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A Fast and Exact Simulation for CIR Process

Permanent Link: http://ufdc.ufl.edu/UFE0043921/00001

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Title: A Fast and Exact Simulation for CIR Process
Physical Description: 1 online resource (78 p.)
Language: english
Creator: Shao, Anqi
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

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Subjects / Keywords: cir -- heston -- simulation
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We present a fast and exact simulation method for the CIR process. Traditional simulation method relies on an algorithm to generate a non-central chi square random ariable, which is quite slow when the degrees of freedom is less than 1, and the non-centrality parameter is large. Thus it is only applicable when people are just interested in simulating the process along a few time points, for example, European options prices. But for some exotic options which depends on the process on a lot of time points, for example Asian option prices, this method is very slow and inefficient. In this paper we analyze the algorithm to see its limitation, and propose a new algorithm which is much faster. This method enables fast and exact simulation of the CIR processon a large number of time grids.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Anqi Shao.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Yan, Liqing.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-11-30

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043921:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043921/00001

Material Information

Title: A Fast and Exact Simulation for CIR Process
Physical Description: 1 online resource (78 p.)
Language: english
Creator: Shao, Anqi
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: cir -- heston -- simulation
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We present a fast and exact simulation method for the CIR process. Traditional simulation method relies on an algorithm to generate a non-central chi square random ariable, which is quite slow when the degrees of freedom is less than 1, and the non-centrality parameter is large. Thus it is only applicable when people are just interested in simulating the process along a few time points, for example, European options prices. But for some exotic options which depends on the process on a lot of time points, for example Asian option prices, this method is very slow and inefficient. In this paper we analyze the algorithm to see its limitation, and propose a new algorithm which is much faster. This method enables fast and exact simulation of the CIR processon a large number of time grids.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Anqi Shao.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Yan, Liqing.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-11-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043921:00001


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AFASTANDEXACTSIMULATIONFORCIRPROCESSByANQISHAOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012AnqiShao 2

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Tomyfamily 3

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ACKNOWLEDGMENTS FirstIwouldliketothankmyadvisorDr.Yanforhiskindlyhelpandconstantencouragementthroughouttheresearchprocess.I'dalsoliketothankDr.Rao,Dr.Hager,Dr.McCulloughandDr.Qiufortheirinterestinmyresearch.IappreciatetheopportunityUFmathdepartmenthasofferedmeduringthoseyears.It'sbeenapleasureandanhonortostudyhere,andI'velearntalotsinceIcamehere.FinallyI'dliketothankmyfamilyfortheirconstantsupport,bothmentallyandphysically.Icouldnotimaginetonishthiswithoutallthehelpsfromthosepeoplethatcareaboutme. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTIONTOCIRMODELANDSIMULATION .............. 10 1.1TheTermStructureOfInterestRates .................... 10 1.2ModelsOfTheShort-TermInterestRate ................... 14 1.3IntroductiontoCIRModel ........................... 17 1.4SimulationIssue ................................ 19 2EULEREISCRETIZATIONANDRELATEDISSUES ............... 23 2.1IntroductionToEuler-MaruyamaScheme .................. 23 2.2ConvergenceOrder .............................. 24 2.3EulerSchemeForCIRProcess ........................ 26 3CIRPROCESSEXACTSIMULATION ....................... 29 3.1TraditionalSimulationMethod ......................... 29 3.2SomeDrawbacksInSimulatingCIRProcess ................ 33 3.3MainResult ................................... 35 4INTRODUCTIONTOHESTONMODELANDSIMULATION ........... 37 4.1OptionPricingTheory ............................. 37 4.2IntroductionToStochasticVolatilityModel .................. 43 4.3IntroductionToHeston'sModelAndSimulationIssue ............ 46 5SIMULATIONRESULTSANDAPPLICATIONSINOPTIONPRICING ..... 53 5.1SimulationResultsForCIRInterestRateModel ............... 53 5.2SimulationResultsForHestonModel .................... 61 6CONCLUSION .................................... 74 REFERENCES ....................................... 75 BIOGRAPHICALSKETCH ................................ 78 5

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LISTOFTABLES Table page 5-1ParametertableforCIRinterestmodelsimulation. ................ 54 5-2EstimatedexpectationofinterestrateattimeT=1intestCaseI.Numbersinparenthesesaresamplestandarddeviations. .................. 54 5-3EstimatedexpectationofinterestrateattimeT=1intestCaseII.Numbersinparenthesesaresamplestandarddeviations. .................. 55 5-4EstimatedexpectationofinterestrateattimeT=1intestCaseIII.Numbersinparenthesesaresamplestandarddeviations. .................. 55 5-5EstimatedbondpriceintestCaseI.Numbersinparenthesesaresamplestandarddeviations. ...................................... 58 5-6EstimatedbondpriceintestCaseII.Numbersinparenthesesaresamplestandarddeviations. ................................. 58 5-7EstimatedbondpriceintestCaseIII.Numbersinparenthesesaresamplestandarddeviations. ................................. 58 5-8ParametertableforHestonmodelsimulation.Inallcasesr=0,V(0)=andX(0)=100. ................................... 62 5-9EstimatedEuropeancalloptionpriceintestCaseI.Numbersinparenthesesaresamplestandarddeviations. .......................... 63 5-10EstimatedEuropeancalloptionpriceintestCaseII.Numbersinparenthesesaresamplestandarddeviations. .......................... 66 5-11EstimatedEuropeancalloptionpriceintestCaseIII.Numbersinparenthesesaresamplestandarddeviations. .......................... 69 5-12EstimateddiscreteAsiancalloptionpriceintestCaseIII.Numbersinparenthesesaresamplestandarddeviations. .......................... 72 6

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LISTOFFIGURES Figure page 1-1Normalyieldcurve .................................. 13 1-2Invertedyieldcurve ................................. 13 1-3ThreesamplepathsofdifferentOU-processeswith=1,=1.2,=0.3 ... 16 1-4AnexampleofCIRprocesswith=0.01,b=1and=0.1 .......... 18 4-1ImpliedVolatilitySurface ............................... 44 5-1ComparisonbetweendifferentschemesinestimatingE[R(1)]intestCaseI .. 56 5-2ComparisonbetweendifferentschemesinestimatingE[R(1)]intestCaseII 56 5-3ComparisonbetweendifferentschemesinestimatingE[R(1)]intestCaseIII 57 5-4ComparisonbetweendifferentschemesinestimatingE(e)]TJ /F7 7.97 Tf 8 6.42 Td[(R10R(t)dt)intestCaseI ......................................... 59 5-5ComparisonbetweendifferentschemesinestimatingE(e)]TJ /F7 7.97 Tf 8 6.42 Td[(R10R(t)dt)intestCaseII ........................................ 60 5-6ComparisonbetweendifferentschemesinestimatingE(e)]TJ /F7 7.97 Tf 8 6.42 Td[(R10R(t)dt)intestCaseIII ........................................ 60 5-7ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseI(K=70) ................................. 64 5-8ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseI(K=100) ................................ 64 5-9ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseI(K=140) ................................ 65 5-10ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseII(K=70) ................................ 67 5-11ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseII(K=100) ................................ 67 5-12ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseII(K=140) ................................ 68 5-13ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseIII(K=70) ................................ 70 5-14ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseIII(K=100) ............................... 70 7

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5-15ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseIII(K=140) ............................... 71 5-16ComparisonbetweendifferentschemesinestimatingdiscreteAsianoptionpriceintestCaseIII(K=70) ............................. 72 5-17ComparisonbetweendifferentschemesinestimatingdiscreteAsianoptionpriceintestCaseIII(K=100) ............................ 73 5-18ComparisonbetweendifferentschemesinestimatingdiscreteAsianoptionpriceintestCaseIII(K=140) ............................ 73 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyAFASTANDEXACTSIMULATIONFORCIRPROCESSByAnqiShaoMay2012Chair:LiqingYanMajor:MathematicsWepresentafastandexactsimulationmethodfortheCIRprocess.Traditionalsimulationmethodreliesonanalgorithmtogenerateanon-centralchisquarerandomvariable,whichisquiteslowwhenthedegreesoffreedomislessthan1,andthenon-centralityparameterislarge.Thusit'sonlyapplicablewhenpeoplearejustinterestedinsimulatingtheprocessalongafewtimepoints,forexample,Europeanoptionsprices.Butforsomeexoticoptionswhichdependsontheprocessonalotoftimepoints,forexampleAsianoptionprices,thismethodisveryslowandinefcient.Inthispaperweanalyzethealgorithmtoseeitslimitation,andproposeanewalgorithmwhichismuchfaster.ThismethodenablesfastandexactsimulationoftheCIRprocessonalargenumberoftimegrids.NumericalresultsonoptionpricingbasedonHestonmodelisratherencouraging,comparedwithothersimulationschemes. 9

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CHAPTER1INTRODUCTIONTOCIRMODELANDSIMULATIONThischapterbrieyreviewsbasictheoryaboutinterestrates,thetermstructureofinterestrates,andintroducedthefamousshortterminterestratemodel:CIRmodelinnance,aswellashowtosimulateaCIRprocessingeneral.Twocommonlyusedmethodsintheliteraturearementioned. 1.1TheTermStructureOfInterestRatesThetermstructureofinterestrates,alsoknownastheyieldcurve,measurestherelationshipamongtheyieldsondefault-freesecuritiesthatdifferonlyintheirtermtomaturity.Thestudyofthisfunctionalrelationshiphaslongbeenofinteresttoeconomists.Thetermstructureimpliesthemarket'santicipationsoffutureeventsbyofferingacompletescheduleofinterestratesacrosstime.Hence,anexplanationofthetermstructuregivesusawaytounderstandandextractthisinformation.Wecanthenusethisinformationtomakepredictionsabouthowchangesintheunderlyingvariableswillaffecttheyieldcurve.Interestratesandtheirdynamicsprovideprobablythemostcomputationallydifcultpartofthemodernnancialtheory.Themodernxedincomemarketincludesnotonlybondsbutallkindsofderivativesecuritiessensitivetointerestrates.Moreoverinterestratesareimportantinpricingallothermarketsecuritiessincetheyareusedintimediscounting.Interestratesarealsoimportantonthecorporatelevelsincemostinvestmentdecisionsarebasedonsomeexpectationsregardingalternativeopportunitiesandthecostofcapital-bothdependontheinterestrates.Theinterestratemarketiswherethepriceofmoneyisset-howmuchdoesitcosttohavemoneytomorrow,moneyinayear,moneyintenyears?Thepriceofmoneyoveratermdependsnotonlyonthelengthoftheterm,butalsoonthemoment-to-momentrandomuctuationsoftheinterestratemarket. 10

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Themostbasicinterestratecontractisanagreementtopaysomemoneynowinexchangeforapromiseofreceivinga(usually)largersumlater.Ingeneral,theworthofsuchacontractwilldependonfactorsotherthanjustthetimevalueofmoney,suchasthecreditworthinessofthepromisor,etc.Wearesolelyconcernedwiththetimevalueofmoneyfordefault-freeborrowing.Thebasiccontractonlyrequirestwonumberstodescribeit-itslength,ormaturity,whichrecordswhenwearetoreceivethelaterpayment,andtheratioofthesizeofthatpaymenttoourinitialpayment.SupposethematuritydateisT,andwepayP(0,T)initiallytoreceiveonedollarattimeT.ThepromiseofadollarattimeTcouldberegardedasanasset,whichwillhavesomeworthatanytimetbeforeT.Thisassetiscalledadiscountbond,andthepriceP(0,T)isitspriceattimezero.ButitcanhaveadifferentpriceatanyothertimetuptomaturityT,callit,say,P(t,T).ThispriceP(t,T),thevalueattimetofreceivingadollarattimeT,isaprocessintime.Realmarketsdonothaveasingleinterestrate.Instead,theyhavebondsofdifferentmaturities,somepayingcouponsandothersnotpayingcoupons.Fromthesebonds,yieldstodifferentmaturitiescanbeimplied.Inpractice,frommarketdataonecanultimatelydeterminepricesofzero-couponbondsforanumberofdifferentmaturitydates.Eachofthesebondshasayieldspecictoitsmaturity,whereyieldisdenedtobetheconstantcontinuouslycompoundinginterestrateoverthelifetimeofthebondthatisconsistentwithitsprice.Inourexample,givenadiscountbondpriceP(t,T)attimet,theyieldR(t,T)isgivenby:R(t,T)=)]TJ /F4 11.955 Tf 10.49 8.08 Td[(logP(t,T) T)]TJ /F3 11.955 Tf 11.96 0 Td[(t.orequivalently,P(t,T)=e)]TJ /F9 7.97 Tf 6.59 0 Td[(R(t,T)(T)]TJ /F9 7.97 Tf 6.59 0 Td[(t).Inoursimpleexample,weassumeazero-couponbondwithfacevalueequalto1.Theformulaaboveimpliesthatcapitalequaltothepriceofthebond,investedata 11

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continuouslycompoundedinterestrateequaltotheyield,would,overthelifetimeofthebond,resultinanalpaymentofthefacevalue.Inreallife,insteadofhavingasingleinterestrate,realmarketshaveayieldcurve,whichonecanregardeitherasafunctionofnitelymanyyieldsplottedversustheircorrespondingmaturitiesormoreoftenasafunctionoftimeobtainedbyinterpolationfromthenitelymanymaturity-yieldpairsprovidedbythemarket.Thedifferenceinyieldsatdifferentmaturitiesreectsmarketbeliefsaboutfutureinterestrates.Ifthereisapossibilitythatratesmightbehigherinthefuture,long-termloanswillhavetochargeahigherratethanshort-termones.Typically,thenormalyieldcurvewouldincreasewithmaturity.Andjustasitsnameindicates,thisistheyieldcurveshapethatformsduringnormalmarketconditions,whereininvestorsgenerallybelievethattherewillbenosignicantchangesintheeconomy,suchasininationrates,andthattheeconomywillcontinuetogrowatanormalrate.Duringsuchconditions,investorsexpecthigheryieldsforxedincomeinstrumentswithlong-termmaturitiesthatoccurfartherintothefuture.Inotherwords,themarketexpectslong-termxedincomesecuritiestoofferhigheryieldsthanshort-termxedincomesecurities.Thisisanormalexpectationofthemarketbecauseshort-terminstrumentsgenerallyholdlessriskthanlong-terminstruments;thefartherintothefuturethebond'smaturity,themoretimeand,therefore,uncertaintythebondholderfacesbeforebeingpaidbacktheprincipal.Toinvestinoneinstrumentforalongerperiodoftime,aninvestorneedstobecompensatedforundertakingtheadditionalrisk.Butifcurrentratesarehighandexpectedtofall,theyieldcurvecanbecomeinvertedandlongbondyieldswillbelessthanshortbonds.Theseyieldcurvesarerare,andtheyformduringextraordinarymarketconditionswhereintheexpectationsofinvestorsarecompletelytheinverseofthosedemonstratedbythenormalyieldcurve.Insuchabnormalmarketenvironments,bondswithmaturitydatesfurtherintothefutureareexpectedtoofferloweryieldsthanbondswithshortermaturities.Theinvertedyield 12

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curveindicatesthatthemarketcurrentlyexpectsinterestratestodeclineastimemovesfartherintothefuture,whichinturnmeansthemarketexpectsyieldsoflong-termbondstodecline.Youmaybewonderingwhyinvestorswouldchoosetopurchaselong-termxed-incomeinvestmentswhenthereisaninvertedyieldcurve,whichindicatesthatinvestorsexpecttoreceivelesscompensationfortakingonmorerisk.Someinvestors,however,interpretaninvertedcurveasanindicationthattheeconomywillsoonexperienceaslowdown,whichcausesfutureinterestratestogiveevenloweryields.Beforeaslowdown,itisbettertolockmoneyintolong-terminvestmentsatpresentprevailingyields,becausefutureyieldswillbeevenlower. Figure1-1. Normalyieldcurve Figure1-2. Invertedyieldcurve Buttheyieldcurvejustgiveusanideaoftherateofborrowingforeachtermlength.Itwouldbeconvenientifwecouldgetthecurrent,orinstantaneouscostofborrowinginasinglenumber.Whatwecandoislookatthecurrentrateforinstantaneousborrowing.Thatis,borrowingwhichispaidback(nearly)instantly.Supposeattimetweborrowovertheperiodfromttot+t,wheretisasmalltimeincrement,theratewegetistheyieldR(t,t+t):R(t,t+t)=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(logP(t,t+t) t 13

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Ast!0,wecallthelimittheinstantaneousrate,orshortrate,rt,whichisgivenbyboththeexpressionsrt=R(t,t),andrt=)]TJ /F5 11.955 Tf 15.63 8.09 Td[(@ @TlogP(t,t).Theshortrateisnotonlyanimportantprocessintheinterestratemarket,butmanymodelsarebasedexclusivelyonitsbehavior,withalltheotherbondsextrapolatedfromit. 1.2ModelsOfTheShort-TermInterestRateThetheoryofinterest-ratemodelingwasoriginallybasedontheassumptionofspecicone-dimensionaldynamicsfortheinstantaneousrateprocessrt.Modelingdirectlysuchdynamicsisveryconvenientsinceallfundamentalquantities(ratesandbonds)arereadilydenedastheexpectationofafunctionaloftheprocessrt.Theshorttermrisklessinterestrateisoneofthemostfundamentalandimportantpricesdeterminedinnancialmarkets.Moremodelshavebeenputforwardtoexplainitsbehaviorthanforanyotherissueinnance.Manyofthemorepopularmodelscurrentlyusedbyacademicresearchersandpractitionershavebeendevelopedinacontinuous-timesetting,whichprovidesarichframeworkforspecifyingthedynamicbehavioroftheshort-termrisklessrate.ApartiallistingoftheseinterestratemodelsincludesthosebyMerton(1973)[ 38 ],BrennanandSchwartz(1982)[ 10 ],Vasicek(1977)[ 42 ],Dothan(1978)[ 21 ],Cox,Ingersoll,andRoss(1985)[ 17 ],HullandWhite(1990)[ 29 ],Black,FischerandPiotrKarasinski(1991)[ 8 ].Oneoftheearliestpaperstotacklearbitrage-freepricingofbondsandinterest-ratederivativeswasVasicek(1977)[ 42 ].ThispaperisbestknownfortheVasicekmodelfortherisk-freerateofinterest,R(t)describedbelow.However,Vasicekalsodevelopedamoregeneralapproachtopricingwhichtiesinwithwhatwenowrefertotheasthe 14

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risk-neutral-pricingapproach.Themodelspeciesthattheinstantaneousinterestratefollowsthestochasticdifferentialequation:dR(t)=(b)]TJ /F3 11.955 Tf 11.95 0 Td[(R(t))dt+dW(t),R(0)=r0.Thisdynamicshassomeinterestingpropertiesthatmakethemodelattractive.Mathematically,theequationislinearandcanbesolvedexplicitly,thedistributionoftheshortrateisGaussian,andboththeexpressionsandthedistributionsofseveralusefulquantitiesrelatedtotheinterest-rateworldareeasilyobtainable.WecansolvetheSDEtoobtain:R(t)=R(0)e)]TJ /F13 7.97 Tf 6.58 0 Td[(t+b(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F13 7.97 Tf 6.59 0 Td[(t)+e)]TJ /F13 7.97 Tf 6.59 0 Td[(tZt0esdW(s).Vasicek'smodelwastherstonetocapturemeanreversion,anessentialcharacteristicoftheinterestratethatsetsitapartfromothernancialprices.Asopposedtostockpricesforinstance,interestratescannotriseindenitely.Thisisbecauseatveryhighlevelstheywouldhampereconomicactivity,promptingadecreaseininterestrates.Similarly,interestratescannotdecreasebelow0.Asaresult,interestratesmoveinalimitedrange,showingatendencytoreverttoalongrunvalue.Inparticular,Vasicek'smodelexhibitsmeanreversion,whichmeansthatiftheinterestrateisabovethelongrunmean(r>b),thenthecoefcientmakesthedriftbecomenegativesothattheratewillbepulleddowninthedirectionofr.Likewise,iftherateislessthanthelongrunmean(r
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Figure1-3. ThreesamplepathsofdifferentOU-processeswith=1,=1.2,=0.3 Themaindisadvantageisthat,underVasicek'smodel,itistheoreticallypossiblefortheinterestratetobecomenegative,anundesirablefeature.SincefromthedistributionofR(t),it'seasytoseethatR(t)isnormallydistributedwithmeanandvariancegivenby:E(R(t))=R(0)e)]TJ /F13 7.97 Tf 6.58 0 Td[(t+b(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F13 7.97 Tf 6.58 0 Td[(t)Var(R(t))=2 2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(2t)Thisimpliesthat,foreachtimet,therateR(t)canbenegativewithpositiveprobability.ThegeneralequilibriumapproachdevelopedbyCox,IngersollandRoss(1985)[ 17 ]ledtotheintroductionofasquare-rootterminthediffusioncoefcientoftheinstantaneousshortratedynamics.Theresultingmodelhasbeenabenchmarkformanyyearsbecauseofitsanalyticaltractabilityandthefactthat,contrarytotheVasicek(1977)[ 42 ]model,theinstantaneousshortrateisalwayspositive.HullandWhite(1990)[ 29 ]proposedanevenmoregeneralmodelbyconsideringatime-varyingparameterintheVasicekmodel.Later,BlackandKarasinski(1991)[ 7 ]assumedthatthelogarithmln(R(t))oftheinstantaneousshortrateevolvesaccordingtoageneralizedVasicek 16

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modelwithtime-dependentcoefcients.Inthenextsection,we'llfocusonthecelebratedCIRshortratemodel. 1.3IntroductiontoCIRModelTheCox-Ingersoll-Ross(CIR)[ 17 ]modelisadiffusionprocesssuitableformodelingthetermstructureofinterestrates.Itwasintroducedin1985byJohnC.Cox,JonathanE.IngersollandStephenA.RossasanextensionoftheVasicekmodel.ThesimplestversionofthismodeldescribesthedynamicsoftheinterestrateX(t)asasolutionofthefollowingstochasticdifferentialequation(SDE): dX(t)=(b)]TJ /F3 11.955 Tf 11.96 0 Td[(X(t))dt+p X(t)dW(t),X(0)=x00 (1) for>0,b>0,>0andastandardBrownianmotionW.Thisprocesshassomeappealingpropertiesfromanappliedpointofview,forexample,theinterestratestaysnon-negative,andiselasticallypulledtowardsthelong-termconstantvaluebataspeedcontrolledby(mean-reverting).Thosepropertiesareattractiveinmodelingreal-lifeinterestrates.Inparticular,thecondition2b2wouldensurethattheoriginisinaccessibletotheprocess,sothatwecangrantthatX(t)remainspositive.Intuitively,whentherateisatalowlevel(closetozero),thestandarddeviationp X(t)alsobecomesclosetozero,whichdampenstheeffectoftherandomshockontherate.Consequently,whentherategetsclosetozero,itsevolutionbecomesdominatedbythedriftfactor,whichpushestherateupwards(towardsequilibrium).Theinterestratebehaviorimpliedbythisstructurethushasthefollowingempiricallyrelevantproperties:(i)Negativeinterestratesareprecluded.(ii)Iftheinterestratereacheszero,itcansubsequentlybecomepositive.(iii)Theabsolutevarianceofthe 17

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interestrateincreaseswhentheinterestrateitselfincreases.(iv)Thereisasteadystatedistributionfortheinterestrate. Figure1-4. AnexampleofCIRprocesswith=0.01,b=1and=0.1 TheSDE(1-1)isnotexplicitlysolvable,hencethetractabilityoftheCIRmodelisnotasgoodastheVasicekmodelinthisregard.Nevertheless,thetransitiondensityfortheprocessisknown.BasedonresultsofFeller[ 25 ],Coxetal.[ 17 ]notedthatthedistributionofX(t)givenX(u)forsomeu
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AlthoughCIRmodelismainlyusedinnanceinmodelinginterestrates,itshouldbenotedthatthisprocesshasothernancialapplications.Forexample,thestochasticvolatilityofthestockprice(Heston)[ 27 ]andthecreditspread(BrigoandAlfonsi)[ 11 ]. 1.4SimulationIssueFinancialmodelsusuallyspecifythedynamicsofthestatevariables,e.g.,stockprice,volatilityandinterestrate,asstochasticdifferentialequations(SDE).IftheseSDEsyieldclosedformsolutions,thenMonteCarlosimulationcanbeusedtogenerateanunbiasedestimatorofthepriceofaderivativesecurity.Basicallywegeneratemanysamplepathsofthestatevariableandcomputethepayoffofthederivativeforeachpath.Discountingandaveragingoverallpathsgivesanestimatorofthederivativeprice.TheerrorintheMonteCarloestimatorcanbecalculatedusingthecentrallimittheoremandconvergestozeroasthenumberofsamplepathsusedincreases.However,inmostcases,theSDEsthatdenethedynamicsofthestatevariablesdonotyieldclosedformsolutions.Inthiscase,it'sstillpossibletouseMonteCarlosimulationtocomputederivativeprices.Butrstwehavetodiscretizethetimeintervalandsimulatingthestateprocessdynamicsonthisdiscretetimegrid.However,theapproximationofcontinuoustimeprocessesbydiscretetimeprocessesintroducesbiasintothesimulationestimator.OnedrawbackoftheCIRprocessisthattheSDE(1-1)isnotexplicitlysolvable.Inpracticalusageofsuchmodels(e.g.topriceoptions)weareoftenfacedwiththeproblemofsimulatingaCIRprocess.Ingeneraltherearetwowaystodoit,namely,exactsimulationmethodsandapproximationschemes.Thereareprosandconsassociatedwitheachmethod.Thedrawbackofexactsimulationmethodsisthecomputationtimethattheyrequire.Exactsimulationingeneralrequiresmoretimethanasimulationwithapproximationschemes(Uptoafactor10).Henceitshouldbeusedtocomputeexpectationsthatdependonthevaluesoftheprocessatjustafewxedtimes.Onthecontrary,forexpectationsthatdependsonallthepath(such 19

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asintegrals)discretizationschemesshouldbepreferred.Ontheotherhand,thedrawbackofapproximationschemesingeneralisthebiastheyintroducedintotheestimator.Sincethemagnitudeofthebiasisunknown,it'sdifculttoobtainvalidcondenceintervals.AndanotherseriousproblemwithdiscretizationschemesofCIRprocess,ingeneralanysquare-rootdiffusionprocess,isthesquare-rootitself,whichhasunboundedderivativesnearzero.Therefore,discretizationschemesthat(explicitlyorimplicitly)involvethederivativesofthecoefcients-evenwhentheyassurethepositivityofapproximation-usuallylosetheiraccuracynearzero,especially,forlarge.Thelargeris,themoreconcentratednearzerothevaluedistributionsofCIRprocessare.Therearewaystogetaroundthisproblem,we'lltalkaboutthemindetailinChapter2.Therstmethodistheexactsimulationmethod,whichisbasedonthetransitionprobabilitydensityfunctionoftheCIRprocess.InCox'spaper[ 17 ],itwasnotedthatthedistributionofX(t)givenX(u)forany0
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whenonehastosimulatetheprocessjustatonetime(orfewtimes),forexampletocomputeEuropeanoptionspriceswithaMonte-Carloalgorithm.Onthecontrary,theyaredrasticallytooslowifonehastosimulatealongatime-grid,asitisthecasetocalculatepath-dependentoptionsprices.ThesecondoneisthepopularEulerdiscretization.Eulerschemecanalwaysbeusedtoapproximatethepathsoftheinterestrateprocessonadiscretetimegrid.Andit'singeneralfasterthantheexactsimulation,whichwillbementionedlater.Thedrawbackisthatit'sanapproximationofcontinuoustimeprocessbydiscretetimeprocess,henceintroducesbiasintothesimulationestimator.Andinpractice,manytimestepsmaybenecessarytoreducethebiastoanacceptablelevel.AnotherproblemisthatwhendiscretizingaCIRprocess,oringeneralasquare-rootdiffusionprocess,asimpleEulerschememaynotbewell-denedbecauseitcanleadtonegativevaluesforwhichthesquarerootisnotdened.ConsiderthefollowingmoststraightforwardEulerschemeonthetimeinterval[0,T]foraCIRprocessX(t):^X(ti+1)=^X(ti)+(b)]TJ /F4 11.955 Tf 13.65 2.66 Td[(^X(ti))[ti+1)]TJ /F3 11.955 Tf 11.95 0 Td[(ti]+q ^X(ti)(W(ti+1))]TJ /F3 11.955 Tf 11.95 0 Td[(W(ti))with^X(t0)=x0>0,>0,b>0,>0canleadtonegativevaluessincetheGaussianincrementisnotboundedfrombelow.Thus,thissimpleschemeisnotwelldened.Tocorrectthisproblem,DeelstraandDelbaen[ 19 ]haveproposedthefulltruncationscheme:^X(ti+1)=^X(ti)+(b)]TJ /F4 11.955 Tf 13.65 2.66 Td[(^X(ti)+)[ti+1)]TJ /F3 11.955 Tf 11.95 0 Td[(ti]+q ^X(ti)+(W(ti+1))]TJ /F3 11.955 Tf 11.96 0 Td[(W(ti))whileDiopproposedin[ 6 ]proposedthereectionscheme:^X(ti+1)=j^X(ti)+(b)]TJ /F4 11.955 Tf 13.65 2.66 Td[(^X(ti))[ti+1)]TJ /F3 11.955 Tf 11.95 0 Td[(ti]+q ^X(ti)(W(ti+1))]TJ /F3 11.955 Tf 11.95 0 Td[(W(ti))j 21

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Also,Alfonsi[ 1 ][ 2 ][ 3 ]presentedseveralimplicitschemesandhigherorderschemes.Inthenextchapter,we'llbrieyreviewsomebasicpropertiesofEulerscheme,andtheproblemswithsimulatinganCIRprocesswithEulerschemes. 22

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CHAPTER2EULEREISCRETIZATIONANDRELATEDISSUESThissectionpresentsthesimplestapproximation,Eulerscheme,toacontinuous-timestochasticprocess.Thismethodisalwayseasytoimplementandalmostapplicableeverywhere,butit'snotalwaysaccurateenoughtomeetpracticalneeds.MostofthefollowingintroductioncomesfromGlasserman'sbook[ 26 ]. 2.1IntroductionToEuler-MaruyamaSchemeTheEuler-MaruyamaSchemeisamethodfortheapproximatenumericalsolutionofastochasticdifferentialequation(SDE).ItisasimplegeneralizationoftheEulermethodforordinarydifferentialequationstostochasticdifferentialequations.WeconsideraprocessXsatisfyingastochasticdifferentialequationoftheform dX(t)=a(X(t))dt+b(X(t))dW(t) (2) withinitialconditionX(0)=x0,somexedrealnumber.Wewon'tgetintodetailsabouttheuniquenessandexistenceconditionsoftheSDE,butessentiallythecoefcientfunctionsaandbareassumedtosatisfysometechnicalconditions.Giventhefunctionsaandb,thestochasticprocessX(t)isasolutionoftheSDE(2-1)ifX(t)solvestheintegralequationX(t))]TJ /F3 11.955 Tf 11.96 0 Td[(X(0)=Zt0a(X(s))ds+Zt0b(X(s))dW(s)AfamousexampleisthesocalledgeometricBrownianmotion,whichisusedtomodeltheevolutionofassetprices.It'saprocesssatisfyingthefollowingSDE:dS(t)=S(t)dt+S(t)dW(t)WecanactuallysolvethisequationandgetS(t)=S(0)expfW(t)+()]TJ /F10 7.97 Tf 13.15 4.71 Td[(1 22)tg. 23

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Inpractice,however,manySDE'sarenotexplicitlysolvableliketheaboveone.Hencewecan'talwaysgetananalyticalsolutionX(t)toagivenSDE.ButwecouldalwaysgetapproximatenumericalsolutionofthegivenSDE.ThebasicideaisessentiallyfromEulerSchemeforordinarydifferentialequation(ODE).Inparticular,let'suse^X(t)todenoteatime-discretizedapproximationtoX(t).Supposewediscretizetheinterval[0,T]:lett=T=Nandtn=nt,n=0,1,2,...,N.TheexactsolutiononthetimegridwouldbeX(tn+1)=X(tn)+Ztn+1tna(X(s))ds+Ztn+1tnb(X(s))dW(s)TheEuler-Maruyamaapproximationonthetimegrid0=t0
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Denition(Strongerrorcriterion)Givenasequenceofdiscretetimeapproximationf^X(0),^X(h),^X(2h),...gtoacontinuoustimeprocessX,whereh=T=nforaxedtimeT.Wesay^XconvergesstronglytoXifitconvergesinL1,i.e.,iflimn!1E(j^X(nh))]TJ /F3 11.955 Tf 11.96 0 Td[(X(T)j)=0Wesaythatadiscretization^Xhasstrongorderofconvergence>0ifEk^X(nh))]TJ /F3 11.955 Tf 11.95 0 Td[(X(T)kchforsomevectornormk.k,someconstantcandforhsufcientlysmall.ThestrongerrorcriterionmeasuresthedeviationbetweentheindividualvaluesofXandtheapproximation^X.Incontrast,aweakerrorcriterionlookslikethefollowing:Denition(Weakerrorcriterion)Givenasequenceofdiscretetimeapproximationf^X(0),^X(h),^X(2h),...gtoacontinuoustimeprocessX,whereh=T=nforaxedtimeT.Wesay^XconvergesweaklytoXiflimn!1E(f(^X(nh)))=E(f(X(T)))Wesaythatadiscretization^Xhasweakorderofconvergence>0ifjE(f(^X(nh))))]TJ /F3 11.955 Tf 11.95 0 Td[(E(f(X(T)))jchforsomeconstantcandforhsufcientlysmall,forallfwhosederivativesoforder0,1,...,2+2arepolynomiallybounded.(Afunctiongispolynomiallyboundedifjg(x)jk(1+jxjq)forsomeconstantskandqandallx2R)Forapplicationsinderivativespricing,wereallyonlycareabouttheweakerrorcriteria.Wewouldliketoensurethatoptionprices(whichareexpectations)computedfrom^X(t)areclosetopricescomputedfromX(t).Wearenotreallyconcernedaboutthespecicpathsofthetwoprocesses. 25

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Undermodestconditions,eventhesimpleEulerschemeconvergesasthetimestephdecreasestozero.Wecomparedifferentdiscretizationschemesbasedontherateatwhichtheyconverge.Alargevalueofimpliesfasterconvergencetozeroofthediscretizationerror.It'softenthecasethatforthesamescheme,thestrongorderofconvergenceissmallerthantheweakorderofconvergence.Forexample,theEulerschemetypicallyhaveastrongorderof1=2,butitoftenachievesaweakorderof1.StrongerconditionsarerequiredfortheEulerschemetohaveweakorder1.In[ 34 ]theauthorsrequirethefunctionsaandbbefourtimescontinuouslydifferentiablewithpolynomiallyboundedderivatives.Butgoodaccuracyonsmoothfunctionsmaynotbedirectlyrelevanttoourintendedapplications:theCIRprocesshasasquarerootfunction,whichisnotLipschitzian.ThisposedaproblemfortheordinaryEulerscheme.We'lldiscussaboutthisprobleminthenextsection. 2.3EulerSchemeForCIRProcessUsinganEulerdiscretizationtosimulateCIRprocessgivesrisetotheproblemthatwhiletheprocessitselfisguaranteedtobenonnegative,thediscretizationisnot.ThemaindifcultywhendiscretizingtheCIRprocessislocatedat0,wherethesquare-rootisnotLipschitzian.Generalschemes,suchastheEulerschemeortheMilsteinschemeareingeneralnotwelldenedbecausetheycanleadtonegativevaluesforwhichthesquarerootisnotdened.InIstvanandMiklos(2011)[ 30 ]theauthorsestablishedtheresultofaconvergencespeedestimateforEulerschemescorrespondingtoSDEswith1=2-Holdercontinuousdiffusioncoefcients.ThisresultcouldbeappliedtoCIRprocess,sincethediffusioncoefcientofCIRprocessfailstobeLipschitzcontinuousneartheorigin.HenceclassicalresultsontherateofstrongconvergenceforthecorrespondingEulerschemedonotapplyinthiscase.Ingeneral,theyshowedthattheconvergencerateforEulerschemescorrespondingtoSDEswith1=2-Holdercontinuousdiffusioncoefcients 26

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(CIRasaspecialcase)withoutanyrestrictionsontheparametersshouldbe1=lnn.Inparticular,xT>0andconsidertheSDEdX(t)=b(t,X(t))dt+(t,X(t))dW(t),X(0)=ontheinterval[0,T],whereW(t)isastandardBrownianmotion,isindependentofW(t),andthecoefcientssatisfythefollowingconditions:Assumption:,f,g:[0,T]R!Raremeasurable;g(t,)ismonotonedecreasing;b=f+gandthereexistK>0,2[0,1=2]and2(0,1]suchthatforallt2[0,T]andx,y2<,j(t,x))]TJ /F5 11.955 Tf 11.95 0 Td[((t,y)jKjx)]TJ /F3 11.955 Tf 11.96 0 Td[(yj1 2+,jf(t,x))]TJ /F3 11.955 Tf 11.96 0 Td[(f(t,y)jKjx)]TJ /F3 11.955 Tf 11.95 0 Td[(yj,jg(t,x))]TJ /F3 11.955 Tf 11.96 0 Td[(g(t,y)jKjx)]TJ /F3 11.955 Tf 11.96 0 Td[(yj,andjb(t,0)j+j(t,0)jK.Undertheaboveassumption,thefollowingtheoremgivestheconvergenceorderofEulerScheme:Theorem:LettheassumptionaboveholdandletEjj1+2<1.ThenthereisaconstantCdependingonlyonK,T,andEjj1+2suchthatEjX())]TJ /F3 11.955 Tf 11.95 0 Td[(Xn()jC lnn,if=0.EjX())]TJ /F3 11.955 Tf 11.95 0 Td[(Xn()jC(1 n+1 n=2),if2(0,1 2].foralln2andforeverystoppingtimeT.HenceforCIRprocess,whichisthecasewhen=0,withoutanyrestrictionontheparameters,aslowrateofO(1=lnn)isestablished. 27

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AlsoinAlfonsi'spaper[ 1 ]theauthorproposedseveralimplicitdiscretizationschemesfortheCIRprocess.Butforlargevaluesof,forexample,when2>>4b,noneoftheschemeseemstobeefcientduetolargediscretizationbias.Inthenextchapter,we'lllookatanewmethodtosimulateanCIRprocessexactly.Themethodisexact,hencethereisnodiscretizationerrors. 28

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CHAPTER3CIRPROCESSEXACTSIMULATIONInthissection,weintroducethetraditionalexactsimulationmethodofaCIRprocess,andshowwhythesocalledPoissonmethodisnotefcientinsomecases.Andwendanewmethodtosimulateanon-centralchi-squaredrandomvariable,whichismuchmoreefcientthanthePoissonmethod. 3.1TraditionalSimulationMethodTheCIRprocessisaMarkovprocesswithcontinuouspathsdenedbythefollowingstochasticdifferentialequation(SDE):dX(t)=(b)]TJ /F3 11.955 Tf 11.95 0 Td[(X(t))dt+p X(t)dW(t),X(0)=x00for>0,b>0,>0andastandardBrownianmotionW.Asalreadymentionedinchapter1,theCIRprocesshassomeappealingpropertiesandwasoriginallyusedtomodelinterestrateinnance.It'swellknownthatthereisawaytogettheexactdistributionoftheCIRprocessX(t),givenX(u)foru
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parameter0.Thesymbol2d(0)denotesarandomvariablewiththisdistribution;theprimein0d2()emphasizesthatthissymbolreferstothenoncentralcase.Thecumulativedensityfunction(CDF)ofacentralchi-squaredistributionisgivenbyP(2d(0)x)=1 2d=2\(d=2)Zx0e)]TJ /F9 7.97 Tf 6.59 0 Td[(z=2z(d=2))]TJ /F10 7.97 Tf 6.59 0 Td[(1dz,where\(.)denotesthegammafunction\(z)=R10tz)]TJ /F10 7.97 Tf 6.58 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(tdtand\(n)=(n)]TJ /F4 11.955 Tf 12.14 0 Td[(1)!ifnisapositiveinteger.Thisexpressiondenesavalidprobabilitydistributionforalld>0andthusextendsthedenitionof2d(0)tonon-integerd.Forintegerdandconstantsa1,...,ad,thedistributionof dXi=1(Zi+ai)2 (3) isnoncentralchi-squarewithddegreesoffreedomandnon-centralityparameter=Pdi=1a2i.Let0d2()denoteanon-centralchi-squaredrandomvariablewithddegreesoffreedomandnon-centralityparameter.Theprobabilitydensityfunction(pdf)ofthisrandomvariableis g(,d,x)=e)]TJ /F16 5.978 Tf 7.78 3.26 Td[( 21Xj=0( 2)j j!e)]TJ /F15 5.978 Tf 7.79 3.26 Td[(x 2 \(d 2+j)xd 2+j)]TJ /F10 7.97 Tf 6.58 0 Td[(1 2d 2+j. (3) anditscharacteristicfunction(c.f.)isEexpfit0d2()g=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(2it))]TJ /F15 5.978 Tf 7.79 3.26 Td[(d 2expf 1)]TJ /F3 11.955 Tf 11.96 0 Td[(i2t)]TJ /F5 11.955 Tf 13.15 8.09 Td[( 2g.Thiseasilyimpliesitsadditivity,namely0d12(1)+0d22(2)d=02d1+d2(1+2).fortwoindependentnoncentralchi-squarerandomvariableswithdegreesoffreedomsd1,d2andnoncentralityparameters1,2,respectively.Therefore,inparticular,wehave 0d2()d=012()+2d)]TJ /F10 7.97 Tf 6.58 0 Td[(1(0),ford>1 (3) 30

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Therefore,togenerate0d2(),d>1,wecangenerate2d)]TJ /F10 7.97 Tf 6.59 0 Td[(1(0)andanindependentstandardnormalrandomvariableZandset 0d2()=(Z+p )2+2d)]TJ /F10 7.97 Tf 6.59 0 Td[(1(0). (3) Thus,samplingfromanoncentralchi-squareddistributionisreducedtosamplingfromanordinarychi-squaredandanindependentnormalwhend>1.Andthismethodisingeneralquiteefcient.Butforany0
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Wemaythereforesample0d2()byrstgeneratingaPoissonrandomvariableNwithmean=2andthen,conditionalonN,samplingachi-squarerandomvariablewithd+2Ndegreesoffreedom.Thismethodcanbeusedtosample0d2()when0
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3.2SomeDrawbacksInSimulatingCIRProcessLet'susethealgorithmabovetosimulateaCIRprocess.Inparticular,weonlyconsiderthecasewhend<1,sincethat'swhenwehavetousethePoissonmethodforsimulation.SupposewewanttosimulateaCIRprocessX(t)alongnequallyspacedtimepointsonthetimeinterval[0,T].Inthiscase,givenaninitialvalueX(0),wecouldusethetransitiondistributionofX(t)tosimulatearandomvectordistributedaccordingtothelawof(X(T=n),X(2T=n),...,X(T))inductively.Lett0,t1,t2,...,tntodenotethetimepoints0,T=n,2T=n,...,T.StartingfromX(0),whichisgiven,wecouldgeneratethewholesamplepathoftheCIRprocessalongthetimegridtiiteratively.Ingeneral,givenX(ti)]TJ /F10 7.97 Tf 6.59 0 Td[(1),theconditionaldistributionofX(ti)isc0d2((i)),wherec=2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F13 7.97 Tf 6.59 0 Td[(t) 4,(i)=4e)]TJ /F13 7.97 Tf 6.59 0 Td[(t 2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F13 7.97 Tf 6.59 0 Td[(t)X(ti)]TJ /F10 7.97 Tf 6.59 0 Td[(1).Heret=ti)]TJ /F3 11.955 Tf 12.31 0 Td[(ti)]TJ /F10 7.97 Tf 6.58 0 Td[(1=T n.It'seasytoseethatthevalueof(i)dependsonX(ti)]TJ /F10 7.97 Tf 6.59 0 Td[(1)andt.ThisfactturnsouttobethemajorprobleminsimulatingaCIRprocesswithd<1usingthePoissonmethodalongmanytimepoints.SupposewewanttouseMonteCarlomethodtoapproximateapath-dependentoption(forexample,AsianoptionorLookbackoption),assumingtheunderlyingassetpriceprocessfollowstheHestonstochasticvolatilitymodel.Inparticular,thepayofffunctionofastandardEuropean-stylediscreteAsiancalloptionis(1 n+1nXi=0Sti)]TJ /F3 11.955 Tf 11.95 0 Td[(K)+whereasthepayofffunctionofaxedstrikediscreteLookbackcalloptionis(max0inSti)]TJ /F3 11.955 Tf 11.96 0 Td[(K)+HereStdenotestheunderlyingassetpriceattimet,Kisthestrikeprice.AtypicalMonteCarlomethodinvolvessimulatingtheassetpriceonthetimegrid0=t0
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payofffunctionvalue.ThentheMonteCarloestimationoftheoptionpriceisgivenbythemeanvalueofthediscountedpayoffvaluesalongthosedifferentpathsgenerated.Thus,inordertocomputethepriceofapath-dependentoptionusingMonteCarlomethod,weareinterestedinsimulatinganassetpriceprocessalongalotoftimepoints.Inparticular,Heston'smodeluseaCIRprocesstomodeltheevolutionofstochasticvolatility.Henceinordertosimulatetheassetpriceprocess,werstneedtosimulateaCIRprocessalongthosetimepoints.Asmentionedabove,thefactthatthevalueof(i)dependsonX(ti)]TJ /F10 7.97 Tf 6.59 0 Td[(1)andtposesaprobleminsimulatingtheCIRprocess.Thereasonisthatwhentheneedednumberoftimepointsnisbig,t=T=nisverysmall.Inthiscase(i)=4e)]TJ /F13 7.97 Tf 6.59 0 Td[(t 2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F13 7.97 Tf 6.59 0 Td[(t)X(ti)]TJ /F10 7.97 Tf 6.59 0 Td[(1)wouldbehugesinceit'sadecreasingfunctionoft,andactuallygoesto1ast!0.Thisposesaseriousproblemforthesimulation.Becausewhend<1,whichiscommonforHestonmodel,inordertogenerateanon-centralchi-squarerandomvariable0d2((i)),weneedtorstgenerateaPoissonrandomvariablewithmean(i)=2.ThenconditionalonthevalueofthePoissonrandomvariableN,generateacentralchi-squarerandomvariable2d+2N(0).Normally,inordertocalculatethepriceofapath-dependentoption,thenumberoftimepointsweneedisbig.InthiscasethealgorithmwillspendahugeamountoftimegeneratingPoissonrandomvariablesateachtimepoint,since(i)=2ishuge.Hence,althoughtheoreticallythisalgorithmcouldstillbeusedtosimulateaCIRprocessatmultipletimepoints,it'snotefcientifthenumberoftimepointsinvolvedisbig.InthiscasepeopleprefertouseEulerSchemetogeneratethesamplepathinstead,althoughitoftengenerateresultswhicharebiasedandnotaccurate.BecausethePoissonmethodissimplytooslow.Inthenextsection,we'llintroduceanewmethodtogenerateanon-centralchi-squarerandomvariable,whichavoidsthisproblem. 34

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3.3MainResultThefollowingtheoremprovidesanewmethodtogenerateanon-centralchisquarerandomvariablebasedonacentralchisquarerandomvariable.TheoremAnoncentralChi-squarer.v2d()canbeexpressedby 2d()d=2d(0)+Y(,Z,~Z,U) (3) where2d(0)hasagammadistributionG(d=2,2),Uhasauniformdistributionon[0,1],Zand~Zarestandardnormalrandomvariables.Allfourrandomvariablesareindependent,and Y(,Z,~Z,U)=8><>:0,:if+2ln(U)0.(Z+p +2ln(U))2+~Z2,:if+2ln(U)>0. (3) Proof:Theprobabilitydensityfunctiong(,d,x)of2d()isgivenby g(,d,x)=e)]TJ /F16 5.978 Tf 7.78 3.26 Td[( 21Xj=0( 2)j j!e)]TJ /F15 5.978 Tf 7.79 3.26 Td[(x 2 \(d 2+j)xd 2+j)]TJ /F10 7.97 Tf 6.58 0 Td[(1 2d 2+j.fromthisdensityfunction,wehave e 2g(,d,x)=g(0,d,x)+1Xj=1( 2)j j!e)]TJ /F15 5.978 Tf 7.78 3.26 Td[(x 2 \(d 2+j)xd 2+j)]TJ /F10 7.97 Tf 6.59 0 Td[(1 2d 2+j.whereg(0,d,x)=e)]TJ /F15 5.978 Tf 7.79 3.25 Td[(x 2 \(d 2)xd 2)]TJ /F10 7.97 Tf 6.59 0 Td[(1 2d 2isthedensityfunctionof2d(0).Now,differentiatingbothsidesw.r.tyields @ @(e 2g(,d,x))=1 21Xj=1(=2)j)]TJ /F10 7.97 Tf 6.59 0 Td[(1 (j)]TJ /F4 11.955 Tf 11.95 0 Td[(1)!g(0,d+2j,x)=1 21Xj=0(=2)j j!g(0,d+2+2j,x)=1 2e 2g(,d+2,x). 35

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Therefore,integratingbothside,wegetthefollowingrelationship: e 2g(,d,x)=g(0,d,x)+1 2Z0et 2g(t,d+2,x)dt.Multiplyingbothsidesbye)]TJ /F16 5.978 Tf 7.78 3.26 Td[( 2,thenlettings=e1 2(t)]TJ /F13 7.97 Tf 6.59 0 Td[(),wehave: g(,d,x)=e)]TJ /F16 5.978 Tf 7.78 3.26 Td[( 2g(0,d,x)+1 2e)]TJ /F16 5.978 Tf 7.78 3.26 Td[( 2Z0et 2g(t,d+2,x)dt=e)]TJ /F16 5.978 Tf 7.78 3.25 Td[( 2g(0,d,x)+Z1e)]TJ /F16 5.978 Tf 6.96 2.34 Td[( 2g(+2lns,d+2,x)dsfromwhichweclaimthat 2d()d=8><>:2d(0),:if+2ln(U)0.2d+2(+2lnU),:if+2ln(U)>0.Byadditivity,2d+2(+2lnU)d=2d(0)+(Z+p +2lnU)2+~Z2,thetheoremfollows.Usingthistheorem,wecouldgenerateanon-centralchi-squarerandomvariableusingacentralchi-squarerandomvariable,pluspossiblytwoindependentGaussianrandomvariablesandauniformrandomvariable.Themostimportantthingisbygeneratingtherandomvariablethisway,weavoidtheproblemofsimulatingaPoissonrandomvariableinaCIRprocesssimulation.Andit'smuchmoreefcientwhenwe'reinterestedinsimulatingtheCIRprocessalongalargenumberoftimepoints,sincethealgorithmdoesn'tdependonthenumberoftimepointsnthewayPoissonrandomvariabledoesintheoldalgorithm.Theoldalgorithmwillbeveryslowwithlargevalueof,whereasthespeedofthisalgorithmisindependenttothevalueof.Henceit'sbothbiasfreeandquick.ThisalgorithmismostsuitableinsimulatingstochasticvolatilityinHestonmodel. 36

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CHAPTER4INTRODUCTIONTOHESTONMODELANDSIMULATIONThischapterbrieydiscussesbasicoptionpricingtheory.Inparticular,Heston'sstochasticvolatility(SV)modelisintroducedandhowtheexactsimulationmethodcanbeusedtopriceoptions. 4.1OptionPricingTheoryOptionshaveexistedforalongtime.Itwasn'tuntilpublicationoftheBlack-Scholes(1973)[ 9 ]optionpricingformulathatatheoreticallyconsistentframeworkforpricingoptionsbecameavailable.ThatframeworkwasadirectresultofworkbyRobertMertonaswellasBlackandScholes.Sincethenthederivativesmarkethasgrownintoamulti-trilliondollarindustry.Optionshavebecomeimportanttoindustry,particularlyastheycanbeusedtohedgeoutrisk.Optionpricinginitselfhasbecomeanimportantresearcharea.Numerousoptionpricingmodelswereproposed,eachonetryingtogeneralizetheBlack-SholesframeworkbyrelaxingoneormoreoftheunrealisticrestrictionsoftheoriginalBlack-Sholesmodel.AnOptionisamajornancialderivative,itgivestheholderofthatoptiontheright,nottheobligationtotradeaxedamountofunderlingassetatanagreed-uponpriceonthematuritydate(Europeanoption)oranytimeonorbeforethematuritydate(Americanoption).Acalloptiongivestheholdertheright,butnotobligationtobuy,aputoptiongivestheholdertheright,butnotobligationtosell.Therearetwosidesofeveryoptioncontract.Ononesideistheinvestorwhohastakenthelongposition(i.e.,hasboughttheoption).Ontheothersideistheinvestorwhohastakenashortposition(i.e.,hassoldorwrittentheoption).ThemoststraightforwardnancialderivativesareEuropeancalloptionandEuropeanputoption.Thepriceinthecontractisknownasthestrikeprice;thedateinthecontractisknownasthematurity.Thepayofffromalong 37

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positioninaEuropeancalloptionis:max(S(T))]TJ /F3 11.955 Tf 11.95 0 Td[(K,0)whereTisthematurity,S(T)isthestockpriceatmaturity,andKisthestrikeprice.Hence,thebuyerofaEuropeancalloptionhopesthepriceoftheunderlyingstockwouldriseaboveKatthematurity,inwhichcasehewouldusetherighttobuythestockatKandmakeaprot.ButifthestockpricewoulddropbelowKatmaturity,hedoesn'thavetheobligationtoexercisetheoption.Similarly,thepayofffromalongpositioninaEuropeanputoptionis:max(K)]TJ /F3 11.955 Tf 11.95 0 Td[(S(T),0)Althoughit'sfairlyeasytounderstandthepayoffstructureofdifferenttypesofoptions,topricethemisquiteadifferentstory.Thequestionofhowoptionsshouldbepricedhadbeenthesubjectoflongintellectualdebates,startingfromtheearlysixties.OptionpricingtheorytracesitsroottoBachelier(1900)whousedBrownianmotiontomodeloptionsonFrenchgovernmentbonds.ThisworkanticipatedbyveyearsEinstein'sindependentuseofBrownianmotioninphysics.Somehow,theresearchdidn'tpickupuntil1960's.InSamuelson(1965)[ 39 ],heconsideredlong-termequityoptions,andusedgeometricBrownianmotiontomodeltherandombehavioroftheunderlyingstock.Baseduponthis,hemodeledtherandomvalueoftheoptionatexercise.Unfortunately,Samuelson'sformulawaslargelyarbitrary.Itofferednomeansforabuyerandsellerwithdifferentriskaversionstoagreeonapriceforanoption.IntheseminalpaperbyBlackandScholes(1973)[ 9 ],theyderivedapartialdifferentialequationforvaluingclaimscontingentonatradedunderlyingasset.Theequationisgeneral.Byapplyingdifferentboundaryconditions,itcanbesolvedtopriceanysuchcontingentclaim.BlackandScholesappliedtheboundaryconditionsforaEuropeancalloptiononanon-dividend-payingstockandobtainedtheirfamousoptionpricingformula.Thekeyideabehindthederivationwastoperfectlyhedgetheoptionby 38

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buyingandsellingtheunderlyingassetinjusttherightwayandconsequentlyeliminaterisk.Thishedgeiscalleddeltahedgingandisthebasisofmorecomplicatedhedgingstrategies.IntheoriginalBlack-Scholesmodel,theyassumedthatthederivative'sunderlyingpricefollowsastandardmodelforgeometricBrownianmotion:dS(t)=S(t)dt+S(t)dWtwhereisaconstantdrift(expectedreturn)ofthesecuritypriceS(t),istheconstantvolatility,andWtisastandardBrownianmotion.Thisstochasticdifferentialequation(SDE)hasthefollowingexplicitsolution:S(t)=S(0)e()]TJ /F8 5.978 Tf 7.79 3.25 Td[(1 22)t+WtUnderthoseassumptions,theyderivedapartialdifferentialequation,nowcalledtheBlack-Scholesequation,whichgovernsthepriceoftheoptionovertime.Thekeyideabehindthederivationwastoperfectlyhedgetheoptionbybuyingandsellingtheunderlyingassetinjusttherightwayandconsequentlyeliminaterisk.Supposeweconstructthefollowingportfolio:=V(S,T))]TJ /F4 11.955 Tf 11.95 0 Td[(SHereV(S,T)denotesavalueofonelongoptionpositionandistheamountshortintheunderlying.Thevalueofthehedgedportfoliowillchangebyd=dV(S,T))]TJ /F4 11.955 Tf 11.95 0 Td[(dSUsingIto'sformula,wehavedV=@V @tdt+@V @SdS+1 22S2@2V @S2dt 39

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Plugin,thechangeintheportfoliowillbe:d=(@V @t+1 22S2@2V @S2)dt+(@V @S)]TJ /F4 11.955 Tf 11.96 0 Td[()dSHence,Theriskintheportfolioisremovedif@V @S)]TJ /F4 11.955 Tf 11.96 0 Td[(=0Therefore,thequantityischosenas=@V @SIftheportfolioisdeltahedgedcontinuously(dynamichedgingstrategy),thenarisk-lessportfolioisconstructedwithadynamicsgivenbyd=(@V @t+1 22S2@2V @S2)dtSincethereisnostochasticterm,isarisk-freeinvestmentandhencemustofferthesamereturnasanyotherrisk-freeinvestment.Therefore,fromtheno-arbitrageconditiond=rdtItisimportanttonotethattheportfoliorepresentsaself-nancing,replicatingandhedgingstrategy.Itreplicatesarisk-freeinvestmentanditishedgedsinceithasnostochasticcomponent.Makingsubstitutionstoequationitisobtainedthat(@V @t+1 22S2@2V @S2)dt=r(V)]TJ /F3 11.955 Tf 11.96 0 Td[(S@V @S)dtSimplifyingtheaboveequation,weshowedthatthepayoffofanycontingencyclaimV(s,t)mustsatisfythefollowingpartialdifferentialequation(Black-Scholesequation):@V @t+1 22S2@2V @S2+rS@V @S=rV 40

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Coupledwiththeappropriateterminalandboundaryconditions,theygotclosed-formsolutionsforEuropeancallandputoptions.Thevalueofacalloptionforanon-dividendpayingunderlyingstockintermsoftheBlack-Scholesparametersis:C(S,t)=N(d1)S)]TJ /F3 11.955 Tf 11.95 0 Td[(N(d2)Ke)]TJ /F9 7.97 Tf 6.59 0 Td[(r(T)]TJ /F9 7.97 Tf 6.59 0 Td[(t)d1=logS K+(r+2 2)(T)]TJ /F3 11.955 Tf 11.95 0 Td[(t) p T)]TJ /F3 11.955 Tf 11.95 0 Td[(td2=logS K+(r)]TJ /F13 7.97 Tf 13.15 4.71 Td[(2 2)(T)]TJ /F3 11.955 Tf 11.96 0 Td[(t) p T)]TJ /F3 11.955 Tf 11.96 0 Td[(t=d1)]TJ /F5 11.955 Tf 11.96 0 Td[(p T)]TJ /F3 11.955 Tf 11.95 0 Td[(tThepriceofacorrespondingputoptionis:P(S,t)=N()]TJ /F3 11.955 Tf 9.3 0 Td[(d2)Ke)]TJ /F9 7.97 Tf 6.58 0 Td[(r(T)]TJ /F9 7.97 Tf 6.59 0 Td[(t))]TJ /F3 11.955 Tf 11.96 0 Td[(N()]TJ /F3 11.955 Tf 9.3 0 Td[(d1)SwhereN(.)isthecumulativedensityfunctionofthestandardnormaldistribution,T)]TJ /F3 11.955 Tf 12.22 0 Td[(tisthetimetomaturity,Sisthespotpriceoftheunderlyingasset,Kisthestrikeprice,ristheriskfreerate,andisthevolatilityofreturnsoftheunderlyingasset.Alternatively,thereisanotherwaytondthepriceofderivatives,namelyriskneutralpricing.Thisisoneofthemostimportantprinciplesinderivativevaluation.Riskneutralpricingisapowerfulmethodforcomputingpricesofderivativesecurities.Thebasicideaofriskneutralpricingisthatgivenarealisticmarketmodelforthestockpriceprocess,wecanconstructariskfreeportfoliothatconsistsonlytheunderlyingstockandriskfreebond.IfthereexistsanequivalentmeasureQunderwhichthediscountedstockpriceprocessisamartingale(therisk-neutralmeasure),thenbasedonMartingaleRepresentationTheorem,wecouldperfectlyreplicateanycontingencyclaimbasedonthestock.TherealworldprobabilitymeasureP,surprisingly,hasnothingtodowithoptionpricing.FindingthisparticularequivalentmeasureQreliesonGirsanovtheorem.Afterwegetthisperfectreplicatingscheme,it'seasytoargue,basedonnoarbitragepricingtheory,thattheonlyarbitrage-freepriceoftheclaimattimezeroistheexpectedvalueofthediscountedpayoffoftheclaimundertherisk-neutral 41

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measureQ.Inparticular,foranEuropeancalloption,thepayoffatmaturityTisgivenby(S(T))]TJ /F3 11.955 Tf 12.16 0 Td[(K)+,whereKisthestrikeprice.Hence,risk-neutralpricingtellsusthatthepriceoftheEuropeancallattime0is:EQ[e)]TJ /F9 7.97 Tf 6.59 0 Td[(rT(S(T))]TJ /F3 11.955 Tf 11.96 0 Td[(K)+]SinceBlackandScholespublishedtheirseminalarticleonoptionpricingin1973,therehasbeenvastexplosionsoftheoreticalandempiricalinvestigationonoptionpricing.PeoplestarttorealizethattheBlack-Scholesmodeldisagreeswithrealityinanumberofways,somesignicant.Forexample,empiricalstudieshaveshownthatanasset'slog-returndistributionisnon-Gaussian.Itischaracterizedbyheavytailsandhighpeaks(leptokurtic).Thereisalsoempiricalevidenceandeconomicargumentsthatsuggestthatequityreturnsandimpliedvolatilityarenegativelycorrelated(alsotermedtheleverageeffect).ThisdeparturefromnormalityplaguestheBlack-Scholes-Mertonmodelwithmanyproblems.AnotherproblemisthattheBlack-Scholesmodelassumesconstantvolatility.IftheBlack-Scholesmodelheld,thentheimpliedvolatilityforaparticularstockwouldbethesameforallstrikesandmaturities.Inpractice,thevolatilitysurface(thethree-dimensionalgraphofimpliedvolatilityagainststrikeandmaturity)isnotat.Thetypicalshapeoftheimpliedvolatilitycurveforagivenmaturitydependsontheunderlyinginstrument.Thisisthefamousvolatilitysmileeffects,whichisalong-observedpatterninwhichat-the-moneyoptionstendtohavelowerimpliedvolatilitiesthanin-orout-of-the-moneyoptions.Finally,Black-Scholesmodelassumescontinuouspathsofthestockprice,whereasinreallife,jumpsinequitypricesfrequentlyhappen.Inthelasttwodecades,optionpricinghaswitnessedanexplosionofnewmodelsthateachrelaxsomeoftherestrictiveBlack-Scholes(BS)(1973)assumptions.Examplesinclude: 42

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(i)thestochastic-interest-rateoptionmodelsofMerton(1973)[ 38 ]andAminandJarrow(1992)[ 33 ];(ii)thejump-diffusion/purejumpmodelsofBates(1991)[ 18 ],MadanandChang(1996)[ 20 ],andMerton(1976)[ 37 ];(iii)theconstant-elasticity-of-variance(CEV)modelofCoxandRoss(1976)[ 31 ];(iv)thestochasticvolatilitymodelsofHeston(1993)[ 27 ],HullandWhite(1987a)[ 28 ],MelinoandTurnbull(1990)[ 36 ],Scott(1987)[ 41 ],SteinandStein(1991)[ 23 ],andWiggins(1987)[ 43 ];(v)generalLevymodelsofBandorff-Nielsen,O.(1998)[ 5 ],Eberlein,E.,U.KellerandK.Prause(1998)[ 22 ],Carr,P.,H.Geman,D.B.MadanandM.Yor(2002)[ 13 ],Carr,P.andL,Wu(2003)[ 15 ].Inthenextsection,we'llfocusononeoftheapproachesabove,namely,stochasticvolatilitymodels. 4.2IntroductionToStochasticVolatilityModelThevolatilityofanancialassetisdenedasthestandarddeviationperunitoftimeofthecontinuouslycompoundedassetreturns.Itisanimportantinputtomanynancialdecisionssuchasassetallocation,optionpricing,andriskmanagement.StochasticvolatilitymodelsareoneapproachtoresolveashortcomingoftheBlack-Scholesmodel.Inparticular,Black-Scholesmodelassumesthattheunderlyingvolatilityisconstantoverthelifeofthederivative,andunaffectedbythechangesinthepriceleveloftheunderlyingsecurity.Theobservedmarketreturnsdisplaynonconstantvolatility,clusteringandarenotnormaldistributed.Theimpliedvolatilityisthevolatilityoftheunderlyingwhich,whensubstitutedintoBlack-Scholesformula,givesatheoreticalpriceequaltothemarketprice.Theimpliedvolatilitiesofoptionsinthemarketshowdependenceonstrikeandtimetoexpirationandarenotconstant.IftheBlack-Scholes 43

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modeliscorrect,thenalloptionsonthesameunderlyingassetshouldgivethesameimpliedvolatility.However,Black-Scholesimpliedvolatilitiesusuallyvaryacrossstrikepricesandacrossmaturities.TakinglimitationsofBlack-Scholesintoaccount,onemightseetheapparentbenetsofusingnon-constantvolatilitymodels.OneoftheappealingfeaturesoftheBlack-Scholesmodelisthatityieldsclosed-formanalyticformulaesformanydifferentkindsofoptions.Itassumesaconstantvolatility,whichisunobservableinthemarket.Inpractice,peopleoftenuseimpliedvolatilitytomeasureoption'srelativevalue.Impliedvolatilityisthevolatilitythat,whenusedinaparticularpricingmodel,yieldsatheoreticalvaluefortheoptionequaltothecurrentmarketpriceofthatoption.Thus,iftheassumptionofconstantvolatilitywereright,theimpliedvolatilityusingBlack-Scholesmodelshouldbethesameacrossallpossiblestrikesandmaturities.Butextensiveempiricalevidencesduringthepastdecadeshaveshownthatit'snotthecase.Inparticular,fromthefollowinggraph,it'seasytoseethattheimpliedvolatilitysurfaceisfarfromat. Figure4-1. ImpliedVolatilitySurface Tomodelthestochasticevolutionofvolatilityovertime,anassumptionaboutthestochasticprocessthatgovernsitsdynamicsneedstobemade.Thespecicationoftheprocessshouldbebasedonitsabilitytoexplainmostoftheempiricalregularities 44

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ofvolatility.Severalempiricalstudieshaveinvestigatedthetime-seriespropertiesofvolatility.Themainndingsincludes:(i)Fattails.Sincetheearlysixties,manyempiricalstudieshavefoundthattheprobabilitythatextremeeventswilloccurisgreaterthanthecorrespondingprobabilitycalculatedunderthenormaldistribution.Inotherwords,theempiricaldistributionofreturnsexhibitsexcesskurtosis;itaccumulatesmoreprobabilitymassinthetails(i.e.,fattails)thanthenormaldistributiondoes.Thisiscalledaleptokurticdistribution.(ii)MeanReversioninVolatility.Empiricalstudieshavefoundthatvolatilityoscillatesaroundaconstantvalue.Thisphenomenonistermedmeanreversion,indicatingthatvolatilitytendstoreverttoalong-runmean.(iii)LeverageEffect.In1976,FischerBlacknotedthatthereisanegativerelationshipbetweenvolatilityandpricechanges.Thisphenomenonistermedleverageeffect.Christie(1982)[ 16 ]attributedtheleverageeffecttothefactthatadropinstockpricestendstoincreasetheleverageoftherm,whichinturnsincreasesitsriskasthisismeasuredbyvolatility.(iv)ClusteringEffect.Anycasualobservationofnancialtimeseriesrevealsclustersofhighandlowvolatilityepisodes.Mandelbrot(1963)[ 35 ]andFama(1965)[ 24 ]reportedevidencethatperiodsofhigh(low)volatilityarefollowedbyperiodsofhigh(low)volatility.Mandelbrothascalledthisphenomenontheclusteringeffectofvolatility.Ineithercase,thesignofchangesfromoneperiodtothenextisunpredictable.Volatilityclusteringsuggeststhepresenceofautocorrelationinvolatilitychanges.Afterthestockmarketcrashin1987,itwasevidentthatvolatilityofthestockreturnscouldnotbetreatedasaconstantparameter.HullandWhite'smodel(1987)[ 28 ]wasoneoftheearliestandsimpleststochasticvolatilitymodelsthatwasintroducedinthesameyearthestockmarketcrashed.Inparticular,theyconsideredthefollowingdynamics:dS(t)=S(t)dt+p V(t)S(t)dW1(t) 45

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dV(t)=aV(t)dt+bV(t)dW2(t)wheredW1tdW2t=dt,anda,barepositiveconstants.Inthiscase,thevolatilityt=p V(t)isageometricBrownianmotion.Scott(1989)[ 40 ]consideredthecaseinwhichthelogarithmofthevolatilityisameanrevertingprocessandthiswasfurtherdevelopedbySteinandStein(1991)[ 23 ].Heassumedthestockpricefollows:dS(t)=S(t)dt+exp(V(t))S(t)dW1(t)dV(t)=a(b)]TJ /F3 11.955 Tf 11.96 0 Td[(V(t))dt+cdW2(t)wheredW1(t)dW2(t)=dt,anda,bandcarepositiveconstants.Inthiscase,thelog-volatilityV(t)=log(t)isanOrnstein-Uhlenbeck(OU)process.In1993Hestonintroducedamodelwherethevolatilityisrelatedtoameanrevertingsquarerootprocess,commonlyknownasCIRprocess.ThemeanrevertingsquarerootprocesswasrstintroducedbyCox,IngersollandRoss(1985)[ 17 ]toimitatethebehaviorofrisk-freeinterestrate.HestonoffersastochasticvolatilitymodelthatisnotbasedontheBSformula.ItprovidesaclosedformsolutionforthepriceofaEuropeancalloptionwhenthespotassetiscorrelatedwithvolatility(Heston1993)[ 27 ].ThesefeaturesmadetheHestonmodelpopularforpricingplainvanillaoptions.Inthenextsectionwe'llintroduceHeston'smodel. 4.3IntroductionToHeston'sModelAndSimulationIssueHestonmodel(1993)isthemostsuccessfulstochasticvolatilitymodelthatattemptstocapturethesmileeffectobservedinimpliedvolatilitiesofliquidlytradedoptions,andtofulllthegapoftheunrealisticconstantvolatilityassumedintheBlack-Scholesmodel[ 9 ].InHestonmodel,variances,notvolatilities,arespecied 46

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tofollowamean-revertingsquarerootprocess.Thisisaprocessthatiswidelyappliedinnance,forexample,theCIRshortratemodel(1985)[ 17 ],andafne-structuremodel.Sinceitsinceptionin1993,theHestonstochasticvolatilitymodelhasreceivedagrowingattentionamongpractitionersandacademics.ThemodelproposedbyHeston[ 27 ]extendstheBlackandScholes[ 9 ]modelandincludesitasaspecialcase.Heston'ssettingtakesintoaccountnon-lognormaldistributionoftheassetsreturns,leverageeffect,importantmean-revertingpropertyofvolatilityanditremainsanalyticallytractable.ItrelaxestheconstantvolatilityassumptionintheclassicalBlack-Scholesmodelbyincorporatinganinstantaneousshorttermvarianceprocess.Assuch,adecent(thoughnotall)numberofsmileandskewpatternscanbebuiltintovolatilitysurfacesbyarelativelyrestrictednumberofparameters.ThebasicHeston'smodeldoesnotmodelstochasticvolatilitiesdirectly,butstochasticvariances.TheprocessspecifyingthevarianceV(t)isidenticaltotheonethatCox-Ingersoll-Ross(1985)[ 17 ]applyforshortinterestrate,andiscalledmean-revertingsquarerootprocess.Themean-reversionisadesiredpropertyforstochasticvolatilityorvarianceandiswelldocumentedbymanyempiricalstudies.Furthermore,itassumesthatS(t),thestockpriceprocessandthevarianceprocessevolveaccordingtothefollowingtwoSDEsundertherisk-neutralmeasure:dS(t)=rS(t)dt+p V(t)S(t)[dW1(t)+p 1)]TJ /F5 11.955 Tf 11.95 0 Td[(2dW2(t)],dV(t)=()]TJ /F3 11.955 Tf 11.96 0 Td[(V(t))dt+p V(t)dW1(t).Therstequationgivesthedynamicsofthestockprice:Stdenotesthestockpriceattimet,ristheriskneutraldrift,p V(t)isthevolatility.Thesecondequationgivestheevolutionofthevariancewhichfollowsthesquare-rootdiffusionprocess:isthelong-termlevelthatvariancegraduallyconvergesto.representsthespeedofmeanreversion,andisaparameterwhichdeterminesthevolatilityofthevarianceprocess.W1(t)andW2(t)aretwoindependentBrownianmotionprocesses,andrepresentsthe 47

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instantaneouscorrelationbetweenthereturnprocessandthevolatilityprocess.Iftheparametersobeythecondition22(knownastheFellercondition)thentheprocessV(t)isstrictlypositive.Typically,thecorrelationisnegative,pointingtothefactthatadown-moveinthestockpriceiscorrelatedwithanup-moveinthevolatility(leverageeffect).ItisworthwhilementioningthatthevarianceprocessV(t)isnoncentrallychi-squaredistributed.Infact,V(t)jV(s)c0d2()wherec=2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F13 7.97 Tf 6.58 0 Td[((t)]TJ /F9 7.97 Tf 6.58 0 Td[(s)) 4,d=4 2,=4e)]TJ /F13 7.97 Tf 6.59 0 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(s) 2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F13 7.97 Tf 6.58 0 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(s))V(s).Basedonthepropertiesofanon-centralchi-squaredistribution,V(t)hasthefollowingtwoconditionalmoments:E(V(t)jV(s))=+(V(s))]TJ /F5 11.955 Tf 11.95 0 Td[()e)]TJ /F13 7.97 Tf 6.59 0 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(s),Var(V(t)jV(s))=V(s)(2 )(e)]TJ /F13 7.97 Tf 6.59 0 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(s))]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(2(t)]TJ /F9 7.97 Tf 6.58 0 Td[(s))+(2 2)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F13 7.97 Tf 6.59 0 Td[((t)]TJ /F9 7.97 Tf 6.58 0 Td[(s))2.If,andsatisfythefollowingcondition:2>2,V0>0,itcanbeshownthatvariancesV(t)arealwayspositiveandthevarianceprocessisthenwell-denedundertheabovecondition.Hestonistherstonewhoproposesaclosed-formpriceforastandardEuropeancallwhenusingasquare-rootvolatilityprocessbyinvertingthecharacteristicfunctionseenasaFouriertransform.Butthereisnoclose-formpriceforpath-dependentexoticoptions,wheretheoptionpayoffdependsonthewholepathoftheunderlyingassetprice.ManypracticalapplicationsofmodelswithHeston-dynamicsinvolvethepricing 48

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andhedgingofpath-dependentoptions,which,inturn,nearlyalwaysrequirestheintroductionofMonteCarlomethods.GivensomearbitrarysetofdiscretetimesftigNi=1,wewanttogeneraterandompathsofthepair(S(t),V(t))alongthosediscretetimes.Thiswouldberequired,forinstance,inthepricingofpath-dependentsecuritieswithpayoutfunctionsthatdependonobservationsofS(t)atagivennitesetofdates.Inordertodothat,itsufcestobeabletogenerate(S(t+),V(t+))given(S(t),V(t)).AfewpreviouslyproposedtechniquesforupdatingS(t)andV(t)fromtimettotimet+includeafulltruncationschemebasedonanaiveEulerscheme,Kahl-JackeldiscretizationschemeinvolvinganimplicitMilsteinschemefortheV-processandtheirIJKdiscretizationforthestockprocess,Broadie-KayaschemebasedonanexactdistributionofS(t),andAnderson'sQEscheme.ThenaiveEulerschemewithafulltruncationxwouldtakethefollowingform:ln^S(t+)=ln^S(t)+(r)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2^V(t)+)+q ^V(t)+(Z1+p 1)]TJ /F5 11.955 Tf 11.95 0 Td[(2Z2)p ,^V(t+)=^V(t)+()]TJ /F4 11.955 Tf 13.82 2.65 Td[(^V(t)+)+q ^V(t)+Z1p whereZ1andZ2areindependentstandardGaussianvariables,andthenotationx+=max(x,0).ItsmaincharacteristicisthattheprocessforVisallowedtogobelowzero,atwhichpointtheprocessforVbecomesdeterministicwithanupwarddriftof.ThebiasoftheMonteCarloestimatorforthestockpriceusingthisnaiveEulerschemebecomesreallybigwhenisrelativelybigand2<<2.Kahl-Jackel(2006)[ 32 ]suggestsdiscretizingtheV-processusinganimplicitMilsteinscheme,andcoupledwiththeirIJKdiscretizationforthestockprocess.Theyproposedthefollowingscheme:ln^S(t+)=ln^S+(r)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 4(^V(t+)+^V(t)))+q ^V(t)Z1p +1 2(q ^V(t+)+q ^V(t))p 1)]TJ /F5 11.955 Tf 11.96 0 Td[(2Z2p +1 4(Z21)]TJ /F4 11.955 Tf 11.95 0 Td[(1), 49

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^V(t+)=^V(t)++q ^V(t)Z1p +1 42(Z21)]TJ /F4 11.955 Tf 11.96 0 Td[(1) 1+ButonethingaboutthisschemeisthatitonlyresultsinpositivepathsfortheVprocessif4>2.Inpractice,thisrestrictionisrarelysatised.ThusthisschemeforVwillproducenegativevalueswithsubstantialprobability.Toxthis,weuseasimilarapproachlikethefulltruncationEulerscheme.Basicallywhenever^V(t)dropsbelowzero,weusethenaiveEulerscheme(deterministicwithanupwarddriftof)toget^V(t+).Otherwise,usetheaboveschemefor^V(t)andmakesuretouse^V(t+)+and^V(t)+insamplingln^S(t+).Broadie-Kaya(2006)[ 12 ]usedthePoissonmethodtosampleV(t+)directly.Basically,theschemeinvolvessamplingfromaPoissondistribution,andthensamplefromacentralchi-squaredistributionwithitsdegreeoffreedomparameterdeterminedbytheoutcomeofthePoissondraw.AswealreadymentionedinsimulatingCIRprocess,thismethodofsamplinganon-centralchisquarerandomvariablewithdegreesoffreedomlessthan1isnotefcient.Afterthis,toobtainabias-freeschemeforsamplingthestockpriceprocess,rstintegratetheSDEforV(t),andgetV(t+)=V(t)+Zt+t()]TJ /F3 11.955 Tf 11.96 0 Td[(V(u))du+Zt+tp V(u)dW1(u)orwecouldwriteitasZt+tp V(u)dW1(u)=)]TJ /F10 7.97 Tf 6.58 0 Td[(1(V(t+))]TJ /F3 11.955 Tf 11.96 0 Td[(V(t))]TJ /F5 11.955 Tf 11.95 0 Td[(+Zt+tV(u)du).Now,byIto'sformula,thelog-stockpricesatisesthefollowingSDE:dlnS(t)=(r)]TJ /F4 11.955 Tf 13.16 8.09 Td[(1 2V(t))dt+p V(u)dW1(u)+p 1)]TJ /F5 11.955 Tf 11.95 0 Td[(2p V(u)dW2(u)Inintegralform,wehavelnS(t+)=lnS(t)+r+ (V(t+))]TJ /F3 11.955 Tf 11.95 0 Td[(V(t))]TJ /F5 11.955 Tf 11.96 0 Td[() 50

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+( )]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2)Zt+tV(u)du+p 1)]TJ /F5 11.955 Tf 11.95 0 Td[(2Zt+tp V(u)dW2(u)afterpluggingintheexpressionofRt+tp V(u)dW1(u)fromearlier.SinceV(u)andW2(u)areindependent,thedistributionofRt+tp V(u)dW2(u)giventhepathgeneratedbyV(t)isnormalwithmean0andvarianceRt+tV(u)du.Nowit'sclearthatconditionalonV(t+)andRt+tV(u)du,thedistributionoflnX(t+)isGaussianwitheasilycomputablerstandsecondmoments.TheyexaminedthecharacteristicfunctionofRt+tV(u)duandnumericallyFourier-inverttogetthedistributionfunction,whichisverycomplicatednumerically.AsmentionedinAnderson'spaper,ratherthanusingFouriermethods,simplywriteZt+tV(u)du[1V(t)+2V(t+)]forcertainconstants1and2.Therearemultiplewaysforsetting1and2,thesimplestbeingtheEuler-likesetting:1=1,2=0.Oracentraldiscretization:1=2=1 2.WeuseournewmethodtosimulatethepathofV(t),andusetheaboveapproachinsteadofBroadieandKaya'sapproachtoapproximateRt+tV(u)du.Afterpluggingintheapproximation,wehave:ln^S(t+)=ln^S(t)+r+ (^V(t+))]TJ /F4 11.955 Tf 13.83 2.66 Td[(^V(t))]TJ /F5 11.955 Tf 11.96 0 Td[()+( )]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2)(1^V(t)+2^v(t+))+p p 1)]TJ /F5 11.955 Tf 11.96 0 Td[(2q 1^V(t)+2^V(t+)Z=ln^X(t)+K0+K1^V(t)+K2^V(t+)+q K3^V(t)+K4^V(t+)ZwhereZisastandardGaussianrandomvariable,independentof^V,andK0,...,K4aregivenbyK0=(r)]TJ /F5 11.955 Tf 13.15 8.08 Td[( ),K1=1( )]TJ /F4 11.955 Tf 13.15 8.08 Td[(1 2))]TJ /F5 11.955 Tf 13.68 8.08 Td[( 51

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K2=2( )]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2)+ ,K3=1(1)]TJ /F5 11.955 Tf 11.96 0 Td[(2),K4=2(1)]TJ /F5 11.955 Tf 11.96 0 Td[(2)OurmethodtosimulatethestockpricefollowsthelineofAnderson'smethod.ButinsteadoftheQEschemeinhispapertoapproximatethedistributionofvolatilityprocess,weuseournewresulttogenerateV-processprecisely.Andbasedonthat,generatethepathforstockpriceusingtheEulerschemeinAnderson'spaper. 52

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CHAPTER5SIMULATIONRESULTSANDAPPLICATIONSINOPTIONPRICINGInthissection,weshowoursimulationresultsonournewexactmethod.Specically,wesimulateaCIRprocesswithdifferentsetofparametersusingbothournewmethodandseveralEulerschemes.WealsosimulatetheHestonmodelforthestockpricewithdifferentsetsofparametersusingbothournewmethodandtraditionalEulerscheme.We'llalsouseittopriceEuropeanoptionsandcomparethemwithexactoptionprices. 5.1SimulationResultsForCIRInterestRateModelTheCIRmodeldescribesthedynamicsoftheinterestrateR(t)asasolutionofthefollowingstochasticdifferentialequation(SDE): dR(t)=(b)]TJ /F3 11.955 Tf 11.96 0 Td[(R(t))dt+p R(t)dW(t),R(0)=r00 (5) for>0,b>0,>0andastandardBrownianmotionW.ForsmallR(t),thenon-centralityparameterapproacheszero,andthedistributionofR(t+t)becomesproportionaltothatofanordinary(central)chi-squaredistributionwithd=4b=2degreesoffreedom.Werecallthatthedensityofacentralchi-squaredistributionwithddegreesoffreedomisf(x;d)=1 2d=2\(d=2)e)]TJ /F9 7.97 Tf 6.59 0 Td[(x=2xd=2)]TJ /F10 7.97 Tf 6.59 0 Td[(1.Thus,forparameterswhichsatisfytheconditiond=4b=2<<2,thepresenceofthetermxd=2)]TJ /F10 7.97 Tf 6.58 0 Td[(1impliesthat,forsmallR(t),thedensityofR(t+t)willbeverylargearound0.ThisisthemainreasonthattraditionalEulermethodfailstobeaccurateanymore.TotestourexactmethodtosimulateaCIRprocess,weusethefollowing3differentcasesintheparametertable: 53

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Table5-1. ParametertableforCIRinterestmodelsimulation. ParametersCaseICaseIICaseIII 21.210.10.20.4b0.40.20.1R(0)0.30.10.05 Herethedegreesoffreedomd=0.04,0.11,0.16,respectively,alllessthan1.Thetimeintervalweconsideris[0,1].WeuseMonteCarlomethodtoapproximateE(R(1)),andcomparetheresultsfromthenewexactmethod,thePoissonmethodandtwoEulerSchemes(truncationandreection).ThetheoreticalmeanE(R(1))=0.310,0.118,0.067,respectively.WeruntheEulerSchemeswithtimepointsn=1,5,7,10,14,20,30,50,andm=10000samplepaths.ThePoissonmethodandournewmethod,ontheotherhand,sincetheyareexact,wererunwithm=10000samplepathsandjustn=1timepoint. Table5-2. EstimatedexpectationofinterestrateattimeT=1intestCaseI.Numbersinparenthesesaresamplestandarddeviations. Euler1Euler2PoissonNew 10.624(0.008)0.909(0.007)0.320(0.011)0.306(0.011)1/50.559(0.011)0.974(0.122)1/70.530(0.011)0.905(0.012)1/100.525(0.012)0.844(0.012)1/140.482(0.011)0.786(0.012)1/200.480(0.012)0.749(0.012)1/300.442(0.012)0.707(0.012)1/500.421(0.012)0.630(0.012) 54

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Table5-3. EstimatedexpectationofinterestrateattimeT=1intestCaseII.Numbersinparenthesesaresamplestandarddeviations. Euler1Euler2PoissonNew 10.217(0.003)0.316(0.002)0.119(0.004)0.121(0.004)1/50.202(0.004)0.335(0.004)1/70.200(0.004)0.321(0.004)1/100.194(0.004)0.304(0.004)1/140.184(0.004)0.291(0.004)1/200.177(0.004)0.268(0.004)1/300.172(0.004)0.249(0.004)1/500.167(0.004)0.231(0.004) Table5-4. EstimatedexpectationofinterestrateattimeT=1intestCaseIII.Numbersinparenthesesaresamplestandarddeviations. Euler1Euler2PoissonNew 10.129(0.002)0.186(0.001)0.068(0.002)0.066(0.002)1/50.124(0.002)0.211(0.003)1/70.118(0.002)0.194(0.003)1/100.113(0.002)0.188(0.003)1/140.110(0.002)0.174(0.003)1/200.106(0.002)0.169(0.003)1/300.100(0.002)0.156(0.003)1/500.095(0.002)0.143(0.003) Obviouslyinthoseparticularcases,thereisalargebiaswithbothEulermethods,sincetheinterestrateprocessstaysaround0mostofthetime.Theresultfromthenewmethodisunbiasedandmuchmoreaccurate.ThePoissonmethodisalsoveryaccurateinthosecases,butwhenngetsbigger,itwouldbemuchslower. 55

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Figure5-1. ComparisonbetweendifferentschemesinestimatingE[R(1)]intestCaseI Figure5-2. ComparisonbetweendifferentschemesinestimatingE[R(1)]intestCaseII 56

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Figure5-3. ComparisonbetweendifferentschemesinestimatingE[R(1)]intestCaseIII Forthesamesetofparameters,wealsotriedtoestimatethebondpriceE(e)]TJ /F7 7.97 Tf 8 6.42 Td[(R10R(t)dt).Here,sinceanintegralisinvolved,we'llapproximatetheintegralbyasummation,whichrequiresustosimulatetheprocessonalotoftimepoints.Sincethereisnoanalyticalresultfortheexpectation,weestimateitusingtheexactnewmethodwith106samplepathsand103timepointstoestimatetheintegral.Andtheestimatedbondpricesare0.821,0.913,0.948,respectivelyinthosethreecases.Withm=10000samplepathsandn=10,14,20,30,50,100timepoints,wecomparedtheperformanceoftheEulerschemes,thePoissonmethodandtheexactmethod: 57

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Table5-5. EstimatedbondpriceintestCaseI.Numbersinparenthesesaresamplestandarddeviations. Euler1Euler2PoissonNew 1/100.749(0.003)0.608(0.002)0.817(0.002)0.833(0.003)1/140.749(0.003)0.632(0.002)0.821(0.002)0.829(0.003)1/200.767(0.003)0.657(0.002)0.819(0.002)0.823(0.003)1/300.771(0.003)0.682(0.002)0.820(0.002)0.826(0.003)1/500.776(0.003)0.701(0.002)0.825(0.002)0.823(0.003)1/1000.780(0.003)0.722(0.002)0.819(0.003)0.821(0.003) Table5-6. EstimatedbondpriceintestCaseII.Numbersinparenthesesaresamplestandarddeviations. Euler1Euler2PoissonNew 1/100.876(0.002)0.820(0.001)0.913(0.001)0.915(0.001)1/140.879(0.002)0.827(0.001)0.912(0.001)0.913(0.001)1/200.884(0.002)0.840(0.001)0.911(0.001)0.914(0.001)1/300.886(0.002)0.853(0.001)0.913(0.001)0.913(0.001)1/500.891(0.002)0.862(0.001)0.912(0.001)0.913(0.001)1/1000.893(0.001)0.871(0.001)0.912(0.001)0.913(0.001) Table5-7. EstimatedbondpriceintestCaseIII.Numbersinparenthesesaresamplestandarddeviations. Euler1Euler2PoissonNew 1/100.921(0.001)0.883(0.001)0.949(0.001)0.949(0.001)1/140.925(0.001)0.888(0.001)0.947(0.001)0.948(0.001)1/200.928(0.001)0.896(0.001)0.948(0.001)0.947(0.001)1/300.930(0.001)0.905(0.001)0.948(0.001)0.948(0.001)1/500.933(0.001)0.909(0.001)0.949(0.001)0.948(0.001)1/1000.936(0.001)0.919(0.001)0.948(0.001)0.948(0.001) 58

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Fromtheabovetables,it'seasytoseethatbothEulerschemesstillhavebiasinpricingthebondprice,whereasthePoissonmethodandournewexactmethodismuchmoreaccurate.Butintermsofspeed,Poissonmethodismuchslowerthanournewexactmethod,evenonly100timepointsareusedintheapproximationoftheintegral.Ournewexactmethodremainsconstantspeed,nomatterhowmanytimepointswe'reinterestedin. Figure5-4. ComparisonbetweendifferentschemesinestimatingE(e)]TJ /F7 7.97 Tf 7.99 6.42 Td[(R10R(t)dt)intestCaseI 59

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Figure5-5. ComparisonbetweendifferentschemesinestimatingE(e)]TJ /F7 7.97 Tf 7.99 6.42 Td[(R10R(t)dt)intestCaseII Figure5-6. ComparisonbetweendifferentschemesinestimatingE(e)]TJ /F7 7.97 Tf 7.99 6.42 Td[(R10R(t)dt)intestCaseIII 60

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5.2SimulationResultsForHestonModelTheHestonmodelassumesthatSt,thestockpriceandthevarianceprocessesevolveaccordingtothefollowingtwoSDEsundertherisk-neutralmeasure:dSt=rStdt+p VtSt[dW(1)t+p 1)]TJ /F5 11.955 Tf 11.95 0 Td[(2dW(2)t],dVt=()]TJ /F3 11.955 Tf 11.95 0 Td[(Vt)dt+p VtdW(1)t.Therstequationgivesthedynamicsofthestockprice:Stdenotesthestockpriceattimet,ristheriskneutraldrift,p Vtisthevolatility.Thesecondequationgivestheevolutionofthevariancewhichfollowsthesquare-rootdiffusionprocess:isthelong-termlevelthatvariancegraduallyconvergesto.representsthespeedofmeanreversion,andisaparameterwhichdeterminesthevolatilityofthevarianceprocess.W(1)tandW(2)taretwoindependentBrownianmotionprocesses,andrepresentstheinstantaneouscorrelationbetweenthereturnprocessandthevolatilityprocess.Totestournewexactmethod,weturntothepricingofEuropeanoptionsintheHestonmodel.Thisconstitutesastandardtestcase,aspricescanbecomputedwithgreatprecisioneitherfromtheanalyticalresultinchapter4,orfromtheFFTmethodinCarrandMadan(1998)[ 14 ].WeconsideracalloptionCmaturingattimeTwithstrikeK;lettheexactoptionpriceattime0beC(0).WeuseourexactmethodtosimulateV(T),andusetheapproachinAnderson[ 4 ]toapproximateX(T)with^X(T).Thenwecangettheapproximation^C(0)totheoptionpricebycomputingtheexpectation^C(0)=E((^X(T))]TJ /F3 11.955 Tf 11.96 0 Td[(K)+).Inordertoestimate^C(0),weuseMonteCarlomethods.Specically,wedrawNindependentsamplesof^X(1)(T),^X(2)(T),...,^X(N)(T)usinganequidistanttime-gridwith 61

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xedstep;then^C(0)1 NNXi=1(^X(i)(T))]TJ /F3 11.955 Tf 11.95 0 Td[(K)+.AndweusetheFFTmethodinCarrandMadan(1998)[ 14 ]tocomputeexactpriceC(0).Thisvalueisusedasabenchmark,andbasedonthat,wecompareournewmethodwithtraditionalEulerscheme.Weconsiderthreedifferentsetsofparameters,whicharelistedinthefollowingtable: Table5-8. ParametertableforHestonmodelsimulation.Inallcasesr=0,V(0)=andX(0)=100. ParametersCaseICaseIICaseIII 10.910.50.31-0.9-0.5-0.3T10155V(0),0.040.040.09 Foreachsetofparametersabove,wecomparetheoptionpriceusingthreedifferentstrikes(K=70,K=100,K=140).Theanalyticvalueissettobeabenchmarkforcomparison.AnditiscomputedusingtheFFTmethodinCarrandMadan(1998)[ 14 ].Inournumericalresults,weusethefollowingdiscretizationschemes:thefulltruncationEulerscheme;thePoissonmethod;andournewscheme.Tokeepthesamplestandarddeviationlow,alltestswererunusingahighnumberofpaths,N=105.Andthevaluesoftimesteprangefrom1=32yearto1year. 62

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Table5-9. EstimatedEuropeancalloptionpriceintestCaseI.Numbersinparenthesesaresamplestandarddeviations. EulerPoissonNewAnalytic K=70 139.722(0.118)36.136(0.071)36.126(0.071)35.8561/238.262(0.095)35.937(0.071)36.037(0.071)1/437.039(0.082)35.905(0.071)35.890(0.071)1/836.230(0.077)35.997(0.071)35.860(0.071)1/1636.159(0.074)35.852(0.071)35.809(0.071)1/3235.980(0.073)35.899(0.071)35.727(0.071) K=100 119.313(0.092)13.308(0.042)13.340(0.042)13.0071/216.738(0.066)13.158(0.042)13.160(0.042)1/415.145(0.054)13.120(0.042)13.186(0.042)1/814.208(0.048)13.106(0.042)13.065(0.042)1/1613.618(0.045)13.075(0.042)13.138(0.042)1/3213.302(0.043)13.036(0.042)12.990(0.042) K=140 14.514(0.060)0.273(0.008)0.270(0.008)0.3301/22.245(0.033)0.279(0.008)0.281(0.008)1/41.043(0.018)0.286(0.008)0.301(0.008)1/80.565(0.011)0.289(0.008)0.294(0.008)1/160.401(0.009)0.291(0.008)0.292(0.008)1/320.343(0.009)0.293(0.008)0.309(0.008) 63

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Figure5-7. ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseI(K=70) Figure5-8. ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseI(K=100) 64

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Figure5-9. ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseI(K=140) 65

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Table5-10. EstimatedEuropeancalloptionpriceintestCaseII.Numbersinparenthesesaresamplestandarddeviations. EulerPoissonNewAnalytic K=70 141.369(0.219)37.012(0.141)37.058(0.140)36.6021/239.954(0.194)37.047(0.140)36.941(0.149)1/438.328(0.167)36.981(0.139)37.090(0.142)1/837.649(0.155)36.908(0.142)36.994(0.145)1/1637.570(0.177)37.127(0.139)36.906(0.139)1/3237.469(0.147)36.836(0.138)37.133(0.138) K=100 123.931(0.263)16.768(0.132)16.543(0.130)16.2031/220.991(0.295)16.637(0.148)16.498(0.122)1/418.578(0.145)16.578(0.127)16.619(0.134)1/817.641(0.149)16.620(0.134)16.770(0.169)1/1617.264(0.135)16.443(0.165)16.680(0.168)1/3216.785(0.123)16.386(0.122)16.414(0.124) K=140 111.727(0.265)5.146(0.112)5.118(0.100)4.8751/28.207(0.149)5.203(0.143)5.174(0.142)1/46.627(0.122)5.184(0.133)5.103(0.098)1/85.832(0.108)5.174(0.145)5.246(0.132)1/165.599(0.130)5.101(0.102)5.175(0.113)1/325.288(0.120)5.011(0.110)5.142(0.113) 66

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Figure5-10. ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseII(K=70) Figure5-11. ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseII(K=100) 67

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Figure5-12. ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseII(K=140) 68

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Table5-11. EstimatedEuropeancalloptionpriceintestCaseIII.Numbersinparenthesesaresamplestandarddeviations. EulerPoissonNewAnalytic K=70 141.456(0.243)38.579(0.174)38.970(0.196)38.7211/239.828(0.202)38.986(0.192)38.978(0.183)1/439.602(0.197)38.678(0.182)38.814(0.192)1/839.257(0.231)38.643(0.215)38.864(0.182)1/1638.938(0.187)38.884(0.179)38.554(0.183)1/3238.408(0.191)38.732(0.187)38.639(0.187) K=100 126.677(0.382)22.188(0.150)22.109(0.339)21.7481/223.913(0.184)22.104(0.171)21.559(0.167)1/423.019(0.177)21.438(0.177)21.559(0.163)1/822.271(0.168)21.549(0.175)21.582(0.160)1/1622.125(0.164)21.704(0.124)21.723(0.156)1/3221.877(0.161)21.733(0.159)21.692(0.169) K=140 114.445(0.199)9.665(0.123)9.887(0.140)9.9811/212.350(0.175)9.736(0.127)9.656(0.130)1/411.028(0.157)10.012(0.130)9.718(0.129)1/810.705(0.150)10.076(0.176)10.020(0.185)1/1610.384(0.148)9.865(0.156)10.083(0.139)1/3210.081(0.148)9.902(0.155)9.817(0.127) 69

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Figure5-13. ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseIII(K=70) Figure5-14. ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseIII(K=100) 70

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Figure5-15. ComparisonbetweendifferentschemesinestimatingEuropeanoptionpriceintestCaseIII(K=140) Fromtheabovetablesandgraphs,wecouldseethatthenewmethodismoreefcientandaccuratethanthefulltruncationEulerScheme,whichhasalargebiasinallthreecases.Andasexpected,Poissonmethodisprettyaccuratebuttakesamuchlongertimetoyieldtheresultthanournewmethod.Andasthetime-stepgetsner,itwillbecomeevenmuchslower.Finally,welookatpriceofdiscreteAsianoptionsunderHeston'smodel.Usingtheparametersincasethree,weuseMonteCarlomethodtoestimatethepriceofadiscreteAsiancalloptionontimeinterval[0,5].UnfortunatelythereisnoanalyticalformulafordiscreteAsiancalloptionunderHeston'smodel.WethereforeapproximatethepriceusingourMonteCarlosimulationwithm=106andn=1000,anduseitasabenchmark.OnceagainthestrikepricesareK=70,100,140.Wewouldusem=10000andn=12,30,52,100,252timepointsinthesimulation.Inthiscase,Poissonmethodwouldbedrasticallytooslowtouseinsimulation.Ournewalgorithmturnsouttobebothaccurateandefcient. 71

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Table5-12. EstimateddiscreteAsiancalloptionpriceintestCaseIII.Numbersinparenthesesaresamplestandarddeviations. EulerPoissonNewAnalytic K=70 1/1233.094(0.357)32.672(0.339)32.920(0.328)32.9071/3032.952(0.349)33.092(0.332)33.307(0.352)1/5232.712(0.333)33.099(0.340)33.281(0.352)1/10032.543(0.326)32.987(0.337)32.885(0.340)1/25232.771(0.333)32.876(0.334)32.674(0.359) K=100 1/1213.533(0.294)13.004(0.278)12.770(0.275)12.7071/3012.887(0.259)12.633(0.263)12.819(0.264)1/5212.687(0.258)13.024(0.306)12.471(0.293)1/10012.580(0.250)12.905(0.274)12.738(0.270)1/25212.952(0.267)12.802(0.267)12.904(0.261) K=140 1/123.645(0.192)3.741(0.245)3.360(0.218)3.1221/303.543(0.200)3.253(0.183)3.031(0.186)1/523.081(0.180)2.759(0.139)3.179(0.170)1/1002.895(0.159)3.095(0.183)3.320(0.244)1/2523.259(0.179)3.231(0.202)3.098(0.184) Figure5-16. ComparisonbetweendifferentschemesinestimatingdiscreteAsianoptionpriceintestCaseIII(K=70) 72

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Figure5-17. ComparisonbetweendifferentschemesinestimatingdiscreteAsianoptionpriceintestCaseIII(K=100) Figure5-18. ComparisonbetweendifferentschemesinestimatingdiscreteAsianoptionpriceintestCaseIII(K=140) 73

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CHAPTER6CONCLUSIONInthispaper,afastandefcientwayofsimulatingCIRprocessisintroduced.Weprovedatheoremthatcouldbeusedtogenerateanon-centralChi-squarerandomvariable.AndthewaywegenerateitavoidstheproblemwiththetraditionalmethodthatinvolvesPoissonrandomvariablegeneration.OurnewmethodisexactcomparedwithEulerschemes,andfastcomparedwithPoissonmethod.Especiallyinthecasewhend<<1andweneedtosimulateaCIRprocessalongmanytimepoints,traditionalPoissonmethodisveryslowsincethenon-centralityparameterisbig.Ontheotherhand,ournewmethodforgeneratingaCIRprocessisindependentofthenumberoftimepointsneededinthesimulation,thusmuchquickerthanthePoissonmethod.Weproposedseveralnumericalteststotestouralgorithmandtheresultshaveshownthatourmethodisveryefcientandsuitabletobeusedinthisparticularcase. 74

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REFERENCES [1] Alfonsi,A.OnthediscretizationschemesfortheCIRprocesses.Montecarlomethodsandapplications11(2005).4:355. [2] .Asecond-orderdiscretizationschemefortheCIRprocess:applicationtotheHestonmodel.preprint(2008). [3] .HighorderdiscretizationschemesfortheCIRprocess:ApplicationtoafnetermstructureandHestonModels.Math.Comput.79(2010).269:209. [4] Anderson,L.EfcientsimulationoftheHestonstochasticvolatilitymodel.Journalofcomputationalnance11(2007):1. [5] Bandorff-Nielsen,O.ProcessofNormalInverseGaussianType.FinanceandStochastics2(1998):41. [6] Berkaoui,A.,Bossy,M.,andDiop,A.EulerschemeforSDEswithnon-Lipschitzdiffusioncoefcient:strongconvergence.ESAIMProbabilityandStatistics12(2008).1:1. [7] Black,F.andKarasinski,P.BondandOptionpricingwhenShortratesareLognormal.FinancialAnalystsJournal(1991):52. [8] Black,FischerandKarasinski,Piotr.Bondandoptionpricingwhenshortratesarelognormal.FinancialAnalystsJournalJuly/August(1991):52. [9] Black,FischerandScholes,Myron.ThePricingofOptionsandCorporateLiabilities.JournalofPoliticalEconomy81(1973).3:637. [10] Brennan,MichaelJ.andSchwartz,EduardoS.Anequilibriummodelofbondpricingandatestofmarketefciency.JournalofFinancialandQuantitativeAnalysis17(1982):75. [11] Brigo,D.andAlfonsi,A.CreditdefaultswapcalibrationandderivativespricingwiththeSSRDstochasticintensitymodel.FinanceandStochastics9(2005).1:29. [12] Broadie,M.andKaya,O.Exactsimulationofstochasticvolatilityandotherafnejumpdiffusionprocesses.OperationalResearch54(2006):217C231. [13] Carr,P.,Geman,H.,Madan,D.B.,andYor,M.Thenestructureofassetreturns:anempiricalinvestigation.JournalofBusiness75(2)(2002):305. [14] Carr,P.andMadan,D.OptionValuationusingtheFastFourierTransform.JournalofComputationalFinance2(1998):61. [15] Carr,P.andWu,L.Thenitemomentlogstableprocessandoptionpricing.JounalofFinance58(2)(2003):753. 75

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[16] Christie,A.A.Thestochasticbehaviorofcommonstodckvariances.JournalofFinancialEconomics10(1982):407. [17] Cox,J.C,Ingersoll,J.E.,andRoss,S.A.Atheoryofthetermstructureofinterestrates.Econometrica53(1985):385. [18] David,Bates.Thecrashof87:Wasitexpected?Theevidencefromoptionsmarkets.JournalofFinance46(1991):1009. [19] Deelstra,G.andDelbaen,F.Convergenceofdiscretizedstochastic(Interestrate)processeswithstochasticdriftterm.Appl.stochasticmodelsdataanal.14(1998):77. [20] Dilip,MadanandChang,Eric.TheVarianceGammaModelandOptionPricing.EuropeanFinanceReview2(1998):79. [21] Dothan,UriL.Onthetermstructureofinterestrates.JournalofFinancialEconomics6(1978):59. [22] Eberlein,E.,Keller,U.,andPrause,K.Newinsightsintosmile,mispricing,andvalueatrisk:thehyperbolicmodel.JournalofBusiness71(3)(1998):371. [23] Elias,SteinandStein,Jeremy.Stockpricedistributionswithstochasticvolatility.ReviewofFinancialStudies4(1991):727. [24] Fama,EugeneF.TheBehaviorofStockMarketPrices.JournalofBusiness38(1965):34. [25] Feller,W.TwoSingularDiffusionProblems.AnnalsofMathematics54(1951):173. [26] Glasserman,P.MonteCarlomethodsinnancialengineering.NewYork:Springer-Verlag,2003. [27] Heston,S.Aclosed-formsolutionofoptionswithstochasticvolatilitywithapplicationstobondandcurrencyoptions.Thereviewofnancialstudies6(1993).2:327. [28] Hull,JohnandWhite,Alan.Thepricingofoptionswithstochasticvolatilities.JournalofFinance42(1987):281. [29] .Pricinginterest-ratederivativesecurities.ReviewofFinancialStudies3(1990):573. [30] Istvan,G.andMiklos,R.AnoteonEulerapproximationsforSDEswithHodercontinuousdiffusioncoefcients.stochasticprocessesandtheirapplications121(2011).10:2189. [31] John,CoxandRoss,Stephen.Thevaluationofoptionsforalternativestochasticprocesses.JournalofFinancialEconomics3(1976):145. 76

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[32] Kahl,C.andJackel,P.FaststrongapproximationMonte-Carloschemesforstochasticvolatilitymodels.J.Quantit.Finance6(2006):513. [33] Kaushik,AminandJarrow,Robert.Pricingoptionsonriskyassetsinastochasticinterestrateeconomy.MathematicalFinance2(1992):217. [34] Kloeden,P.E.andPlaten,E.Numericalsolutionofstochasticdifferentialequations.Berlin:Springer-Verlag,1992. [35] Mandelbrot,B.B.Thevariationofcertainspeculativeprices.JournalofBusiness36(1963):394. [36] Melino,AngeloandTurnbull,Stuart.Pricingforeigncurrencyoptionswithstochastic.JournalofEconometrics45(1990):239. [37] Merton,R.C.Optionpricingwhenunderlyingstockreturnsarediscontinuous.JournalofFinancialEconomy3(1976):125. [38] Merton,RobertC.Theoryofrationaloptionpricing.BellJournalofEconomicsandManagementScience4(1973):141. [39] Samuelson,P.A.ProofthatProperlyAnticipatedPricesFluctuateRandomly.IndustrialManagementReview6(1965):41. [40] Scott,E.andTucker,A.L.Predictingcurrencyreturnvolatility.JournalofBankingandFinance13(1989).6:839. [41] Scott,Louis.Optionpricingwhenthevariancechangesrandomly:Theory,estimators,andapplications.JournalofFinancialandQuantitativeAnalysis22(1987):419. [42] Vasicek,O.Anequilibriumcharacterizationofthetermstructure.JournalofFinancialEconomics5(1977):177. [43] Wiggins,James.Optionvaluesunderstochasticvolatilities.JournalofFinancialEconomics19(1987):351. 77

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BIOGRAPHICALSKETCH AnqiShaowasborninUrumqi,Chinain1982.HemovedtoYantaiwithhisfamilyin1994andattendedNo.1highschoolthere.Hewenttothedepartmentofmathematicsin1999,andgothisbachelor'sdegreefromShandongUniversityin2003.HethenwenttograduateschoolatLehighUniversityin2004,andtransferredtoUniversityofFloridain2005.HereceivedMasterofScienceinmathematicsin2008andMasterofStatisticsin2009.Hecompletedhisdoctorateofphilosophyintheareaofmathematicalnanceinthesummerof2012. 78