.32310.39)10012.94313.05013.03713.03213.03113.031(12.90612.98)10515.81315.95515.93015.92215.92015.920(15.77315.853)11018.982519.08619.09019.08319.07919.078(18.9419.025) 0.4906.1026.1156.1026.0986.0976.097(6.0696.13)958.23458.2798.2668.2598.2568.255(8.2068.263)10010.742510.79710.79410.78710.78510.784(10.71110.774)10513.62313.67613.66713.66513.66413.665(13.58813.658)11016.8216.89416.87916.87416.87316.873(16.78216.858) 0.2903.4513.4703.4663.4643.4633.463(3.4353.467)955.32955.3575.3505.3485.3485.348(5.315.349)1007.70957.7587.7507.7487.7477.747(7.6867.733)10510.61410.65510.64810.64510.64410.644(10.58710.641)11013.9514.00213.99813.99613.99513.994(13.9213.98)

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Americancalloptionwithstrike=100,r=0:03,q=0:05,=0:1,=4:0,=0,=0:2,v0=0:2,(boundarycalculatedusingconstantvolatility(SAmethod)modelwith=p TS0BM35102550 19011.130511.31011.25111.24511.24411.245(11.09411.167)9513.53613.73113.70713.69613.69113.690(13.49613.576)10016.19216.44216.38416.37016.36716.367(16.14916.235)10519.01919.31219.27319.26619.26519.265(18.97319.065)11022.076522.38222.40422.38522.37622.374(22.02822.125) 0.8909.7499.8389.8099.7999.7979.796(9.7159.783)9512.119512.22412.20212.18412.17812.176(12.08212.157)10014.725514.86214.82714.81814.81614.816(14.68514.766)10517.597517.72217.73217.71617.70817.706(17.55417.641)11020.68220.87320.82920.82420.82320.823(20.63620.728) 0.6908.07658.1638.1308.1198.1168.115(8.0478.106)9510.356510.43610.41510.40710.40510.405(10.32310.39)10012.94313.04713.01413.00212.99812.998(12.90612.98)10515.81315.89815.88815.88015.87715.877(15.77315.853)11018.982519.05619.03719.02619.02319.022(18.9419.025) 0.4906.1026.1186.0976.0906.0896.087(6.0696.13)958.23458.2698.2568.2468.2428.242(8.2068.263)10010.742510.78710.77110.76610.76410.764(10.71110.774)10513.62313.66813.65113.64213.63913.638(13.58813.658)11016.8216.84616.84416.83916.83616.837(16.78216.858) 0.2903.4513.4713.4663.4623.4613.461(3.4353.467)955.32955.3535.3475.3445.3435.343(5.315.349)1007.70957.7537.7437.7397.7397.738(7.6867.733)10510.61410.64710.63710.63310.63110.631(10.58710.641)11013.9513.98613.98113.97713.97613.976(13.9213.98)

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Americancalloptionwithstrike=100,r=0:03,q=0:05,=0:1,=4:0,=0,=0:2,v0=0:2,(boundarycalculatedusingconstantvolatility(SAmethod)modelwith=p TS0BM35102550 19011.130511.28711.22911.21911.21711.217(11.09411.167)9513.53613.70813.67013.65813.65413.653(13.49613.576)10016.19216.39216.35016.32916.32316.321(16.14916.235)10519.01919.28519.23119.21119.20819.207(18.97319.065)11022.076522.34222.31122.30322.30022.300(22.02822.125) 0.8909.7499.8279.7909.7809.7779.776(9.7159.783)9512.119512.21112.17412.15612.15112.150(12.08212.157)10014.725514.85514.79614.78314.78114.781(14.68514.766)10517.597517.69117.68117.66717.66217.661(17.55417.641)11020.68220.79920.79820.78020.77220.770(20.63620.728) 0.6908.07658.1468.1188.1078.1038.102(8.0478.106)9510.356510.41510.40210.39310.39010.388(10.32310.39)10012.94313.02312.99312.98012.97512.974(12.90612.98)10515.81315.88815.85015.84415.84515.844(15.77315.853)11018.982519.01519.00318.98918.98218.982(18.9419.025) 0.4906.1026.1036.0916.0846.0816.081(6.0696.13)958.23458.2648.2498.2388.2338.232(8.2068.263)10010.742510.77310.75910.75310.75010.750(10.71110.774)10513.62313.65313.63213.62213.61913.619(13.58813.658)11016.8216.83116.81616.81116.81016.810(16.78216.858) 0.2903.4513.4723.4643.4603.4593.458(3.4353.467)955.32955.3565.3485.3425.3405.339(5.315.349)1007.70957.7477.7397.7357.7337.733(7.6867.733)10510.61410.63810.62610.62310.62210.622(10.58710.641)11013.9513.98013.96913.96413.96313.962(13.9213.98)

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AitSahlia,F.,Carr,P.,1997.Americanoptions:Acomparisonofnumericalmethods.NumericalMethodsinFinance. AitSahlia,F.,Lai,T.,Winter1999.Acanonicaloptimalstoppingproblemforamericanoptionsanditsnumericalsolution.JournalofComputationalFinance3(2),33. AitSahlia,F.,Lai,T.,2001.Exerciseboundariesandefcientapproximationstoamericanoptionpricesandhedgeparameters.JournalofComputationalFinance4,85. Amin,K.,1991.Onthecomputationofcontinuoustimeoptionpricesusingdiscreteapproxima-tions.JournalofFinancialQuantitativeAnalysis26(4),477. Baba,N.,Gallardo,P.,May2008.OTCderivativesmarketactivityinthesecondhalfof2007.MonetaryandEconomicDepartment,BankforInternationalSettlements,http://www.bis.org/publ/otc hy0805.pdf. Black,F.,Scholes,M.,1973.Thepricingofoptionsandcorporateliabilities.JournalofPoliticalEconomy81,637. Boyle,P.,Broadie,M.,Glasserman,P.,1997.MonteCarlomethodsforsecuritypricing.JournalofEconomicDynamicsandControl21(8-9),1267. Boyle,P.P.,1986.Optionvaluationusingathree-jumpprocess.InternationalOptionsJournal3,7. Brennan,M.J.,Schwartz,E.S.,1977.Thevaluationoftheamericanputoption.JournalofFinance32,449. Broadie,M.,Detemple,J.,Ghysels,E.,Torres,O.,2000.Americanoptionswithstochasticdividendsandvolatility:Anonparametricinvestigation.JournalofEconometrics94,53. Broadie,M.,Detemple,J.B.,September2004.Optionpricing:Valuationmodelsandapplica-tions.ManagementScience50(9),1145. Broadie,M.,Glasserman,P.,2004.Astochasticmeshmethodforpricinghigh-dimensionalamericanoptions.JournalofComputationalFinance7(4),35. Carr,P.,Jarrow,R.,Myeni,R.,1992.Alternativecharecterizationsofamericanputoptions.MathematicalFinance2. Carriere,J.,1996.Valuationoftheearly-exercisepriceforderivativesusingsimulationsandsplines.Insurance:MathematicsandEconomics19,19. Chiarella,C.,Ziogas,A.,2005.Pricingamericanoptionsunderstochasticvolatility.In:11thAnnualConferenceonComputinginEconomicsandFinance,SocietyforComputationalEconomics,Washington,USA,June2005. 71

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Samuelson,P.A.,1965.Rationaltheoryofwarrantpricing.IndustrialManagementReview6,13. Talay,D.,1982.Howtodiscretizestochasticdifferentialequations.LectureNotesinMathemat-ics972:Springer-Verlag. Taylor,H.M.,1967.Evaluatingacalloptionandoptimaltimingstrategyinthestockmarket.ManagementScience14,111. Tsitsiklis,J.,VanRoy,B.,1999.Optimalstoppingofmarkovprocesses:Hilbertspacetheory,approximationalgorithms,andanapplicationtopricinghigh-dimensionalnancialderivatives.IEEETransactionsonAutomaticControl44. Tsitsiklis,J.N.,Roy,B.V.,2001.RegressionmethodsforpricingcomplexAmerican-styleoptions.IEEETransactionsonNeuralNetworks12(4),694. Tzavalis,E.,Wang,S.,2003.Pricingamericanoptionsunderstochasticvolatility:Anewmethodusingchebyshevpolynomialstoapproximatetheearlyexerciseboundary.DepartmentofEconometrics,QueenMary,UniversityofLondonWorkingPaper(488). WilliamH.Press,BrianP.Flannery,S.A.T.W.T.V.,1992.NumericalRecipesinC:TheArtofScienticComputing,2ndEdition.CambridgeUniversityPress. 73

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SuchandanGuhawasborninKolkata,India,in1980toMrs.HeeraGuhaandMr.SatyendraNathGuha.HenishedhisschoolinginKolkataandwenttotheIndianInstituteofTechnology,KharagpurforhisB.Tech.inmanufacturingscienceandengineeringandM.Tech.inindustrialengineeringandmanagement.SinceAugust2004,hehasbeenpursuinghisPh.D.inthedepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridaasagraduatealumnifellow.Hisareaofresearchismathematicalnanceandheiscurrentlyworkingonpricingofderivatives. 74