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American Option Pricing in a Jump-Diffusion Model

Permanent Link: http://ufdc.ufl.edu/UFE0025116/00001

Material Information

Title: American Option Pricing in a Jump-Diffusion Model
Physical Description: 1 online resource (52 p.)
Language: english
Creator: Berros, Jeremy
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: american, diffusion, engineering, european, finance, jump, model, option, pricing, stochastic
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Many alternative models have been developed lately to generalize the Black-Scholes option pricing model in order to incorporate more empirical features. Brownian motion and normal distribution have been used in this Black-Scholes option-pricing framework to model the return of assets. However, two main points emerge from empirical investigations: (i) the leptokurtic feature that describes the return distribution of assets as having a higher peak and two asymmetric heavier tails than those of the normal distribution, and (ii) an empirical phenomenon called 'volatility smile' in option markets. Among the recent models that addressed the aforementioned issues is that of Kou (2002), which allows the price of the underlying asset to move according to both Brownian increments and double-exponential jumps. The aim of this thesis is to develop an analytic pricing expression for American options in this model that enables us to efficiently determine both the price and related hedging parameters.
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 Jeremy Berros.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: AitSahlia, Farid.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0025116:00001

Permanent Link: http://ufdc.ufl.edu/UFE0025116/00001

Material Information

Title: American Option Pricing in a Jump-Diffusion Model
Physical Description: 1 online resource (52 p.)
Language: english
Creator: Berros, Jeremy
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: american, diffusion, engineering, european, finance, jump, model, option, pricing, stochastic
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Many alternative models have been developed lately to generalize the Black-Scholes option pricing model in order to incorporate more empirical features. Brownian motion and normal distribution have been used in this Black-Scholes option-pricing framework to model the return of assets. However, two main points emerge from empirical investigations: (i) the leptokurtic feature that describes the return distribution of assets as having a higher peak and two asymmetric heavier tails than those of the normal distribution, and (ii) an empirical phenomenon called 'volatility smile' in option markets. Among the recent models that addressed the aforementioned issues is that of Kou (2002), which allows the price of the underlying asset to move according to both Brownian increments and double-exponential jumps. The aim of this thesis is to develop an analytic pricing expression for American options in this model that enables us to efficiently determine both the price and related hedging parameters.
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 Jeremy Berros.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: AitSahlia, Farid.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0025116:00001


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Firstofall,IwouldliketothankDr.AitSahliaforsharinghispassionwithme.Itwasarealpleasuretoworkwithhim.Then,Iwanttothankmybelovedanceewhosupportedmealongmywork.Andnally,IamverygratefultomyparentswhoinculcatedthevirtuesandtoolsthatallowmetobeasIamandtoliveasIlive. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 9 1.1Literature .................................... 9 1.2OrganizationoftheThesis ........................... 10 2KOU'SDOUBLE-EXPONENTIALJUMP-DIFFUSIONMODEL ........ 12 2.1ModelFormulation ............................... 12 2.2PropertiesoftheModel ............................ 14 2.3ComparisonwithOtherModels ........................ 14 2.3.1TheConstantElasticityofVarianceModel .............. 15 2.3.2TheNormalJump-DiusionModel .................. 15 2.3.3ModelsBasedont-Distribution .................... 16 2.3.4StochasticVolatilityModels ...................... 16 2.3.5AneJump-DiusionModels ..................... 17 2.3.6ModelsBasedonLevyProcesses .................... 18 2.4LeptokurticFeature ............................... 18 2.5EquilibriumforGeneralJump-DiusionModels ............... 19 2.6TheVolatilitySmile .............................. 23 3AMERICANOPTIONPRICINGINKOU'SMODEL .............. 27 3.1HhFunction ................................... 27 3.2EuropeanCallandPutOptions ........................ 28 4ANALYTICRESULTFORAMERICANOPTIONPRICING .......... 30 4.1MainResult ................................... 30 4.2ProofoftheResult ............................... 33 5CONCLUSION .................................... 36 5.1ImplicationoftheMainResult ........................ 36 5.2LimitationsoftheModel ............................ 36 5.3ScopeofFurtherResearch ........................... 37 APPENDIX ADERIVATIONOFTHERATIONALEXPECTATIONS ............. 38 5

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...................... 41 REFERENCES ....................................... 50 BIOGRAPHICALSKETCH ................................ 52 6

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Figure page 2-1Comparisonbetweendensityg(solidline)ofLeptokurticfeatureandnormaldistributiondensity(dottedline)withsamemeanandvariance ......... 25 2-2MidmarketandModel-ImpliedVolatilitiesforJapaneseLIBORCapletsinMay1998 .......................................... 26 3-1TheHhfunctionforn=1;3;5withtheSteepestCurveforn=5andtheFlattestCurveforn=1 ............................... 29 7

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22t+W(t)N(t)i=1Vi(2{2)NotethatE(Y)=p 1q 2,Var(Y)=pq(p 1+q 2)2+(p 21+q 22)and 13

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S(t) p p 2e(xt)1xt21t p

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2e(xt)2xt+22t p 1q 2t;Varg(G)=2t+(pq1 21+q 22)t+p 1q 22t(1t);where'(:)isthestandardnormaldensityfunction.YouhavearepresentationofdensityfunctionginFig.2-2 19

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(t)=1dt+1dW1(t)+d24N(t)Xi=1(eVi1)35(2{6)wheretheeVi0areanyindependentidenticallydistributed,nonnegativerandomvariables.Tosimplicitymatters,allthreesourcesofrandomness,thePoissonprocessN(t),thestandardBrownianmotionW1(t),andthejumpsizeeV,areassumedtobeindependent.Althoughitisintuitivelyclearthat,generallyspeaking,theassetpricep(t)shouldfollowasimilarjump-diusionprocessasthatofthedividendprocess(t),acarefulstudyoftheconnectionbetweenthetwoisneeded.Thisisbecausep(t)and(t)maynothavesimilarjumpdynamics.Furthermore,derivingexplicitlythechangeofparametersfrom(t)top(t)alsoprovidessomevaluableinformationabouttheriskpremiumsembeddedinjump-diusionmodels.TheworkherebuildsuponandextendsthepreviousworkbyNaikandLee(1990)inwhichthespecialcasethateVihasalognormaldistributionisinvestigated.AnotherdierenceisthatNaikandLee(1990)requirethattheassetpayscontinuousdividendsandthereisnooutsideendowmentprocess,whileheretheassetpaysnodividendsandthereisanoutsideendowmentprocess.Consequently,thepricingformulaearedierenteveninthecaseoflognormaljumps.Forsimplicity,asinNaikandLee(1990),KouonlyconsiderstheutilityfunctionofthespecialformsU(c;t)=etc 20

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221(1)(2)(1)1;wherethenotation(a)1means(a)1:=Eh(eV)(a)1i:AswillbeeninProposition1(Kou,2002),thisassumptionguaranteesthatinequilibriumthetermstructureofinterestrateispositive.Proposition1.Suppose(a1)1<1.(1)LettingB(t,T)bethepriceofazerocouponboundwithmaturityT,theyieldr:=1 (Tt))log(B(t;T)isaconstantindependentofT, 221(1)(2)(1)1>0(2{8)(2)LetZ(t):=ertUc((t);t)=er((t))1.ThenZ(t)isamartingaleunderP, 21

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221(2)+1(+1)1:(2{13)If(14)issatised,thenunderP, 22

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Comparisonbetweendensityg(solidline)ofLeptokurticfeatureandnormaldistributiondensity(dottedline)withsamemeanandvariance 25

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MidmarketandModel-ImpliedVolatilitiesforJapaneseLIBORCapletsinMay1998 26

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2n+1 2;1 2;1 2x2) 2n)x1F1(1 2n+1;3 2;1 2x2) (1 2+1 2n))where1F1istheconuenthypergeometricfunction.Atree-termrecursionisalsoavailablefortheHhfunction: 27

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22;;e;ep;e1;e2log(K=S(0));TKerTr1 22;;;p;1;2log(K=S(0));Twhereep=p 28

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TheHhfunctionforn=1;3;5withtheSteepestCurveforn=5andtheFlattestCurveforn=1 29

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p

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p niHhi(c) 32

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p e +2 22c++ for>0,6=0,andn1;andIn(c;;;)=ec niHhi(c) p e +2 22c for>0,<0,andn1

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p 35

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((t))1(A{1)Usingthefactthat(T) 221(Tt)+1(1)(W1(T)W1(t))N(T)Yi=N(t)+1eV1iE0@N(t)Yi=N(t)+1eV1i1A=1Xj=0e(Tt)[(Tt)]j 2211 221(1)2(1)1Notethatitimplies 221t+1(1)(W1(t))N(t)Yi=1eV1i=((0))1exp1 221(1)2(1)1t+1(1)(W1(t))N(t)Yi=1eV1i

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(Uc((t);t))=erTEZ(T)

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2=1(B{4)andasx!1 2dt2n 44

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niHhi(c)+ n+1 e +2 22c++ (B{7)(2)If<0and<0,thenforallx1 niHhi(c) n+1p e +2 22c (B{8)Toprovethisresult,Koumakesusofintegrationbypartstoobtain:Case1.>0and6=0.Since,foranyconstantandn0,exHhn(x)!0asx!1thanksto(B-4),integrationbypartsleadstoIn=1 Z1cexHhn1(c)dxInotherwords,wehavearecursion,forn0,In=(ec)Hhn(c)+( )In1withI1=p e +2 22c++

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iHhni(c+)+ n+1I1=ec niHhi(c+)+ n+1p e +2 22c++ wherethesumoveranemptysetisdenedtobezero.Case2.<0and<0.Inthiscase,wemustalsohave,forn0andanyconstant<0;exHhn(x)!0asx!1,thanksto(B-5).Usingintegrationbyparts,weagainhavethesamerecursion,forn0;In=(ec=)Hhn(c)+(=)In1,butwithadierentinitialconditionI1=p exp +2 Solvingityields(B-8),forn1.Finally,wesumthedoubleexponentialandthenormalrandomvariablesPropositionB.3.(Kou,2002)Supposef1;2;:::gisasequenceofi.i.d.exponentialrandomvariableswithrate>0,andZisanormalvariablewithdistributionN(0;2).Thenforeveryn1,wehave:(1)Thedensityfunctionsaregivenby: +(B{9) +(B{10) 46

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+dt

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p

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Abramowitz,M.,I.A.Stegun.1972.HandbookofMathematicalFunction.10thPrinting,U.S.NationalBureauofStandards,Washington,D.C. AitSahlia,F.,L.Imhof,T.L.Lai.2003.Fastandaccuratvaluationofamericanbarrieroptions.ComputationalFinance7129{145. AitSahlia,F.,L.Imhof,T.L.Lai.2004.Pricingandhedgingofamericanknock-inoptions.TheJournalofDerivatives1244{50. AitSahlia,F.,T.L.Lai.1998.Randomwalkdualityandthevaluationofdiscretelookbackoptions.AppliedMathematicalFinance5277{340. AitSahlia,F.,T.L.Lai.2001.Exerciseboundariesandecientapproximationstoamericanoptionpricesandhedgeparameters.JournalofComputationalFinance485{103. AitSahlia,F.,T.L.Lai.2007.Acanonicaloptimalstoppingproblemforamericanoptionsanditsnumericalsolution.JournalofRisk1085{100. AitSahlia,F.,A.Runnemo.2007.Acanonicaloptimalstoppingproblemforamericanoptionsunderadoubleexponentialjump-diusionmodel.JournalofRisk1085{100. Amin,K.1993.Jumpdiusionoptionvaluationindiscretetime.JournalofFinance481833{1863. Andersen,L.,J.Andreasen.2000.Volatilityskewsandextensionsofthelibormarketmodel.Appl.Math.Finance71{32. Barberis,N.,A.Schleifer,R.Vishny.1998.Amodelofinvestorsentiment.JournalFinancialEconomy49307{343. Barndor-Nielsen,O.E.,N.Shepard.2001.Non-gaussianornstein-uhlenbeckbasedmodelsandsomeofthierusesinnancialeconomics(withdiscussion).J.Roy.Statist.Soc.,Ser.B63167{241. Baroe-Adesi,G.,R.E.Whaley.1987.Ecientanalyticapproximationofamericanoptionvalues.JournalofFinance42301{320. Carr,P.,H.German,M.YorD.B.Madan.2003.Stochasticvolatilityforlevyprocesses.MathematicalFinance13345{382. Carr,P.,R.Jarrow,R.Myneni.1992.Alternativecharacterizationsofamercianputoptions. Due,D.,J.Pan,K.Singleton.2000.Transfromanalysisandoptionpricingforanejump-diuions.Econometrica681343{1376. 50

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Hull,J.C.2000.Options,Futures,andOtherDerivatives.PearsonEducation,Inc.,UpperSaddleRiver,N.J. Kou,S.G.2002.Ajump-diusionmodelforoptionpricing.ManagementScience481086{1101. Kou,S.G.,H.Wang.2003.Firstpassagetimesforajumpdiusionprocess.AdvancedAppliedProbability35504{531. Kou,S.G.,HuiWang.2004.Optionpricingunderadoubleexponentialjumpdiusionmodel.ManagementScience501178{1192. Lamberton,D.,B.Lapeyre.1997.IntroductionauCalculStochastiqueAppliquealaFinance.ellipses,32rueBargue,Paris. Lucas,R.E.1978.Assetpricesinanexchangeeconomy.Econometrica461429{1445. McKean,H.P.1965.Afreeboundaryproblemfortheheatequationarisingfromaprobleminmathematicaleconomics.IndustrialManagementReview633{39. Merton,R.C.1973.Thetheoryofrationaloptionpricing.BellJournalEcomonicManagementScience4141{183. Merton,R.C.1974.Onthepricingofcorporatedebt:Theriskstructureofinterestrate.JournalofFinance29449{469. Merton,R.C.1976.Optionpricingwhenunderlyingstockreturnsarediscountinuous.JournalFinancialEconomy3125{144. Naik,V.,M.Lee.1990.Generalequilibriumpricingofoptionsonthemarketportfoliowithdiscontinuousreturns.ReviewofFinancialStudies3493{521. Peskir,G.,A.Shiryaev.2006.OptimalStoppingandFree-BoundaryProblems.Birkhauser,Zurick,Germany. Pham,H.1997.Optimalstopping,freeboundary,andamericanoptioninajump-diusionmodel.AppliedMathematicsandOptimization35145{164. 51

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JeremyBerroswasborninCompiegne,France.Heearnedhisbachelor'sdegreeinMathematics,PhysicandEngineeringSciencefrom"PierreetMarieCurieCollege".In2008hebeganhisMasterofScienceintheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFlorida.Hereceivedhismaster'sdegreefromtheUniversityofFloridainthesummerof2009. 52



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PAGE 15

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PAGE 16

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