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8/11/2019 Appunti1
1/7
Finance
Optionpricin
g:theBlack&Scholesmodel
Ro
ber
toRen
`o
[email protected]
Dipar
timento
diEconom
iaPo
litica
,Un
iversi
t`adiSiena
Novem
ber
,20
02
p
.1/42
Introductiontofinance
FinanceisasubsetofEco
nomics,whichisquitepeculiarfortwo
reasons:
Alotofdataareavailable
......
.....
thanmodelsareverysophisticated
.
Novem
ber
,2002
p
.2/42
Introductio
ntofinance
Tradedassetsinfinancialmark
etsare:
stocks(equity)
bond
currencies(foreignexchange)
commodities:gold
,cotton,
etc.
derivatives
Novem
ber
,2002
p
.3/42
Derivatives
Derivativesarefinancialcontractswhosepayoffatfuturedates
dependsuponthevalueofanotherasset,calledunderlying
.
Themostheavilytrad
edassetsarefuturesandoptions.
Novem
ber
,20
02
p
.4/42
Futures
Afuturescontractgivesyoutheobligationtobuyanassetina
futuredate.Ininstitutiona
lmarkets,thecounterpartshavetopay
aguaranteewhensigning
thecontract.Theresnoinitialprice.
Futuresaremoreliquidthantheirunderlying.
Futuresaredifferentfrom
forwardbecauseofcontractual
differencesregardingguaranteesandthewaythepayoffis
delivered.
Novem
ber
,2002
p
.5/42
Options
Anoptionisafinancialcontractwhichprovidestherighttobuy
(sell)anassetinafuturedatea
tagivenprice.
Calloption:
C
T
S
T
K
Putoption:
P
T
K
S
T
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ber
,2002
p
.6/42
-
8/11/2019 Appunti1
2/7Absenceofar
bitrageandcompletenes
s
Weassumethatinthemarketisisnotpossibletorealizea
positivepayoffwithnorisk.
Suchacontractwouldbean
arbitragecontract.Th
isassumptioniscalledabsenceofarbitrage
.
Amarketissaidtobecompleteifitispossibletoreplicateany
derivativewithabuy-and-sellstrategyaccomplishedwiththe
underlying.
Novem
ber
,20
02
p
.7/42
Perf
ectmarket
Wewillworkinmarketin
which:
theresnoarbitrage
therearenotransactioncosts
assetscanbeshort-selledandboughtinthedesired
quantities
theresnodefaultrisk
Novem
ber
,2002
p
.8/42
MoneyMarketAccount
TheMoneyMarketaccountisamodelofarisklessasset,whose
payoffiscertain
dB
t
r
t
r
t
B
t
dt
shortrate,spotrate
Theshortrateisthestatevariableadoptedininterestratemodels.
Novem
ber
,2002
p
.9/42
MoneyMarketAccount
Ifr
t
iddeterministi
c,theevolutionofB
t
isdeterministic,the
solutionoftheordina
rydifferentialequationis
B
t
B
0
exp
t0
r
s
ds
Ifr
t
isconstant,itcoincideswiththeyieldtomaturity.
Novem
ber
,200
2
p.1
0/42
FIB30timeseries
Letslookthetimeseriesofanasset(FIB302000-2
001
,daily
data):
Novem
ber
,2002
p
.11/42
Asset
pricing
Thepriceofafinancialassetis
modeledviaastochastic
differentialequation:
dS
t
S
t
tS
t
S
dt
t
S
t
S
dW
t
drift
volatility
Novem
ber
,2002
p
.12/42
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8/11/2019 Appunti1
3/7
TheBlac
kandScholesmodel
IntheBlackandScholesmodeltherearetwoassetsSandB
,
whichfollowthefollowingdynamics:
dB
t
rB
t
dt
dS
t
S
t
dt
S
t
dW
t
r
areconsta
nts!
Sistheriskyasset,tipycallyastock.
Bistheriskless
asset.
TheBlackandScholesmodelcanbeusedtovaluederivatives.
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ber
,200
2
p.1
3/42
Contingen
tclaimvaluation
Letusconsidertheproble
mofpricingaderivativeissuedint
whichpromisesapayoffa
tT
tgivenby
S
T
.
Forexample
,fortheCalloption
S
T
S
T
K
.
Let
s
F
s
S
s
bethederivativepriceattimet
s
T.
Novem
ber
,2002
p
.14/42
Contingentclaimvaluation
Itoslemmaallowsustocomputetheevolutionof
t
S
d
t
t
t
dt
t
dW
t
t
Ft
SF
S
1 2
2S2
2F
S2
F
t
S
F
S
F
Novem
ber
,2002
p
.15/42
Portfolio
ConsideraportfoliocomposedbytheassetSandthederivative.
Let:
uSbethefractio
nofwealthinvestedinS.
ubethefractio
nofwealthinvestedin
Thevalueoftheportfolioisthengivenby:
V t
uS
t
VS
t
u
t
V
t
then:
dV
V uS
u
dt
V
uS
u
dW
t
IfuS
u
0the
portfolioisriskless.
Novem
ber
,200
2
p.1
6/42
Arbitr
ageandPDE
Nowweusetheassumptionofabsenceofarbitrage;sincethe
portfolioisriskless,
itsdrifthastobeequaltor.
uS
u
r
SolvingforuSandu
,kee
pinginmindthatuS
u
1wehave
Ft
rSFs
1 2
2S2
2F
S2
rF
0
F
T
S
S
T
WethenfoundaPDEforthepriceofanyderivative.
Novem
ber
,2002
p
.17/42
Arbitrage
Someconsiderations
Ourresultsdependsonly
ontheabsenceofarbitrage.Thisis
notastonishing,
sincethe
valuationequationforFneedsthe
dynamicsofStobecomp
uted
.
Thisresultisvalidonlyif
thederivativeisactivelytraded.
Thisresultisstillvalidif
anddependont
S.
Novem
ber
,2002
p
.18/42
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8/11/2019 Appunti1
4/7
RiskNeutralvaluation
Letslookforaprobabilisticinterpretationofthisresult
.
TheFeynman-Kacfo
rmulasaysthatthesolutionofthePDEis
givenby:
F
tS
t
er
T
t
EI
X
T
t
whereXisthesolutionoftheSDE:
d
X
rrXdt
XdW
t
X
t
S
t
TheonlydifferenceinthediffusionofXwithrespecttoSisthat
thedriftisgivenbyr
insteadof.
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ber
,200
2
p.1
9/42
RiskNeutralValuation
IfXwouldbethepriceof
anasset,thaninvestorsarerisk-neutral
Indeed
,investorwouldequallytradeX
,whichhasdriftrbutis
uncertain,
andB
,whichhasdriftrandnouncertainity.
Actually
,investorsarerisk-averse.
Theytakeriskypositionsonly
ifthempromiseanexcess
return.
Novem
ber
,2002
p
.20/42
Girsanovtheorem
DenotebyPtheprobabilityun
derwhichWt
isaBrownian
motion.
Girsanovtheoremsaysthatfor
anyprocess
t
,then
B
t
W
t
t 0
s
dsisaBrownianmotionunderthe
probabilityPwithdensity
L
t
exp
t0
s
dW
s
1 2
t0
s
2ds
withrespecttoP
.
Novem
ber
,2002
p
.21/42
R
iskPremium
Wethencall
theprobabilityunderwhichW
isaBrownian
motion.
Weinterpret
XastheassetSundertheprobability
.
iscalledriskneutralprobability.FollowingGirsanovs
theorem,
therelation
betweenW
andWis:
W
t
W
t
r
r
t
Riskpremium:exces
sreturnperriskunit
.
Riskismeasuredvia
volatility
.
Novem
ber
,200
2
p.2
2/42
RiskNeutralValuation
Then,
followingFeynman-Kacformula
,wecanwrite:
F
t
S
t
er
T
t
EI
S
T
t
Letusconsiderthediscou
ntedprice:
P
s
S s
F
T
S
T
er
T
s
Theaboveformulasaysth
at,
t
s
T:
P
t
EI
P
s
t
Suchaprocessiscalleda
martingale,and
isalsocalled
equivalentmartingalemea
sure
.
Novem
ber
,2002
p
.23/42
RiskNeutralProbabilityandarbitrage
TherelationwefoundfortheB
lackandScholesmodelismuch
moregeneral:
Fundamentaltheoremofmathematicalfinance:
Inthemarket,
therearenoarbitrageopportunitiesifandonlyif
thereexistsaprobability
wh
ichisequivalenttoPunderwhich
discountedpricesaremartingalesunder
.
Themarketiscompleteifandonlyif
isunique.
Theabovereasoningsshowtha
ttheBlackandScholesmodelis
completeandfreeofarbitrage.
Novem
ber
,2002
p
.24/42
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8/11/2019 Appunti1
5/7
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8/11/2019 Appunti1
6/7
The
hedgingstrategy
Sinceforthestrategy
u
u0theportfoliovalueisgivenby:
dV
t
V
t
u
0
t
r
u
t
dt
V
t
u
t
t
dW
t
wehave:
u
t
SF
SF
u0
t
Ft
1 2
2S2
2F
S2
rF
Ifweasku0
u
1wehave:
Ft
rSF
S
1 2
2S2
2F
S2
rF
0
whichisthePDEfor
theoptionprice!
Novem
ber
,200
2
p.3
1/42
Th
eGreeks
Thepricesensitivitywith
respecttotheparametersismeasured
bytheso-calledGreeks:
F
S
2F
S2
Fr
Ft
F
Novem
ber
,2002
p
.32/42
he
dging
Whenwehedgeanoptioninth
eBlackandScholesmodel,
we
have:
u
SF
thanisexactlythenumberofstocksSthatIhavetobuyfora
givennumberofoptionsF
,inordertohavearisklessportfolio.
Followingthat,
intheBlackan
dScholesmodelhedgingiscalled
hedging.
Novem
ber
,2002
p
.33/42
Blackand
Scholes:theempirics
TheBlackandScholesmodelisparamountlyused
,startingfrom
theearlySeventies.I
tisthenaturalbenchmarkforanyoption
pricingmodel.
Itowe
sitsfortunetoitssimplicity:itsthe
simplestdiffusionmo
delwithNormalreturns.
Nowweshouldasko
urselvesifitsassumptionarerespectedby
financialmarkets,andconsequentlyifitcanbeusedsafelyornot.
Novem
ber
,200
2
p.3
4/42
Areinterestratesconstant?
Interestratesvarythrough
time:
Novem
ber
,2002
p
.35/42
Isvolatilit
yconstant?
Volatilityisnotconstant.
Novem
ber
,2002
p
.36/42
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7/7
ArereturnsNormal?
Lookingatthetimes
eriesitseemsnot.
...
Novem
ber
,200
2
p.3
7/42
ArereturnsNormal?
Indeedtheyarenot,thedistributionisleptokurtic(fattails):
Novem
ber
,2002
p
.38/42
Arereturns
independent?
TheACFofreturnsisnullafteroneday!Thisiscoherentwith
theassumptionsofindependen
treturns,but.
..
Novem
ber
,2002
p
.39/42
Arereturnsindependent?
TheACFoftheabsolutevalueofreturnsshowsdependencefor
verylongperiods(1-2months).
Novem
ber
,200
2
p.4
0/42
Impliedvolatilityandmoneyness
TheBlackandScholesformulaimpliesthatimpliedvolatility
doesnotdependonthemo
neynessS
K.
Impliedvolatilityshouldb
ethasameforCalloptionatthemoney
(S
K
1),inthemoney(S
K
1),outofthemoney(S
K
1).
Sucharesultwouldbean
importantpointmadebytheBlackand
Scholesmodel,
irrespectiv
elyoftheempiricalfactswhichit
cannotaccountfor.
Novem
ber
,2002
p
.41/42
Smile
effect
Impliedvolatilitydependsonm
oneyness:
Novem
ber
,2002
p
.42/42
