Stochastic Calculus for Finance I: The Binomial Asset Pricing Model (Springer Finance)

200 Pages·2005·7.87 MB·English

by  Steven E. Shreve

SpringFeirn ance
EditorBioaalr d
M.A vellaneda
Barone-Adesi
G.
M.B roadie
M.H.AD.a vis
Derman
E.
Kliippelberg
C.
Kopp
E.
Schachermayer
W.
Springer
NewY ork
Berlin
Heidelberg
HongK ong
London
Milan
Paris
Tokyo
SpringFeirn ance
SprinFgienra nicsae p rogrammoefb ookasi meadts tudeanctasd,e miacnsd,
practitwioornkeironsng i ncreastiencghlnyi cal aptpotr hoaean cahleyossf i s
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StevEe.Sn h reve
Stochastic
Calcuflours
FinanIc e
TheB inomAisasle t
PricMiondge l
Wit3h3 F igures
Steven Shreve
E.
DepartmentM athematiSccaile nces
of
CarnegiMee lloUn niversity
PittsburPgAh .
15213
USA
[email protected]
Scanv ond er
DeutschFeinl iale
ders taatlichen
Bauerschaft
(KOLX03'a)
Mathematics Subject (C2l00a0s)6s:0i -f0i61c0,aH tIiOo6.n0 J659.1 828
LibraorfyC o ngreCsast aloging-in-PDuabtlai cation
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ISBN0 -387-401(00a·l8pk a.p er)
I.F inance-Mathemmaotdieclasl- Textbo2o.Sk tso.c haasntailcy sis­
TextbooksT.i tle.S pringer finance.
I. II.
HG 06.S5270 03
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332''.5019 22-dc22 2003063342
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ISBN0 -387-40100·8 Printoenda cid-fpraepee r.
2004 Springer-VNeerwlY aogr kL.L C
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To my students
Preface
Origionf T hisT ext
Thitse xhtase volfvreodmm a themcaotuircissnte hsMe as teorfS cieinnc e
ComputaFtniiaonn(caMelS CpFrogaratCm a rneMgeilelU onni veTrhsei ty.
contoeftn hti s hbaobsoe keu) ns esdu ccuelslwsyift h stwnhdoesmneat tsh ­
ematbiaccsk grcoounnsdoi fsc atlsc ulcuasl caunldu psr-obbaasbeTidhl ei ty.
texgitv perse csitsaet emoefrn estuslp tlsa,u siabrigluimteaynn etdsv ,es no me
proobfusmt,o riem protnat,li yntueei xtpilvandaetvieolnaosnp dre edfi ned
throculgashs reoxopme rwiietnthche mi ast earripear lo YiEdxeedr.cc iosne­s
cluedvee crhya pStoemroe.ft heesxet etnhdte h eoarnydo thearrdser awn
formp racticali nqp uraonbtliefitmnasat nicvee.
Thefi rstth rcehea ptoefVr osl uImh ea vbee euns eidna h a-lsfemester
courisnte h MeS CFp rog.rT ahmef uVloll uImh ea bse euns eidna full­
semesctoeurri snte hC ea rneMgeilelB oanc heplroorg'risanC m o mptuotinaal
FianncVeo.l uImIwe as develtoops eudp porth atlhfr-eseec moeusritsnee rs
the MpSrCoFg ram.
Dedication
Sinictiesn cepitn1i 99o4nt, h Cea rneMgeilelM oans ter's ipnCr oomgprua­m
tatiFoninaanlhc aegs r aduhautneddr oefsd tsu deTnhtess.e pwehoohp alvee,
comfer oamv arieetdyu coafta inodpn raeolsf siboancaklg rohuanvbdees e,n
aj otyo t ea.cT hhehya vbee en etaolg eearr n, asking sqtuiemsut­ions that
lattehdi nking, wtoour nkdienrgs tthhameana rdtd e rbioattlhh eoretically
anpdr actically, antdh eo fitneocnfal durdseiiqtoutineoo spntiMaicalnsn gy.
camfer otmh fien anicnedt ursayn,wd e rger aciinos uhsa rtihnekgin ro wledge
inw ays etnhhaatn tchecedl assroom experience for all.
Thitse xatn dm yo wns toorfke n owlehdagvbeee nefigtreeda ftrloym
MSCF I MSCF
interactwiiotnhts h e studenatnsd, contintuoel earfnr omt he
VIII Preface
alumInt ia.kt eh iosp porttuoen xiptryge rsast ittotu hdeess teu deanntds
fomresrt utdsbe ynd edictahtwiiosnr gtk ot hem.
Acknowledgments
Converswaittsiheo vnespr eaolpi lnec,l umdyci onllge agDuaevsHi eda atnhd
DmitKrrya omvk,h avien fluetnhciteesdx Ltu.k aKsrzu rke amdu cohft he
manuscarnpidrp otv induemde rcooumsm eanntcdso rrecOttihosentrsu .d ents
anfda uclthya vpeo tienodu etr rionra sns du ggeismtrpeodv emoefen atrsl ier
darftosft hwiosr Sko.m oeft heasreJc o natAhnadne rBsoognd,Da ony tchi­
novS,t even GSielaJlnoi nsAepnsai,te o,Kl airi okAl,n drKzreaju PseetL,ru k­
saSn,e rgMeyaygc hilNoivcR,ka is musIsseaSnao,cn Miasns,i mTaos s-aSnolet,
DavWihdi atkearn Udw eW ystIunps .o me casoefts h,e sues ederrasar fltise r
havseu ggeesxteerdco ires xeasm palnedts h,e ciorn tribaurtaeci konnosw l­
edgaetda ppropriaitnte h ep otienxtts. To alilnt htdehe ovseel ­who aided
opmeonftt h tiesx It ,am mo sgtr autle.f
Duritnhcger eaotfti hotines x ttha,eu thwoarps a rtsiuaplploybr ytt heed
NatioSncaileF nocuen dautnidoenr DgMrSa-n9t8s0 2D4M6S4-1,300 814a,n d
DMS-190391A1n.yo pinifionndsi,na gndsc ,on cluosrri eocnosm mendations
expreisnts hemidas t earriteah lo osfte h aeu thaonrdd o n onte cesrseaflreiclty
thvei eowfts h Neat ioSncaile Fnocuen dation.
PittsbPuerngnhas,ny liUvaS,A StevEe.Sn hr eve
Decem2b0e0r3
Contents
TheB inomNio-aAlr bitrPargiec iMnogd el. . . . .1 . . . . . . . . . .
1 .
1.1On e-PeriBoidn omiMaold el . . . . . . . . 1. . . . . . . . . . . . . . . . . . . . .
.
1.2Mu ltiperBiiondo miMaold el. . . . . . . . 8. . . . . . . . . . . . . . . . . . . . .
.
1.3Co mputatioCnoanls iderat.i on.s . . . . . .1 5. . . . . . . . . . . . . . . . . . . .
1.4Su mmar.y . . . . . . . . . . . .1 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5No tes. . . . . . . . . . . . . 2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
1.6Ex ercis.e s . . . . . . . . . . . 2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
2 ProbabiTlhietoyro ynC oiTno sSsp ac.e . . ... .... ... . . . .. 2.5
.
2.1Fi niPtreo babilSiptayc es. . . . . . . . .2 5. . . . . . . . . . . . . . . . . . . . . . .
.
2.2Ra ndomV ariablDeiss,t ributainodnE sx,p ectation.s . 27
... ... ..
2.3C onditioEnxaple ctat.i.o n.s . . . . .. .. .. .. .. .. . . . . . . . . . . . . . 31
.
2.4M artingal.e s. . .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . 36
.
2.5M arkovP rocess.e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6S ummary. . . . . . . . . . . . 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
2. N7o tes. . . . .. .. .. .. .. .. .. .. ..5 4.. .. .. .. . . . . . . . . . . . . . . . . . . .
.
2.8E xercis.es . .. .. . . ... .. .. . . ... . .. .. . ... . . ... . . ... .. ... . 5 .4
.
3 StatPer ices 61
...............................................
3.1Ch angoef M easure. . 61
. ...................................
3.2R adon-NikodDyemr ivatPirvoec es.s . ... .... .. .. .. . .. . .. .. 6 5.
3.3C apitAasls ePtr iciMnogd el . .. . . .. .. . . .. . .. . .. .. .. ... ..7 0.
.
3.4S ummary. . . . . . ... ... . . . .. . ... ... . .. ... . . . .. ... . ... 8.0. . .
.
3.5N ote.s ... . . .. . ... .... . .... . ... . . . . ... .... . . .. . . .. .. 8.3 . . .
3.6E xerci.s.e s . .. .. .... ..... ..... . .. . . ... .. . . ... . . . .. . .. 8..3 .
4 AmericDaenr ivaSteicveu ri.t i.e s. . . .. .. ..8 9.. .. .. .. .. .. . . . .
.
4.1In troducti.on. .... 89
.. .. . .. ...............................
4.2No n-Path-DepenAdmeenrti caDne rivati.v..e.s . 90
............
4.3S toppiTnigm es . . . . . . . . . . . 96. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
4.4Ge nerAamle ricaDne rivati.ve.s ..... .101
... ........... . ..... .
Contents
X
4.5A mericaCna lOlp tio.ns. ................1.1..1. .............
4.6Su n1ma.ry. ......................1.1..3. ...................
47. Note.s .........................1.1.5. .....................
4.8Ex erci.ses . . .. . . . . .1 15
.... ..... .. ..... ... ........... ... ....
5 Random Wal.k. . .. 119
... .. .. ................... ..............
5.1In troduction. . . . . .1 19
...... ............. ............... ....
5.2Fi rsPta ssage Times. . . . .. 120
..... ... ..... ..... .... .........
5.3R eflectPiroinn ci.p.le. A..n... ............1.2.7. ...............
5.4Pe rpetuAamle ricaPnu t: Example . . 129
...................
5.5Su mmar.y. ......................1..3.6. ...................
5.6No te.s .........................1.3.8. .....................
5.7Ex erci.se.s. .....................1.3..8. ...................
Intetr-eRsate-DepAesnsdee.tn.st . .... . 143
6 .... ..... ... ..... ..
6.1In troduc.ti.on. . .......143
. ... ...................... .. . .. . .
6.2Bi nomiMaold elf orI ntereRsatt e.s ............1.4..4. ........
6.3F ixed-IncoDmeer ivatives . .. . 145
............. ................
6.4Fo rwarMde asur.es. ..................1.6.0.. ...............
6.5Fu ture.s. .......................1.6.8. ....................
6.6Su mmary .. . . . .... .173
... ....................... .... . ..... .
6.7l'\ ot.es. ..............................1.7.4. ...............
6.8Ex ercises .. . . .. ..147
...... . . . . ..... .................... ... .
ProooffFu ndamentParlo perotfi es
ConditiEoxnpaelc tatio.ns . 177
..... ...........................
Refere.n.ces. . . . ..181
.... ........................................ .
Index 185
..........................................................
Introduction
Background
Bya wardHianrgrM ya rkowWiitlzl,Si ahma rapnedM, e rtMoinl lteh1re9 90
NobePlr iiznEe c onomtihcNeso ,b Perli Czoem mitbtreeo utgohw to rldwide
attentthifeoa nct th atth per evifoouryste ya hrasds eetnh eem ergeonfc e
an ews cientific tdhi"est chieopoflrfi iynn aen,cT eh.it"sh eory attempts to
undershtoawfin nda ncialw moarhrkok,wte otm sa kteh emmo reeffi cieanntd,
howt hey shoruelgdu lIabttee e xdp.l aainnds enhances trhoel e important
themsaer kpeltasiy nc apital aalnldro icsraket diuocntto if oaicnl iteacto­e
nomiacc tivWiittyh.ol uots iitnasgp plictaopt riaocntas ipceacoltf ts r ading
andr egulatthiteoh ne,oo rffiy n anhcaseb ecoimnec reasmiantghleym atical,
tot hpeo itnhtap tr oblienfim nsa nacrene o wd rivriensge ianrm caht hematics.
HarrMya rkow1it9z5P'2hs . Dt.h ePsoifrsot lSieol ecltaiitodhgn er ound­
worfko trh mea thematthiecooarflfiy n anMcaer.k owdievtezl oapn eodt ion
ofm earne tuarnndc ovarifaoncrco emsm osnt octkhsaa tl lohwiemtd o q uan­
titfyh ceo ncoefp" td iversiifinac m aatrikoHenet"s . h owheodwt oc ompute
thmee arne tuarnndv ariafron cgeai vpeonr tfaonldai rog utehdai tn vestors
shouhlodlo dn ltyh opsoero tlfiwohso vsaer iainsmci en imaamlo naglp lo rtfo­
liwoist ahg ivmeena rne tuArlnt.h otuhgleha nguoaffig nea nncoewi nvolves
stoch(aIsttcoia)cl cumlaunsa,g emoefrn itsi kna q uantimfiaanbnlieestr h e
underltyhienmogeft hmeo dern atnhdep orrayc otfiq cuea ntifitnaatnicvee.
In1 699R,o beMretr tionnt rodsutcoecdh casatlicciu nltutosh set uodfy
finanMceer.t woasn m otivabtyte hdde e sitroue n dershtoawpn rdi caerse
seitn fi nancial marikste htcesl ,as swihciaclh econoomf"i ecqsu iq­uestion
libriaunmdi, nl" a tpearp ehreus s etdh mea chinoefsr tyo chcasatlicctu ol us
begiinnv estiogfta htiiisso snu e.
Att hsea mtei mase M ertowno'rsak n dw itMhe rtoasns'iss taFnicse­,
cheBrl aacnkdM yroSnc howleerdsee velotphiecniegrl eborpattiepodrn i cing
fromulTah.iw so rwko nt h1e 979N obePlr iizneE conomIitpc rso.v iad ed
satissfyoilnugtt ioa onin m portparnatc tpircoabllt ehmao,tf fi ndianf ga ir
prifcoear E uropceaaolnpl t i(oin.t eh.re,i gthobt u yo nseh aorfea g iven