oBinomialrees金融工程华东师范大学汤银才

Introductionto

BinomialTrees

Chapter91Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullASimpleBinomialModel

ofStockPriceMovementsInabinomialmodel,thestockprice

attheBEGINNINGofaperiod

canleadtoonly2stockprices

attheENDofthatperiod2Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullOptionPricing

BasedontheAssumptionofNoArbitrageOpportunities

Procedures:

EstablishaportfolioofstockandoptionValuethePortfolionoarbitrageopportunitiesnouncertaintyatmaturitynoriskwiththeportfolio

risk-freeinterestearnedValuetheoptionRisk-freeinterest=valueofportfoliotoday3Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullASimpleBinomialModel:

Example

Astockpriceiscurrently$20Inthreemonthsitwillbeeither$22or$18StockPrice=$22StockPrice=$18Stockprice=$204Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullStockPrice=$22OptionPrice=$1StockPrice=$18OptionPrice=$0Stockprice=$20OptionPrice=?ACallOptionA3-monthcalloptiononthestockhasastrikepriceof$21.Figure9.1(P.202)5Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullConsiderthePortfolio: LONGDshares SHORT1calloption Figure9.1becomes Portfolioisrisklesswhen22D–1=18DorD=0.2522D–118DSettingUpaRisklessPortfolioS0=206Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullValuingthePortfolio

(withRisk-FreeRate12%)Therisklessportfoliois:LONG

0.25shares

SHORT1call

optionThevalueoftheportfolioin3monthsis

22*0.25-1=4.50=18*0.25Thevalueoftheportfoliotodayis

4.50e-0.12*0.25=4.36707Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullValuingtheOptionTheportfoliothatis: LONG

0.25shares

SHORT1call

option

isworth4.367Thevalueofthesharesis

5.000=0.25*20Thevalueoftheoptionistherefore

0.633=5.000-4.3678Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullGeneralizationConsideraderivative

thatlastsfortimeTand

thatisdependentonastockFigure9.2(P.203)S0uƒuS0dƒdS0ƒ9Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullGeneralization(continued)Considertheportfoliothatis: LONG

shares

SHORT1derivative

Figure9.2becomes TheportfolioisrisklesswhenS0uD–ƒu=S0d

D–ƒd

orwhenS0uD–ƒuS0

dD–ƒdS0-f10Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullGeneralization(continued)ValueoftheportfolioattimeT

is

S0u

D–ƒuValueoftheportfoliotodayis

(S0u

D–ƒu)e–rTAnotherexpressionfortheportfoliovaluetodayis

S0

D–fHence,

ƒ=S0

D–(S0u

D–ƒu

)e–rT

11Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullGeneralization

(continued)SubstitutingforDweobtainƒ=[pƒu+(1–p)ƒd]e–rT

where12Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullGeneralization(continued):

ProofwithanExampleThisisknownastheNoArbitragemethodologyInourearlierexamplef=0.633and=0.25Iff<0.633,e.g.f=0.60

==>

S0-f=0.25*20-0.6=4.4>4.367

t=0 ST=18ST=22

Buycall -0.600 0 1

Sell

Shares 5.000-18*0.25=-4.50-22*0.25=-5.50

Lend4.367atr -4.3674.50

4.50

NetFlows 0.033

00•13Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullGeneralization(continued):

ProofwithanExampleIff>0.633,e.g.f=0.65

==>

S0-f=0.25*20-0.65=4.35<4.367

t=0 ST=18 ST=22

Buy

Shares -5.00018*0.25=4.5022*0.25=5.50

Borrow4.367atr 4.367-4.50

-4.50

Sellcall 0.650

0 -1

NetFlows 0.017

00•14Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullIrrelevanceofaStock’s

ExpectedReturnWhenwearevaluinganoptionintermsofthe underlyingstocktheexpectedreturnonthe stockisirrelevantThisisbecauseinourformula

f=S0-(S0u-fu)

e-rT

fdoesnotinvolvetheprobabilityofthestock movingupordownItdoesnotmatterifwesaytheprobabilityofan

increaseis50%or80%wegetthesameresult•15Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullIrrelevanceofaStock’sE(R)

Proof:(continued)Let’scallputheprobabilityofanincreaseinthestockpriceandpd=1-pu

theprobabilityofastockdecrease

S0-f=[pu(S0u-fu)+pd(S0d-fd)]e-kT

wherekistheappropriateratefortheriskinvolvedHowever,ischosensuchthat

S0u-fu=S0d-fdandweknowthatpd=1-puSubstituting,

S0-f=[pu(S0u-fu)+(1-pu)(S0u-fu)]e-kT

=(S0u-fu)e-rT

assincethisisrisk-free,k=rNopu’sorpd’sleft,thusprobabilityofstockincreaseisirrelevant•16Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullIrrelevanceofaStock’sE(R)

(continued)TheprobabilityofanincreaseinthestockpriceisirrelevantbecauseoptionsareredundantsecuritiesInourtwo-stepmodels,weformarisk-lessportfoliowithstockandtheoptionThus,thereturn/pay-offfromtheoptionisoffsetbythereturnonthestockandtheportfolioreturnisthesameinbothstatesThus,nomatterwhattheprobabilityofastockincrease,theansweristhesame•17Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullPseudo-ProbabilitiesHowever,wecansolveforanimpliedorpseudo-probabilitythatwillgiveusthecorrectanswerUsing

inf=S0-(S0u-fu)

e-rTweobtain

where

•18Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullStockExpectedReturnusing

Pseudo-Probabilities

Whatisthestock’sexpectedreturnusingthesepseudo-probabilities?

Thus,theexpectedreturnonthestockisthe

risk-freerateeventhoughthereturnhassome variance.Hence,weareinarisk-neutralworld.

•19Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullRisk-NeutralValuationThevariabledoesNOTappearinthesolutionto

theoptionvalueusingtheBinomialMethodThus,thesolutionisindependentofallvariables

affectedbyriskpreferenceThesolutionisthereforethesameinarisk-free

worldasitisintherealworldHence,wecanassumethattheworldisrisk-

neutralThisleadstotheprincipleofrisk-neutralvaluation•20Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullRis嘉k-N及eut巧ral剧Va钓lua休tio想n(Co循nti颠nue闹d)ƒ=兽[pƒu+(1蒙–p)ƒd]e-rTThe令va炎ria改ble卵spand锻(1–p)c授an筹be继int蒙erp矿ret础ed之as响therisk碍-neu构tral阴pro叫babi扁liti旬esof荷up糖an粗dd帅own梢mo晒vem挠ent铅sThe吧va借lue蜘of运a五der成iva到tiv售ei约si肥ts宅exp值ect涛ed辫pay厕off键in信arisk跟-neu两tral坊wor叉lddis缺cou乘nte辟da顿tt蕉herisk蜂-fre卵era炊teFig绝ure吴9.皮2b芦eco座mesS0uƒuS0dƒdS0ƒp1-p21Opt墨ion谱s,咽Fut况ure伸s,议and遗Ot杏her宾De径riv冷ati娘ves,4t停hed男itio莫n©射2000浸by幼Joh晨nC.索Hul伤lOri系gin伤al尚Exa稼mpl直e钱Rev地isi崭tedFig冤ure巷9.践2b唤eco伐mesSinc丛episa存ris详k-ne轮utra胃lpr盈obab迅ilit领y枝20e0.1纸2*0曾.25=2才2p+18健(1-p)p=0级.65赤23管Alt宰ern和ati舒vel仰y,纯we求can丢us贼et蛮he准for盯mul副aS0u=22ƒu=1S0d=18ƒd=0S0ƒp(1–p)22Opti标ons,洋Fut炸ures箩,an知dOt薪her领Deri解vati驴ves,4t伏hed愤itio鸽n©杏2000军by洗Joh氏nC.胳Hul叉lOrig暗inalExa僚mpl鹿eRev身isi良tedFigu奋re9各.2航bec闲omesS0u=洒22似fu=1锡S0=20歌f字S0d=离18混fd=0Theval宴ueoft渡heopt卫ionis疾e-0.1缺2*0.娇25[0.6牌523*晃1+珍0.34阔77*0夏]=疾0.63悬30.65230.3477•23Opt出ion故s,捡Fut笛ure吼s,肾and烦Ot列her想De稳riv鸦ati科ves,4质th荐edi钟tio宗n©过20意00熔by夹Jo糕hn稼C.甘Hul员lATw茎o-St令epE明xamp狂le20221824.219.816.2Not迫et衔he蜜cha衬nge烫in老th邻ef常orm颗ula弄fo吸rpwith找am怨ulti售-ste侦ptr击eeFigu浪re9袜.3Each骗tim莲est钉epi开s3劲mont江hs,让ris陷sti攀ll浆12%.24Opti吉ons,斤Fut含ures纸,an婚dOt林her莲Deri昼vati就ves,4t劳hed碍itio荡n©劣2000共by叹Joh惹nC.餐Hul胶lValu东ing攻aC驴all甚Opti滋onFigu喝re9热.4,阶X=2氧1Valu恨eat蒸nod摄eB龄=e-0.1储2*0.委25(0.宪652栽3*3溉.2住+0穿.34壁77*连0)疼=2背.02谦57Valu敢eat欺nod锣eA咬=e-0.泼12*泉0.2花5(0.6肝523*连2.02挪57+削0.3宰477*唇0)=惨1.2要823201.28念23221824.谊23.219.80.016.20.02.0韵2570.0ABCDEF25Opti请ons,准Fut齿ures粘,an勒dOt抛her克Deri测vati胆ves,4涉th姨edi恋tio漂n©楼20叨00甘by控Jo陆hn适C.凤Hul购lGene睬rali纵zati吨onFig约ure窑9.刑6(获P.2唇09)S0fS0ufuS0dfdS0u2fuuS0d2fddS0udfud26Opt劈燕ion辅s,蒙Fut鲁ure凯s,冷and叮Ot药her便De些riv沈ati以ves,4t河hed黎itio到n©新2000杠by磨Joh厦nC.战Hul谊lAPut量Opti星onExam呈pleFig描ure谱10衡.7罢,X=52剩,u=1.交2,d=0.双8,r=5贯%,衔andT=2f=e-0.域05*晶2(0.替628滨22*0+缺2*0护.62呈82*中0.3包718域*4+辜0.3耽7182*20笔)=4困.19移237204843220601.414740

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