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Mathematical modeling and analysis of options with jump-diffusion volatility

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Title:
Mathematical modeling and analysis of options with jump-diffusion volatility
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Andreevska, Irena
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University of South Florida
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Subjects / Keywords:
Derivative pricing
Volatility
European option
American option
Partial integro-differential equation
Compound Poisson process
Fourier transform
Laplace transform
Mean-reversion
Dissertations, Academic -- Mathematics and Statistics -- Doctoral -- USF   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

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Abstract:
ABSTRACT: Several existing pricing models of financial derivatives as well as the effects of volatility risk are analyzed. A new option pricing model is proposed which assumes that stock price follows a diffusion process with square-root stochastic volatility. The volatility itself is mean-reverting and driven by both diffusion and compound Poisson process. These assumptions better reflect the randomness and the jumps that are readily apparent when the historical volatility data of any risky asset is graphed. The European option price is modeled by a homogeneous linear second-order partial differential equation with variable coefficients. The case of underlying assets that pay continuous dividends is considered and implemented in the model, which gives the capability of extending the results to American options. An American option price model is derived and given by a non-homogeneous linear second order partial integro-differential equation. Using Fourier and Laplace transforms an exact closed-form solution for the price formula for European call/put options is obtained.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2008.
Bibliography:
Includes bibliographical references.
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Mode of access: World Wide Web.
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by Irena Andreevska.
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Title from PDF of title page.
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Document formatted into pages; contains 76 pages.

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usfldc doi - E14-SFE0002343
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Mathematical modeling and analysis of options with jump-diffusion volatility
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ABSTRACT: Several existing pricing models of financial derivatives as well as the effects of volatility risk are analyzed. A new option pricing model is proposed which assumes that stock price follows a diffusion process with square-root stochastic volatility. The volatility itself is mean-reverting and driven by both diffusion and compound Poisson process. These assumptions better reflect the randomness and the jumps that are readily apparent when the historical volatility data of any risky asset is graphed. The European option price is modeled by a homogeneous linear second-order partial differential equation with variable coefficients. The case of underlying assets that pay continuous dividends is considered and implemented in the model, which gives the capability of extending the results to American options. An American option price model is derived and given by a non-homogeneous linear second order partial integro-differential equation. Using Fourier and Laplace transforms an exact closed-form solution for the price formula for European call/put options is obtained.
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Dissertation (Ph.D.)--University of South Florida, 2008.
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System requirements: World Wide Web browser and PDF reader.
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Adviser: Professor Yuncheng You, Ph.D.
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Derivative pricing.
Volatility.
European option.
American option.
Partial integro-differential equation.
Compound Poisson process.
Fourier transform.
Laplace transform.
Mean-reversion.
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u http://digital.lib.usf.edu/?e14.2343



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MathematicalModelingandAnalysisofOptions withJump-DiusionVolatility by IrenaAndreevska Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematicsandStatistics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:ProfessorYunchengYou,Ph.D. Wen-XiuMa,Ph.D. MarcusMcWaters,Ph.D. CarolWilliams,Ph.D. DateofApproval: April9,2008 Keywords:Derivativepricing,Volatility,Europeanoption,Americanoption,partial integro-dierentialequation,compoundPoissonprocess,Fouriertransform,Laplace transform,mean-reversion c Copyright2008,IrenaAndreevska

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Dedication Tomysister,whotaughtmehowtolearnandlovemath,andtomyhusbandforall thelove,supportandencouragementhehasgivenmewhileworkingonmy dissertation.

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Acknowledgments Severalpeopledeservemysinceregratitudefortheirunconditionalsupport,withoutwhomIwouldhavenotbeenabletocompletethisdissertation.Firstandforemost,Iwouldliketothankmyresearchandthesisadvisor,Dr.YunchengYou,for hisgeneroushelpandguidance.Hisvaluablecommentsandideasalwayskeptmeon therighttrack.Iamalsogratefulforthesupportofmyothercommitteemembers, Dr.Wen-XiuMa,Dr.MarcusMcWaters,andDr.CarolWilliams.Iwishtothank Dr.MyungKim,fortakingontheobligationofchairingmydefensecommittee.And last,Iwouldliketoexpressmygratitudeforhavingunderstanding,andsupportive, familyandfriends.

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TableofContents Abstractiv 1Introduction1 2Black-ScholesTheoryofDerivativePricing4 2.1StochasticCalculusDenitions,NotionsandTheorems........4 2.2Black-ScholesDerivativePricingModel.................7 2.3PricingAmericanOptionsIntheBlack-ScholesSettings........13 2.4SeveralExistingExtensionsoftheBlack-ScholesModel........18 3ModelingOptionswrittenonStockswithJump-DiusionVolatility23 3.1FormulationoftheEuropeanOptionsPricingModel.........23 3.2FormulationoftheAmericanOptionsPricingModel..........33 4ClosedFormSolutionofaPureDiusionOptionPricingModel39 4.1IntegralTransformsofthePureDiusionPricingModel.......40 4.2SolutionoftheTransformedPureDiusionPricingModel......43 4.3InverseIntegralTransformsofthePureDiusionPricingFormula..47 5ClosedFormSolutionofaJumpDiusionModelforEuropeanOptions55 5.1IntegralTransformsoftheJump-DiusionPricingModel.......55 5.2SolutionofthetransformedJump-DiusionPricingPIDE.......57 5.3InverseIntegralTransformsoftheJump-DiusionPricingFormula..61 i

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6OptionGreeks66 6.1Delta....................................66 6.2Gamma..................................68 6.3Theta...................................69 6.4Vega....................................71 7ResultsandConclusion73 References75 AbouttheAuthorEndPage ii

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ListofFigures ListofFiguresiii 1.1aHighreturnof39.1%ofS&P500duringtherst10monthsof1987 followedbyadropof20.4%onOctober19,1987leavingtheindexprice almostunchangedfromitslevelatthebeginingoftheyear bDropof82%inCompanyXYZ'sstockpriceinjustoneweek...2 2.1AnAmericancallonanassetpayingcontinuousdividendsisaliveonly withinthedomain f S; : S 2 [0 ;S C ; 2 ;T ] g .........15 2.2AnAmericanputonanassetpayingcontinuousdividendsisaliveonly withinthedomain f S; : S 2 S P ; 1 ; 2 ;T ] g ........16 2.3VolatilitysmileforS&P500calloptions.TheS&P500indexonJune 21,2006is$1252 : 20,withrateofreturn r =0 : 97%,andmaturitydate August23,2006..............................18 2.4aDailyreturnsforS&P500indexpricesbetween07/01/04and06/30/07. bDailyreturnsofCompanyXYZstockpricesbetween07/01/04and 06/30/07..................................19 iii

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MathematicalModelingandAnalysisofOptions WithJump-DiffusionVolatility IrenaAndreevska ABSTRACT Severalexistingpricingmodelsofnancialderivativesaswellastheeectsof volatilityriskareanalyzed.Anewoptionpricingmodelisproposedwhichassumes thatstockpricefollowsadiusionprocesswithsquare-rootstochasticvolatility.The volatilityitselfismean-revertinganddrivenbybothdiusionandcompoundPoisson process.Theseassumptionsbetterreecttherandomnessandthejumpsthatare readilyapparentwhenthehistoricalvolatilitydataofanyriskyassetisgraphed. TheEuropeanoptionpriceismodeledbyahomogeneouslinearsecond-orderpartial dierentialequationwithvariablecoecients.Thecaseofunderlyingassetsthatpay continuousdividendsisconsideredandimplementedinthemodel,whichgivesthe capabilityofextendingtheresultstoAmericanoptions.AnAmericanoptionprice modelisderivedandgivenbyanon-homogeneouslinearsecondorderpartialintegrodierentialequation.UsingFourierandLaplacetransformsanexactclosed-form solutionforthepriceformulaforEuropeancall/putoptionsisobtained. iv

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1Introduction Theinitialfocusofthedissertationistoanalyzethecurrentpricingmodelsofnancialderivatives,includingtheeectsofvolatilityriskanduncertainty.Mostoption pricingschemesformulatedtodatehavebeenbasedontheclassicalBlack-Scholes theory.BlackandScholeshavemodeledthestockpricewithastochastic dierentialequationdrivenbyageometricBrownianmotionandhavequantiedthe riskbyaconstantvolatilityparameter.Theconstantvolatilityassumptionisfrequentlyinvalidintheworldmarkets.Thereareanumberofextensionsoftheoriginal Black-Scholespricingmodelthathavebeenpursuedinpracticeandintheliterature whichconsiderotherformsofvolatility:timedependent,timeandstatedependent, andstochastic-discreteorcontinuous.PopularcontinuousstochasticvolatilitymodelsareoeredbyHullandWhitewhomodelthevolatilityasasquare-root functionthatfollowsgeometricBrownianmotion,ScottandSteinandStein useamean-revertingOUprocesstodescribethevolatilityfunctiontherst onewithexponentialandthesecondwithanabsolutevaluefunction.Allofthese researchersassumedthatthepriceoftheunderlyingassetanditsvolatilityareuncorrelated.Hestonreleasesthisassumptionwhenoeredamodelthatusesa square-rootvolatilityfunctionandavolatilityparameterthatfollowstheCIRprocessandallowsanon-zerocorrelationbetweenthestockandthevolatility.Allof thementionedextensionsassumecontinuouspathsofthestockprices.Inpractice, assetpricesoccasionallyjump.Atypicalexampleisthe1987marketcrash.Adaily moveof20%asintheS&P500isunlikelyinthelognormalmodelseeFigure1.1 a.Evenbeforethe"BlackMonday"thedayofthemarketcrash,Mertonin1976 accountsfornonlognormalbehaviorbyaddingdiscretejumpstotheassetpriceand 1

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keepingthevolatilityconstant.However,afterthemarketcrash,thereweremore attemptsamongresearcherstobuildmodelsthatallowlargemarketmovements,also knownasreturnswith"fattails",andatthesametimekeeptherandomnessofthe volatility.Batesin1996oeredajump-diusionoptionpricemodelwithastockprice thatfollowsajump-diusionprocessandstochasticvolatilitydrivenbyaBrownian motion.Jumpsinreturnscanexplainlargemovementstosomeextent,howeverthe impactofthesejumpsismomentary;today'sjumpinreturnshasnothingtodowith thefuturedistributionofreturns,andlargemovementswerepresentbothbeforeand afterOctober19,1987.Also,anegativedropof65%inonedayinthestockpriceof CompanyXYZrequiresreallyhighvolatilitythatthepurediusivevolatilitymodel cannotproduceseeFigure1.1b.Thus,theproposedmodelinthisdissertation Figure1.1:aHighreturnof39.1%ofS&P500duringtherst10monthsof1987followed byadropof20.4%onOctober19,1987leavingtheindexpricealmostunchangedfromits levelatthebeginingoftheyear bDropof82%inCompanyXYZ'sstockpriceinjustoneweek accountsforjumpsinvolatility.Itisbasedonastockpricestochasticdierential equationdrivenbyaBrownianmotionandavolatilitythatfollowsstochasticdierentialequationdrivenbybothBrownianmotionandacompoundPoissonprocess,in ordertobetterreecttherandomnessandthejumpsthatarereadilyapparentwhen thehistoricalvolatilitydataofanystockorriskyassetisgraphedorwhenlookingat thebehaviorovertimeofimpliedvolatilities.IntheBlack-Scholesmodelthemarket iscompletesothederivativescanbeperfectlyhedgedwiththeunderlyingasset,only. However,instochasticvolatilitymodelsthereismorethanonesourceofrandomness andsoperfecthedgingisimpossible.Thus,theuseofabenchmarkoptiontohedge 2

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theintendedoptionisnecessary.UsingthishedgingtechniqueinChapter3,pricing modelsforbothEuropeanandAmericanoptionsarederived.Thepricingmodels aregivenbyalinearsecond-orderpartialintregro-dierentialequation,therstone homogeneousandthesecondonewithanonhomogeneoustermthataccountsforthe extraprivilegesthattheAmericanoptionsoer.InChapter4wederiveanexact solutionofthehomogeneousPDEforthepurediusioncaseortheso-calledHeston's modelbyusingFourierandLaplacetransforms.Thesolutionobtainedisidentical totheonethatHestonprovides,towhichhearrivesbyguessingitsform.InChapter5thehomogeneousPDEinthejump-diusioncaseissolved.Thecalculationof theGreeksandtheirapplicationtofewinvestmentstrategiesaregiveninChapter6. Usefulstochasticcalculusdenitionsandtheoremsaswellasabriefintroductionto theBlack-ScholespricingmodelaregiveninChapter2. 3

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2Black-ScholesTheoryofDerivativePricing 2.1StochasticCalculusDenitions,NotionsandTheorems Inthissectionwediscussseveralstochasticprocessesandtheirproperties,widelyused inthenextchapter.Wealsodenequadraticvariationandcovariationprocessesfor semimartingalesandtheirproperties,aswellasthemulti-dimensionalIt^oformula. Denition2.1.1 Areal-valuedstochasticprocess B t isa standardBrownianmotion ifitsatisesthefollowingproperties: i B 0 =0 ; ii B t isacontinuousfunctionof t almostsurely; iii B t hasindependentnormallydistributedincrements: B t )]TJ/F27 11.9552 Tf 11.956 0 Td [(B s N ;t )]TJ/F27 11.9552 Tf 11.955 0 Td [(s ,for s 0 where N t isacountingprocesswithintensity ,and f J k ;k =1 ; 2 ;::: g areindependentidenticallydistributedrandomvariables,withdistribution G ,whicharealso independentof N t Intherestofthissectionassumethat X and Y aresemimartingalessuchthat X )]TJ/F19 11.9552 Tf 9.299 0 Td [(= Y )]TJ/F19 11.9552 Tf 9.298 0 Td [(=0. 4

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Denition2.1.3 The quadraticvariation processof X ,denotedby [ X;X ] ,is denedby [ X;X ] t = X t 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 Z t 0 X s dX s : .1.1 Example2.1.1 UsingthedenitionaboveandthedenitionoftheIt^ointegralit canbeshownthat[ t;t ]=0and[ B;B ] t = t where B isaBrownianmotion. Denition2.1.4 The pathbypathcontinuouspart [ X;X ] c of [ X;X ] isdened by [ X;X ] t =[ X;X ] c t + X 0 s t X s 2 : .1.2 Example2.1.2 Forthesemimartingale X = B + C ,where B isaBrownianmotion and C isacompoundPoissonprocess,thequadraticvariationisgivenby[ X;X ] t = t + N t X k =1 J 2 k ; sincethecontinuouspartis[ X;X ] c =[ B;B ]= t andthejumppartis X 0 s t X s 2 = X 0 s t C s 2 = N t X k =1 J 2 k : Denition2.1.5 The covariation processof X and Y isdenedbythefollowing polarizationidentity [ X;Y ]= 1 2 [ X + Y;X + Y ] )]TJ/F19 11.9552 Tf 11.955 0 Td [([ X;X ] )]TJ/F19 11.9552 Tf 11.955 0 Td [([ Y;Y ] : .1.3 Theorem2.1.6 If X isaquadraticpurejumpsemimartingale,thatis [ X;X ] c =0 and Y isanarbitrarysemimartingale,then [ X;Y ] t = X Y + X 0 s t X s Y s : Example2.1.3 ThetheoremaboveimpliesthatthecovariationofaBrownianmotion B andacompoundPoissonprocess C iszero,[ B;C ]=0,since B hasnojumpterm B s =0,for0
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TheIt^oformulaextendsthechangeofvariableformulaoftheclassicalcalculusto stochasticintegralswithsemimartingaleintegrators.Wewillusetheone-dimensional It^oformulatodevelopthestockpricemodelintheBlack-Scholessetting,aswellasthe multi-dimensionalIt^oformulatodevelopthestockpricemodelwithjump-diusive volatility. Theorem2.1.7 Multi-dimensionalIt^oformula. If X isavectorof d semimartingalesand g : R d R hascontinuoussecondorderpartialderivatives,then i g X isasemimartingale,and iitheintegralformoftheIt^oformulais g X t )]TJ/F27 11.9552 Tf 11.955 0 Td [(g X = d X i =1 Z t 0+ @g @x i X s )]TJ/F19 11.9552 Tf 9.299 0 Td [( dX i s + 1 2 d X i;j =1 Z t 0+ @ 2 g @x i @x j X s )]TJ/F19 11.9552 Tf 9.299 0 Td [( d [ X i ;X j ] c s + X 0
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Y isavectorin R d 0 consistingof d 0 real-valuedsemimartingales. Example2.1.4 Thestochasticdierentialequation dX t = cX t dt + X t dB t ;t 2 [0 ; ] ; .1.5 hasauniquesolution X t = X exp c )]TJ/F27 11.9552 Tf 11.956 0 Td [( 2 = 2 t + B t ; called geometricBrownianMotion process. 2.2Black-ScholesDerivativePricingModel SuggestedbySamuelsonandusedbyBlackandScholes,thepriceofarisk-freeasset bond t attime t canbedescribedbyanordinarydierentialequation d t = r t dt; where r istheinterestrateforlendingorborrowingmoney.Thepriceofarisky assetstock S t attime t withconstantrateofreturn ,constantvolatility andinnitesimalincrementsofBrownianmotion dW t ,ismodeledbyastochasticdierential equation dS t = S t dt + S t dW t : .2.6 WecanjustifyandnanciallyinterpretthisSDEsimplybydividingtheequation abovebythestockprice S t .Thentherighthandsideoftheinnitesimalreturn dS t =S t hasareturnterm dt andariskyterm dW t .UsingExample2.1.4weobtain thatthepriceoftheriskyasset S t attime t isgivenby S t = S exp )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 = 2 t + W t ; where S 0 istheinitialstockprice. 7

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Derivatives or contingentclaims arecontractsrelatedtoanunderlyingasset.We aremainlyinterestedinEuropeanandAmericanoptions. A EuropeanCallOption isacontractthatgivestheholdertheright,butnotthe obligation,tobuyoneunitofanunderlyingassetforapredetermined strikeprice K onthe maturity date T .ThepayofunctionoftheEuropeancalloptionis h S T = max S T )]TJ/F27 11.9552 Tf 11.955 0 Td [(K; 0= S T )]TJ/F27 11.9552 Tf 11.956 0 Td [(K + where S T istheunderlyingassetpriceatmaturity T A EuropeanPutOption isacontractthatgivestheholdertheright,butnotthe obligation,toselloneunitofanunderlyingassetforapredetermined strikeprice K onthe maturity date T .ThepayofunctionoftheEuropeanputoptionis h S T = max K )]TJ/F27 11.9552 Tf 11.955 0 Td [(S T ; 0= K )]TJ/F27 11.9552 Tf 11.955 0 Td [(S T + : An AmericanCall Put Option isacontractthatgivestheholdertheright,but nottheobligation,tobuyselloneunitofanunderlyingassetforapredetermined strikeprice K atanytimeofone'schoicebeforetheoption'sexpirationdate T .The time atwhichtheoptionisexercisediscalledthe exercisetime BlackandScholeshavederivedapartialdierentialequationthatholdsforthe priceofanyderivativeonanon-dividendpayingstock.Thederivationisbased onthemaineconomicprinciples:no-arbitrageandthecreationofrisklessportfolio. Theprincipleof no-arbitrage saysthatinaperfectlyliquidmarketitispossibleto buyandsellanynitequantityoftheunderlyingassetatanytimethereexistno opportunitiestoearnarisk-freeprotfreelunch.Also,theyhaveassumedthatthe tradingiscontinuousintime,andtherearenotransactioncostsortaxes. Supposethereexistsapricingfunction P t;S t ofaEuropeanoptionwithmaturity T andapayofunction h S T withenoughregularitythatwecanapplytheone8

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dimensionalIt^oformulaandobtain: dP t;S t = S t @P @S + @P @t + 1 2 2 S 2 t @ 2 P @S 2 dt + S t @P @S dW t : .2.7 Constructaportfoliobyholdingoneoptionandselling a t unitsoftheriskyasset S t Thevalueofthisportfolioattime t is: = P )]TJ/F27 11.9552 Tf 11.955 0 Td [(aS: Thechangeofthevalueoftheportfolioinasmalltimeinterval dt isgivenby: d = dP )]TJ/F27 11.9552 Tf 11.955 0 Td [(adS: Notethatwedonotdierentiate a = a t;S becauseitisbeingxedduringthistime interval.Substituting.2.6and.2.7intheequationabove,weobtain: dP )]TJ/F27 11.9552 Tf 11.955 0 Td [(adS = S @P @S + @P @t + 1 2 2 S 2 @ 2 P @S 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(aS dt + S @P @S )]TJ/F27 11.9552 Tf 11.955 0 Td [(aS dW: Choosing a = @P @S calleddelta-hedgeratioweeliminatetheriskypartthatcomes fromthepresenceoftheBrownianmotionincrement,asahedgingstrategy.Since wehaveassumedthatthereisnoarbitrageopportunity,theportfoliomustgrowata risk-freerate,hence d = r dt .NowtheequationaboveresultsintheBlack-Scholes PDE @P @t + 1 2 2 S 2 @ 2 P @S 2 + rS @P @S )]TJ/F27 11.9552 Tf 11.955 0 Td [(rP =0 ; 0 t T; 0 S< 1 ; .2.8 withaterminalcondition P T;S = h S T .Theliteraturepresentstwoapproachesin solvingtheBlack-ScholesPDE:themartingaleapproachandtheapproachofreductiontoaheatequation.InthissectionwearesolvingthisequationusingaFourier transform.First,set S = e x and = T )]TJ/F27 11.9552 Tf 11.955 0 Td [(t ,thenthePDE.2.8becomes @P @ = 1 2 2 @ 2 P @x 2 + r )]TJ/F19 11.9552 Tf 13.15 8.088 Td [(1 2 2 @P @x )]TJ/F27 11.9552 Tf 11.955 0 Td [(rP; 0 T; )-222(1
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withinitialcondition P ;x = h x = 8 < : max f e x )]TJ/F27 11.9552 Tf 11.955 0 Td [(K; 0 g forcalloptions max f K )]TJ/F27 11.9552 Tf 11.955 0 Td [(e x ; 0 g forputoptions : DenetheFouriertransform F [ f x ]tobe F = 1 2 Z 1 f x e i!x dx: Thenthefollowingpropertieshold F @f @t = @F @t ; F @f @x = )]TJ/F27 11.9552 Tf 9.298 0 Td [(i!F; F @ 2 f @x 2 = )]TJ/F27 11.9552 Tf 9.298 0 Td [(! 2 F: ApplyingaFouriertransformwithrespectto x toequation.2.9andusingthe propertiesabove,thefollowinglinearPDEisobtained: @ ^ P @ = 1 2 i! 2 )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(1 2 2 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(i!r )]TJ/F27 11.9552 Tf 11.955 0 Td [(r ^ P whosesolutionisgivenby ^ P ;! = C 1 e 1 2 i! 2 )]TJ/F26 5.9776 Tf 7.782 3.258 Td [(1 2 2 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(i!r )]TJ/F28 7.9701 Tf 6.587 0 Td [(r : .2.10 Theconstant C 1 canbedeterminedusingtheinitialcondition C 1 = F [ h x ] : NowapplyingtheinverseFourierTransformto.2.10weget P ;x = F )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 h F [ h x ] e 1 2 i! 2 )]TJ/F26 5.9776 Tf 7.782 3.258 Td [(1 2 2 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(i!r )]TJ/F28 7.9701 Tf 6.587 0 Td [(r i : Setting G = e 1 2 i! 2 )]TJ/F26 5.9776 Tf 7.782 3.259 Td [(1 2 2 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(i!r )]TJ/F28 7.9701 Tf 6.587 0 Td [(r ,theinverseFouriertransform g x = F )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 [ G ] 10

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canbecalculatedusingtheproperty F )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 h e )]TJ/F28 7.9701 Tf 6.586 0 Td [(! 2 i = r e )]TJ/F28 7.9701 Tf 6.587 0 Td [(x 2 = 4 andcompletingthesquarein G G = e )]TJ/F29 5.9776 Tf 7.782 3.258 Td [( 2 2 + 1 2 i r )]TJ/F29 5.9776 Tf 5.757 0 Td [( 2 2 2 e )]TJ/F28 7.9701 Tf 6.586 0 Td [( 2 +2 r 2 8 2 ; yielding g x = r 2 2 e )]TJ/F34 5.9776 Tf 7.782 16.587 Td [( x )]TJ/F29 5.9776 Tf 6.952 2.76 Td [( 2 )]TJ/F26 5.9776 Tf 5.756 0 Td [(2 r 2 2 2 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(r : Bytheconvolutiontheorem,foraEuropeancalloption,wehave P ;x = F )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 [ F [ h x ] F [ g x ]] = 1 2 Z 1 h x )]TJ/F27 11.9552 Tf 11.955 0 Td [(w g w dw = 1 p 2 Z x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [(e x )]TJ/F28 7.9701 Tf 6.586 0 Td [(w )]TJ/F27 11.9552 Tf 11.955 0 Td [(K e )]TJ/F34 5.9776 Tf 7.782 16.587 Td [( w )]TJ/F29 5.9776 Tf 6.952 2.76 Td [( 2 )]TJ/F26 5.9776 Tf 5.756 0 Td [(2 r 2 2 2 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(r dw = e x p 2 Z x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K e )]TJ/F34 5.9776 Tf 7.782 16.586 Td [( w + 2 +2 r 2 2 2 2 dw )]TJ/F27 11.9552 Tf 20.721 8.088 Td [(e )]TJ/F28 7.9701 Tf 6.586 0 Td [(r p 2 Z x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K Ke )]TJ/F34 5.9776 Tf 7.782 16.586 Td [( w )]TJ/F29 5.9776 Tf 6.951 2.76 Td [( 2 )]TJ/F26 5.9776 Tf 5.756 0 Td [(2 r 2 2 2 2 dw since h x )]TJ/F27 11.9552 Tf 11.955 0 Td [(w =0when w x )]TJ/F19 11.9552 Tf 11.955 0 Td [(ln K .Setting d 1 = 1 p x )]TJ/F19 11.9552 Tf 11.956 0 Td [(ln K + r + 1 2 2 ; d 2 = d 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( p andsubstituting x =ln S and = T )]TJ/F27 11.9552 Tf 12.761 0 Td [(t weobtainaclosedformsolutionofthe Black-ScholesPDEthatrepresentsapriceofaEuropeancalloption P call t;S = S d 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(Ke )]TJ/F28 7.9701 Tf 6.586 0 Td [(r T )]TJ/F28 7.9701 Tf 6.586 0 Td [(t d 2 ; .2.11 whereisthecumulativestandardnormaldistributionfunction.Usingthe put-call 11

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parity arelationshipbetweenputandcalloptionswiththesamematurity T andthe samestrikeprice K P call t;S )]TJ/F27 11.9552 Tf 11.955 0 Td [(P put t;S = S )]TJ/F27 11.9552 Tf 11.955 0 Td [(Ke )]TJ/F28 7.9701 Tf 6.587 0 Td [(r T )]TJ/F28 7.9701 Tf 6.586 0 Td [(t ; .2.12 theEuropeanputoptionpricingformulacanbeobtained P put t;S = Ke )]TJ/F28 7.9701 Tf 6.587 0 Td [(r T )]TJ/F28 7.9701 Tf 6.586 0 Td [(t )]TJ/F27 11.9552 Tf 9.299 0 Td [(d 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(S )]TJ/F27 11.9552 Tf 9.299 0 Td [(d 1 : Itisworthpointingoutthatthedriftterm doesn'tappearintheB-Spricing formula,whichmeansthevalueoftheoptiondoesn'tdependontheinvestors'risk preferences.Thereasonforthisistheperfecthedgingstrategywhichallowscomplete eliminationoftherisk.Thisobservationofrisk-neutralityisamajorbreakthroughin theoptionpricingtheory. TheBlack-Scholesformulacanalsobederivedusingtherisk-neutralpricingmethod, takingthepriceoftheoptiontobetherisk-neutralexpectedvaluediscountedatthe risk-freeinterestrate: P t;S = e )]TJ/F28 7.9701 Tf 6.586 0 Td [(rT E Q [ h S T ] where Q isthesocalled equivalentmartingalemeasure ,aprobabilitymeasureequivalenttotheobjectiveprobability P ,underwhichithediscountedprice ~ S t = e )]TJ/F28 7.9701 Tf 6.587 0 Td [(rt S t isamartingale,andiitheexpectedvalueofthediscountedpayoofaderivative givesitsno-arbitrageprice.Next,wewillndtherisk-neutralmeasure Q : d ~ S t = d )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [(e )]TJ/F28 7.9701 Tf 6.586 0 Td [(rt S t = )]TJ/F27 11.9552 Tf 9.299 0 Td [(re )]TJ/F28 7.9701 Tf 6.586 0 Td [(rt S t dt + e )]TJ/F28 7.9701 Tf 6.586 0 Td [(rt dS t = )]TJ/F27 11.9552 Tf 11.955 0 Td [(r ~ S t dt + ~ S t dW t : For ~ S t tobeamartingalewewillabsorbthedrifttermintothemartingaleterm: d ~ S t = ~ S t dW t + )]TJ/F27 11.9552 Tf 11.955 0 Td [(r dt 12

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where = )]TJ/F28 7.9701 Tf 6.587 0 Td [(r iscalledthe marketpriceofassetrisk 1 orthe Sharperatio theratio oftheriskpremiumtovolatility.Dene dW t = dW t + t; then d ~ S t = ~ S t dW t isamartingale.UsingtheGirsanov'stheorem,auniqueequivalentmartingalemeasure Q willbeobtained: d Q d P = exp )]TJ/F19 11.9552 Tf 10.494 8.087 Td [(1 2 Z T 0 2 dt )]TJ/F32 11.9552 Tf 11.956 16.272 Td [(Z T 0 dW t = exp )]TJ/F19 11.9552 Tf 10.494 8.088 Td [(1 2 2 T )]TJ/F27 11.9552 Tf 11.955 0 Td [(W T : ThestockpriceSDEundertherisk-neutralmeasureisobtainedbyreplacingthereal worldrateofreturn withtherisk-freeinterestrate r : dS t = rS t dt + S t dS t : .2.13 Therateofreturnundertherisk-neutralmeasureshouldequaltherealrateofreturn minusthetotalassetrisk )]TJ/F27 11.9552 Tf 13.151 8.087 Td [( )]TJ/F27 11.9552 Tf 11.955 0 Td [(r = r: 2.3PricingAmericanOptionsIntheBlack-ScholesSettings In1973,MertonrelaxedoneoftheassumptionsintheBlack-Scholesmodelbyconsideringanassetpayingcontinuousdividendsatrate q .Thedividendpaymentreduces thegrowthofthestockpricefrom S t to S t e )]TJ/F28 7.9701 Tf 6.587 0 Td [(q T )]TJ/F28 7.9701 Tf 6.587 0 Td [(t sothatthepricingmodelof options ondividend-payingstock becomes @P @t + 1 2 2 S 2 @ 2 P @S 2 + r )]TJ/F27 11.9552 Tf 11.955 0 Td [(q S @P @S )]TJ/F27 11.9552 Tf 11.956 0 Td [(rP =0 ; 0 t T: .3.14 1 Ifthestatevariable X t followstheprocess dX t = P X t dt + X t dB P t where W P t isaBrownianmotion undertheobjectiveprobabilitymeasure P andthereexistsequivalentprobabilitymeasure Q suchthat X t under Q isgivenby dX t = Q X t dt + X t dB Q t ,thenthemarketpriceofriskprocessisdenedby \050 X t =[ X t ] )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 [ P X t )]TJ/F54 8.9664 Tf 9.215 0 Td [( Q X t ]. 13

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Usingtheappropriateboundaryconditions,thepriceofaEuropeancalloptionona dividend-payingassetcanbeobtained P div:call t;S;K = Se )]TJ/F28 7.9701 Tf 6.587 0 Td [(q T )]TJ/F28 7.9701 Tf 6.586 0 Td [(t d 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(Ke )]TJ/F28 7.9701 Tf 6.586 0 Td [(r T )]TJ/F28 7.9701 Tf 6.586 0 Td [(t d 2 ThepriceofaEuropeanputoptioncanbeeasilydeterminedusingtheput-callparity foroptionsonadividend-payingstock P call t;S;K )]TJ/F27 11.9552 Tf 11.955 0 Td [(P put t;S;K = Se )]TJ/F28 7.9701 Tf 6.587 0 Td [(q T )]TJ/F28 7.9701 Tf 6.586 0 Td [(t )]TJ/F27 11.9552 Tf 11.956 0 Td [(Ke )]TJ/F28 7.9701 Tf 6.586 0 Td [(r T )]TJ/F28 7.9701 Tf 6.587 0 Td [(t : ThismodelbecomesextremelyusefulwhenextendingthepricingresultstoAmericanoptions.Whentheunderlyingassetpaysnodividends,anearlyexerciseofthe Americancalloptionisalwaysundesirable,andhereiswhy:rst,theprivilegeofan earlyexerciseoftheAmericanoptions,inadditiontoalltherightsthattheEuropean optionshave,makestheAmericanoptionsworthatleasttheirEuropeancounterpart, P A T;S;K P E T;S;K : .3.15 Thisextracostiscalledan earlyexercisepremium .Second,thePut-Callparity implies P E call T )]TJ/F27 11.9552 Tf 11.956 0 Td [(t;S;K = S t )]TJ/F27 11.9552 Tf 11.956 0 Td [(K | {z } exercisevalue + P E put T )]TJ/F27 11.9552 Tf 11.956 0 Td [(t;S;K | {z } insuranceagainstS T < K + K )]TJ/F27 11.9552 Tf 11.955 0 Td [(e )]TJ/F28 7.9701 Tf 6.587 0 Td [(r T )]TJ/F28 7.9701 Tf 6.587 0 Td [(t | {z } timevalueofmoneyonK >S t )]TJ/F27 11.9552 Tf 11.955 0 Td [(K; .3.16 sinceboth,thevalueoftheputoptionandthetimevalueonthestrikeK,arepositive forall tS t )]TJ/F27 11.9552 Tf 11.955 0 Td [(K; whichmeansthatifweexercisetheAmericancallpriortotime T wewillreceive S t )]TJ/F27 11.9552 Tf 12.108 0 Td [(K whichislessthan P A call T )]TJ/F27 11.9552 Tf 12.109 0 Td [(t;S;K ,theamountthatwewouldreceiveifwe justselltheAmericancall.However,earlyexerciseofanAmericanputoptionon 14

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anon-dividendpayingassetmightbepreferable.Onceagain,wedemonstratethat usingtheput-callparityimplication P put T )]TJ/F27 11.9552 Tf 11.955 0 Td [(t;S;K = P call T )]TJ/F27 11.9552 Tf 11.956 0 Td [(t;S;K )]TJ/F27 11.9552 Tf 11.955 0 Td [(S t + Ke )]TJ/F28 7.9701 Tf 6.587 0 Td [(r T )]TJ/F28 7.9701 Tf 6.587 0 Td [(t : .3.17 Arguingasbefore,theputwillneverbeexercisedaslongas P A put T )]TJ/F27 11.9552 Tf 9.93 0 Td [(t;S;K >K )]TJ/F27 11.9552 Tf 9.931 0 Td [(S t Thisinequalityandrelationship.3.17imply P call T )]TJ/F27 11.9552 Tf 11.955 0 Td [(t;S;K >K )]TJ/F27 11.9552 Tf 11.955 0 Td [(e )]TJ/F28 7.9701 Tf 6.586 0 Td [(r T )]TJ/F28 7.9701 Tf 6.586 0 Td [(t : Thismeansthatwhenthetimevalueof K exceedstheinsurancevalueoftheput whenacompanyisgoingbankruptthecallvaluebecomesalmostvaluelesswecannot ruleoutearlyexercise. Wewouldliketoexerciseoptionsearlybecausewewanttoreceivesomething soonerratherthanlater.Whenweexercisecalloptionswereceivestocksowhen thestockpaysdividendsitisnormalthatwewouldpreferearlyexercise.When exercisingputoptionswereceiveanamount K ,sowhenexercisingputsearlywecan earninterestonthisamount.Ifwelookattheinterestasadividendoncash,we maysaydividendsaretheonlyreasonforearlyexercise.Ithasbeenshownthatearly Figure2.1:AnAmericancallonanassetpayingcontinuousdividendsisaliveonlywithin thedomain f S; : S 2 [0 ;S C ; 2 ;T ] g exerciseofoptionsonadividendpayingassetisoptimalonlywhentheassetprice S 15

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atagiventimetoexpiry ,risesaboveforcallsorfallsbelowforputssomecritical assetvalue S called optimalexerciseprice Figures2.1and2.2.Thecollection ofthesecriticalassetvaluesformsacurveknownas optimalexerciseboundary ,which wewilldenoteby a .If S isaknownfunction,theAmericanoptionpricing problembecomesaboundaryvalueproblemwithtimedependentboundary. Figure2.2:AnAmericanputonanassetpayingcontinuousdividendsisaliveonlywithin thedomain f S; : S 2 S P ; 1 ; 2 ;T ] g Theput-callparitydoesn'tholdforAmericanoptions,howeverausefulput-call symmetryrelationforthepricesoftheAmericancallandputoptionsaswellasa relationfortheiroptimalexercisepriceshavebeenestablished.Theyaregivenwith theexpressions P A ;S ; K;r;q = P A ;K ; S;q;r ; .3.18 S C ; r;q = K 2 S P ; q;r : ToderivepricingformulasforAmericanoptionsonadividend-payingassetwewill considertheeectsofacontinuous 1 dividendyieldataconstantrate q .Thepriceof anAmericanoptionismodeledwiththeboundaryvalueproblem @P @ )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(1 2 2 S 2 @ 2 P @S 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [( r )]TJ/F27 11.9552 Tf 11.955 0 Td [(q S @P @S + rP =0 ; .3.19 intheboundedregion0 S a forcallsand S a forputs,and0 T 1 Fordiscretedividendscheck[14] 16

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subjecttotheinitialandboundaryconditions P call ;S =max S )]TJ/F27 11.9552 Tf 11.955 0 Td [(K ;P put ;S =max K )]TJ/F27 11.9552 Tf 11.955 0 Td [(S P call ; 0=0 ;P put ; 0=0 P call ;a = a )]TJ/F27 11.9552 Tf 11.955 0 Td [(K;P put ;a = K )]TJ/F27 11.9552 Tf 11.955 0 Td [(a lim S a @P call @S =1 ; lim S a @P put @S = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 : Whenever S>a ,theAmericancallvalueissimplyitsintrinsicvalue S )]TJ/F27 11.9552 Tf 11.983 0 Td [(K ,and inthecaseoftheAmericanput,for Sa rK )]TJ/F27 11.9552 Tf 11.955 0 Td [(qS; if S a foraputoption. ThesolutionoftheAmericanoptionpricingPDEinBlack-Scholessettingcan beobtainedbyrstsubstituting S by e x ,thentakingFouriertransformofthePDE abovewithrespecttothespatialvariableandnallyapplyingDuhamel'sprinciple. Thedetailswillnotbepresentedhere,sincetheextensionofthisresultinajumpdiusionvolatilitysettingisleftforafuturework. 17

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2.4SeveralExistingExtensionsoftheBlack-ScholesModel TheBlack-ScholesmodelisbasedonageometricBrownianmotionanditisveryusefulasarstapproximationofthepricechange.Theproblemswiththismodelarethe assumptionsthatthevolatilityiskeptconstant,tradingtakesplacecontinuouslyin timeandthatthestockpricedynamicshasacontinuoussamplepathwithprobability one.Therehavebeenseveralattemptsamongtheresearcherstorelaxtheseassumptions,bydeningalternativestochasticprocessesforthestockpriceand/orspecifying deterministicorstochasticmodelsforthestockpricevolatility.AwellknowndiscrepancybetweentheBlack-Scholesoptionpricesandthemarkettradedoptionpricesis the smile/skewcurve obtainedwhenthe impliedvolatilityI isgraphedagainstthe strikeprice K .The impliedvolatilityI isaquantityusedtocomparecertainmodel predictionsandobservations,andisdenedtobethevalueofthevolatilityparameter which,whenpluggedintheBSformula,theobservedmarketpriceandtheBSoption pricecoincide: P BS t;S ; K;T;I = P obs : Thesmileeectshowsthattheimpliedvolatilitiesofmarketpricesarenotconstant Figure2.3:VolatilitysmileforS&P500calloptions.TheS&P500indexonJune21,2006 is$1252 : 20,withrateofreturn r =0 : 97%,andmaturitydateAugust23,2006 18

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butdependonthestrikepriceandthematuritytime,andisillustratedinFigure 2.3withdatatakenfromS&P500indexoptions.Anotherassumptionmadebythe Black-Scholesmodelisthatthestockreturnsarenormallydistributed.However, empiricalstudiesareshowingthatthetruedistributionismoreskewedthanthe normaldistributionandithasfattertailsseeFigure2.4. Figure2.4:aDailyreturnsforS&P500indexpricesbetween07/01/04and06/30/07. bDailyreturnsofCompanyXYZstockpricesbetween07/01/04and06/30/07. OneattempttomodifytheBlack-Scholesmodelsoitreectstherealmarket behaviorisdonebyassumingthatvolatilityisapositivedeterministicfunctionof timeandstockprice,wheretheSDEmodelingthestockpriceis dS t = S t dt + t;S t S t dW t : Dierentchoicesof t;S t haveprovidedseveralmodelsintheliterature.Worth mentioningisthe ConstantElasticityofVariance modelbyHullinitially suggestedbyCoxandRoss,inwhich t;S t = S )]TJ/F28 7.9701 Tf 6.587 0 Td [( t ,0 1.The modelachievesthevolatilitysmileeect,butthedisadvantageisthatthestockprice andthevolatilitychangesareperfectlycorrelated.Anotherspecialcaseisthe TimeDependentVolatility model,sothat t;S t = t .Inthiscasetheoptionpricecan becomputedusingtheBlack-Scholesformulawithvolatilityparameter p 2 ,where 2 = 1 T )]TJ/F27 11.9552 Tf 11.955 0 Td [(t Z T t 2 d 19

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isthetimeaveragedvolatility.Underthismodel,alloptionswiththesamematurity andxedtime t havethesamevolatilitysothesmileeectisnotpresent.However, theimpliedvolatilityvarieswithtimetomaturity,since 2 isdierentfordierent maturitytimes.Ingeneral,theproblemwithalldeterministicvolatilityfunction modelsisthestabilityofthefunctionovertime:itmayworkwiththisweek'sdata butthenextweek'sdatawillsuggestcompletelydierentvolatilityfunction. Theempiricalstudiesofstockpricesalsorevealthattheestimatedvolatilityhas "random"behavior.Axforthisaswellasthefat-tailedreturnsisthestochastic volatilitySVmodeling.IntheSVmodelstheassetpriceisgivenbytheSDE dS t = S t dt + t S t dW t ; .4.21 where t isapositivefunctionofastochasticprocess Y t .Byletting Y t bedriven byasecondBrownianmotion Z t weareachievingthe"notperfect"correlationof thevolatilityandthestockprice.However,thismakesthemarketincompleteand thus,morecomplicatedcalculationsandmodelderivationarerequired.Monitoring thebehaviorofhistoricalandimpliedvolatilitiessuggestthatthevolatilitytendsto goupforacertainperiodoftime,thendropsdownforsimilartimespan,thengoes upagain,andsoon.Inotherwords,thevolatilityrevertsarounditsmean,whichis modeledby dY t = m )]TJ/F27 11.9552 Tf 11.955 0 Td [(Y t dt + :::dZ t : Intheequationabove iscalledtherateofmeanreversion, m isthelong-runmean of Y ,and Z t isaBrownianmotionsuchthat corr W t ;Z t = .Thereareeconomic argumentsforanegativecorrelationbetweenvolatilityandassetprices.Inmost modelsitistakentobeaconstantbetween )]TJ/F19 11.9552 Tf 9.298 0 Td [(1and1.ThereareseveralSVmodels studiedindetailintheliteratureandusedinpractice.Allmodelsmentionedbelow useequation.4.21fortheassetprice,butusedierentvolatilityfunctionsdriven bydierentstochasticprocesses: 1. Hull-White .ThisistherstSVmodelintheliterature,inwhichthe 20

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volatilityfunctionis f y = p y anditisdrivenbyageometricBrownianmotion dY t = c 1 Y t dt + c 2 Y t dZ t : AssuminguncorrelatedBrownianmotions,HullandWhitehavederivedaclosedformsolutionforthismodel. 2. Scott and Stein-Stein .Bothmodelsassumeavolatilitydrivenby amean-revertingOrnstein-Uhlenbeckprocess dY t = m )]TJ/F27 11.9552 Tf 11.955 0 Td [(Y t dt + dZ t ; butdierentvolatilityfunctions;Scotthasused f y = e y andSteinandStein haveused f y = p y .Inbothcasesaclosed-formsolutionhasbeenderived, assuminguncorrelatedBrownianmotions, corr W t ;Z t =0. 3. Heston and Ball-Roma .TheCox-Ingersoll-RossCIRprocessis usedasavolatilitydrivingprocess dY t = m )]TJ/F27 11.9552 Tf 11.956 0 Td [(Y t dt + p Y t dZ t ; and f y = p y asavolatilityfunction.Thisformulationhastheadvantageof strictlypositivevolatilityaslongas m )]TJ/F28 7.9701 Tf 11.774 5.256 Td [( 2 2 0Bu[7].BallandRomahave consideredthecaseofuncorrelatedBrownianmotions.TheHeston'smodelis themostpopularintheliteratureandinpractice,sinceitassumesacorrelationoftheBrownianmotions, corr W t ;Z t = where 2 [ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1]andgivesa closed-formsolutionbyguessingtheformofthepricingformulaandtheformof thecharacteristicfunctionsoftherisk-neutralprobabilitiesusedinthepricing formula.Modelsthatoerclosedformsolutionsaremoreattractiveforthemarketmakerssincetheyneedlesscomputationtime.AlthoughHeston'spricing formulainitsclosedformleavesaninniteintegraltobesolvedbyanumerical method,itisstillmuchfasterthanotherSVmodels. 21

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Thestochasticvolatilitymodelscorrecttheconstantvolatilityassumptioninthe Black-Scholesmodelandgenerateleptokurticreturndistribution,buttheydon'taccountforpossiblejumpsinassetpriceandvolatility.Largepricechangesarepresent inthenancialmarkets.Theycanbecausedbyspeculations,companiesearningscall, expectationofanewproduct,ortherecentbadjudgmentofrealestatelenders.These largepricemovementscannotbegeneratedbypurediusionprocesses.Bates[18] ndsthatpurediusionmodelswillneedimplausiblyhighvolatilitylevelstoexplain them.Eraker[20]concludesthataddingjumpsinthestockpriceSDEcanexplain somebutnotallofthemarketmovementspresentbeforeandafterthe1987market crash.InhisjointworkwithJohannesandPolsonheinvestigatestheperformanceofmodelswithjumpsinpricesandvolatilitypointingoutthesignicanceof jumpsinvolatility. Alloftheaboveinitiatedthenecessityofincorporatingjumpsinthepricingmodel andderivingaclosedformpricingformula.Thederivationprocessandtheresults arepresentedinthenextthreechapters. 22

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3ModelingOptionswrittenonStockswithJump-Diffusion Volatility 3.1FormulationoftheEuropeanOptionsPricingModel Themarket"jump"phenomenaareoftenbestmodeledasvolatilityjumpprocesses [2].Consideringthisfactaswellastherandomcharacteristicofthevolatility,itseems naturaltoproposeapricingmodelwithassetpricedrivenbyageometricBrownian motion dS t = S t dt + t S t dW t .1.1 wherethevolatilityisapositiverealfunction t = f Y t drivenbyamean-reverting jump-diusionprocess Y t : dY t = m )]TJ/F27 11.9552 Tf 11.955 0 Td [(Y t dt + p Y t dZ t + dC t : .1.2 Theprocesses W t and Z t arecorrelatedBrownianmotions, corr W t ;Z t = isthe rateofmean-reversion, m isthelong-termvolatilitymean, isthevolatilityofthe volatilityprocess.Theprocess C t = N t X k =1 J k ;t> 0,isacompoundPoissonprocess withintensity ,and J k ;k =1 ; 2 ;::: areindependentidenticallydistributedrandom variableswithdistribution J k .Thesumof dN t jumpsisthecompoundPoisson process dC t thatusuallyhassymbolicnotation JdN t = dN t X k =1 J k : 23

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Theinnitesimalmomentsofthejumpprocessare E [ JdN t ]= E [ J ] dt and Var [ JdN t ]= E [ J 2 ] dt .Also,weassumethatthediusionprocessesareuncorrelated tothejumpprocess,thatis corr W t ;C t =0and corr Z t ;C t =0. Thepriceoftheoption P isafunctionof t S t and Y t .Assumingthat P t;S t ;Y t hascontinuoussecondorderpartialderivativesandusingthedierentialformofthe multi-dimensionalIt^oformulagiveninTheorem2.1.7,wehave: dP t;S t ;Y t = @P @t dt + @P @S dS t + @P @y dY t + 1 2 @ 2 P @S 2 d [ S;S ] c t + @ 2 P @S@y d [ S;Y ] c t + 1 2 @ 2 P @y 2 d [ Y;Y ] c t + P Y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P Y )]TJ/F27 11.9552 Tf 13.151 8.088 Td [(@P @y Y dN t ; .1.3 where dN t = 8 < : 0withprobability p =1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(dt ,jumpdoesn'toccur 1withprobability p = dt ,jumpoccurs : Proposition3.1.1 Thecontinuouspartofthequadraticvariationof S ,where S is givenbySDE.1.1is [ S;S ] c = S 2 + Z t 0 S 2 f 2 Y d: .1.4 Proof. First,notethat S t hasnojumpterm,hence[ S;S ] c t =[ S;S ] t .Next,the stochasticintegralequationthatcorrespondstoSDE.1.1isgivenby S t = S | {z } D + Z t 0 S d | {z } A t + Z t 0 S dW | {z } M t ;t 0 Bythelinearityofquadraticvariation,wehave [ S;S ]=[ D;D ]+[ D;A ]+[ D;M ]+[ A;D ]+[ A;A ]+[ A;M ]+[ M;D ]+[ M;A ]+[ M;M ]. Allcovariationswitharguments D and A or D and M arezerobecause D = S isa constantand A =0= M .Thequadraticvariation[ A;A ]=0because d [ t;t ]=0 24

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and[ A;M ]=[ M;A ]=0since d [ t;W ]= d [ W;t ]=0seeExample2.1.1.Thus [ S;S ] c =[ D;D ]+[ M;M ] ; equivalentto [ S;S ] c = S 2 + Z t 0 S 2 f 2 Y d [ W;W ] | {z } d = S 2 + Z t 0 S 2 f 2 Y d: Proposition3.1.2 Thecontinuouspartofthequadraticvariationof Y ,where Y is givenbytheSDE.1.2is [ Y;Y ] c = Y 2 + 2 Z t 0 Y d: .1.5 Proof. TheStochasticIntegralEquationthatcorrespondstoSDE.1.2isgivenby Y t = Y | {z } D + Z t 0 m )]TJ/F27 11.9552 Tf 11.955 0 Td [(Y d | {z } A t + Z t 0 p Y dZ | {z } M t + Z t 0 dC | {z } F t : .1.6 Since D isaconstantand A = M = F =0,thecovariations[ D;A ],[ D;M ], [ D;F ],[ A;D ],[ M;D ],[ F;D ]equal0.TheresultobtainedinExample2.1.1implies [ A;A ]=0 d [ t;t ]=0and[ A;M ]=[ M;A ]=0 d [ t;Z ]= d [ Z;t ]=0.InExample 2.1.3wehaveshownthatthecovariationofaBrownianmotionandacompound Poissonprocessiszero,thus[ M;F ]=[ F;M ]=0.FromTheorem2.1.6itfollows that[ A;F ]=[ F;A ]=0.Finally,usingtheresultsaboveaswellasthelinearityof quadraticcovariationweobtain [ Y;Y ]=[ D;D ]+[ M;M ]+[ F;F ] : 25

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However,theprocess F t isapurejumpso[ F;F ] c =0.Therefore [ Y;Y ] c = Y 2 + 2 Z t 0 Y d [ Z;Z ] = Y 2 + 2 Z t 0 Y d: Proposition3.1.3 Thecontinuouspartofthecovariationof S and Y ,where S and Y aregivenby.1.1and.1.2,respectively,is [ S;Y ] c = S Y + Z t 0 S p Y f Y t d: .1.7 Proof. TheSIEofthesumof S and Y isgivenby S + Y t = S + Y | {z } D + Z t 0 [ S + m )]TJ/F27 11.9552 Tf 11.955 0 Td [(Y ] d | {z } A t + Z t 0 S dW | {z } M 1 t + Z t 0 p Y dZ | {z } M 2 t + Z t 0 dC | {z } F t : Usingthesameargumentsasintheprevioustwopropositions,wehave [ D;A ]=[ D;M 1 ; 2 ]=[ D;F ]=[ A;D ]=[ M 1 ; 2 ;D ]=[ F;D ]=[ A;A ]=[ A;M 1 ; 2 ]= [ M 1 ; 2 ;A ]=[ M 1 ; 2 ;F ]=[ F;M 1 ; 2 ]=[ A;F ]=[ F;A ]=0,giving [ S + Y;S + Y ]=[ D;D ]+[ M 1 ;M 1 ]+[ M 2 ;M 2 ]+2[ M 1 ;M 2 ]+[ F;F ] = S + Y 2 + Z t 0 S 2 f 2 Y d [ W;W ] | {z } d + 2 Z t 0 Y d [ Z;Z ] | {z } d +2 Z t 0 S f Y p Y d [ W;Z ] | {z } d + Z t 0 d [ C;C ] ; 26

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Since F t isapurejumpprocess,thecontinuouspartofitsquadraticvariationis0. Hence [ S + Y;S + Y ] c = S + Y 2 + Z t 0 S 2 f 2 Y d +2 Z t 0 S f Y p Y d + 2 Z t 0 Y d: Thisresult,theresultsoftheprevioustwopropositionsandthepolarizationidentity, imply [ S;Y ] c = S Y + Z t 0 S f Y p Y d: Dierentiatingequations.1.4,.1.5and.1.7weobtainthefollowingexpressionsfor d [ S;S ] c t d [ Y;Y ] c t and d [ S;Y ] c t : d [ S;S ] c t = S 2 f 2 y dt d [ Y;Y ] c t = ydt and d [ S;Y ] t = S p yf y dt Equation.1.3thenbecomes dP t;S t ;Y t = @P @t dt + @P @S dS t + @P @y dY t + 1 2 S 2 f 2 y @ 2 P @S 2 dt + S p yf y @ 2 P @S@y dt + 1 2 2 y @ 2 P @y 2 dt + P y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P y )]TJ/F27 11.9552 Tf 11.955 0 Td [(J @P @y dN t : .1.8 IntheBlack-Scholescaseitissucienttohedgewiththeunderlyingasset,only, becausethereisasinglesourceofrandomness.However,inthejump-diusionvolatilitycasewetrytohedgewiththeunderlyingassetandanotherbenchmarkoption writtenonthesameunderlyingassetjustwitheitherlaterexpirationdateordierent strikeprice.Hence,wecreateaportfoliowithonecalloption P a t unitsofstock and b t unitsofthebenchmarkoption Q withsamepayofunctionas P ,andconsider itsvalue = P )]TJ/F27 11.9552 Tf 11.955 0 Td [(a t S t )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t Q: 27

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Assumingtheportfolioisself-nancing,thechangeintheportfoliovalueinasmall timeinterval dt is d = dP )]TJ/F27 11.9552 Tf 11.955 0 Td [(a t dS t )]TJ/F27 11.9552 Tf 11.956 0 Td [(b t dQ: .1.9 Usingequation.1.8oncefor dP andthenfor dQ weobtain d = @P @t dt + @P @S dS t + @P @y dY t + 1 2 S 2 f 2 y @ 2 P @S 2 dt + S p yf y @ 2 P @S@y dt + 1 2 2 y @ 2 P @y 2 dt + P y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P y )]TJ/F27 11.9552 Tf 11.955 0 Td [(J @P @y dN t )]TJ/F27 11.9552 Tf 11.955 0 Td [(a t dS t )]TJ/F27 11.9552 Tf 11.956 0 Td [(b t @Q @t dt )]TJ/F27 11.9552 Tf 11.956 0 Td [(b t @Q @S dS t )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t @Q @y dY t )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(1 2 b t S 2 f 2 y @ 2 Q @S 2 dt )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t S p yf y @ 2 Q @S@y dt )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(1 2 b t 2 y @ 2 Q @y 2 dt )]TJ/F27 11.9552 Tf 11.956 0 Td [(b t Q y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(Q y )]TJ/F27 11.9552 Tf 11.955 0 Td [(J @Q @y dN t : Substitutingfor dS t and dY t andgroupingthetermsweobtain: d = @P @t + S @P @S + m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y @P @y + 1 2 S 2 f 2 y @ 2 P @S 2 + S p yf y @ 2 P @S@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 13.15 0 Td [(a t S )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t @Q @t )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t S @Q @S )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y @Q @y )]TJ/F19 11.9552 Tf 13.15 8.087 Td [(1 2 b t S 2 f 2 y @ 2 Q @S 2 )]TJ/F27 11.9552 Tf 13.151 0 Td [(b t S p yf y @ 2 Q @S@y )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(1 2 b t 2 y @ 2 Q @y 2 dt + Sf y @P @S )]TJ/F27 11.9552 Tf 11.955 0 Td [(a t )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t @Q @S dW t + p y @P @y )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t @Q @y dZ t + @P @y )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t @Q @y dC t + P y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P y )]TJ/F27 11.9552 Tf 11.955 0 Td [(J @P @y )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t Q y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t Q y + b t J @Q @y dN t : .1.10 Toeliminatetheriskthatcomesfromthediusiontermswechoosesuitablevalues for a t and b t b t = @P @y @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 and a t = @P @S )]TJ/F27 11.9552 Tf 13.15 8.088 Td [(@Q @S @P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 : .1.11 Bytheprincipleofnoarbitrage d = r dt = r P )]TJ/F27 11.9552 Tf 12.045 0 Td [(a t S t )]TJ/F27 11.9552 Tf 12.045 0 Td [(b t Q dt .Substitutingthe righthandsideofthisequationin.1.10andtheexpressionsfor a t and b t givenby .1.11,weobtain 28

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r P )]TJ/F27 11.9552 Tf 11.955 0 Td [(S @P @S )]TJ/F27 11.9552 Tf 11.955 0 Td [(S @Q @S @P @y @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F27 11.9552 Tf 13.151 8.088 Td [(@P @y @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 Q # dt = @P @t + S @P @S + m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y @P @y + 1 2 S 2 f 2 y @ 2 P @S 2 + S p yf y @ 2 P @S@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(S @P @S )]TJ/F27 11.9552 Tf 13.15 8.088 Td [(@Q @S @P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F27 11.9552 Tf 13.151 8.088 Td [(@Q @t @P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(S @Q @S @P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F27 11.9552 Tf 13.15 0 Td [( m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y @Q @y @P @y @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(1 2 S 2 f 2 y @ 2 Q @S 2 @P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F27 11.9552 Tf 13.15 0 Td [(S p yf y @ 2 Q @S@y @P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(1 2 2 y @ 2 Q @y 2 @P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 # dt + P y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P y )]TJ/F27 11.9552 Tf 11.955 0 Td [(J @P @y )]TJ/F32 11.9552 Tf 11.955 16.857 Td [( Q y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(Q y )]TJ/F27 11.9552 Tf 11.955 0 Td [(J @Q @y @P @y @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 # dN t : Aftercancelingoutsomeoftheterms,theequationabovebecomes r P )]TJ/F27 11.9552 Tf 11.955 0 Td [(S @P @S )]TJ/F27 11.9552 Tf 11.955 0 Td [(S @Q @S @P @y @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F27 11.9552 Tf 13.151 8.088 Td [(@P @y @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 Q # dt = @P @t + 1 2 S 2 f 2 y @ 2 P @S 2 + S p yf y @ 2 P @S@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 13.151 8.088 Td [(@Q @t @P @y @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F19 11.9552 Tf 13.682 8.088 Td [(1 2 S 2 f 2 y @ 2 Q @S 2 @P @y @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(S p yf y @ 2 Q @S@y @P @y @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F19 11.9552 Tf 13.682 8.088 Td [(1 2 2 y @ 2 Q @y 2 @P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 # dt + P y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P y )]TJ/F19 11.9552 Tf 11.955 0 Td [( Q y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(Q y @P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 # dN t : Multiplyingthelastequationby @P @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 andmovingalltermscontaining P onthe leftandalltermscontaining Q ontherighthandside,weobtain 29

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@P @t + rS @P @S + 1 2 S 2 f 2 y @ 2 P @S 2 + S p yf y @ 2 P @S@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(rP @P @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 dt +[ P y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P y ] @P @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 dN t = @Q @t + rS @Q @S + 1 2 S 2 f 2 y @ 2 Q @S 2 + p ySf y @ 2 Q @S@y + 1 2 2 y @ 2 Q @y 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [(rQ @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 dt +[ Q y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(Q y ] @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 dN t : Takingexpectationovertheprobabilitydistributionofjumpsweobtain @P @t + rS @P @S + 1 2 S 2 f 2 y @ 2 P @S 2 + S p yf y @ 2 P @S@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(rP @P @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 dt + E [ P t;S;y + J )]TJ/F27 11.9552 Tf 11.956 0 Td [(P t;S;y ] @P @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 dt = @Q @t + rS @Q @S + 1 2 S 2 f 2 y @ 2 Q @S 2 + p ySf y @ 2 Q @S@y + 1 2 2 y @ 2 Q @y 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [(rQ @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 dt + E [ Q t;S;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(Q t;S;y ] @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 dt; .1.12 where E [ P t;S;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P t;S;y ]= Z 1 0 [ P t;S;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P t;S;y ] J dJ: .1.13 Theleftandrighthandsidesofequation.1.12areidentical,exceptthattherst oneisafunctionof P ,only,andtheotheroneisafunctionof Q ,only.Then,there mustbeafunction k t;S;y suchthat @P @t + rS @P @S + 1 2 S 2 f 2 y @ 2 P @S 2 + S p yf y @ 2 P @S@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(rP + E [ P t;S;y + J )]TJ/F27 11.9552 Tf 11.956 0 Td [(P t;S;y ]= )]TJ/F27 11.9552 Tf 9.299 0 Td [(k t;S;y @P @y : Theterminfrontof @P @S isthedrifttermofthestockpriceSDEundertherisk-neutral probabilitymeasure.Sotheterminfrontof @P @y shouldbethedrifttermoftheSDE 30

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thatcorrespondstothevolatilityfunctionintherisk-neutralworld.Thisimpliesthat thefunction k t;S;y isadierenceoftherealworlddrifttermandthetotalmarket priceofvolatilityrisk\050 y : k t;S;y = m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y )]TJ/F19 11.9552 Tf 11.955 0 Td [(\050 y p y: Cheridito,FilipovicandKimmelfordetailssee[5]havemodeledthemarketprice ofriskassociatedwiththevolatilitydrivenbyaprocessgivenbyequation.1.2as \050 y = p y forsomeconstant with =0possibleandintheoryisdetermined bythebenchmarkoption Q t;S t ;Y t .Alloftheaboveprovesthefollowingtheorem: Theorem3.1.4 ThepriceofaEuropeancalloptionwrittenonastockdrivenby SDE.1.1andvolatilitythatfollowsSDE.1.2,withstrikepriceKandmaturity Tismodeledwiththefollowingterminal-boundaryvalueproblem @P @t + rS @P @S +[ m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y )]TJ/F27 11.9552 Tf 11.955 0 Td [(y ] @P @y + 1 2 S 2 f 2 y @ 2 P @S 2 + S p yf y @ 2 P @S@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(rP + E [ P t;S;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P t;S;y ]=0 ; .1.14 P T;S;y =max S )]TJ/F27 11.9552 Tf 11.955 0 Td [(K; 0 ; lim S !1 [ P t;S;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(S ]=0 ; P t; 0 ;y =0 ; P t;S; 0=0 ; .1.15 where 0 S< 1 ; 0 y< 1 ; 0 t T ,and E [ P t;S;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P t;S;y ]= Z 1 [ P t;S;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P t;S;y ] J dJ istheexpectedvalueofthechangeintheoptionpricewithrespecttothejumpprobabilitydistributionfunction. 31

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Whensolvingthisboundaryvalueproblem,wewillnditconvenienttoexpress itinthefollowingform: Theorem3.1.5 Theterminal-boundaryvalueprobleminTheorem3.1.4forEuropeancalloptionisequivalenttothefollowinginitial-boundaryvalueproblem: @P @ = r )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(1 2 f 2 y @P @x +[ m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y )]TJ/F27 11.9552 Tf 11.956 0 Td [(y ] @P @y + 1 2 f 2 y @ 2 P @x 2 + p yf y @ 2 P @x@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(rP + E [ P ;x;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P ;x;y ] ; .1.16 P ;x;y =max e x )]TJ/F27 11.9552 Tf 11.956 0 Td [(K; 0 ; .1.17 lim x P ;x;y =0 ; lim x !1 [ P ;x;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(e x ]=0 ; P ;x; 0=0 ; .1.18 where
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3.2FormulationoftheAmericanOptionsPricingModel WhendiscussingtheAmericancalloptionsintheBlack-Scholessettingwementioned thatforanon-dividendpayingassettheycanbetreatedasEuropeanoptions.Thus, whenmodelingAmericanoptionsitmakessensetoconsiderdividendpayingunderlyingassets. Assumethattheunderlyingasset S t paysdividendsatacontinuousrate q dS t = )]TJ/F27 11.9552 Tf 11.955 0 Td [(q S t dt + f Y t S t dW t .2.19 withvolatilitythatfollowsajump-diusionprocess dY t = m )]TJ/F27 11.9552 Tf 11.956 0 Td [(Y t dt + p Y t dZ t + J t dN t ; .2.20 where W t and Z t arecorrelateddiusionprocesseswithcorrelation and N t isa Poissonprocessnotcorrelatedtotheprevioustwo.Denethedividendprocesstobe dD t = qS t dt; .2.21 thenthetimechangeintheportfolio,initiallygivenbyequation.1.9,becomes d = dP )]TJ/F27 11.9552 Tf 11.955 0 Td [(a t dS t )]TJ/F27 11.9552 Tf 11.955 0 Td [(a t dD t )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t dQ: .2.22 Ifweowntheunderlyingasset,wereceivedividends,andviseversa,ifweshortthe assetweneedtopaydividends.The a t dD t termaccountsforthis. ApplyingIt^o'slemmaforboth, dP and dQ ,weobtain: 33

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d = @P @t + )]TJ/F27 11.9552 Tf 11.955 0 Td [(q S @P @S + m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y @P @y + 1 2 S 2 f 2 y @ 2 P @S 2 + S p yf y @ 2 P @S@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(a t )]TJ/F27 11.9552 Tf 11.955 0 Td [(q S )]TJ/F27 11.9552 Tf 11.955 0 Td [(a t qS t )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t @Q @t )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t )]TJ/F27 11.9552 Tf 11.955 0 Td [(q S @Q @S )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y @Q @y )]TJ/F19 11.9552 Tf 10.494 8.087 Td [(1 2 b t S 2 f 2 y @ 2 Q @S 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [(b t S p yf y @ 2 Q @S@y )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(1 2 b t 2 y @ 2 Q @y 2 dt + Sf y @P @S )]TJ/F27 11.9552 Tf 11.955 0 Td [(a t )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t @Q @S dW t + p y @P @y )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t @Q @y dZ t +[ P y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P y )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t Q y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(b t Q y ] dN t : .2.23 Wecaneliminatetheriskthatcomesfromthediusiontermsbypurchasing/selling a t and b t sharesofstockandoptions,respectively,wheretheseparametersaregiven by b t = @P @y @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 and a t = @P @S )]TJ/F27 11.9552 Tf 13.15 8.088 Td [(@Q @S @P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 : .2.24 Bytheprincipleofnoarbitrage,whichmathematicallyisgivenby d = r dt ,and substitutingtheexpressionsfor a t and b t in.2.23weobtain r P )]TJ/F27 11.9552 Tf 11.956 0 Td [(S @P @S )]TJ/F27 11.9552 Tf 11.956 0 Td [(S @Q @S @P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F27 11.9552 Tf 13.151 8.088 Td [(@P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 Q # dt = @P @t + 1 2 S 2 f 2 y @ 2 P @S 2 + S p yf y @ 2 P @S@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(qS @P @S )]TJ/F27 11.9552 Tf 11.955 0 Td [(qS @Q @S @P @y @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F27 11.9552 Tf 10.494 8.088 Td [(@Q @t @P @y @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(1 2 S 2 f 2 y @ 2 Q @S 2 @P @y @Q @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F27 11.9552 Tf 9.299 0 Td [(S p yf y @ 2 Q @S@y @P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(1 2 2 y @ 2 Q @y 2 @P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 # dt + P y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P y )]TJ/F19 11.9552 Tf 11.955 0 Td [( Q y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(Q y @P @y @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 # dN t : Rearrangingthetermsintheequationaboveandtakingexpectationovertheprobabilitydistributionofjumps,weobtain 34

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@P @t + r )]TJ/F27 11.9552 Tf 11.955 0 Td [(q S @P @S + 1 2 S 2 f 2 y @ 2 P @S 2 + S p yf y @ 2 P @S@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(rP + E [ P y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P y ] g @P @y )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 dt = @Q @t + r )]TJ/F27 11.9552 Tf 11.955 0 Td [(q S @Q @S + 1 2 S 2 f 2 y @ 2 Q @S 2 + p ySf y @ 2 Q @S@y + 1 2 2 y @ 2 Q @y 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [(rQ + E [ Q y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(Q y ] g @Q @y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 dt: UsingsameargumentsasintheEuropeanoptioncase,wecanshowthefollowing proposition Proposition3.2.1 Thepriceofanoptionwrittenonastockwithcontinuousyield q describedbySDE.2.19andvolatilitydrivenbySDE.2.20,ismodeledwiththe partialintegro-dierentialequation @P @t + r )]TJ/F27 11.9552 Tf 11.955 0 Td [(q S @P @S +[ m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y )]TJ/F27 11.9552 Tf 11.956 0 Td [(y ] @P @y + 1 2 S 2 f 2 y @ 2 P @S 2 + S p yf y @ 2 P @S@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(rP + E [ P t;S;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P t;S;y ]=0 ; .2.25 where E [ P t;S;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P t;S;y ]= Z 1 0 [ P t;S;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P t;S;y ] J dJ istheexpectedvalueofthechangeintheoptionpricewithrespecttothejumpprobabilitydistributionfunction. WecandeneamodelforthepriceofanAmericanoptionbyaddingboundary conditionstothepropositionabove.Analogoustotheconstantvolatilitysetting Section2.3,thefunction P t;S;y satisesafreeboundaryproblem.Inthejumpdiusionvolatilitycasethefreeboundarybecomesasurfacebecauseitdependson additionalspatialvariable y .Let a y;t betheearlyexerciseboundaryattime t and volatilitylevel y thepathofcriticalstockpricesatwhichearlyexerciseoccurs,then 35

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thefollowingpropositionholds: Proposition3.2.2 ThepriceofanAmericanoptionwrittenonastockthatpays dividendsatacontinuousrate q andgivenbySDE.2.19,withvolatilitydrivenby SDE.2.20,ismodeledwithPIDE.2.25intheregion 0 t T 0 S a y;t and 0 y< 1 .TheboundaryconditionsintheAmericanoptionscaseare P T;S;y = 8 < : max S )]TJ/F27 11.9552 Tf 11.955 0 Td [(K; 0 ; foracalloption max K )]TJ/F27 11.9552 Tf 11.955 0 Td [(S; 0 ; foraputoption ; P t;a y;t ;y = 8 < : a y;t )]TJ/F27 11.9552 Tf 11.955 0 Td [(K; foracalloption K )]TJ/F27 11.9552 Tf 11.955 0 Td [(a y;t ; foraputoption lim S a y;t @P @S = 8 < : 1 ; foracalloption )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; foraputoption : .2.26 ThelasttwoboundaryconditionsareprovidedbyFouqueetal.[6]toensurecontinuity of P and @P @S Let S = e x and = T )]TJ/F27 11.9552 Tf 11.955 0 Td [(t ,thenitiseasytoshowthefollowingresult: Proposition3.2.3 ThefreeboundaryvalueprobleminProposition.2.2for P t;S;y isequivalentto @P @ = r )]TJ/F27 11.9552 Tf 11.956 0 Td [(q )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(1 2 f 2 y @P @x +[ m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y )]TJ/F27 11.9552 Tf 11.955 0 Td [(y ] @P @y + 1 2 f 2 y @ 2 P @x 2 + p yf y @ 2 P @x@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(rP + Z 1 [ P ;x;y + J )]TJ/F27 11.9552 Tf 11.956 0 Td [(P ;x;y ] J dJ; .2.27 intheboundeddomain
PAGE 44

P ;a y; ;y = 8 < : ln a y; )]TJ/F27 11.9552 Tf 11.955 0 Td [(K; foracalloption K )]TJ/F19 11.9552 Tf 11.956 0 Td [(ln a y; ; foraputoption ; lim x ln a y; @P @x = 8 < : a y; ; foracalloption )]TJ/F27 11.9552 Tf 9.299 0 Td [(a y; ; foraputoption : .2.28 Theorem3.2.4 ThehomogeneousPIDEinequation.2.27intheboundeddomain
PAGE 45

Z ln a y; e i!x @P @ )]TJ/F32 11.9552 Tf 11.956 16.856 Td [( r )]TJ/F27 11.9552 Tf 11.955 0 Td [(q )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(1 2 f 2 y @P @x )]TJ/F19 11.9552 Tf 11.955 0 Td [([ m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y )]TJ/F27 11.9552 Tf 11.956 0 Td [(y ] @P @y )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(1 2 f 2 y @ 2 P @x 2 )]TJ/F27 11.9552 Tf 9.298 0 Td [( p yf y @ 2 P @x@y )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(1 2 2 y @ 2 P @y 2 + rP )]TJ/F27 11.9552 Tf 9.298 0 Td [( Z 1 [ P ;x;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P ;x;y ] J dJ dx =0 : Since R ln a y; dx = R 1 dx )]TJ/F32 11.9552 Tf 12.198 9.631 Td [(R 1 ln a y; dx wecanwritetheexpressionabove asfollows F @P @ )]TJ/F32 11.9552 Tf 11.955 16.857 Td [( r )]TJ/F27 11.9552 Tf 11.955 0 Td [(q )]TJ/F19 11.9552 Tf 13.15 8.087 Td [(1 2 f 2 y @P @x )]TJ/F19 11.9552 Tf 11.956 0 Td [([ m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y )]TJ/F27 11.9552 Tf 11.955 0 Td [(y ] @P @y )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(1 2 f 2 y @ 2 P @x 2 )]TJ/F27 11.9552 Tf 12.487 0 Td [(f y p y @ 2 P @x@y )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(1 2 2 y @ 2 P @y 2 + rP )]TJ/F27 11.9552 Tf 11.955 0 Td [( Z 1 [ P ;x;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P ;x;y ] J dJ = Z 1 ln a y; e i!x @P @ )]TJ/F32 11.9552 Tf 11.956 16.857 Td [( r )]TJ/F27 11.9552 Tf 11.955 0 Td [(q )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(1 2 f 2 y @P @x )]TJ/F19 11.9552 Tf 11.955 0 Td [([ m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y )]TJ/F27 11.9552 Tf 11.955 0 Td [(y ] @P @y )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(1 2 f 2 y @ 2 P @x 2 )]TJ/F27 11.9552 Tf 12.487 0 Td [(f y p y @ 2 P @x@y )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(1 2 2 y @ 2 P @y 2 + rP )]TJ/F27 11.9552 Tf 11.955 0 Td [( Z 1 [ P ;x;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P ;x;y ] J dJ dx: When x> ln a y; thepriceoftheAmericancalloptionissimplythepayo, P ;x;y = e x )]TJ/F27 11.9552 Tf 12.642 0 Td [(K ,andso @P @ = @P @y = @ 2 P @y 2 = @ 2 P @x@y =0, @P @x = e x and @ 2 P @x 2 = e x Thus,theintegralontherighthandsideoftheequationabovebecomes Z 1 ln a y; e i!x qe x )]TJ/F27 11.9552 Tf 11.955 0 Td [(rK dx = Z 1 e i!x H x )]TJ/F19 11.9552 Tf 11.956 0 Td [(ln a y; qe x )]TJ/F27 11.9552 Tf 11.955 0 Td [(rK dx: .2.31 TakingtheinverseFouriertransformwewillobtain.2.29. TheAmericanputoptionnonhomogeneousPDE.2.30canbederivedsimilarly. Notethatthenon-homogeneousterm H x )]TJ/F19 11.9552 Tf 11.955 0 Td [(ln a y; qe x )]TJ/F27 11.9552 Tf 11.955 0 Td [(rK inPIDE3.2.29 arisesonlywhenthecalloptionisexercisedearly,thatis x> ln a y; ,inwhichcase theholderreceivesdividendsbyowningthestock S qe x ,andpaysinterestof rK for borrowing K 38

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4ClosedFormSolutionofaPureDiffusionOptionPricingModel Beforewemovetodeterminingaclosedformsolutiontothejumpdiusionmodel describedinthepreviouschapter,let'sfocusontheHeston'smodelinwhichthe underlyingassetpricefollowsthediusionprocess dS t = S t dt + p Y t S t dW t .0.1 andthevolatilityisdrivenbymean-revertingdiusionprocess Y t : dY t = m )]TJ/F27 11.9552 Tf 11.955 0 Td [(Y t dt + p Y t dZ t : .0.2 Settingthejumpamplitude J equalto0inthederivationprocessgivenintheprevious chapter,wewillobtainaPDEthatdescribesthepriceofanoptionwrittenonan assetdrivenbySDEs.0.1and4.0.2: @P @ = r )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(1 2 y @P @x +[ m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y )]TJ/F27 11.9552 Tf 11.955 0 Td [(y ] @P @y + 1 2 y @ 2 P @x 2 + y @ 2 P @x@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(rP; .0.3 where
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4.1IntegralTransformsofthePureDiusionPricingModel Assumethat R 1 j P ;x;y j dx< 1 ,andthatthefunction P ;x;y isofexponential order,thatislim y !1 P ;x;y e )]TJ/F28 7.9701 Tf 6.586 0 Td [(By =0forsomerealnumber B Dene ^ P ;!;y tobetheFouriertransformof P ;x;y withrespectto x : ^ P ;!;y F [ P t;x;y ]= Z 1 P ;x;y e i!x dx: Thenthefollowingpropertieshold F @P @x = )]TJ/F27 11.9552 Tf 9.299 0 Td [(i! ^ P ;!;y ; F @ 2 P @x 2 = )]TJ/F27 11.9552 Tf 9.299 0 Td [(! 2 ^ P ;!;y ; F @P @ = @ ^ P @ ; F @P @y = @ ^ P @y ; and F @ 2 P @y 2 = @ 2 ^ P @y 2 : Also, F @ 2 P @x@y = Z 1 e i!x @ 2 P @x@y dx = @ @y Z 1 e i!x @P @x dx = @ @y )]TJ/F27 11.9552 Tf 9.299 0 Td [(i! ^ P = )]TJ/F27 11.9552 Tf 9.298 0 Td [(i! @ ^ P @y : .1.4 ApplyingtheFouriertransformtoequation.0.3andusingtheaboveproperties, weobtain: @ ^ P @ = i! 1 2 y )]TJ/F27 11.9552 Tf 11.955 0 Td [(r ^ P +[ m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y )]TJ/F27 11.9552 Tf 11.955 0 Td [(y ] @ ^ P @y )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(1 2 2 y ^ P )]TJ/F27 11.9552 Tf 11.956 0 Td [(i!y @ ^ P @y + 1 2 2 y @ 2 ^ P @y 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(r ^ P whichisequivalenttothePDE @ ^ P @ = 1 2 2 y @ 2 ^ P @y 2 +[ m )]TJ/F19 11.9552 Tf 11.955 0 Td [( + + i! y ] @ ^ P @y )]TJ/F19 11.9552 Tf 11.955 0 Td [([ r + i!r + 1 2 )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [(! 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! y ] ^ P: .1.5 Thedomainof y is[0 ; 1 ,sowewillapplytheLaplacetransformtoequation.1.5withrespectto y .Let ~ P betheLaplacetransformof ^ P 40

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~ P L h ^ P ;!; i = Z 1 0 ^ P ;!;y e )]TJ/F28 7.9701 Tf 6.587 0 Td [(y dy; providedthattheintegralconvergeswhen y !1 .Wewillndusefulthefollowing propertiesoftheLaplacetransform: L @ ^ P @ # = @ ~ P @ ; L @ ^ P @y # = ~ P ;!; )]TJ/F19 11.9552 Tf 14.589 3.022 Td [(^ P ;!; 0 ; and L @ 2 ^ P @y 2 # = 2 ~ P ;!; )]TJ/F27 11.9552 Tf 11.956 0 Td [( ^ P ;!; 0 )]TJ/F27 11.9552 Tf 13.151 8.088 Td [(@ ^ P @y ;!; 0 : However,when y =0,thepriceoftheunderlyingassetlosestheriskytermandit becomescompletelydeterministic.Webuyoptionsinordertoeliminateorreduce theriskassociatedwithbuying/sellingriskyassets.Thus,inthecaseofnoriskitis naturaltoassumethatthepriceoftheoptionis0, P ;x; 0=0.Then,theFourier transformof P ;x; 0is ^ P ;!; 0=0.Now,thepropertiesabovearesimpliedto L @ ^ P @y # = ~ P ;!; ; and L @ 2 ^ P @y 2 # = 2 ~ P ;!; )]TJ/F27 11.9552 Tf 13.15 8.088 Td [(@ ^ P @y ;!; 0 : Also,itiseasytoseethat L h y ^ P ;!;y i = )]TJ/F28 7.9701 Tf 10.494 4.707 Td [(@ ~ P @ bydierentiating ~ P = Z 1 0 ^ P ;!;y e )]TJ/F28 7.9701 Tf 6.586 0 Td [(y dy; withrespectto : @ ~ P @ = @ @ Z 1 0 ^ P ;!;y e )]TJ/F28 7.9701 Tf 6.587 0 Td [(y dy = Z 1 0 ^ P ;!;y @ @ e )]TJ/F28 7.9701 Tf 6.587 0 Td [(y dy = )]TJ/F32 11.9552 Tf 11.291 16.273 Td [(Z 1 0 ye )]TJ/F28 7.9701 Tf 6.586 0 Td [(y ^ P ;!;y dy = L h y ^ P i : ThelastpropertywillbeveryhelpfulincalculatingtheLaplacetransformsof y @ ^ P @y and y @ 2 ^ P @y 2 .Usingintegrationbyparts,weobtain: 41

PAGE 49

L y @ ^ P @y # = Z 1 0 ye )]TJ/F28 7.9701 Tf 6.587 0 Td [(y @ ^ P @y dy = ye )]TJ/F28 7.9701 Tf 6.587 0 Td [(y ^ P 1 0 )]TJ/F32 11.9552 Tf 11.955 16.272 Td [(Z 1 0 e )]TJ/F28 7.9701 Tf 6.586 0 Td [(y ^ Pdy + Z 1 0 ye )]TJ/F28 7.9701 Tf 6.587 0 Td [(y ^ Pdy = L h y ^ P i )]TJ/F19 11.9552 Tf 14.589 3.022 Td [(~ P = )]TJ/F27 11.9552 Tf 9.299 0 Td [( @ ~ P @ )]TJ/F19 11.9552 Tf 14.589 3.022 Td [(~ P; and L y @ 2 ^ P @y 2 # = Z 1 0 ye )]TJ/F28 7.9701 Tf 6.587 0 Td [(y @ 2 ^ P @y 2 dy = ye )]TJ/F28 7.9701 Tf 6.587 0 Td [(y @ ^ P @y 1 0 )]TJ/F32 11.9552 Tf 11.955 16.273 Td [(Z 1 0 e )]TJ/F28 7.9701 Tf 6.587 0 Td [(y @ ^ P @y dy + Z 1 0 ye )]TJ/F28 7.9701 Tf 6.587 0 Td [(y @ ^ P @y dy = L @ ^ P @y # + L y @ ^ P @y # = )]TJ/F19 11.9552 Tf 9.299 0 Td [(2 ~ P )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 @ ~ P @ : Havingtheseresults,whenapplyingLaplacetransformtoequation.1.5,weobtain therst-orderlinearPDE: @ ~ P @ = )]TJ/F19 11.9552 Tf 10.494 8.088 Td [(1 2 2 2 @ ~ P @ +2 ~ P + m ~ P + + + i! @ ~ P @ + ~ P )]TJ/F19 11.9552 Tf 11.955 0 Td [( r + i!r ~ P + 1 2 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! @ ~ P @ whichsimpliesto @ ~ P @ + 1 2 2 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [( + + i! )]TJ/F19 11.9552 Tf 13.15 8.088 Td [(1 2 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! @ ~ P @ = m )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 + + + i! )]TJ/F27 11.9552 Tf 11.955 0 Td [(r )]TJ/F27 11.9552 Tf 11.955 0 Td [(i!r ~ P: .1.6 Recallthatatmaturity,when =0,thepriceoftheoptionis P ;x;y = h x ,and oncewetaketheFouriertransformwithrespectto x andtheLaplacetransformwith respectto y weobtain 42

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~ P ;!; = L [ F [ h x ]]= L h ^ h i = ^ h L [1]= ^ h : 4.2SolutionoftheTransformedPureDiusionPricingModel Proposition4.2.1 Thegeneralsolutionofthepartialdierentialequation @w @x + @w @y = f x;y w; .2.7 isgivenby w = exp Z x f t;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(x + t dt y )]TJ/F27 11.9552 Tf 11.956 0 Td [(x where isanarbitrarycontinuouslydierentiablefunction,andthelowerlimitofthe integralcanbychosenarbitrarily. Proof. UsingtheLeibnizintegralrulewecalculate @w @x and @w @y @w @x = @ @x exp Z x f t;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(x + t dt y )]TJ/F27 11.9552 Tf 11.955 0 Td [(x = f x;y )]TJ/F32 11.9552 Tf 11.955 16.273 Td [(Z x f 0 t;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(x + t dt e R x f t;y )]TJ/F28 7.9701 Tf 6.587 0 Td [(x + t dt y )]TJ/F27 11.9552 Tf 11.955 0 Td [(x )]TJ/F27 11.9552 Tf 11.955 0 Td [(e R x f t;y )]TJ/F28 7.9701 Tf 6.586 0 Td [(x + t dt 0 y )]TJ/F27 11.9552 Tf 11.955 0 Td [(x @w @y = @ @y exp Z x f t;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(x + t dt y )]TJ/F27 11.9552 Tf 11.955 0 Td [(x = Z x f 0 t;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(x + t dte R x f t;y )]TJ/F28 7.9701 Tf 6.587 0 Td [(x + t dt y )]TJ/F27 11.9552 Tf 11.955 0 Td [(x + e R x f t;y )]TJ/F28 7.9701 Tf 6.586 0 Td [(x + t dt 0 y )]TJ/F27 11.9552 Tf 11.955 0 Td [(x Addingupthepartialderivatives,wehave @w @x + @w @y = f x;y exp Z x f t;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(x + t dt y )]TJ/F27 11.9552 Tf 11.955 0 Td [(x | {z } w : 43

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Proposition4.2.2 Theclosed-formsolutionofthepartialdierentialequation @ ~ P @ + 1 2 2 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [( + + i! )]TJ/F19 11.9552 Tf 13.15 8.088 Td [(1 2 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! @ ~ P @ = m )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 + + + i! )]TJ/F27 11.9552 Tf 11.955 0 Td [(r )]TJ/F27 11.9552 Tf 11.955 0 Td [(i!r ~ P .2.8 withinitialcondition ~ P ;!; = ^ h isgivenby ~ P ;!; = ^ h [2 + i ] 2 )]TJ/F29 5.9776 Tf 7.782 3.258 Td [(m 2 e h m + )]TJ/F29 5.9776 Tf 5.756 0 Td [( 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(r )]TJ/F28 7.9701 Tf 6.587 0 Td [( + i m + 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(r! )]TJ/F28 7.9701 Tf 6.587 0 Td [( i [ + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i + i + i + + i! ] f ; ;! )]TJ/F27 11.9552 Tf 11.956 0 Td [( 0 ; .2.9 where 0 f ; ;! = and = aredenedbelow. f ; ;! = i 2 )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [(e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 + + i + i + + i! + e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i [ + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! ] 2 m 2 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ; .2.10 0 = A ;! = i e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 [ + i 2 + + + i! 2 ] 2 f + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i [ + i + i + + i! ] g ; .2.11 = r 1 2 n 2 2 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 11.956 0 Td [( + 2 + p G o ; = 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [(2 + r 2 n 2 2 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [( + 2 + p G o ; .2.12 and G = 2 2 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [( + 2 2 + 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 + 2 : .2.13 44

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Proof. ThefollowingtransformationwillleadtoarstorderPDEwithcoecients 1infrontofthepartialderivativesofPDE.2.8 = Z d 1 2 2 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [( + + i! )]TJ/F25 7.9701 Tf 13.151 4.707 Td [(1 2 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! = i p a + ib ln p a + ib )]TJ/F27 11.9552 Tf 11.955 0 Td [(i 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [( )]TJ/F27 11.9552 Tf 11.956 0 Td [( )]TJ/F27 11.9552 Tf 11.956 0 Td [(i! p a + ib + i 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [( )]TJ/F27 11.9552 Tf 11.956 0 Td [( )]TJ/F27 11.9552 Tf 11.956 0 Td [(i! .2.14 where a = a = 2 2 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [( + 2 and b = b = 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 + .Let p a + ib = + i; .2.15 thenwecannd p a + ib explicitlybysquaring.2.15andthensolvingthesystem ofequations 8 < : 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [( 2 = a = 2 2 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [( + 2 2 = b = 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 + Thesolutionofthissystemis: = r 1 2 a p a 2 + b 2 = b q 2 a p a 2 + b 2 : .2.16 or,intermsof asfollows: = r 1 2 n 2 2 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [( + 2 + p G o ; = 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [(2 + 2 ; .2.17 G = 2 2 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [( + 2 2 + 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 + 2 .2.18 wherethe signshavebeenreplacedwith+'sin.2.17andareappropriately chosenaccordingtoavailableinformationthatweneed and tobereal valuedfunctions. GoingbacktosimplifyingPDE.2.8usingthe -transformation,wecalculate 45

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and d d intermsof ,obtainingthefollowingexpressions: = i + i )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i + i 2 + e i + i + + + i! 2 ; .2.19 d d = 2 + i 2 e i + i 2 + e i + i 2 : ThenPDE.2.8becomes @ ~ P @ + @ ~ P @ = i m )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 + i )]TJ/F27 11.9552 Tf 11.956 0 Td [(e i + i 2 + e i + i + m 2 + + i! )]TJ/F27 11.9552 Tf 11.955 0 Td [(r )]TJ/F27 11.9552 Tf 11.956 0 Td [(ir! ~ P: .2.20 ThisPDEisofthesameformasPDE.2.7sotoobtainitssolutionwecanapply Proposition4.2.1.Choosingthelowerboundoftheintegralinthispropositiontobe 0willsimplifythedeterminationofthearbitraryfunction )]TJ/F27 11.9552 Tf 12.205 0 Td [( .Thus,weneed tocalculatetheintegral Z 0 i m )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 2 + i )]TJ/F27 11.9552 Tf 11.956 0 Td [(e i )]TJ/F28 7.9701 Tf 6.586 0 Td [( + t + i 1+ e i )]TJ/F28 7.9701 Tf 6.587 0 Td [( + t + i + m + + i! 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(r )]TJ/F27 11.9552 Tf 11.955 0 Td [(ir! dt = m + + i! 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(r )]TJ/F27 11.9552 Tf 11.955 0 Td [(ir! + i m )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 + i 2 +ln 1+ e i )]TJ/F28 7.9701 Tf 6.586 0 Td [( + i 1+ e i + i 2 m 2 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 : Setting =0makestheintegralaboveequalto0andusingtheinitialcondition intermsof ,thearbitraryfunction becomeseasytodetermine: = ^ h 2 + e i + i i + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i + i + + + i! + e i + i : .2.21 Finally,theexactsolutionofPDE.2.8isgivenby ~ P ;!; = ^ h 2 e 2 m + m )]TJ/F29 5.9776 Tf 5.756 0 Td [( m )]TJ/F29 5.9776 Tf 5.756 0 Td [( 2 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(r + i m! + m )]TJ/F29 5.9776 Tf 5.756 0 Td [( 2 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(r! + e i )]TJ/F28 7.9701 Tf 6.586 0 Td [( + i i + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i )]TJ/F28 7.9701 Tf 6.586 0 Td [( + i + + + i! + e i )]TJ/F28 7.9701 Tf 6.586 0 Td [( + i 1+ e i )]TJ/F28 7.9701 Tf 6.587 0 Td [( + i 1+ e i + i 2 m )]TJ/F29 5.9776 Tf 5.756 0 Td [( 2 2 ; .2.22 46

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andsubstitutingtheexpression.2.14for intheexpressionabove,weobtain formula.2.9,whichprovestheproposition. 4.3InverseIntegralTransformsofthePureDiusionPricingFormula IntheprevioussectionwefoundthesolutionoftherstorderPDE.2.8.However, thissolutiondoesn'tgiveusthepriceoftheoption.TondthepriceofaEuropean callputoptionswewillneedtoapplyaninverseLaplacetransformwithrespectto andaninverseFouriertransformwithrespectto inthepricingformula.2.9. TheinverseLaplacetransformisgivenby f t = 1 2 i Z + i 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(i 1 e st ^ f s ds; .3.23 where issuchthatthecontourofintegrationistotheright-handsideofanysingularitiesof ^ f s ,ortranslatedintermsof ^ P ;!;y theinversionformulaabovewill read: ^ P ;!;y = ^ h [2 + i ] 2 )]TJ/F26 5.9776 Tf 7.782 3.259 Td [(2 m 2 e h m + )]TJ/F29 5.9776 Tf 5.756 0 Td [( 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(r )]TJ/F28 7.9701 Tf 6.587 0 Td [( + i m + 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(r! )]TJ/F28 7.9701 Tf 6.587 0 Td [( i [ + i )]TJ/F27 11.9552 Tf 11.956 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i + i + i + + i! ] 1 2 i Z + i 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(i 1 e y f ; ;! )]TJ/F27 11.9552 Tf 11.955 0 Td [( 0 d: .3.24 Thenumberofsingularitiesinthefunctionabovewilldependonthevalueof 2 m 2 )]TJ/F19 11.9552 Tf 10.754 0 Td [(1, thepowerofthefunctiondenedby f ; ;! .Bu[7]explainsthatinorderfor thevolatilityprocess p y with y drivenby3.1.2tobepositive,fortheparameters ;m and theinequality m 2 2 musthold.Thismakestheexponentof f ; ;! nonnegative.Thisimpliesthatthefunction ~ P ;!; hasonesingularitythatisalso asimplepoleat = 0 .Tocalculatetheintegralin.3.24weusethe Cauchy IntegralFormula thatstates Lemma4.3.1 If f s isanalyticfunctionwithinandonasimpleclosedcurve C and 47

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s 0 isanypointinteriorto C ,then f s 0 = 1 2 i I C f s s )]TJ/F27 11.9552 Tf 11.955 0 Td [(s 0 ds whereCistraversedinthepositivecounterclockwisesense. Corollary4.3.2 TheinverseLaplaceTransformof ~ P ;!; givenby.2.9is equalto ^ P ;!;y = ^ h e A ;! y + [ B + iC ] 2 + i + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i [ + i + i + + i! ] 2 m 2 ; .3.25 where A ;! isdenedasin.2.11and B and C aregivenby: B = m + + 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(r C = m )]TJ/F27 11.9552 Tf 11.956 0 Td [( 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [(r!; .3.26 and ^ h istheFouriertransformoftheboundaryconditiondenedthroughoutthe section. Proof. Thefunction f ; ;! isanalyticeverywhereandsoisanyexponential function.Thustheproduct e y f ; ;! isanalyticeverywhere.Then,applyingthe CauchyIntegralFormula, theintegralin.3.24becomes: 1 2 i Z + i 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(i 1 e y f ; )]TJ/F27 11.9552 Tf 11.955 0 Td [( 0 d = e y 0 f 0 ; ;! : Anexpressionfor f 0 ; ;! canbeeasilycalculatebypluggingin 0 givenby.2.11 in.2.10 f 0 ; ;! = 4 + i 2 e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i [ + i + i + + i! ] 2 m 2 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 : 48

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Thenew,simpliedformof ^ P ;!;y in.3.24isgivenby.3.25. Now,wepresenttheexactsolutionofthepartialdierentialequation.0.3: Theorem4.3.3 ThepriceofaEuropeanCalloptiononanunderlyingassetthatfollowsthediusionprocess.0.1drivenbymean-revertingdiusionvolatility.0.2 canbecalculatedusingtheformula P ;x;y = e x P 1 ;x;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(KP 2 ;x;y ; .3.27 where x =ln S t P 1 ;x;y = 1 2 + 1 Z 1 0 < e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K ^ g + i ; )]TJ/F27 11.9552 Tf 9.299 0 Td [(i! d!; .3.28 P 2 ;x;y = 1 2 + 1 Z 1 0 < e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K ^ g ; ;y )]TJ/F27 11.9552 Tf 9.298 0 Td [(i! d!; .3.29 ^ g ; ;y = e A ;! y + [ B + iC ] 2 + i + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i [ + i + i + + i! ] 2 m 2 : .3.30 = and = aredenedasin.2.17, A ; isdenedasin.2.11 and B and C areasin.3.26. Proof. Usingthedenitionof^ g ; ;y wecanwrite.3.25as ^ P ;!;y = ^ h ^ g ; ;y Then,tondthepriceoftheoptionweneedtotakeinverseFouriertransformofthis equationwithrespectto .Let g x ; betheinverseFouriertransformof^ g ; ;y g x ; = 1 2 Z 1 e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i!x ^ g ; d!: .3.31 49

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Notethat,if g x ; isaprobabilitydensityfunction,then^ g ; ;y willbeitscharacteristicfunction.Also,itiseasytocheckthatatmaturity,i.e.when =0, ^ g ;0=1,andso g x ;0= 1 2 Z 1 e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i!x = x ; where x istheDiracdeltafunction.Bydenition, x 0and R 1 x dx =1, soitcanbeinterpretedasaprobabilitydensityfunction,andsocan g x Next,usingtheconvolutiontheoremweobtain: P ;x;y = F )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 h ^ h ^ g ; ;y i = Z 1 h x )]TJETq1 0 0 1 357.754 518.33 cm[]0 d 0 J 0.478 w 0 0 m 6.652 0 l SQBT/F27 11.9552 Tf 357.754 511.509 Td [(x g x ; d x where h x istheterminalcondition.Thus,fortheEuropeancalloptioncase,we have: P ;x;y = Z 1 max e x )]TJETq1 0 0 1 244.957 425.491 cm[]0 d 0 J 0.359 w 0 0 m 4.767 0 l SQBT/F28 7.9701 Tf 244.957 420.804 Td [(x )]TJ/F27 11.9552 Tf 11.955 0 Td [(K; 0 g x ; d x = e x 2 Z 1 Z x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K e )]TJETq1 0 0 1 268.715 391.23 cm[]0 d 0 J 0.359 w 0 0 m 4.767 0 l SQBT/F28 7.9701 Tf 268.715 386.543 Td [(x )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x ^ g ; d xd! )]TJ/F27 11.9552 Tf 14.203 8.088 Td [(K 2 Z 1 Z x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! x ^ g ; d xd!: Denote P 1 ;x;y = 1 2 Z 1 Z x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K e )]TJETq1 0 0 1 350.314 316.51 cm[]0 d 0 J 0.359 w 0 0 m 4.767 0 l SQBT/F28 7.9701 Tf 350.314 311.823 Td [(x )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! x ^ g ; d xd!; and P 2 ;x;y = 1 2 Z 1 Z x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x ^ g ; d xd!; andcalculatetheseintegrals.Fortherstintegral,wewillchangetheintegration variablefrom to + i anddenote^ g 1 =^ g + i whereneeded,sothatintegral 50

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P 1 becomes P 1 = 1 2 Z 1 ^ g + i lim e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! )]TJ/F27 11.9552 Tf 11.955 0 Td [(e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K i! d! = 1 2 lim Z 1 0 e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! ^ g 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i! ^ g 1 )]TJ/F27 11.9552 Tf 9.298 0 Td [(! i! d! + 1 2 Z 1 0 e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K ^ g + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i! x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K ^ g )]TJ/F27 11.9552 Tf 9.299 0 Td [(! + i )]TJ/F27 11.9552 Tf 9.298 0 Td [(i! d!: Similarly,forthesecondintegral P 2 ,wehave: P 2 = 1 2 lim Z 1 0 e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! ^ g )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i! ^ g )]TJ/F27 11.9552 Tf 9.299 0 Td [(! i! d! )]TJ/F32 11.9552 Tf 12.487 16.272 Td [(Z 1 0 e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K ^ g )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K ^ g )]TJ/F27 11.9552 Tf 9.299 0 Td [(! i! d! Inordertomoveonwiththeintegration,weneedthefollowingfacts: 1.^ g )]TJ/F27 11.9552 Tf 9.299 0 Td [(! = ^ g )]TJ/F27 11.9552 Tf 9.298 0 Td [(! = ; sinceall -termsin aresquared ; )]TJ/F27 11.9552 Tf 9.298 0 Td [(! = )]TJ/F27 11.9552 Tf 9.299 0 Td [(! [ 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 + ] )]TJ/F27 11.9552 Tf 9.298 0 Td [(! = )]TJ/F27 11.9552 Tf 9.299 0 Td [( ; A )]TJ/F27 11.9552 Tf 9.299 0 Td [(! ; = i )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [(e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i )]TJ/F28 7.9701 Tf 6.587 0 Td [(i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 [ )]TJ/F27 11.9552 Tf 11.955 0 Td [(i 2 + + )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! 2 ] 2 f )]TJ/F27 11.9552 Tf 11.956 0 Td [(i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! + e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i )]TJ/F28 7.9701 Tf 6.587 0 Td [(i [ )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + i + )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! ] g = )]TJ/F27 11.9552 Tf 9.298 0 Td [(i )]TJ/F27 11.9552 Tf 5.479 -9.683 Td [(e i )]TJ/F28 7.9701 Tf 6.587 0 Td [(i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 [ )]TJ/F27 11.9552 Tf 11.955 0 Td [(i 2 + + )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! 2 ] 2 f )]TJ/F27 11.9552 Tf 11.956 0 Td [(i + i + )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! + e i )]TJ/F28 7.9701 Tf 6.586 0 Td [(i [ )]TJ/F27 11.9552 Tf 11.955 0 Td [(i )]TJ/F27 11.9552 Tf 11.956 0 Td [(i + )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! ] g = A ; B )]TJ/F27 11.9552 Tf 9.298 0 Td [(! = 2 m + m + m )]TJ/F27 11.9552 Tf 9.298 0 Td [(! 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(r = B )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(2 m 2 C )]TJ/F27 11.9552 Tf 9.298 0 Td [(! = )]TJ/F27 11.9552 Tf 9.299 0 Td [(m! )]TJ/F27 11.9552 Tf 11.955 0 Td [(m )]TJ/F27 11.9552 Tf 9.298 0 Td [(! 2 + r! = )]TJ/F27 11.9552 Tf 9.299 0 Td [(C )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(2 m 2 Combiningalloftheseweget 51

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^ g )]TJ/F27 11.9552 Tf 9.298 0 Td [(! = e )]TJ/F26 5.9776 Tf 7.782 4.025 Td [(2 im )]TJ/F29 5.9776 Tf 5.756 0 Td [(i 2 e A )]TJ/F28 7.9701 Tf 6.587 0 Td [(! y + [ B )]TJ/F28 7.9701 Tf 6.587 0 Td [(iC ] [2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(i ] 2 m 2 f )]TJ/F27 11.9552 Tf 11.955 0 Td [(i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + )]TJ/F27 11.9552 Tf 11.956 0 Td [(i! + e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i )]TJ/F28 7.9701 Tf 6.587 0 Td [(i [ )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + i + )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! g 2 m 2 = e A y + [ B )]TJ/F28 7.9701 Tf 6.586 0 Td [(iC ] [2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(i ] 2 m 2 f )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + i + )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! + e i )]TJ/F28 7.9701 Tf 6.586 0 Td [(i [ )]TJ/F27 11.9552 Tf 11.955 0 Td [(i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! g 2 m 2 = ^ g 2.^ g )]TJ/F27 11.9552 Tf 9.299 0 Td [(! + i = ^ g + i Usingthefactthatcomplexconjugateofacompositionoffunctionsisacompositionofthecomplexconjugateofthefunctionsandusingthepreviousfact, weget ^ g + i = ^ g + i = ^ g )]TJ/F27 11.9552 Tf 11.955 0 Td [(i =^ g )]TJ/F27 11.9552 Tf 9.299 0 Td [(! + i 3.Nowitisclearthat e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K g )]TJ/F27 11.9552 Tf 9.298 0 Td [(i! = e i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K g )]TJ/F27 11.9552 Tf 9.299 0 Td [(! i! ; and e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K g + i )]TJ/F27 11.9552 Tf 9.299 0 Td [(i! = e i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K g )]TJ/F27 11.9552 Tf 9.299 0 Td [(! + i i! 4.Ifthecharacteristicfunction t isknown,Shephard[16]givesaformulato computethedistributionfunction F x byusinganinversiontheorem.The resultis F x = 1 2 )]TJ/F19 11.9552 Tf 16.686 8.088 Td [(1 2 Z 1 0 t e )]TJ/F28 7.9701 Tf 6.587 0 Td [(itx )]TJ/F27 11.9552 Tf 11.955 0 Td [( )]TJ/F27 11.9552 Tf 9.299 0 Td [(t e itx it Usingfacts3.and4.wecancontinuethecomputationof P 1 and P 2 P 1 ;x;y = 1 2 )]TJ/F19 11.9552 Tf 18.216 0 Td [(lim G + 1 2 Z 1 0 2 < e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K ^ g + i )]TJ/F27 11.9552 Tf 9.299 0 Td [(i! d! = 1 2 + 1 Z 1 0 < e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K ^ g + i )]TJ/F27 11.9552 Tf 9.298 0 Td [(i! d!; 52

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P 2 ;x;y = 1 2 )]TJ/F19 11.9552 Tf 18.217 0 Td [(lim G + 1 2 Z 1 0 2 < e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K ^ g )]TJ/F27 11.9552 Tf 9.299 0 Td [(i! d! = 1 2 + 1 Z 1 0 < e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K ^ g )]TJ/F27 11.9552 Tf 9.298 0 Td [(i! d!; where G x isthecumulativedistributionfunctionthatcorrespondstothecharacteristicfunction^ g Note:Thepriceoftheputoptioncanbecalculatedusingtheput-callparity.2.12. When =0,theintegralsintheexpressionsof P 1 and P 2 inthetheoremabove are Z 1 0 < e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K )]TJ/F27 11.9552 Tf 9.299 0 Td [(i! d! = sgn x )]TJ/F19 11.9552 Tf 11.955 0 Td [(ln K 2 ; where sgn x isthesignumfunctiondenedasfollows sgn x = 8 > > > < > > > : 1, x> 0 0, x =0 )]TJ/F19 11.9552 Tf 9.299 0 Td [(1, x< 0 : Thisimplies P 1 ;x;y = P 2 ;x;y = 8 > > > < > > > : 1, e x )]TJ/F27 11.9552 Tf 11.955 0 Td [(K> 0 1 2 e x = K 0, e x )]TJ/F27 11.9552 Tf 11.955 0 Td [(K< 0 : whichconrmsthatatmaturitythepriceofaEuropeancalloptionisexactlythe payo,max e x )]TJ/F27 11.9552 Tf 11.955 0 Td [(K; 0. Theproblemiswell-posedandwebelievethesolutionderivedaboveisunique, butwehaven'tshownitsuniqueness. Theintegrals.3.28and.3.29inthetheoremaboverequirenumericalintegration.ThediscussionforumsabouttheHeston'smodelon"WilmottForums" indicatethatprogrammersareinfavoroftheFastFourierTransformalgorithmor theSimpson'srulewhenitcomestoevaluatingtheseintegrals. 53

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Hestonarrivesatthesameoptionpricingformulaaswedo,butusingdierent solutiontechnique.Heguessesthefunctionalformofthecharacteristicfunctionto be f ;!;y = e C 1 ;! + D 1 ;! y + i!x ; andthenhesolvesfor C 1 ;! and D 1 ;! .Hestatesthatthisguessexploits thelinearityofthecoecientsinthepricingPDE.Theadvantageofoursolution techniqueisthatitdoesn'trequireanyguessing,whichmeansthatothervolatility functionsand/ordierentboundaryconditionscanbeused.Oursolutionapproach includesHeston'ssolutionasaparticularcase. 54

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5ClosedFormSolutionofaJumpDiffusionModelforEuropean Options TodeterminetheoptionpriceonastockthatfollowsadiusionprocesswithjumpdiusionvolatilitydescribedbySDEs.1.1and.1.2weuseanapproachsimilarl tothepurediusioncase. 5.1IntegralTransformsoftheJump-DiusionPricingModel InSection3.1thejump-diusionpricingmodelforEuropeanoptionswasderived: @P @ = r )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(1 2 y @P @x +[ m )]TJ/F27 11.9552 Tf 11.955 0 Td [(y )]TJ/F27 11.9552 Tf 11.955 0 Td [(y ] @P @y + 1 2 y @ 2 P @x 2 + y @ 2 P @x@y + 1 2 2 y @ 2 P @y 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(rP + Z 1 [ P ;x;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P ;x;y ] J dJ: .1.1 InthissectionwewanttosolvethisPIDEintheregion
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F Z 1 [ P ;x;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P ;x;y ] J dJ = Z 1 Z 1 e i!x [ P ;x;y + J )]TJ/F27 11.9552 Tf 11.955 0 Td [(P ;x;y ] J dJdx = Z 1 [ ^ P ;!;y + J )]TJ/F19 11.9552 Tf 14.59 3.022 Td [(^ P ;!;y ] J dJ; weobtain @ ^ P @ = 1 2 2 y @ 2 ^ P @y 2 +[ m )]TJ/F19 11.9552 Tf 11.955 0 Td [( + + i! y ] @ ^ P @y )]TJ/F19 11.9552 Tf 11.955 0 Td [([ r + i!r + 1 2 )]TJ/F27 11.9552 Tf 5.48 -9.683 Td [(! 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! y ] ^ P + Z 1 [ ^ P ;!;y + J )]TJ/F19 11.9552 Tf 14.59 3.022 Td [(^ P ;!;y ] J dJ: .1.2 Asintheprevioussection,wewanttotakeLaplacetransformofPIDE.1.2with respectto y .WecanusethepreviouslycalculatedLaplacetransformsofmostofthe termsinthisequationexceptfortheintegralterms.TheLaplacetransformofthe secondintegralisstraightforward L Z 1 ^ P ;!;y J dJ = Z 1 0 Z 1 e )]TJ/F28 7.9701 Tf 6.586 0 Td [(y ^ P ;!;y J dJd = ~ P ;!; Z 1 J dJ = ~ P ;!; ; since J istheprobabilitydensityfunctionof J andthus R 1 J dJ =1. Setting P ;x;y =0forall y 0and z = y + J intherstintegral,itsLaplace transformbecomeseasytocalculate: L Z 1 ^ P ;!;y + J J dJ = Z 1 J Z 1 e )]TJ/F28 7.9701 Tf 6.587 0 Td [( z )]TJ/F28 7.9701 Tf 6.586 0 Td [(J ^ P ;!;z J dJd = Z 1 e J J Z 1 0 e )]TJ/F28 7.9701 Tf 6.587 0 Td [(z ^ P ;!;z d dJ; since ^ P ;!;y =0, 8 y 0 = ~ P ;!; Z 1 e J J dJ: ThelastintegralisnothingelsebuttheMomentGeneratingfunctionof J .Letitbe 56

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denotedby M .Now,thetransformedequationbecomesrstorderPDE @ ~ P @ + 1 2 2 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [( + + i! )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(1 2 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! @ ~ P @ = M + m )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 + + + i! )]TJ/F27 11.9552 Tf 11.955 0 Td [(r )]TJ/F27 11.9552 Tf 11.955 0 Td [(i!r )]TJ/F27 11.9552 Tf 11.955 0 Td [( ~ P: .1.3 5.2SolutionofthetransformedJump-DiusionPricingPIDE ThedierencebetweenthePDEofthepurediusionmodel.2.8andthePDEof thejump-diusionmodel.1.3isintheexpressionintherighthandside,only,so whensolvingPDE.1.3wetakethesameapproachaswhensolvingPDE.2.8. ThisisdemonstratedinthefollowingProposition. Proposition5.2.1 ThesolutionofPDE.1.3withinitialcondition ~ P ;!; = ^ h isgivenby ~ P ;!; = ^ h [2 + i ] 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(m e [ 2 m + m )]TJ/F28 7.9701 Tf 6.587 0 Td [(r )]TJ/F28 7.9701 Tf 6.587 0 Td [( m )]TJ/F28 7.9701 Tf 6.587 0 Td [( 2 + i m! )]TJ/F28 7.9701 Tf 6.587 0 Td [(r! + m )]TJ/F28 7.9701 Tf 6.586 0 Td [( 2 ] [ + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i + i + i + + i! ] f ; ;! )]TJ/F27 11.9552 Tf 11.956 0 Td [( 0 e )]TJ/F29 5.9776 Tf 5.756 0 Td [(i + i I 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(I 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(I 3 + I 4 ; .2.4 where 0 f ; ;! = and = aredenedasinProposition4.2.2,and I 1 ; 2 ;!; = Z 1 e iJ [ + i )]TJ/F29 5.9776 Tf 5.756 0 Td [(i + + i! ] 2 E 1 a 1 ; 2 J dJ; and I 3 ; 4 ;!; = Z 1 e )]TJ/F29 5.9776 Tf 5.756 0 Td [(iJ [ + i + i + + i! ] 2 E 1 a 3 ; 4 J dJ: Thefunction E 1 istheExponentialintegral,denedby E 1 z = Z 1 z e )]TJ/F28 7.9701 Tf 6.586 0 Td [(u u du; j arg z j <; andthevariables a 1 ;a 2 ;a 3 and a 4 aredenedby 57

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a 1 = 2 iJ + i [ + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 ] e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i 2 f + i + i + + i! )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 +[ + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 ] e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i g ; a 2 = iJ 2 [ + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 ] ; a 3 = )]TJ/F19 11.9552 Tf 9.298 0 Td [(2 iJ + i [ + i + i + + i! )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 ] 2 f + i + i + + i! )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 +[ + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 ] e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i g ; a 4 = )]TJ/F27 11.9552 Tf 10.494 8.088 Td [(iJ 2 [ + i + i + + i! )]TJ/F27 11.9552 Tf 11.956 0 Td [( 2 ] : Proof. Weusethe -transformationcalculatedin.2.14 = i + i ln + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [( )]TJ/F27 11.9552 Tf 11.955 0 Td [( )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! + i + i 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [( )]TJ/F27 11.9552 Tf 11.955 0 Td [( )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! : Then,equation.1.3canberewrittenas @ ~ P @ + @ ~ P @ = i m )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 + i )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i + i 2 + e i + i + m 2 + + i! )]TJ/F27 11.9552 Tf 9.299 0 Td [(r )]TJ/F27 11.9552 Tf 11.955 0 Td [(i!r )]TJ/F27 11.9552 Tf 11.955 0 Td [( + M i + i )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i + i 2 + e i + i + + + i! 2 ~ P: .2.5 Byproposition.2.1thesolutionofthisequationcanbecalculatedas ~ P ;!; = exp Z 0 RHS )]TJ/F27 11.9552 Tf 11.955 0 Td [( + t dt )]TJ/F27 11.9552 Tf 11.955 0 Td [( ; where RHS isthefunctionontherighthandsideofPDE.2.5without ~ P andis anarbitraryfunctionthatcanbedeterminedusingtheinitialcondition.Sincewhen =0, exp )]TJ 5.48 -0.053 Td [(R 0 RHS )]TJ/F27 11.9552 Tf 11.956 0 Td [( + t dt =1,for weobtain = ^ h 2 + e i + i i + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i + i + + + i! + e i + i : 58

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Fortheintegraltermwehave Z 0 RHS )]TJ/F27 11.9552 Tf 11.955 0 Td [( + t dt = m + + i! + i m )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 + i 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [(r )]TJ/F27 11.9552 Tf 11.955 0 Td [(ir! )]TJ/F27 11.9552 Tf 11.955 0 Td [( +ln 1+ e i )]TJ/F28 7.9701 Tf 6.587 0 Td [( + i 1+ e i + i 2 m )]TJ/F29 5.9776 Tf 5.756 0 Td [( 2 2 + Z 0 M i + i )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i )]TJ/F28 7.9701 Tf 6.587 0 Td [( + t + i 2 + e i )]TJ/F28 7.9701 Tf 6.586 0 Td [( + t + i + + + i! 2 dt; thus,thesolutionofequation.2.5isgivenby ~ P ;!; = ^ h 2 e 2 m + m )]TJ/F29 5.9776 Tf 5.756 0 Td [( m )]TJ/F29 5.9776 Tf 5.756 0 Td [( 2 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(r )]TJ/F28 7.9701 Tf 6.587 0 Td [( + i m! + m )]TJ/F29 5.9776 Tf 5.756 0 Td [( 2 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(r! + e i )]TJ/F28 7.9701 Tf 6.587 0 Td [( + i i + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i )]TJ/F28 7.9701 Tf 6.586 0 Td [( + i + + + i! + e i )]TJ/F28 7.9701 Tf 6.586 0 Td [( + i 1+ e i )]TJ/F28 7.9701 Tf 6.587 0 Td [( + i 1+ e i + i 2 m )]TJ/F29 5.9776 Tf 5.756 0 Td [( 2 2 e R 0 M i + i )]TJ/F29 5.9776 Tf 5.756 0 Td [(e i )]TJ/F29 5.9776 Tf 5.756 0 Td [( + t + i 2 + e i )]TJ/F29 5.9776 Tf 5.756 0 Td [( + t + i + + + i! 2 dt : Usingthedenitionofthefunction M ,fortheintegralabovewehave Z 0 M i + i 2 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i + i )]TJ/F28 7.9701 Tf 6.587 0 Td [( + t 1+ e i + i )]TJ/F28 7.9701 Tf 6.586 0 Td [( + t + + + i! 2 dt = Z 1 J e J + + i! 2 Z 0 e iJ + i 2 1 )]TJ/F29 5.9776 Tf 5.757 0 Td [(e i + i )]TJ/F29 5.9776 Tf 5.756 0 Td [( + t 1+ e i + i )]TJ/F29 5.9776 Tf 5.756 0 Td [( + t dtdJ = Z 1 e J + + i! 2 J Z ub J lb J 2 2 Je z 2 z + iJ + i 2 z )]TJ/F27 11.9552 Tf 11.956 0 Td [(iJ + i dzdJ; .2.6 where ub J = iJ + i 2 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i + i 1+ e i + i ; and lb J = iJ + i 2 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i + i )]TJ/F28 7.9701 Tf 6.587 0 Td [( 1+ e i + i )]TJ/F28 7.9701 Tf 6.587 0 Td [( : .2.7 Thelastintegralisobtainedbysubstitutingthewholeexpressionintheexponentby z = iJ + i 2 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i + i )]TJ/F28 7.9701 Tf 6.586 0 Td [( + t 1+ e i + i )]TJ/F28 7.9701 Tf 6.586 0 Td [( + t ; 59

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andcalculating dt tobe dt = )]TJ/F19 11.9552 Tf 9.298 0 Td [(2 2 J iJ + i + 2 z iJ + i )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 z dz: Also,forthelastintegralin.2.6wehave Z ub J lb J 2 2 Je z 2 z + iJ + i 2 z )]TJ/F27 11.9552 Tf 11.955 0 Td [(iJ + i dz = Z ub J lb J i 2 e z + i 2 z + iJ + i dz )]TJ/F32 11.9552 Tf 11.955 16.273 Td [(Z ub J lb J i 2 e z + i 2 z )]TJ/F27 11.9552 Tf 11.955 0 Td [(iJ + i dz = e iJ + i 2 i + i Z ub 1 J lb 1 J e )]TJ/F28 7.9701 Tf 6.587 0 Td [(z z dz )]TJ/F27 11.9552 Tf 13.297 8.088 Td [(e )]TJ/F29 5.9776 Tf 5.756 0 Td [(iJ + i 2 i + i Z ub 2 J lb 2 J e )]TJ/F28 7.9701 Tf 6.587 0 Td [(z z dz; .2.8 where ub 1 J = 2 iJ + i e i + i 2 + e i + i ;lb 1 J = 2 iJ + i e i + i )]TJ/F28 7.9701 Tf 6.587 0 Td [( 2 + e i + i )]TJ/F28 7.9701 Tf 6.587 0 Td [( ; ub 2 J = )]TJ/F19 11.9552 Tf 9.298 0 Td [(2 iJ + i 2 + e i + i ; and lb 2 J = )]TJ/F19 11.9552 Tf 9.298 0 Td [(2 iJ + i 2 + e i + i )]TJ/F28 7.9701 Tf 6.586 0 Td [( : UsingthedenitionofExponentialintegral,thelastexpressionin.2.8canbe writtenas e iJ + i 2 i + i E 1 2 iJ + i e i + i )]TJ/F28 7.9701 Tf 6.587 0 Td [( 2 + e i + i )]TJ/F28 7.9701 Tf 6.586 0 Td [( )]TJ/F84 11.9552 Tf 11.955 0 Td [(E 1 2 iJ + i e i + i 2 + e i + i )]TJ/F27 11.9552 Tf 13.297 8.087 Td [(e )]TJ/F29 5.9776 Tf 5.756 0 Td [(iJ + i 2 i + i E 1 )]TJ/F19 11.9552 Tf 9.298 0 Td [(2 iJ + i 2 + e i + i )]TJ/F28 7.9701 Tf 6.587 0 Td [( )]TJ/F84 11.9552 Tf 11.955 0 Td [(E 1 )]TJ/F19 11.9552 Tf 9.298 0 Td [(2 iJ + i 2 + e i + i : .2.9 Thisresultwillgiveusanexpressionfor ~ P ;!; ~ P ;!; = ^ h 2 e h m + )]TJ/F29 5.9776 Tf 5.756 0 Td [( 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(r )]TJ/F28 7.9701 Tf 6.587 0 Td [( + + i m + 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(r! )]TJ/F28 7.9701 Tf 6.587 0 Td [( i + e i )]TJ/F28 7.9701 Tf 6.587 0 Td [( + i i + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(e i )]TJ/F28 7.9701 Tf 6.587 0 Td [( + i + + + i! + e i )]TJ/F28 7.9701 Tf 6.587 0 Td [( + i 1+ e i )]TJ/F28 7.9701 Tf 6.587 0 Td [( + i 1+ e i + i 2 m )]TJ/F29 5.9776 Tf 5.756 0 Td [( 2 2 e )]TJ/F29 5.9776 Tf 5.756 0 Td [(i + i I 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(I 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(I 3 + I 4 ; .2.10 60

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with I 1 ;!; = Z 1 e iJ [ + i )]TJ/F29 5.9776 Tf 5.756 0 Td [(i + + i! ] 2 E 1 2 iJ + i e i + i )]TJ/F28 7.9701 Tf 6.586 0 Td [( 2 + e i + i )]TJ/F28 7.9701 Tf 6.587 0 Td [( J dJ; I 2 ;!; = Z 1 e iJ [ + i )]TJ/F29 5.9776 Tf 5.756 0 Td [(i + + i! ] 2 E 1 2 iJ + i e i + i 2 + e i + i J dJ; I 3 ;!; = Z 1 e )]TJ/F29 5.9776 Tf 7.782 4.025 Td [(iJ [ + i + i + + i! ] 2 E 1 )]TJ/F19 11.9552 Tf 9.299 0 Td [(2 iJ + i 2 + e i + i )]TJ/F28 7.9701 Tf 6.587 0 Td [( J dJ; and I 4 ;!; = Z 1 e )]TJ/F29 5.9776 Tf 7.782 4.025 Td [(iJ [ + i + i + + i! ] 2 E 1 )]TJ/F19 11.9552 Tf 9.299 0 Td [(2 iJ + i 2 + e i + i J dJ: .2.11 When.2.9and.2.11areexpressedintermsof weobtainexactly.2.4. 5.3InverseIntegralTransformsoftheJump-DiusionPricingFormula Proposition5.2.1givesthesolutionofthetransformedjump-diusionpricingmodel. Inordertoobtaintheoptionpricingformulaofthejump-diusionmodelintermsof x and y weneedtoapplytheinverseLaplacetransformwithrespectto andthe inverseFouriertransformwithrespectto Theorem5.3.1 TheinverseLaplacetransformof ~ P ;!; givenby.2.4is ^ P ;!;y = ^ h e A ;! y + [ B + iC ] e )]TJ/F29 5.9776 Tf 12.029 3.258 Td [(i + i I 0 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(I 0 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(I 0 3 + I 0 4 2 + i + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i [ + i + i + + i! ] 2 m 2 ; .3.12 where A ;! B and C aregivenby: A ;! = i e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 [ + i 2 + + + i! 2 ] 2 f + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i [ + i + i + + i! ] g ; B = m + + 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(r C = m )]TJ/F27 11.9552 Tf 11.956 0 Td [( 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [(r!: .3.13 61

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Also, I 0 1 ;I 0 2 ;I 0 3 and I 0 4 areallfunctionsof and/or andaregivenby: I 0 1 ; 2 = Z 1 e iJ 2 [ + i )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + + i! ] E 1 a 0 1 ; 2 J dJ; and I 0 3 ; 4 = Z 1 e )]TJ/F29 5.9776 Tf 8.094 3.258 Td [(iJ 2 [ + i + i + + i! ] E 1 a 0 3 ; 4 J dJ; with a 0 1 ;a 0 2 ;a 0 3 and a 0 4 denedas a 0 1 = iJ 2 [ + i )]TJ/F27 11.9552 Tf 11.956 0 Td [(i + + i! ] ; a 0 2 ;! = 2 iJ + i [ + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! ] 2 f + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! +[ + i + i + + i! ] e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i g ; a 0 3 = )]TJ/F27 11.9552 Tf 10.494 8.088 Td [(iJ 2 [ + i + i + + i! ] ; a 0 4 ;! = )]TJ/F19 11.9552 Tf 9.299 0 Td [(2 iJ + i [ + i + i + + i! ] e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i 2 f + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! +[ + i + i + + i! ] e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i g : Proof. ApplyingtheLaplaceinversionformulato.2.4weobtain: ^ P ;!;y = ^ h [2 + i ] 2 )]TJ/F26 5.9776 Tf 7.782 3.259 Td [(2 m 2 e h m + )]TJ/F29 5.9776 Tf 5.756 0 Td [( 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(r )]TJ/F28 7.9701 Tf 6.586 0 Td [( + i m + 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(r! )]TJ/F28 7.9701 Tf 6.587 0 Td [( i [ + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i + i + i + + i! ] Z + i 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(i 1 e y f ; ;! e )]TJ/F29 5.9776 Tf 5.756 0 Td [(i + i I 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(I 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(I 3 + I 4 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 0 d; .3.14 where issuchthatthecontourofintegrationistotherightsideofanysingularities of f ; ;! e )]TJ/F29 5.9776 Tf 5.756 0 Td [(i + i I 1 )]TJ/F29 5.9776 Tf 5.756 0 Td [(I 2 )]TJ/F29 5.9776 Tf 5.756 0 Td [(I 3 + I 4 )]TJ/F28 7.9701 Tf 6.586 0 Td [( 0 and I 1 ;I 2 ;I 3 and I 4 arefunctionsof andaredenedas inProposition5.2.1.Usingthesameargumentasinthepurediusioncase,the exponent 2 m 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 0,thusthefunction f ; ;! = i 2 )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [(e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 + + i + i + + i! + e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i [ + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! ] 2 m 2 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 .3.15 isanalytic.Theexponentialfunction e y isalsoanalytic.Apotentialsingularity pointfor e )]TJ/F29 5.9776 Tf 5.756 0 Td [(i + i I 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(I 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(I 3 + I 4 isthevaluefor forwhichthedenominatorin a 2 and a 4 62

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is0,thatis 1 = + i + e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i i 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i )]TJ/F19 11.9552 Tf 13.15 8.088 Td [( + + i! 2 ; however, f 1 ; ;! =0.Hence,wecanapplythe CauchyIntegralFormula .3.1 tocalculatetheintegralin.3.14with 0 = i e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 [ + i 2 + + + i! 2 ] 2 f + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i [ + i + i + + i! ] g .3.16 asasimplepole.Thisimpliesthat Z + i 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(i 1 f ; ;! e y e )]TJ/F29 5.9776 Tf 5.756 0 Td [(i + i I 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(I 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(I 3 + I 4 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 0 d =2 if 0 ; ;! e y 0 e )]TJ/F29 5.9776 Tf 5.756 0 Td [(i + i I 1 0 )]TJ/F28 7.9701 Tf 6.587 0 Td [(I 2 0 )]TJ/F28 7.9701 Tf 6.586 0 Td [(I 3 0 + I 4 0 ; with f 0 ; ;! = 4 + i 2 e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i [ + i + i + + i! ] 2 m 2 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 : Whenweplug 0 'sexpressionabovein I 1 ;I 2 ;I 3 and I 4 ,thesefourwill becomefunctionsof and .Denotingthemby I j 0 = I 0 j ,for j =1 ; 3and I j 0 = I 0 j ;! ,for j =2 ; 4,weobtaintherequiredresult. Theorem5.3.2 ThepriceofaEuropeanCalloptiononanunderlyingassetthat followsthediusionprocess.0.1drivenbymean-revertingjump-diusionvolatility .1.2canbecalculatedusingtheformula P ;x;y = e x P 1 ;x;y )]TJ/F27 11.9552 Tf 11.955 0 Td [(KP 2 ;x;y ; .3.17 where x =ln S t 63

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P 1 ;x;y = 1 2 + 1 Z 1 0 < e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K ^ g + i ; )]TJ/F27 11.9552 Tf 9.299 0 Td [(i! d!; P 2 ;x;y = 1 2 + 1 Z 1 0 < e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K ^ g ; ;y )]TJ/F27 11.9552 Tf 9.298 0 Td [(i! d!; ^ g ; ;y = A ;! y + [ B + iC ] e )]TJ/F29 5.9776 Tf 12.03 3.259 Td [(i + i I 0 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(I 0 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(I 0 3 + I 0 4 2 + i + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i [ + i + i + + i! ] 2 m 2 ; = and = aredenedasin.2.17, A ;! B and C aredened asin.3.13and I 0 1 ;I 0 2 ;! ;I 0 3 and I 0 4 ;! aredenedasinProposition5.3.1. Proof. Theonlydierencebetweenthistheoremandtheorem4.3.3isthedenition ofthefunction^ g ; ;y .Thus,wecanusethesameproofforprovingthistheorem aslongasweshowthat^ g ;0=1andthat^ g )]TJ/F27 11.9552 Tf 9.298 0 Td [(! = ^ g First,notethatattime =0 I 0 2 ;! = I 0 1 and I 0 4 ;! = I 0 3 whichmakes e )]TJ/F29 5.9776 Tf 12.029 3.258 Td [(i + i I 0 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(I 0 2 ;! )]TJ/F28 7.9701 Tf 6.587 0 Td [(I 0 3 + I 0 4 ;! =1 : andso^ g ;0=1. Next,recallthat )]TJ/F27 11.9552 Tf 9.299 0 Td [(! = )]TJ/F27 11.9552 Tf 9.299 0 Td [(! )]TJ/F27 11.9552 Tf 9.298 0 Td [(! = )]TJ/F27 11.9552 Tf 9.298 0 Td [( .Then,itiseasytoshowthat I 0 1 )]TJ/F27 11.9552 Tf 9.299 0 Td [(! = I 0 3 ; I 0 1 = I 0 3 )]TJ/F27 11.9552 Tf 9.298 0 Td [(! ;I 0 2 )]TJ/F27 11.9552 Tf 9.299 0 Td [(! ; = I 0 4 ; and I 0 4 )]TJ/F27 11.9552 Tf 9.298 0 Td [(! ; = I 0 2 ; : UsingtheseresultsaswellasfewresultsthatweprovedinTheorem4.3.3, A )]TJ/F27 11.9552 Tf 9.298 0 Td [(! ; = A ; ;B )]TJ/F27 11.9552 Tf 9.299 0 Td [(! = B )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(2 m 2 ; and C )]TJ/F27 11.9552 Tf 9.298 0 Td [(! = )]TJ/F27 11.9552 Tf 9.298 0 Td [(C )]TJ/F19 11.9552 Tf 13.15 8.088 Td [(2 m 2 ; 64

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weobtain^ g )]TJ/F27 11.9552 Tf 9.299 0 Td [(! = ^ g TherestoftheprooffollowsthatofTheorem4.3.3. 65

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6OptionGreeks Whenmanagingaportfolioofoptions,bothmarketmakersandoptioninvestors areinterestedinassessingthechangeintheoptionpriceanditssensitivitywhen changeinthepriceoftheunderlyingasset,volatilityandinterestrateoccurs.The OptionGreeksareformulasthatexpressthechangeintheoptionpricewhenoneof theparameterschanges;thustheyareconsideredtobemeasuresofriskexposure. Mathematically,theyaresimplythederivativeoftheoptionpricewithrespecttoone inputonlywhiletheothervariablesarekeptconstant. 6.1Delta Deltaisthemostwellknownandthemostimportantoftheoptiongreeks.It measuresthechangeintheoptionpricewhenthestockpriceincreases/decreasesby $1,thatis = @P @S : KnowingtheDeltavalueoftheoptionisimportantforoptiontraders.Ifyou believethatthepriceoftheunderlyingassetwillgouponedollarwithinafewdays andboughtcalloptionsinordertoprepareforthatmove,thedeltaofyourcall optionswilltellyouexactlyhowmuchmoneyyouwillmakewiththat$1surge.The optiondeltathereforehelpsyouplanhowmuchcalloptionstobuyifyouareplanning tocaptureacertainprot. WeusedthisparameterwhenderivingtheBlack-Scholesequationinsection2.2 whereweusedsharesofstocktohedgetheriskassociatedwiththerandomnessin thestockprice.WestartthecalculationoftheofaEuropeancalloptioninthe 66

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jump-diusionmodelbydierentiatingformula.3.17withrespectto x ,andthus wehave @P @S = @P @x @x @S = P 1 + @P 1 @x )]TJ/F27 11.9552 Tf 11.955 0 Td [(Ke )]TJ/F28 7.9701 Tf 6.587 0 Td [(x @P 2 @x ; where P 1 and P 2 aredenedinTheorem5.3.2.Tocalculate @P 1 @x ,changethevariable + i with ,inwhichcase )]TJ/F27 11.9552 Tf 9.298 0 Td [(i! shouldbereplacedby )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(i! .Thenwehave: P 1 ;x;y = 1 2 + 1 Z 1 0 < )]TJ/F27 11.9552 Tf 9.299 0 Td [(e )]TJ/F25 7.9701 Tf 6.587 0 Td [( x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K ^ g ; ;y 1+ i! d!; and @P 1 @x = K e x Z 1 0 < e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K ^ g ; ;y 1+ i! d! + K e x Z 1 0 < i!e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K ^ g ; ;y 1+ i! d! = K e x Z 1 0 < e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K ^ g ; ;y d!: Thecalculationof @P 2 @x doesn'trequireanysubstitutionand,similartotheprevious derivative,weobtain @P 2 @x = 1 Z 1 0 < e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K ^ g ; ;y d!: Thus, @P 1 @x = K e x @P 2 @x ; whichprovesthefollowingresult. Proposition6.1.1 TheDeltaofaEuropeancalloptioninajump-diusionvolatility settingis = @P @S = P 1 ; .1.1 where P 1 anditscomponentsaredenedinTheorem5.3.2. Consideringthefactthat P 1 isaprobabilityfunctionwehavethattheDeltafor aEuropeancallisbetween0and1,whichconrmstwoverylogicalthingsfroma nancialperspective-thepriceofthecallincreasesasthestockpriceincreasesand thechangeinthecallpricecannotbegreaterthanthechangeinthestockprice. 67

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Usingtheput-callparity,theDeltaoftheEuropeanputis P = C )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; .1.2 andhence, P isalwaysnegative.Thismeansthatthepriceoftheputdecreasesas thepriceoftheunderlyingassetincreases,andthatthechangeinthepriceoftheput islessthanthechangeofthestockprice. 6.2Gamma GammameasuresthechangeinDeltaasthestockpricechanges,thusitisdened asthederivativeofwithrespectto S .Thisparameterisimportantbecauseit showshowfastourDeltapositionchangesinrelationtothepriceoftheunderlying asset,however,itisnotnormallyneededforthecalculationofmostoptiontrading strategies.GammaisparticularlyimportantforDeltaneutraltraderswhowantto predicthowtoresettheirDeltaneutralpositionsasthepriceoftheunderlyingstock changes. TherelationshipbetweenthecallandtheputDeltasdenedinequation.1.2 impliesthatthecallandputGammasareequal.Belowwedemonstratehowthey canbecalculatedinthejump-diusionvolatilitycase: @ 2 P @S 2 = @P 1 @S = @P 1 @x @x @S = e )]TJ/F28 7.9701 Tf 6.587 0 Td [(x K e x 1 Z 1 0 < e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K ^ g ; ;y d!: Proposition6.2.1 TheGammaofaEuropeancall/putoptioninajump-diusion volatilitysettingis )]TJ/F28 7.9701 Tf 7.314 -1.793 Td [(C =)]TJ/F28 7.9701 Tf 19.739 -1.793 Td [(P = @ 2 P @S 2 = K e 2 x 1 Z 1 0 < e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K ^ g ; ;y d!; .2.3 where P 1 anditscomponentsaredenedinTheorem5.3.2. 68

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6.3Theta Thetameasureshowfastthepremiumofastockoptiondecayswithtime.This parameterismeasuredindaysandisactiveevenontheweekends,whenthemarkets areclosed.SomeoptiontradingstrategiesthatareparticularlyThetasensitiveare CalendarCallSpreadandCalendarPutSpreadwheretradersneedtomaintainanet positiveThetainordertoensureaprot. Mathematicallyitisdenedasapartialderivativeoftheoptionvaluewithrespect tothetimetoexpiration = T )]TJ/F27 11.9552 Tf 11.955 0 Td [(t .It'seasytoseethat @P 1 @ = K e x 1 Z 1 0 < )]TJ/F27 11.9552 Tf 9.299 0 Td [(e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K @ ^ g ; ;y @ 1+ i! # d!; @P 2 @ = 1 Z 1 0 < )]TJ/F27 11.9552 Tf 9.298 0 Td [(e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K @ ^ g ; ;y @ i! # d!; andso,for @P @ wehave: @P @ = e x @P 1 @ )]TJ/F27 11.9552 Tf 11.955 0 Td [(K @P 2 @ = K Z 1 0 < e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K @ ^ g ; ;y @ i! )]TJ/F27 11.9552 Tf 11.956 0 Td [(! 2 # d!: Tocalculate @ ^ g ; ;y @ weneedtocalculate @A ; @ @I 0 2 ;! @ and @I 0 4 ;! @ .Therstderivativecanbeshowntobe @A ;! @ = 2 + i 2 [ + i 2 + + + i! 2 ] e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i 2 f + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i [ + i + i + + i! ] g 2 ; andfortheothertwowecanapplytheLeibnizintegralruletotheexponentialintegrals andobtain 69

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@I 0 2 @ = Z 1 e iJ [ + i )]TJ/F29 5.9776 Tf 5.756 0 Td [(i + + i! ] 2 @a 0 2 @ E 1 a 0 2 J dJ = )]TJ/F27 11.9552 Tf 9.298 0 Td [(i + i [ + i + i + + i! ] )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! +[ + i + i + + i! ] e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i Z 1 exp iJ [ + i 2 + + + i! 2 ] e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 f + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! +[ + i + i + + i! ] e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i g J dJ; @I 0 4 @ = Z 1 J e )]TJ/F29 5.9776 Tf 5.756 0 Td [(iJ [ + i + i + + i! ] 2 @a 0 4 @ E 1 a 0 4 dJ = i + i [ + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! ] + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! +[ + i + i + + i! ] e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i Z 1 exp iJ [ + i 2 + + + i! 2 ] e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 f + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! +[ + i + i + + i! ] e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i g J dJ: Notethattheintegralfunctioncanbeexpressedintermsof A ;! as e JA ;! .Then forthepartialderivativewithrespectto oftheexponentof^ g ; ;y wehave: @ @ A ;! y + B + iC )]TJ/F27 11.9552 Tf 22.681 8.088 Td [(i + i I 0 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(I 0 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(I 0 3 + I 0 4 = 2 + i 2 [ + i 2 + + + i! 2 ] e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i 2 f + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i [ + i + i + + i! ] g 2 y + B + iC + Z 1 e JA ;! J dJ; .3.4 andhence @ ^ g @ =^ g ; ;y B + iC + Z 1 e JA ;! J dJ + 2 + i 2 [ + i 2 + + + i! 2 ] e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i 2 f + i )]TJ/F27 11.9552 Tf 11.955 0 Td [(i + + i! + e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i + i [ + i + i + + i! ] g 2 y )]TJ/F19 11.9552 Tf 57.975 8.088 Td [(2 im + i [ + i + i + + i! ] e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i 2 f + i )]TJ/F27 11.9552 Tf 11.956 0 Td [(i + + i! +[ + i + i + + i! ] e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i + i g : .3.5 Proposition6.3.1 TheThetaofaEuropeancalloptioninajump-diusionvolatility 70

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settingisgivenby C = @P @ = K Z 1 0 < e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K @ ^ g ; ;y @ i! )]TJ/F27 11.9552 Tf 11.956 0 Td [(! 2 # d!; .3.6 where @ ^ g ; ;y @ isgivenbyexpression.3.5. Usingtheput-callparity,theThetaforEuropeanputoptionsisfoundtobe P = C )]TJ/F27 11.9552 Tf 11.955 0 Td [(rKe )]TJ/F28 7.9701 Tf 6.587 0 Td [(r : 6.4Vega ThoughnotaGreekletter,thismeasurefallsunderthe"Greeks".Vegameasuresthe sensitivityoftheoptionpricetothevolatilityoftheunderlyingasset.Vegaisquoted toshowthetheoreticalpricechangeforevery onepercentagepoint .01changein impliedvolatility.Anincreaseinvolatilityraisesthepriceofboth,callandput, options.OneoptiontradingstrategythatisparticularlyVegasensitiveisStraddle buyingacallandaputwiththesamestrikepriceandtimetoexpiration.This strategyisusedwhenthebuyerbelievesthevolatilitywillbeprettyhigh,andthus thestockpricewillmoveupordown.Ifthestockpricerises,thetraderwillmakea protonthecallandifthestockpricedeclines,thetraderwillmakeaprotonthe purchasedput. Vegaisdenedasapartialderivativeoftheoptionpricewithrespecttothe volatility which,inthemodeldescribedinthispaper,is = p y .Then @P @ =2 p y @P @y : 71

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Since @P 1 @y = K e x 1 Z 1 0 < )]TJ/F27 11.9552 Tf 9.299 0 Td [(e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.587 0 Td [(ln K @ ^ g ; ;y @y 1+ i! # d!; and @P 2 @y = 1 Z 1 0 < )]TJ/F27 11.9552 Tf 9.299 0 Td [(e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K @ ^ g ; ;y @y i! # d!; wehave: @P @y = K Z 1 0 < e )]TJ/F28 7.9701 Tf 6.586 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K @ ^ g ; ;y @y i! )]TJ/F27 11.9552 Tf 11.956 0 Td [(! 2 # d!: Itiseasytoseethat @ ^ g @y = A ;! ^ g ; ;y ; whichgivesusthefollowing: Proposition6.4.1 TheVegaofaEuropeancall/putoptioninajump-diusionvolatilitysettingisgivenby V C = V P = @P @ = 2 K p y Z 1 0 < A !; e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i! x )]TJ/F25 7.9701 Tf 6.586 0 Td [(ln K ^ g ; ;y i! )]TJ/F27 11.9552 Tf 11.955 0 Td [(! 2 d!: .4.7 Note:Becauseofput-callparitytheVegaisthesameforEuropeancallandput optionswithsamestrikepriceandtimetoexpiration. 72

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7ResultsandConclusion Initially,thispaperintroducestheBlack-ScholesmodelsforpricingEuropeanand Americanoptionswrittenonnon-dividendanddividendpayingunderlyingassets. Theconstantvolatilityandthelognormaldistributionofthereturnsassumptionsare arguedtobeinvalidincomparisontorealnancialmarketdata.Thenecessityof havingrandomlychangingvolatilityispresented.Moreover,EuropeanandAmerican optionpricingmodelsarederivedthatallowvolatilitywithjump-diusivebehavior asstatedandprovedinTheorem3.1.4andTheorem3.2.4.TheEuropeanoption pricingmodelisgivenbyahomogeneoussecond-orderlinearpartialdierentialequationwithvariablecoecients,andtheAmericanoptionspricingmodelisgivenbya nonhomogeneousformoftheEuropeanoptionsPDE,reducedfromafree-boundary valueproblem. Thefastpaceofthechangesinnancialmarketsrequiresfastderivativeprice computationmethods.Withthatinmind,weseekaclosedformsolutionforthe Europeanoptionpricingmodel.Hestonarrivedataclosedformsolutionforthepure diusionvolatilitycaseusingthemethodofcharacteristicfunctionsbyguessingtheir form.HisguessisexploitingthelinearityofthecoecientsinthepricingPDE.This meansthatfordierentchoicesofthevolatilityfunctionordierentboundaryconditions,thissolutiontecnhiquecannotbeused.WeusearigorousPDEapproachto determineaclosedformsolutionofthepricingPDE.1.14,anapproachthathas moreexibilitytoaccomodatedierentdetermeningconditions.First,settingthe jumpfrequencyto0,thejump-diusionpricingmodelbecomespurediusion,identicaltotheonethatHestonderives.ThenapplyingaFouriertransformwithrespectto thelogarithmofthestockpricevariable, x ,andaLaplacetransformwithrespectto 73

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thevolatilityvariable, y ,theinitialPDEisreducedtoarst-orderlinearPDE.Next, asolutionofthisPDEisderivedandnallyanexactclosed-formpricingformula isobtained.Usingasimilarapproach,aclosed-formsolutionofthejump-diusion volatilitymodelhasalsobeenderived.ThisisstatedandprovedinTheorem5.3.2. Both,purediusionandjump-diusion,optionpricingformulasinvolveintegralsthat neednumericalevaluation.Thechallengeinevaluatingtheseintegralsiswhatmethod willbethefastest. Marketmakersandinvestorsarenotinterestedindeterminingtheoptionprice only.Theyarealsointerestedinknowingthehedgingparameters.Thelastchapter givesformulasforcalculatingtheoptionGreeksthatarewidelyusedwheninvestment strategiesaremade. Thereareseveralproblemsthatwewouldliketoaddressinsomefuturework. Therstoneistoreleasetheconstantriskrateassumptionanduseastochasticrisk rate.AnotherproblemistodetermineasolutionofthenonhomogeneousPDEand soobtainaclosed-formpricingformulaforAmericanoptions.Allowingjumpsin thestockpricewhilekeepingthejumpsinthevolatilityprocessandndingclosedformsolutionforthismodelisanotherchallengethatwewouldliketoconsider. However,weneedtokeepinmindthateventhoughaddingmoreassumptionsand parameterstothepricingmodelsmightshowamorerealisticpictureofthebehavior ofthenancialmarkets;thisalsoincreasesthepricingcomplexityaswellasthetime requiredforestimationoftheparametersandcalculationofthepriceitself.Proving theuniquenessofthesolutionsobtainedinthisdissertationwillfollowaswell. 74

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References [1]S.Heston,1993,"AClosed-FormSolutionforOptionswithStochasticVolatility withApplicationstoBondandCurrencyOptions", TheReviewofFinancial Studies ,VOL.6,No.2,327-343 [2]J.-P.Fouque,G.Papanicolau,K.R.Sircar,2000,"DerivativesinFinancialMarkets withStochasticVolatility", CambridgeUniversityPress [3]RichardHaberman,"ElementaryAppliedPartialDierentialEquationswith FourierSeriesandBoundaryValueProblems" [4]M.Grigoriu,2002,"StochasticCalculus,ApplicationsinScienceandEngineering", Birkhauser [5]P.Cheridito,D.Filipovic,R.Kimmel,2005,"MarketPriceofRiskSpecications forAneModels:TheoryandEvidence" [6]J.-P.Fouque,G.Papanicolau,K.R.Sircar,2000,"Mean-RevertingStochastic Volatility" [7]RobertBu,"UncertainVolatilityModels:TheoryandApplication", Springer Finance [8]S.Malone,2002,workingpaper"AlternativePriceProcessesforBlack-Scholes: EmpiricalEvidenceandTheory" [9]R.Schobel,J.Zhu,1999,"StochasticVolatilityWithanOrnstein-UhlenbackProcess:AnExtension", EuropeanFinanceReview 75

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[10]G.Yan,F.B.Hanson,2005,"OptionPricingforaStochastic-VolatilityJumpDiusionModelwithLog-UniformJump-Amplitudes", ProceedingsofAmerican ControlConference [11]F.B.Hanson,J.J.Westman,2002,"OptimalConsumptionandPortfolioControl forJump-DiusionStockProcesswithLog-NormalJumpsCorrected", ProceedingsofAmericanControlConference [12]S&P500optionquotes,http://quotes.optionetics.com/optionetics/quote.asp, June21,2006 [13]Optiongreeks,http://optiontradingpedia.com/ [14]Y-K.Kwok,1998,"MathematicalModelsofFinancialDerivatives", Springer [15]C.Chiarella,A.Ziogas,2005,"PricingAmericanOptionsUnderStochastic Volatility" [16]N.G.Shephard,1991,"FromCharacteristicFunctiontoDistributionFunction: ASimpleFrameworkfortheTheory" [17]A.D.Polyanin,V.F.Zaitsev,A.Moussiaux,2002,"HandbookofFirstOrder PartialDierentialEquations" [18]D.Bates,1996,"JumpsandStochasticVolatility:ExchangeRateProcesses implicitinPHLXDeutschMarkOptions", ReviewofFinancialStudies [19]D.Bates,2000,"TheCrashof'87:WasItExpected?TheEvidencefromOptions Markets", TheJournalofFinance [20]B.Eraker,June2004,"DoStockPricesandVolatilityJump?ReconcilicngEvidencefromSpotandOptionPrices", TheJournalofFinance References 76