Nume ical In es iga ion o
Black-Scholes Equa ion
A disse a ion submi ed o he Uni e si y o Dhaka in
pa ial ul illmen o he equi emen s o he deg ee o
Mas e s o Science in Ma hema ics
SUBMITTED BY
Examina ion Roll No. :2804
M.S. Session :2017-2018
Regis a ion No. :2013-512-745
Depa men o Ma hema ics
Facul y o Science
Uni e si y o Dhaka
Janua y, 2020
Acknowledgemen
I am p i ileged o ha e he oppo uni y o con ey my espec and since e g a i ude o
my supe iso P o esso D . Sami Kuma Bhowmik, Depa men o Ma hema ics,
Uni e si y o Dhaka, whose encou aging guidance, inspi a ion and con idence in me
ha e been my bes mo i a ion o accomplish his pape success ully.
I would like o pay my espec and p o ound g a e ulness o Jakobin Alam Khan,
MS: 2016-2017, Depa men o Ma hema ics, whose kind guidance and bene olen
assis ance ha e helped me o be capable o composing his pape success ully.
I con ey my keen ecogni ion and deep indeb edness o all academic and admin-
is a i e s a s o Depa men o Ma hema ics, Uni e si y o Dhaka o hei co dial
assis ance. Finally, I am g a e ul and hank ul o my belo ed pa en s, b o he , iends
and all he well- wishe s o hei inspi a ion and ai h in me, which made i possible
o me o accomplish his a .
i
Abs ac
Black-Scholes Equa ion p o ides a heo e ical es ima e o he p ice o op ions. In
his hesis wo k, we conside bo h Linea and Nonlinea Black-Scholes Model o
he Eu opean op ion p ice. We analy ically app oach he Black-Scholes Equa ion
by he ans o ma ion o he p oblem in o a o wa d con ec ion-di usion equa ion
wi h linea and nonlinea e m. We use Spec al Colloca ion Me hod and Di e en ial
Quad a u e Me hod o Linea Black-Scholes Equa ion and compa e he nume ical
esul s wi h he exac solu ion o he equa ion. Then he Nonlinea Black-Scholes
Equa ion has been in es iga ed. He e, di e en ola ili y models due o he p esence
o ansac ion cos s has been conside ed. We use Fini e-Di e ence-Me hod o sol e
he Nonlinea Black-Scholes Equa ion. Finally we p esen he nume ical esul s o he
Nonlinea Black-Scholes Equa ion. We use MATLAB o nume ical implemen a ion
and LATEX has been used o he w i e up.
ii
Con en s
In oduc ion iii
1
1 Black Scholes Equa ion 1
1.1 In oduc ion................................ 1
1.2 FinancialDe i a i es........................... 1
1.3 Op ionP ice................................ 2
1.3.1 Classi ica ion o Op ion P ice . . . . . . . . . . . . . . . . . . 3
1.3.2 Eu opeanOp ion......................... 4
1.4 Black-Scholes Equa ion . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Limi a ions o Black-Scholes Equa ion . . . . . . . . . . . . . . . . . 5
1.6 De i a ion o Black Scholes Equa ion By Hedging a gumen . . . . . 6
1.7 De i a ion o Black-Scholes Equa ion by Binomial Model . . . . . . . 8
1.8 Pa i yo Pu -Call............................. 9
1.9 Sensi i i yAnalysis............................ 9
2 Linea Black-Scholes equa ion’s Analy ic Solu ion 12
2.1 In oduc ion................................ 12
2.2 Black Scholes Equa ion’s Linea Model . . . . . . . . . . . . . . . . . 12
2.3 Analy icalSolu ion............................ 14
2.4 G aphical Solu ion o he Black-Scholes Equa ion . . . . . . . . . . . 19
3 Linea Black-Scholes Equa ion wi h Spec al Me hod 22
3.1 In oduc ion................................ 22
iii
3.2 Spec al Colloca ion Me hod . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Chebyshe Di e en ia ion Ma ix . . . . . . . . . . . . . . . . . . . . 23
3.4 MATLAB unc ion : ’cheb’ . . . . . . . . . . . . . . . . . . . . . . . . 26
3.5 Nume ical Solu ion o Black-Scholes Equa ion wi h he help o Spec al
Me hod .................................. 29
3.6 S abili y and Con e gence o Spec al Me hod . . . . . . . . . . . . . 31
3.7 Conclusion................................. 32
4 Di e en ial Quad a u e Me hod o Linea Black-Scholes Equa ion 33
4.1 In oduc ion................................ 33
4.2 Di e en ial Quad a u e Me hod (DQM) . . . . . . . . . . . . . . . . 33
4.3 Disc e iza ion o domain and S abili y . . . . . . . . . . . . . . . . . 34
4.4 De i a i es o Space by using Di e en ial Quad a u e Me hod . . . . 35
4.5 Ma lab unc ion : Di e en ial Quad a u e Ma ix . . . . . . . . . . . 37
4.6 Nume ically sol ing Black-Scholes PDE using Di e en ial Quad a u e
Me hod .................................. 39
4.7 Conclusion................................. 41
5 Nonlinea Black Scholes Equa ion and Vola ili y Models 42
5.1 In oduc ion................................ 42
5.2 Nonlinea Black Scholes Equa ion . . . . . . . . . . . . . . . . . . . . 42
5.3 Bounda yCondi ions .......................... 43
5.3.1 Eu opean Call Op ion . . . . . . . . . . . . . . . . . . . . . . 43
5.3.2 Eu opean Pu Op ion . . . . . . . . . . . . . . . . . . . . . . . 43
5.4 Vola ili yModels ............................. 43
5.5 Leland’s Model o Vola ili y . . . . . . . . . . . . . . . . . . . . . . . 45
5.6 Boyle and Vo s Vola ili y Model . . . . . . . . . . . . . . . . . . . . 47
5.7 Pa as and A ellaneda Vola ili y Model . . . . . . . . . . . . . . . . . 47
5.8 Hodges and Neube ge Vola ili y Model . . . . . . . . . . . . . . . . . 47
5.9 Ba les’ and Sone Model . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.10 Risk adjus ed P icing Vola ili y Model . . . . . . . . . . . . . . . . . 51
i
5.11Analy icalSolu ion............................ 52
5.12Conclusion................................. 53
6 Nume ical In es iga ion o Nonlinea Black-Scholes Equa ion 54
6.1 Eu opeanCallop ion........................... 55
6.2 Fini e-Di e ence Me hods . . . . . . . . . . . . . . . . . . . . . . . . 55
6.3 G id .................................... 56
6.4 Di e ence Quo ien s . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.5 Vola ili yModels ............................. 59
6.6 Exis ence and Con e gence . . . . . . . . . . . . . . . . . . . . . . . . 60
6.7 Explici Me hod(Fo wa d-Time Cen al Space) . . . . . . . . . . . . . 63
6.8 Implici Me hod (Backwa d Time-Cen al Time) . . . . . . . . . . . . 65
6.9 C ank-Nicolson .............................. 66
6.10 G aphical Rep esen a ion . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.11Conclusion................................. 73
7 F ac ional Black-Scholes Equa ion 74
7.1 In oduc ion................................ 74
7.2 T ans o med Black-Scholes Equa ion . . . . . . . . . . . . . . . . . . 74
7.3 F ac ional De i a i e . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.4 F ac ional Black-Scholes Equa ion o Time . . . . . . . . . . . . . . . 75
7.5 F ac ional Black-Scholes Equa ion o Space . . . . . . . . . . . . . . . 77
7.6 Conclusion................................. 79
8 Conclusion 80
Bibliog aphy 81
Lis o Figu es
1.1 Eu opean Call Op ion P ice V(S, ) a s ike p ice K= 100 . . . . . . 4
1.2 Eu opean Pu Op ion P ice V(S, ) a s ike p ice K= 100 . . . . . . 5
2.1 Value o he Eu opean call op ion V(S, ) a ime, = 0, =.2, =.4,
=.6, =.8 and =1........................... 20
2.2 Value o he Eu opean call op ion V(S, ) a a ious imes in a su ace
plo .................................... 20
3.1 Eu opean Call Op ion P ice V(S, ) using Spec al Colloca ion Me hod. 30
3.2 The compa ison o Eu opean Call Op ion P ice V(S, ) wi h he Exac
solu ion................................... 30
4.1 Eu opean Call Op ion P ice V(S, ) using Di e en ial Quad a u e Me hod. 40
4.2 Eu opean Call Op ion P ice V(S, ) is compa ed wi h he Exac solu ion. 40
6.1 Value o Eu opean Call Op ion p ice V(S, ) o Nonlinea Black-Scholes
models o a ious ola ili y models o spa ial s ep h=.1 and ime
s ep k=.0001 wi h he C ank-Nicolson Me hod. . . . . . . . . . . . . 68
6.2 The impac o ansac ion cos s Nonlinea Vnonlinea (S, )−Vlinea (S, )
wi h he C ank-Nicolson Me hod. . . . . . . . . . . . . . . . . . . . . 69
6.3 Value o Eu opean Call Op ion p ice V(S, 0) o di e en ola ili y
models s he linea model wi h he C ank-Nicolson Me hod. . . . . . 70
6.4 Value o Eu opean Call Op ion p ice V(S, ) o Nonlinea Black-Scholes
models o a ious ola ili y models o spa ial s ep h=.1 and ime
s ep k=.001 wi h he Implici Me hod. . . . . . . . . . . . . . . . . 71
i
6.5 The Iden i y model ψ(x) = xwi h pa ame e (a=.02) o he (Vnonlinea (S, )−
Vlinea (S, )) plo o spa ial s ep h=.1 and ime s ep k=.001 wi h
heImplici Me hod. ........................... 71
6.6 Value o Eu opean Call Op ion p ice V(S, ) o Nonlinea Black-Scholes
models o a ious ola ili y models o spa ial s ep h=.1 and ime
s ep k=.00001 wi h he Explici Me hod. . . . . . . . . . . . . . . . 72
6.7 The Iden i y model ψ(x) = xwi h pa ame e (a=.02) o he (Vnonlinea (S, )−
Vlinea (S, )) plo o spa ial s ep h=.1 and ime s ep k=.001 wi h
heExplici Me hod. ........................... 72
ii
In oduc ion
Du ing he las se e al decades he ading o op ions and s ocks has encoun e ed a
g owing in e es in bo h scien i ic wo k and e e yday li e. Fische Black and My on
Scholes published a pape i led ’The P icing o Op ions and Co po a e Liabili ies’
in 1973. They showed how he p icing o op ions is uniquely de e mined by hei o -
mula. A e hei publica ion o his pape , he o mula was named a e hem. The
a bi age easoning me hod was used in hei pape which was de eloped by Robe
Me on. They we e gi en he Nobel p ize in 1997 o his Op ion P icing model.
This Black-Scholes Model (BSM) was he i s model o p icing op ions. A e his
model, he e we e mo e models de eloped by he ma hema icians o es ima e s ock
op ion p ice. The e a e some limi a ions o he BSM. Tha is why he classical model
some imes di e wi h he ealis ic p ice o he op ions [3, 44]. The e had been nume -
ous a emp s on imp o ing he BSM [28, 35]. In he Linea BSM, he ola ili y ( he
s anda d de ia ion o he s ock p ices) is assumed o be cons an . This assump ion is
no ealis ic. Tha is why in he Nonlinea Black-Scholes Model (NBSM), he ola il-
i y e m is no cons an . The e o e, NBSM is less e o p one han he linea one.
The e ha e been a ious schemes used o sol e BSM. I one wan s o sol e an ODE o
PDE o high accu acy on a simple domain, and i he da a de ining he p oblem a e
smoo h, hen spec al me hods a e usually he bes ool [67]. This me hod has he
abili y o achie e en digi s o accu acy whe e a ini e di e ence scheme would ge
wo o h ee. This me hod demands less compu e memo y han he al e na i es and
was de eloped by S e en O szag in 1969. Lloyd N. T e e hen’s book named ”Spec al
Me hods in MATLAB” p o ides undamen al ideas and echniques o spec al me h-
ods. This book o e s a ema kable ange o ODE and PDE p oblems which can be
iii
1.3.2 Eu opean Op ion
Eu opean Call Op ion is a con ac be ween wo pa ies whe e a a ixed ime
in he u u e, called expi a ion da e o ma u i y T, he possesso o he op ion may
buy an asse known as he unde lying asse S( ), o a ixed p ice called he s ike o
exe cise p ice K. The o he pa y has he ag eemen o sell he asse i he possesso
wishes o buy i .
We ha e al eady discussed abou he h ee cases o Op ions. F om hem we can
easily deduce ha he alue o he Eu opean Call op ion a expi y, called he pay-o
unc ion:
V(S, T) = max(S−K, 0).
Figu e 1.1: Eu opean Call Op ion P ice V(S, ) a s ike p ice K= 100
Eu opean Pu Op ion is he au ho i y o sell he unde lying asse S( ) a he
ma u i y da e T o he s ike p ice K.
The pay-o unc ion o Eu opean Pu Op ion is:
V(S, T) = max(K−S, 0).
4
Figu e 1.2: Eu opean Pu Op ion P ice V(S, ) a s ike p ice K= 100
1.4 Black-Scholes Equa ion
In he Incep ion o he 1970’s Fishe Black, My on Scholes and Robe Me on de el-
oped he classical Black-Scholes Equa ion(PDE). I was a quin essen ial ad ancemen
in he ield o quan i a i e inance. The e a e wo classes o Black-Scholes PDE
which a e Linea and Nonlinea . Classical model always e e s o linea Black-Scholes
Equa ion.The Black-Scholes PDE is dependen on wo independen a iables which
a e ime and S ock p ice S( ) . The Black-Scholes PDE is o he pa abolic o m. I
is called he backwa d pa abolic o m because o he p esence o ha ing he opposi e
sign on he second de i a i e. The e a e some assump ions in ol ed in Black-Scholes
PDE. As well as , he e a e some limi a ions in his op ion p icing model.
In his wo k, we educe he Black-Scholes PDE in o hea equa ion. Then we use
nume ical schemes o sol e i .
1.5 Limi a ions o Black-Scholes Equa ion
The e a e some limi a ions o Black-Scholes model because o some assump ions
[4,7]:
•Th oughou he op ion’s li e ime, he s ock pays no di idends.
Bu in eali y mos o he en e p ises gi e he paymen o di idends o hei
sha e holde . The e o e, his is a signi ican cons ain o he model.
5
•The e a e no commissions cha ged in his model.
Usually he pa icipan s o he ma ke ha e o pay a commision o buy o sell
op ions. E en loo ade s ha e o pay a small ee. The commission paid by he
indi idual ade ’s is mo e signi ican which can some imes de o m he ou come
o he Black-Scholes Model.
•In e es a e and ola ili y a e known and cons an .
•The ading o asse s is a con inous p ocess.
•T ansac ion cos s a e igno ed.
•No mal dis ibu ion is ollowed o unde lying asse p ice.
•I assumes ha op ions has he simila p ope ies o he Eu opean op ion p ice.
1.6 De i a ion o Black Scholes Equa ion By Hedg-
ing a gumen
We gene ally p esume ha he asse p ice S( ) always ollows a Geome ic B owninan
mo ion:
dS =µSd +σSdW.
o dS =S(µd +σdW).
o dS
S=µd +σdW.
whe e Wis he s ochas ic a iable. Wand dW ope a e as unp edic abili y in s ock
ma ke . And µd wo ks as he expec ed alue whe eas σ2d is he a iance.
F om I o’s Lemma, we can w i e:
dV =∂V
∂ +1
2σ2S2∂2V
∂S2+µS ∂V
∂S d +σS ∂V
∂S dW.
We can ew i e his as:
∆V=∂V
∂ +1
2σ2S2∂2V
∂S2+µS ∂V
∂S ∆ +σS ∂V
∂S ∆W.
6
We can ew i e dS =µSd +σSdW his as:
∆S=µS∆ +σS∆W
Then we a e going o cons uc a po olio Π which can be sel inanced. This po olio
is composed o an amoun ∆ o he unde lying s ock, such ha he po olio is iskless
and one op ion [62, 19, 10]. The e o e, he alue o he po olio a ime is
Π( ) = V( )+∆S( ).(1.1)
Hence we can w i e,
dΠ =dV + ∆dS (1.2)
=(∂V
∂ +µS ∂V
∂S +1
2σ2S2∂2V
∂S2+ ∆µS)d + (σS ∂V
∂S + ∆σS)dW. (1.3)
The e a e wo p ope ies o he po olio. The i s p ope y s a es ha i mus isk-
ee which indica es ha he 2nd e m in ol ing he B ownian Mo ion dW is ze o ,
so ha
σS ∂V
∂S + ∆σS = 0,
which becomes
∆ = −∂V
∂S .
Now we can Subs i u e ∆ in Equa ion (1.2) o ge
dΠ = ∂V
∂ +1
2σ2S2∂2V
∂S2.
The second p ope y is ha he po olio has o ob ain he isk ee a e.
dΠ = Πd (1.4)
=(∂V
∂ +1
2σ2S2∂2V
∂S2)d = (V−∂V
∂S S)d (1.5)
We can easily ea ange (1.5) o ge
∂V
∂ +1
2σ2S2∂2V
∂S2+ S ∂V
∂S − V = 0.
This is called he Black-Scholes Pa ial Di e en ial Equa ion.
7
1.7 De i a ion o Black-Scholes Equa ion by Bino-
mial Model
The s ock p ice a he ime is S de ined as,
u= expσ√d
A he ime +d he he s ock p ice goes up o Su
+d =uS , ha ing he p obabili y
p=exp d −d
u−d
The de i a i e has isk-less alue which gi es a ela ion [19],
Vexp d =pVu+ (1 −p)Vd=p(Vu−V d) + Vd.(1.6)
whe e V=V(S ), Vu=V(Su
+d and Vd=V(Sd
+d ).
Now we will use Taylo se ies expansions o Vu, Vd,exp d , u and dup o o de d .
F om Taylo se ies we ha e,
Vu≈V+∂V
∂S (Su
+d −S ) + 1
2
∂2V
∂S2(Su
+d −S )2+∂V
∂ d (1.7)
=V+∂V
∂S S (u−1) + 1
2
∂2V
∂S2S2
(u−1)2+∂V
∂ d . (1.8)
We simila ly ge ,
Vd≈V+∂V
∂S S (d−1) + 1
2
∂2V
∂S2S2
(d−1)2+∂V
∂ d . (1.9)
The o he expansions a e
exp d ≈1 + d
u≈1 + σ√d +1
2σ2d
d≈1−σ√d +1
2σ2d
He e we ha e (u−1)2= (d−1)2=σ2d . F om which, we can w i e
p(Vu−Vd) = p(u−d)∂V
∂S S (1.10)
= ( d +σ√d −1
2σ2d )∂V
∂S S .(1.11)
8
We will plug in (1.9) and (1.10) in (1.6) and cancel e ms o p oduce
V(1 + d ) = S
∂V
∂S d +V+1
2σ2S2
∂2V
∂S2d +∂V
∂ d
o , d = S
∂V
∂S d +1
2σ2S2
∂2V
∂S2d +∂V
∂ d .
We cancel d om his equa ion o ob ain he Black Scholes PDE
∂V
∂
1
2S2σ2∂2V
∂S2+ S ∂V
∂S − V = 0.
1.8 Pa i y o Pu -Call
We will make a po olio wi h one long Call op ion and one sho Pu op ion [65]:
VPF (S, T) = VECO(S, T)−VEP O(S, T)
So i becomes,
VPF (S, T) = max(S−K, 0) −max(K−S, 0) = S−K.
F om he solu ion o Black-Scholes PDE wi h he inal condi ion, we ha e,
VPF (S, ) = S−Ke (T− ).
The e o e we can say,
VECO(S, )−VEP O(S, ) = S−Ke (T− ).
This is called he pa i y o Call-Pu . As we can see om abo e:
Call −Pu =Asse −Fo wa d.
1.9 Sensi i i y Analysis
Sensi i i y analysis o he Op ion p ice wi h espec o some pa ame e s known as
G eeks [65]:
9
•Del a: ∆ is de ined as:
∆ = ∂V
∂S .
I is he measu emen o he a e o change o he op ion p ice Vwi h espec
o he change in he unde lying asse p ice S.
Del a hedging use his Del a because he isk-less po olio is s able in acco -
dance o:
QS
QV
=−∂V
∂S =−∆.
whe e QV,QSa e Op ions and S ocks numbe s in he Po olio.
Now Del a o Eu opean Call and Pu op ions a e:
∆EC =∂V EC
∂S =N(d1).
and
∆EP =∂V EP
∂S =−N(−d1).
We can obse e ha :
∆EC ∈(0,1) and ∆EP ∈(−1,0)
E alua ion o Del a o ma ke da a ime se ies:
We can igu e ou he implied ola ili y σimplied( ) om he ma ke da a se ies
o he op ion p ice V eal( ) and he unde lying asse p ice S eal( ). We ha e o
sol e:
V eal( ) = VEC (S eal( ), ;σimplied( )).
•Gamma: I is de ined as:
Γ = ∂∆
∂S =∂2V
∂S2
I is he measu emen o he change o Del a o he op ion p ice Vwi h espec
o he change in he asse p ice S:
ΓEC = ΓEP =∂∆EC
∂S =N0(d1)∂d1
∂S =e−1
2d2
1
σp2π(T− )S.
•Rho:
I is he sensi i i y wi h espec o he in e es a e :
ρ=∂V
∂ .
10
So i calcula es he a e o change o he op ion p ice Vwi h espec o he
in e es a e .
•The a:
I is de ined as :
Θ = ∂V
∂ .
So i calcula es he a e o change o he op ion p ice Vwi h espec o he
in e es a e .
•Vega:
Sensi i i y wi h espec o he ola ili y σ:
ν=∂V
∂σ .
So i calcula es he a e o change o he op ion p ice Vwi h espec o he
ola ili y σ.
The Black-Scholes Equa ion is:
∂V
∂ +1
2σ2S2∂2V
∂S2+ S ∂V
∂S − V = 0.
The G eek Model o he Black-Scholes Equa ion is :
Θ + σ2
2S2Γ + S∆− V = 0.
Black-Scholes equa ion is impo an in he wo ld o inancial ma hema ics. In his
chap e , we ha e gi en a desc ip ion o he undamen al pa s o Black-Scholes Equa-
ion.
11
Chap e 2
Linea Black-Scholes equa ion’s
Analy ic Solu ion
2.1 In oduc ion
Black Scholes Equa ion is p ominen in he quan i a i e inance wo ld o he es ima-
ion o he op ion p ice. We discuss a leng h o he analy ical solu ion o he Linea
Equa ion in his chap e . Ou main ocus is o ans o m he Linea Black Scholes
Equa ion in o a hea -di usion equa ion ; Then we sol e he hea equa ion analy -
ically. A e ha , we subs i u e he a iables o ge he solu ion o Black-Scholes
PDE.
2.2 Black Scholes Equa ion’s Linea Model
The Black Scholes pa ial di e en ial equa ion [10] is
∂V
∂ + S ∂V
∂S +1
2σ2S2∂2V
∂S2− V = 0.(2.1)
He e
•V≡V(S, ) is he payo unc ion. I is he p ice o he op ion a a ime . I
is a wo a iable unc ion wi h he asse ’s p ice Sand ime .
•S( ) s ock p ice which he unde lying asse ’s p ice a ime . I is also non
nega i e.
• is he ime . He e is om 0 o T. ( Tis he expi a ion ime.)
12
• is he iskless in e es a e.
•σis he ola ili y.
•Kis he s ike p ice o an op ion.
•µis he d i a e.
In he case o Linea Black-Scholes Equa ion, σ, , µ [65] a e cons an s.
We con e he linea Black-Scholes Equa ion o a hea equa ion. We need o ha e
he ini ial and bounda y condi ions i we wan o ind he solu ion o Black-Scholes
Equa ion. We a e conside ing only he Eu opean Op ion P ice.
Eu opean Op ion P ice: In his Op ion p ice, he domain o ime is,
0≤ ≤T
The domain o S ock p ice Sis,
0< S < ∞
Eu opean Call Op ion: F om [10, 65]:
V(S, T) = max(S−K, 0) o 0< S < ∞
V(0, ) = 0 o ∈[0, T]
V(S, ) = S−Ke− (T− ) o S → ∞
Eu opean Pu Op ion: F om [10, 65]:
V(S, T) = max(K−S, 0) o 0< S < ∞
V(0, ) = Ke− (T− ) o ∈[0, T]
V(S, )=0 o S → ∞
13
Op ion a expi y ime plo ed o e a ange o s ock p ices a e 70 ≤S≤130. We use
he Black Scholes Equa ion’s exac solu ion o e alua e he alue o he op ion p ice
be o e expi a ion.
Figu e 2.1: Value o he Eu opean call op ion V(S, ) a ime, = 0, =.2, =.4,
=.6, =.8 and = 1.
Figu e 2.2: Value o he Eu opean call op ion V(S, ) a a ious imes in a su ace
plo
20
Black-Scholes Equa ion is e ec i e o a ious con ex s o he inancial ield. In
his chap e , we in es iga ed he analy ical solu ion o he Linea Black-Scholes
model. And a g aphical ep esen a ion o he exac solu ion is gi en oo.
21
Chap e 3
Linea Black-Scholes Equa ion
wi h Spec al Me hod
3.1 In oduc ion
We in es iga e Black-Scholes pa ial di e en ial equa ion (PDE) wi h di e en Nu-
me ical schemes. In his chap e we sol e i using spec al me hod. This me hod is
well known o i s supe io e iciency and less ime consump ion. I akes less ime
han o he nume ical schemes such as ini e-di e ence, ini e elemen s and Gela kin
me hod. The spec al colloca ion me hod is applied o Black-Scholes PDE. We dis-
cuss b ie ly abou he Spec al colloca ion me hod. The e is a MATLAB execu ion o
he Spec al Me hod [72]. We use he Chebyshe polynomial and Chebyshe poin s
o he spec al me hod. Also he e is a compa ison o he nume ical calcula ion wi h
he exac solu ion a he end o his chap e .
3.2 Spec al Colloca ion Me hod
Spec al Me hod has he abili y o achie e he exponen ial accu acy o sol e he di -
e en ial equa ions and app oxima ing he solu ion [67]. The accu acy o he Spec al
Me hod is uled by he smoo hness o unc ions.
We use polynomials o app oxima e a unc ion [31]. The spec al colloca ion is also
called pseudospec al me hod. Usually we use Chebyshe and Legend e polynomials.
Le he weigh be wiand colloca ion poin be xi.These weigh s and colloca ion poin s
connec ed o Chebyshe polynomial can be e alua ed as [31]:
22
1. Chebyshe -Gauss-Loba o: xi= cos πi
N, w0=wN=π
2N, wi=π
N
2. ChebyShe -Gauss-Radau: xi= cos 2πi
2N+1, w0=π
2N+1 , wi=2π
2N+1
3. ChebyShe -Gauss: xi= cos 2i+1
2N+2, wi=π
N+1
3.3 Chebyshe Di e en ia ion Ma ix
We use he Chebyshe -Gauss-Loba o poin s.
xj= cos πj
N, j = 0,1,2,··· , N
And Chebyshe colloca ion poin s a e e alua ed in he in e al [−1,1]
We use hese poin s o build up he Chebyshe Di e en ia ion Ma ix. And his
ma ix is used as he di e en ial ope a o .
Now we a e gi en a g id unc ion de ined on he Chebyshe poin s; we acqui e
he disc e e de i a i e gin wo s ages [72]:
•We le qbe he unique polynomial wi h a deg ee ≤Mwi h q(xj) = j,0≤j≤
M
•gj=q0(xj)
The e o e his ope a ion has o be linea . We can ep esen his by he mul iplica ion
o (M+ 1) ×(M+ 1) ma ix [72] , which is deno ed by DM:
g=DM . (3.1)
In he e Mis an a bi a y posi i e in ege .
Now o M= 1. The in e pola ing poin s a e:
x0= 1,
and
x1=−1.
23
The Lag ange o m o he in e pola ing polynomial h oughou he 0and 1is,
q(x) = 1
2(1 + x) 0+1
2(1 −x) 1.
The de i a i e o q(x) is:
q0(x) = 1
2 0−1
2 1.(3.2)
He e M= 1, so he ma ix is 2 ×2 ma ix.
Le ’s deno e aij = [aij] The en ies o he ma ix whe e i, j = 1,2,3,··· , M.
Fo he i s column o he ma ix, we se x0= 1 and ge :
a11 =1
2,
and
a21 =1
2.
Fo he second column o he ma ix, we se x1=−1 and ge :
a12 =−1
2,
and
a22 =−1
2.
Hence he Chebyshe Di e en ia ion Ma ix o M= 1 is:
D1=1
2−1
2
1
2−1
2.
Now o M= 2. The in e pola ing poin s a e:
x0= 1, x1= 0, and x2=−1.
The Lag ange o m o he in e pola ing polynomial h oughou he 0, 1and 2is:
q(x) = 1
2x(1 + x) 0+ (1 + x)(1 −x) 1+1
2x(1 −x) 1.
The de i a i e o q(x) is:
q0(x) = (x+1
2) 0−2x 1+ (x−1
2) 2.(3.3)
24
He e M= 2, so he ma ix is 3 ×3 ma ix. Fo he i s column o he ma ix, we
se x0= 1 and ge :
a11 =x+1
2=3
2,
Fo x1= 0,
a21 =x+1
2=1
2.
Fo x2=−1,
a31 =x+1
2=−1
2.
Fo he Second column o he ma ix, we se x0= 1 and ge :
a12 =−2x=−2,
Fo x1= 0,
a22 =−2x= 0,
Fo x2=−1,
a32 =−2x= 2.
Fo he Thi d column o he ma ix, we se x0= 1 and ge :
a13 =x−1
2=1
2,
Fo x1= 0,
a23 =x−1
2=−1
2,
Fo x2=−1,
a33 =x−1
2=−3
2.
The e o e he Chebyshe Di e en ia ion Ma ix o M= 2 is:
D2=
3
2−21
2
1
20−1
2
−1
22−3
2
.
We can gi e he gene aliza ion o he en ies o DM o a bi a y M[72, 30]
25
Theo em: Le ’s conside o M≥1, we ha e he columns and ows (M+1)×(M+1)
o he Chebyshe Spec al Di e en ia ion Ma ix DM om 0 o M. The en ies o
his ma ix a e:
(DM)00 =2M2+ 1
6,(DM)MM =−2M2+ 1
6,(3.4)
(DM)jj =−xj
2(1 −x2
j), j = 1,2,··· , M −1.(3.5)
(DM)ij =ci
cj
(−1)i+j
(xi−xj), , i 6=j, i, j = 1,2,··· , M −1.(3.6)
whe e
ci=(2i= 0 o M
1elsewhe e.
We can w i e his in Ma ix o m,
DM=
2M2+1
62(−1)j
1−xj··· ··· 2(−1)j
1−xj
1
2(−1)M
−1
2
(−1)i
1−xi−xj
2(1−x2
j)
(−1)i+j
(xi−xj)··· (−1)i+j
(xi−xj)
1
2
(−1)M+i
1+xi
.
.
.(−1)i+j
(xi−xj)−xj
2(1−x2
j)
....
.
..
.
.
.
.
..
.
.......(−1)i+j
(xi−xj)
.
.
.
−1
2
(−1)i
1−xi
(−1)i+j
(xi−xj)··· (−1)i+j
(xi−xj)−xj
2(1−x2
j)
1
2
(−1)M+i
1+xi
−1
2(−1)M−2(−1)M+j
1+xj··· ··· −2(−1)M+j
1+xj−2M2+1
6
.
The j h column o DMca ies he de i a i e o he deg ee Mpolynomial in e polan
qj(x) o he del a unc ion bols e ed a xj, ep esen ed a he g id poin s xi
I we ha e ze o bounda y condi ions, hen e alua ion becomes easie . We can exclude
he i s and las column o DMma ix because hey will be mul iplied by ze o.
3.4 MATLAB unc ion : ’cheb’
AMATLAB unc ion called cheb [72] is used o e alua e he Chebyshe di e en-
ia ion ma ix DM. This unc ion e u ns a ec o xand a ma ix D. The unc ion
26
is gi en below:
% CHEB E alua e Di = Chebyshe di e en ia ion ma ix, x = Chebyshe g id
unc ion [Di ,x] = cheb(M)
i M==0, Di =0; x=1; e u n, end
x = cos(pi*(0:M)/M)’;
c1 = [2; ones(M-1,1); 2].*(-1).^(0:M)’;
X = epma (x,1,M+1);
dX = X-X’;
Di = (c1*(1./c1)’)./(dX+(eye(M+1)));
Di = Di - diag(sum(Di ’));
This MATLAB unc ion does no e alua e DMp ecisely by he (3.4,3.5,3.6) o mulas.
Ins ead his makes he bes use o (3.5) o he o -diagonal en ies bu hen acqui es
he diagonal (3.4,3.5) om he iden i y:
(DM)ii =−
N
X
j=0
j6=i
(DM)ij.
F om his, i is sligh ly easy o p og am. And i c ea es a ma ix wi h be e s abili y
p ope ies [59, 7].
We a e gi ing a ew Chebyshe Di e en ia on Ma ix DMma ices e alua ed
by he MATLAB unc ion cheb:
>> cheb(1)
ans =
0.5000 -0.5000
0.5000 -0.5000
>> cheb(2)
27
ans =
1.5000 -2.0000 0.5000
0.5000 0 -0.5000
-0.5000 2.0000 -1.5000
>> cheb(3)
ans =
3.1667 -4.0000 1.3333 -0.5000
1.0000 -0.3333 -1.0000 0.3333
-0.3333 1.0000 0.3333 -1.0000
0.5000 -1.3333 4.0000 -3.1667
>> cheb(4)
ans =
5.5000 -6.8284 2.0000 -1.1716 0.5000
1.7071 -0.7071 -1.4142 0.7071 -0.2929
-0.5000 1.4142 0 -1.4142 0.5000
0.2929 -0.7071 1.4142 0.7071 -1.7071
-0.5000 1.1716 -2.0000 6.8284 -5.5000
He e we can analyze ha i s wo Ma ices D1and D2ma ches wi h he en ies o
he ma ices we e alua ed analy ically. The e o e his unc ion cheb is accu a e.
28
3.5 Nume ical Solu ion o Black-Scholes Equa ion
wi h he help o Spec al Me hod
We a e no going o sol e he o iginal Black-Scholes PDE. Bu we sol e nume ically
he Hea Equa ion using Spec al Colloca ion Me hod.
uxx =uτ.
A e he nume ical solu ion o Hea Equa ion, he a iables a e back-subs i u ed
o ge he o iginal solu ion o Black-Scholes Equa ion. We can e alua e he second
de i a i e by
d2
dx2=D2
M.
by squa ing he Chebyshe Di e en ia ion Ma ix DM.
F om he equa ion (3.1) we can say:
w=D2
M .
He e D2
Mis an (M+ 1) ×(M+ 1) ma ix which will map a ec o ( 0, 1,··· , M)T
o a ec o (g0, g1,··· , gM)T. We can w i e his in ma ix o m:
g0
g1
.
.
.
.
.
.
gM−1
gM
=
D2
M
0
1
.
.
.
.
.
.
M−1
M
.
Spec al Me hod is used o disc e ize he spa ial domain xin he in e al x∈[−1,1].
We used he cheb unc ion o he second de i a i e. The ime domain τhas no
ex a calcula ion. We ha e a polynomial in e polan q(x). This polynomial has been
calcula ed using he command poly al(poly i (···)). The main algo i hm o ou
code ollows he algo i hm om P og am 34 o [72] whe e he Allen-Cahn equa ion
(u =uxx +u−u3) is sol ed using spec al me hod and Fo wa d Eule . We can
dec ease he e o exponen ially, i we inc ease he deg ee o he polynomial.
29
1. The i s o de de i a i e: F om [45, 53] ,we can exp ess diagonal and o
diagonal en ies o he weigh ing coe icien s o he i s o de .
Diagonal en y: i=j,
A(1)
ij =
N
X
k=1,k6=i
1
(xi−xk)=−
N
X
k=1,k6=i
A(1)
ik .
O -diagonal en y: i6=j,
A(1)
ij =
N
Y
k=1,k6=i
(xi−xk)
(xj−xi)Y
k=1,k6=j
(xj−xk)
=1
(xj−xi)
N
Y
k=1,k6=i
(xi−xk)
Y
k=1,k6=j
(xj−xk)
=1
(xj−xi)·
N
Y
k=1,k6=i,k6=j
(xi−xk)
(xj−xk).
2. Second o de De i a i e
Diagonal en y: i=j,
A(2)
ij =−
N
X
k=1,k6=i
A(2)
ik .
O -diagonal en y: i6=j,
A(2)
ij = 2A(1)
ij A(1)
ii −1
xi−xj.
3. Highe O de De i a i es: As we can see om he i s wo cases, he e is
a ecu ence o mula o he highe de i a i es. Tha is, o e alua e weigh ing
coe icien s o highe o de , we need he weigh ing coe icien s o all he lowe
o de s ela i e o he one we need.
Diagonal en y: i=j,
A(l)
ij =−
N
X
k=1,k6=i
A(l)
ik .
36
O -diagonal en y: i6=j,
A(l)
ij =l A(1)
ij A(l−1)
ii −A(l−1)
ij
(xi−xj)!.
We ha e a Hea Equa ion o sol e. We eplace he second de i a i e in i by he
Di e en ial Quad a u e Ma ix.
DQM = [Aij].
uxx =∂2u
∂x2=D2
QM u.
4.5 Ma lab unc ion : Di e en ial Quad a u e Ma-
ix
Mehme Mu a Al ug Bicak coded he ma lab unc ion ile o Di e en ial Quad a u e
Ma ix [9]. The unc ion ile is gi en below:
unc ion [D]=Di _Quad(N)
% Di e en ial quad a u e ma ix o N poin s based on Loba o g id poin s.
o ii=1:N
X(ii)=0.5*(1-cos((ii-1)*pi/(N-1)));
end
SagTa a =1;
Ta a =0;
o ii=1:N
o jj=1:N
i ii~=jj
a1=(1/(X(jj)-X(ii)));
o k=1:N
i k~=ii & k~=jj
SagTa a =SagTa a *(X(ii)-X(k))/(X(jj)-X(k));
end
end
37
a(ii,jj)=a1*SagTa a ;
SagTa a =1;
end
i ii==jj
o k=1:N
i k~=ii
Ta a =Ta a +(1/(X(ii)-X(k)));
end
end
a(ii,jj)=Ta a ;
Ta a =0;
end
end
end
D=a;
ic
A ew ou pu s o Di e en ial Quad a u e Ma ix p oduced by he unc ion
ile a e gi en below:
>> Di _Quad(1)
ans =
0
>> Di _Quad(2)
ans =
-1 1
-1 1
>> Di _Quad(3)
ans =
-3.0000 4.0000 -1.0000
38
-1.0000 0.0000 1.0000
1.0000 -4.0000 3.0000
>> Di _Quad(4)
ans =
-6.3333 8.0000 -2.6667 1.0000
-2.0000 0.6667 2.0000 -0.6667
0.6667 -2.0000 -0.6667 2.0000
-1.0000 2.6667 -8.0000 6.333
4.6 Nume ically sol ing Black-Scholes PDE using
Di e en ial Quad a u e Me hod
He e, we use Di e en ial Quad a u e Me hod (DQM) o sol e he ans o med Equa-
ion (Hea equa ion) o Black-Scholes PDE.
We e alua e he Di e en ial Quad a u e Ma ix (DQM) wi h he help o [9] o
Chebyshe -Gauss-Loba o G id poin s. This di e en ial quad a u e ma ix con e s
he di e en ial equa ion in o a eigen alue p oblem( y0=A∗y=⇒Dy =Ay).
We sol e he Black-Scholes Equa ion o he domain x∈[−1,1]. Now o ou choice
o Chebyshe -Gauss-Loba o poin s, we ge he g id poin s in he in e al [0,1]. To
con e he in e al om [0,1] o [−1,1], we use 2x−1.
We e alua e Eu opean Call Op ion V(S, ) o he ollowing pa ame e s:
T= 1; = 0.02, σ = 0.2; K= 100.
39
Figu e 4.1: Eu opean Call Op ion P ice V(S, ) using Di e en ial Quad a u e
Me hod.
Then we ha e he compa ison wi h he exac solu ion o Black-Scholes Equa-
ion. And i ma ches wi h he exac solu ion:
Figu e 4.2: Eu opean Call Op ion P ice V(S, ) is compa ed wi h he Exac solu ion.
Now we ha e he able o e o s compa ed o Di e en ial Quad a u e Me hod.
We use in ini y no m a = 0.
40
S ock P ice SDi e en ial Quad a u e Me hod Exac Value E o
137.71278 47.37890 47.37287 0.00603
139.09681 48.74329 48.73885 0.00444
141.90675 50.12376 50.12078 0.00298
143.33294 52.93267 52.93226 0.00042
144.77346 54.36104 54.36175 0.00070
146.22846 55.80532 55.80704 0.00173
147.69808 57.26551 57.26816 0.00265
149.18247 58.74161 58.74510 0.00349
150.68178 60.23366 60.23791 0.00426
152.19616 61.74168 61.74663 0.00494
153.72575 63.26574 63.27130 0.00557
155.27072 64.80588 64.81200 0.00613
156.83122 66.36218 66.36881 0.00663
Table 4.1: Compa ison o Di e en ial Quad a u e Me hod and Exac Solu ion o he
Eu opean Call Op ion P ice V(S, ).
4.7 Conclusion
In oduc ion o Di e en ial Quad a u e Me hod is gi en in his chap e . We buil up
he di e en ial quad a u e ma ix which helped ex ensi ely in ou MATLAB imple-
men a ion o Black-Scholes PDE.
41
Chap e 5
Nonlinea Black Scholes Equa ion
and Vola ili y Models
5.1 In oduc ion
In his chap e we discuss abou Nonlinea Black Scholes Equa ion and se e al ola il-
i y models. A sho and e icien analy ical app oach o he Nonlinea Black Scholes
Equa ion is gi en oo. We ocus on he Eu opean Op ion p ice.
5.2 Nonlinea Black Scholes Equa ion
The linea Black Scholes Equa ion is:
V +1
2σ2S2VSS + SVS− V = 0.(5.1)
He e S( )>0 and ∈(0, T) The e a e some es ic i e assump ions on he linea
Black-Scholes Equa ion which do no ma ch in eali y. Because o ansac ion cos s,[5,
11, 51],la ge in es o p e e ences [27, 29, 64] and and incomple e ma ke s[68]. In his
hesis, we ocus on couple o ansac ion cos models o he nonlinea Black–Scholes
equa ions o Eu opean op ion p ice whe e µ he d i a e is cons an and modi ied
ola ili y unc ion is no cons an ,
˜σ2= ˜σ2( , S, VS, VSS).
So he equa ion (5.1) becomes he nonlinea Black Scholes Equa ion:
V +1
2˜σ2( , S, VS, VSS)S2VSS + SVS− V = 0 (5.2)
whe e S( )>0 and ∈(0, T).
42
5.3 Bounda y Condi ions
We ha e o inalize he di e en ial p oblem by s a ing he e minal and bounad y
condi ions o Eu opean Call and Eu opean Pu Op ion p ice. Only hen we can ind
a unique solu ion o he equa ion (5.2).
Al hough we only ocus on Eu opean Call Op ion P ice in his disse a ion.
5.3.1 Eu opean Call Op ion
Eu opean call op ion is he solu ion o (5.2) on 0 ≤S < ∞, 0 ≤ ≤Twi h he
ollowing ini ial condi ion and bounda y condi ions:
V(S, T) = (S−K)+, o 0≤S < ∞,
V(0, ) = 0,
V(S, )≈S−Ke− (T− ), as S → ∞.
5.3.2 Eu opean Pu Op ion
The alue V(S, ) o he Eu opean pu op ion is he solu ion o 1.2 on 0 ≤S <
∞, 0 ≤ ≤Twi h he ollowing ini ial condi ion and bounda y condi ions:
V(S, T) = (K−S)+, o 0≤S < ∞,
V(0, ) = Ke− (T− ),
V(S, )≈0, as S → ∞.
5.4 Vola ili y Models
In classical Black-Scholes model, he essen ial pa ame e which can no be obse ed
bu is assumed o be cons an is known as he ola ili y σ. Many esea che s ied di -
e en app oaches o imp o e he model by using a modi ied ola ili y unc ion ˜σ(.) o
ake ansac ion cos s in o accoun , big ade s and illiquid ma ke s which is e en u-
ally he cause o he nonlinea i y o (5.2). In his pa , we discuss b ie ly abou couple
o ola ili y models and ocus will be on he ansac ion cos ela ed ola ili y models.
43
1. We can eplace he ola ili y σin he Black-Scholes model by he app oxima ed
ola ili y om he p eceding alues o he unde lying asse . This ola ili y is
called he his o ical ola ili y [32].
2. We can calcula e he modi ied ola ili y om he exac solu ion o he Classical
Black-Scholes Model i he p ice o Op ion and e e y o he pa ame e s a e
known. The implied ola ili y is he alue σ, o which call op ion solu ion
and pu op ion solu ion is ue in compa ison o he eal ma ke da a. We can
e alua e he implied ola ili y om he di e ence be ween he obse ed op ion
p ice V(S, ) ( om s ock ma ke ) and he Black-Scholes exac solu ion whe e
all he pa ame e s a e aken in o accoun om he eali y excep o he implied
ola ili y.
3. Ano he model is de eloped by Hull,Whi e[38] and Hes on[35], ola ili y ollows
he dynamics o a s ochas ic p ocess which is called he s ochas ic ola ili y.
4. A i s we ha e obse ed ola ili ies a each s ock p ice and ime which is
e med as local ola ili y ˜σ= ˜σ(S, ). We can eplace cons an ola ili ies wi h
hem. Dupi e [50] in es iga ed he eliances and exp essed he local ola ili y
as a unc ion o implici ola ili ies.
5. We assume ha e e y secu i y is accessible a any size and any ime o ha any
single ading will no impac he p ice, is no always co ec . Hence,se e al
au ho s ha e modeled illiquid ma ke s and big ade e ec s.F ey and S emme
[23]and a e some ime F ey and Pa ie [28] examined hese esul s on he p ice
and eme ge wi h he ou come
˜σ=σ
1−ρλ(S)SVSS
.(5.3)
whe e ρis cons an ,σ he his o ical ola ili y.
These nonlinea models a e all consis en wi h he linea model i he supplemen a y
pa ame e s o ansac ion cos anish (Le, Ψ(.), M).
The e a e couple o modi ied ola ili y unc ion models,
44
5.5 Leland’s Model o Vola ili y
Leland’s model is one o he in amous implied ola ili y model. By he ade a
disc e e imes [51] was he plan o Leland’s o elax he hedging cond ions. This also
has he po en ial o educe he Po olio adjus men expenses. Leland assumes ha
he ansac ion cos is
ω|∆|S/2.
whe e ωis he whole ip ansac ion cos pe uni money o he ansac ion. ∆ is
he numbe o asse s pu chased (∆ >0) o sold (∆ <0) a p ice S, is p opo ional
o he mone a y alue o he asse s pu chased o sold.
We ake a po olio which eplica es wi h ∆ uni s o he unde lying asse and he
bond B:
Π=∆S+B.
A e ha , a mino change in ime o he size δ he change in he po olio becomes
δΠ=∆δS + Bδ −ω
2|δ∆|S. (5.4)
He e δS is he change in p ice S. The e o e, he i s e m ep esen s he change in
alue. Then we ha e he 2nd which ep esen s he bond g ow h in δ ime. A e
ha we ha e δ∆ which ep esen s he change in he numbe o asse s. Hence, he
las e m becomes he ansac ion cos because o he po olio change.
We use I o’s lemma o he alue o he op ion p ice V:= V(S, ) and ge :
δV =VSδS + (V +σ2
2S2VSS)δ . (5.5)
We assume ha he op ion Vis eplica ed by he po olio Π and hei alues ha e o
be equi alen a all imes and he e can be no isk-less p o i . Because o no-a bi age
a gumen , we ge
δΠ = δV
Equa ing he e ms in he las wo equa ions ,we ge
∆ = VS,
45
unp o ec ed po olio. In his way he po olio is s ill well p o ec ed wi h he Risk
adjus ed P icing Me hodology(RAPM) and he imp o ed ola ili y is,
˜σ2=σ2(1 + 3(C2M
2πSVSS)1/3).
He e M≥0 is he ansac ion cos measu e and C≥0 is he isk p emium measu e.
5.11 Analy ical Solu ion
We sol e (5.2) subjec o he e minal and bounda y condi ions . We change he
a iables [?, 24],
x= ln( S
K),
τ=1
2σ2(T− ),
u(x, τ) = e−xV(S, )
K.
Di e en ia ion gi es,
V =uττ S=−1
2σ2Suτ,
VS=uxxSS+u=ux+u,
VSS =uxxxS+uxxS=1
S(uxx +ux).
Plugging hese alues in o (5.2) ;we ge ,
−1
2σ2Suτ+1
2˜σ2S(uxx +ux) + S(ux+u)− uS = 0.
Now we a e going o mul iply by −2
Sσ2
uτ−˜σ2
σ2(uxx +ux)−Dux= 0.
Whe e D=2
σ2and ˜σ2depends on he ola ili y model. x∈Rand 0 ≤τ≤σ2T
2Now
we plug in he ola ili y models:
˜σ2=σ2(1 + Le.sign(uxx +ux))
o ,
˜σ2=σ2(1 + Ψ(e2 τ
σ2aKex(uxx +ux))
52
o ,
˜σ2=σ2(1 + (e2 τ
σ2aKex(uxx +ux))
o ,
˜σ2=σ2(1 + 3(C2M
2π(uxx +ux))1/3).
5.12 Conclusion
In his chap e , we ha e discussed ai ly abou he di e en ola ili y models. We
ha e gi en sho desc ip ions o he amous ola ili y models. This chap e ended
wi h he analy ical solu ion o he Nonlinea Black-Scholes Model. This solu ion has
simila i ies wi h he Linea one.
53
Chap e 6
Nume ical In es iga ion o
Nonlinea Black-Scholes Equa ion
The e a e nume ous nume ical me hods o sol ing Black–Scholes equa ion o Eu-
opean, Ame ican and Asian op ions. We can also sol e linea Black-Scholes model
using nume ical me hods. Bu he e a e majo di e ence be ween he same nume ical
me hod as he ola ili y becomes a a iable. We show he p ocess and some di e en
app oaches o sol e he ola ili y e m. Also he ola ili y e m is de ined as di e en
unc ions because o he nonlinea i y.
We ha e al eady showed in he p e ious chap e s ha he e a e exac solu ion o
Eu opean call and pu op ions. Bu i we seek mo e accu a e answe which ma ches
wi h he eali y, we ha e o dug a bi deepe which e en ually gi es us he nonlin-
ea Black-Scholes equa ions wi h a ious ola ili y models. Fo hese complica ed
con ac s in gene al se ings, analy ical o mulae a e seldom a ailable and nume ical
me hods ha e o used o sol e he p oblem. These Me hods a e di e en om la ice
me hods which includes binomial and inomial app oxima ions [18], Mon e Ca lo
me hods using he leas -squa e echniques [42], analy ical app oxima ions [1, 14, 52],
ini e-elemen disc e iza ions [74, 47] o ini e-di e ence me hods [2, 13, 17].
We ha e ano he classical me hod which consis s o de eloping o he ee bound-
a y p oblem in o a linea complemen a y p oblem (LCP) and he solu ion by he
P ojec ed Successi e O e Relaxa ion (PSOR) me hod o C ye [20]. And he e was
ano he me hod de eloped called penal y and on – ixing [55]. The unde lying model
changes which is a disad an age in hese me hods.
54
A comple e di e en p ocedu e which is based on a ecu si e calcula ion[37] o he
ea ly exe cise bounda y, es ima ing he bounda y only a some poin s and hen
app oxima ing he whole bounda y by Richa dson ex apola ion. Explici bound-
a y acking algo i hms which a e a ini e-di e ence bisec ion scheme [49] o he
on – acking s a egy o Han and Wu [33].
Ou nume ical in es iga ion is solely ocused on ini e di e ence schemes. We gi e a
de ailed analysis on his scheme.
6.1 Eu opean Call op ion
We a e going o use ini e-di e ence schemes o sol e he con e ed pa ial di e en ial
equa ion (PDE) o Black-Scholes Equa ion,
uτ−˜
σ2
σ2(uxx +ux)−Dux= 0.(6.1)
Whe e x∈Rand 0 ≤τ≤˜
Twi h he espec i e ola ili ies (5.2) subjec o he
ini ial and bounda y condi ions,
u(x, 0) = (1 −e−x)+ o x ∈R,
u(x, τ)=0 as x → ∞,
u(x, τ)≈1−e−Dτ−xas x → ∞.
We a e going o gi e a in oduc ion o ini e di e ence schemes and hen p esen he
nume ical esul s om MATLAB.
6.2 Fini e-Di e ence Me hods
The main goal o ini e-di e ence schemes is o app oxima e he de i a i es o he
di e ence quo ien s and he solu ion o he eme ging disc e e schemes.
A i s , we s a by disc e izing he domain o he con e ed Pa ial di e en ial
Equa ion (PDE) o Black Scholes Equa ion wi h he espec i e ola ili ies. Then
55
we a e going o eplace he de i a i es by app op ia e di e ence quo ien s. Then ou
in es iga ion using ini e-di e ence con inues in e ms o con e gence . We in oduce
classical ini e-di e ence schemes o he Eu opean Call op ion.
6.3 G id
We a e going o ans o m he spa ial domain and he ime domain.
Now x∈Rand τ∈[0,˜
T] by a bounded in e al x∈[−R, R], R > 0. Now we
disc e ize he new compu a ional domain by a uni o m g id (xi, τn) wi h xi=ih
and τn=nk, whe e h > 0 is he spa ial s ep, and k > 0 is he ime s ep. And
i∈[−N, N],−R=−Nh, R =Nh, n ∈[0, M] and ˜
T=Mk.
We deno e he app oxima e solu ion o he pa ial di en ial equa ion o Black-
Scholes in xia ime τnby
Un
i≈u(xi, τn).
And disc e ize he ini ial and bounda y condi ions in he ollowing way:
U0
i= (1 −e−ih)+,
Un
−N= 0,
Un
N= 1 −e−Dnk−Nh.
Now o a mo e i ing ea men o he unbounded spa ial domain x∈R, we a e
going o in oduce a i icial bounda y condi ions [26] o con ine he unbounded
domain o a bounded compu a ional domain.
6.4 Di e ence Quo ien s
The spa ial de i a i e can be app oxima ed wi h o wa d di e ences:
ux(x, τ) = u(x+h, τ)−u(x, τ)
h+O(h).
o backwa d di e ences:
ux(x, τ) = u(x, τ)−u(x−h, τ)
h+O(h).
56
We can sum hese di e ences which esul s in cen al di e ences and we ge :
ux(x, τ) = u(x+h, τ)−u(x−h, τ)
2h+O(h2).
Fo he second spa ial de i a i e we compu e using he Taylo o mula:
uxx(xi, τn) = u(x+h, τ)−2u(x, τ) + u(x−h, τ)
h2+O(h2).
Now we can callback ha a unc ion (h) is in O(h ), i he e exis s a cons an
M > 0,such ha | (h)|≤ M|h |as h→0 . I means ha he quan i y (h) is
bounded by a cons an mul iple o h o su icien ly small h[70]. We call he e o
be ween di e en ial quo ien and di e ence quo ien he unca ion e o .
To disc e ize he pa ial de i a i es o he ans o med Black-Scholes Equa ion,
we a e going o in oduce he ollowing no a ion o he o wa d di e ence quo ien
wi h he spa ial s ep size h:
D+
hUn
i=Un
i+1 −Un
i
h≈ux(xi, τn).
and we lea e ou he e o e m O(h).
Simila ly he backwa d di e ence quo ien wi h espec o he spa ial a iable is
deno ed as:
D−
hUn
i=Un
i−Un
i−1
h≈ux(xi, τn).
So he cen al di e ence quo ien is:
D0
hUn
i=Un
i+1 −Un
i−1
2h≈ux(xi, τn).
whe e unca ion e o o O(h) and O(h2) a e omi ed.
Now o he second spa ial de i a i e we in oduce he s anda d di e ence quo ien :
D2
hUn
i=Un
i+1 −2Un
i+Un
i−1
h2≈uxx(xi, τn).
Whe e he e o e m is O(h2).The cen al di e ences in he ime a iable a e ne e
used in p ac ice because hey always lead he way o bad nume ical schemes, which a e
inhe en ly no s able [76]. Now using he schemes esul s in o a sys em o equa ions
ha can be w i en in a ma ix o m:
AnUn+1 =BnUn+dn.(6.2)
57
whe e
Un= (Un
−N+1, ...., Un
0, ..., Un
N−1)T∈R2N−1.
An=
a0a10 0 ··· 0
a−1.........0
0.............
.
.
0............0
.
.
..........a1
0 0 ··· 0a−1a0
∈R(2N−1)×(2N−1).
Bn=
b0b1b20··· 0
b−1.........0
b−2.............
.
.
0............b2
.
.
..........b1
0 0 ··· b−2b−1b0
∈R(2N−1)×(2N−1).
dn=
b−2Un
−N−1+b−1Un
−N−a−1Un+1
−N
b−2Un
−N
0
.
.
.
0
b2Un
N
b1Un
N+b2Un
N+1 −a1Un+1
N
∈R(2N−1)×(2N−1).
Now he ma ix Anis idiagonal. Hence he esul ing sys ems can be sol ed wi h
linea e o O(N) using he Thomas algo i m [71, 33]. This is comple ed by i s
decomposing he ma ix An=LnRnin o a lowe and an uppe bidiagonal ma ix
and secondly sol ing LnRnUn+1 =BnUn+dnby o wa d and backwa d subs i u ion.
Hence , we sol e LnYn=BnUn+dn o Ynand a e ha we sol e RnUn+1 =Yn
o Un+1.
We need Un
−N−1and Un
N+1 o he ec o dn. Bu hese alues a e no in he g id
poin s ha a e aken in o accoun . They a e he poin s be o e and a e he bounda y
58
condi ions. We de ine he auxilia y o ghos bounda y condi ions [70],
Un
−N−1= 0 and Un
N+1 = 1 −e−Dnk−(N+1)h.(6.3)
And a e his, we a e going o assume ha ,
1
X
i=−1
ai=
2
X
i=−2
bi= 1.
This is sa is ied by any consis en scheme a e no maliza ion o he coe icien s [61].
6.5 Vola ili y Models
We can deal wi h he de i a i es in he ola ili y in di e en ways. We can w i e he
modi ied ola ili y as,
˜
σ2=σ2(1 + s(x, τ)).
He e s(x, τ) deno es he ola ili y co ec ion in xa ime τ. And e m depends on
he i s and second spa ial de i a i es o u. In he pape o Du ing[25], he p oposes
a smoo he app oxima ion o uxx o he ola ili y co ec ion by choosing:
uxx(xi, τn)≈Un
i+2 −2Un
i+Un
i−2
4h2=D2
2hUn
i.
He e we ha e unca ion e o o o de O(h2). We deal wi h he nonlinea i y explici ly
in all he schemes.
Then, om Leland’s Model:
sn
i= 2
π
κ
σ√δ sign(D2
2hUn
i+D0
hUn
i).
Ba les’ and Sone ’s model’s ola ili y co ec ion is :
sn
i= Ψ(eDτn+xia2K(D2
2hUn
i+D0
hUn
i)).
Then ola ili y co ec ion o he case when Ψ(.) is conside ed as he iden i y is:
sn
i=eDτn+xia2K(D2
2hUn
i+D0
hUn
i).
59
Las ly he ola ili y co ec ion o he Risk Adjus ed P icing M hedology(RAPM)
is:
sn
i= 3 C2M
2π(D2
2hUn
i+D0
hUn
i)1/3
.
The e is a complica ion wi h he e alua ion o sn
ia he bounda y, as in he heo y
we equi e Un∈R2N+3 o ha e he abili y o calcula e sn
N−1and sn
−N+1. Bu his
calcula ion demands he need o Un
−N−1and Un
N+1, which a e ou side he domain.
Du ing’s pape [25] has he s a emen ha he impac o he nonlinea i y a he
bounda y is no ha signi ican and i can be neglec ed o la ge R. So we can
assume ha ou auxilia y o ghos bounda y condi ions a e alid and say ha :
sn= (sn
−N+1,··· , sn
0,··· , sn
N−1)T∈R2N−1.
6.6 Exis ence and Con e gence
Ou desi e is o gi e a logical app oxima ion o he sequence o he solu ions o
AnUn+1 =BnUn+dn.
Then hese needs o be sa is ied:
1. A uni o m solu ion Unhas o exis o each n∈[0, M −1].
2. And Un
ihas o con e ge owa ds he exac solu ion o
uτ−˜
σ2
σ2(uxx +ux)−Dux= 0,
as k→0 and h→0.
We a e going o callback he e ms and condi ions o he exis ence and con e gence
o he linea case. In which case he ola ili y co ec ion sn
iis equal o ze o and
Hence An=A,Bn=Band he coe icien s o dna e cons an s.
A uni o m solu ion o he sys em o equa ions exis s [41], i he ma ix Ais egula (a
egula ma ix Ais desc ibed as a squa e ma ix ha o all posi i e in ege n, is
such ha Anhas posi i e en ies.). The scheme (2) con e ges, i i s consis en and
s able. Hence, we desc ibe he e ms consis ency and s abili y.
60
•Consis ency
A scheme Lh,k o o de (a, b) is consis en , i he e exis s a cons an M > 0 such
ha ,
max
i,n |Lh,kun
i|≤ M(ka+hb)
o su icien ly small h, k > 0. In he e, un
iis he exac solu ion o (1) in (xi, τn)
and Lh,k is he ini e di e ence scheme ha excludes he unca ion e o o
o de O(ka+hb).
•S abili y
To es ablish s abili y o (2), we a e going o ake in o accoun compu e e alu-
a ed ec o including he ounding e o s ˆ
Un. So
Aˆ
Un+1 =Bˆ
Un+dn+ n.
whe e ndeno es he ounding e o s. The e o ec o is,
en=ˆ
Un−Un,
ac s in acco ding o
Aen+1 =Ben+ n.
To simpli y his ,we a e going o assume ha e06= 0 which means ha he e is
al eady a ounding e o when e alua ing he ini ial condi ion. Simul aneously,
we a e going o assume ha he ma ix ec o mul iplica ion o ob ain Un+1
wo ks p ecisely, So
n= 0 n∈[0, M −1].
We ha e he e o :
en+1 =A−1Ben= (A−1B)2en−1=··· = (A−1B)n+1e0.
Fo us o ha e a s able sys em, p eceding e o s ha e o be damped and he e o e
we need
(A−1B)n+1e0→0as n → ∞.
61
(a) RAPM model (M=.01, C = 30). (b) Ba les’ and Sone ’s model (a=.02).
(c) ψ(x) = xselec ed as he Iden i y (a=.02). (d) Leland’s model (δ =.01, κ =.05).
Figu e 6.1: Value o Eu opean Call Op ion p ice V(S, ) o Nonlinea Black-Scholes
models o a ious ola ili y models o spa ial s ep h=.1 and ime s ep k=.0001
wi h he C ank-Nicolson Me hod.
The impac o ansac ion cos s which a e modeled by he ola ili ies and e alu-
a ed wi h he C ank-Nicolson ini e di e ence scheme can be seen in he nex igu e,
we plo he di e ence be ween he Eu opean Call Op ion p ice wi h
Vnonlinea (S, )−Vlinea (S, ).
ansac ion cos s and he Eu opean Call Op ion p ice wi hou he ansac ion cos s.
We ge a signi ican p ice de ia ion be ween he classical (linea ) Black-Scholes Model
and he nonlinea model.
68
(a) ψ(x) = xselec ed as he Iden i y (a=.02). (b) Ba les’ and Sone ’s model (a=.02).
(c) RAPM model (M=.01, C = 30). (d) Leland’s model (δ =.01, κ =.05).
Figu e 6.2: The impac o ansac ion cos s Nonlinea Vnonlinea (S, )−Vlinea (S, )
wi h he C ank-Nicolson Me hod.
The di e ence ha we ha e is no symme ic o all he ansac ion cos models.
Bu dec eases close o he expi y da e. And his is he expec ed esul . We ha e
plo ed all he nonlinea models wi h he linea model in igu e (6.3). Now o each
ola ili y model and each di e ence scheme, we compa e he de ia ion o accu acy
o he abo e compu a ion a = 0. We use he in ini y no m l∞. I we educe he
spa ial s ep size o h=.001 o h=.0001, i imp o es he accu acy conside ably.
Howe e , i inc eases he compu a ional ime monumen ally. The de ia ion om he
linea model is e alua ed only o he s ock p ice S( ) = 269 a = 0. We can see
he di e ence be ween he Models om he able.
Vola ili y Model De ia ion om he linea model sn
i= 0
Iden i y .00068
Leland .02945
RAPM .03813
Table 6.1: l∞de ia ion o di e en ola ili y models o he s ock p ice S( ) = 269
a = 0.
69
Figu e 6.3: Value o Eu opean Call Op ion p ice V(S, 0) o di e en ola ili y models
s he linea model wi h he C ank-Nicolson Me hod.
The plo (6.3) shows he p ice o a Eu opean Call op ion o V(S, 0) ha is a
= 0, and how all he ansac ion cos models con e ge. We ha e gi en he algo-
i hms o explici and implici me hods oo. We use hem o implemen hem in o
MATLAB oo. A i s we plo he Eu opean Call Op ion p ice V(S, ) o di e en
ola ili y models in Implici Me hod. Then we gi e a Nonlinea Vs Linea g aph
simila o he C ank-Nicolson in implici me hod. We epea his o explici me hod
oo. They gi e he same esemblance wi h he C ank-Nicolson g aphs. Bu C ank-
Nicolson gi es be e nume ical esul s.
70
(a) ψ(x) = xselec ed as he Iden i y (a=.02). (b) Ba les’ and Sone ’s model (a=.02).
(c) RAPM model (M=.01, C = 30). (d) Leland’s model (δ =.01, κ =.05).
Figu e 6.4: Value o Eu opean Call Op ion p ice V(S, ) o Nonlinea Black-Scholes
models o a ious ola ili y models o spa ial s ep h=.1 and ime s ep k=.001
wi h he Implici Me hod.
Figu e 6.5: The Iden i y model ψ(x) = xwi h pa ame e (a=.02) o he
(Vnonlinea (S, )−Vlinea (S, )) plo o spa ial s ep h=.1 and ime s ep k=.001
wi h he Implici Me hod.
71
(a) ψ(x) = xselec ed as he Iden i y (a=.02). (b) Ba les’ and Sone ’s model (a=.02).
(c) RAPM model (M=.01, C = 30). (d) Leland’s model (δ =.01, κ =.05).
Figu e 6.6: Value o Eu opean Call Op ion p ice V(S, ) o Nonlinea Black-Scholes
models o a ious ola ili y models o spa ial s ep h=.1 and ime s ep k=.00001
wi h he Explici Me hod.
Figu e 6.7: The Iden i y model ψ(x) = xwi h pa ame e (a=.02) o he
(Vnonlinea (S, )−Vlinea (S, )) plo o spa ial s ep h=.1 and ime s ep k=.001
wi h he Explici Me hod.
72
6.11 Conclusion
We ha e ho oughly in es iga ed he Nonlinea Black-Scholes Equa ion nume ically.
A i s we show how he Nonlinea Black-Scholes Equa ion can be ep esen ed as
sys em o equa ions. We use ma cies ins ead o Sys em o equa ions. Then we
gi e g aphical ep esena ion o he solu ion in implici , explici and C ank-Nicolson
Me hod. We gi e a Compa ison be ween he di e en ansac ion cos modeled by
ola ili ies.
73
Chap e 7
F ac ional Black-Scholes Equa ion
7.1 In oduc ion
In quan i a i e inance, op ion p icing is a majo p oblem. The e a e models o he
han Black-Scholes model o p icing op ions which gi e be e esul s; one such
model is known as Fini e Momen Log S able (FMLS) which can be pu in w i ing
as ac ional di usion class equa ion [40]. F ac ional Calculus is he calculus o he
di e en ia ion and in eg a ion whe e he o de is a ac ional numbe [56]. We seek
o disco e he analy ical solu ion o Black-Scholes Equa ion’s F ac ional De i a i es
by he use o Adomian Decomposi ion Me hod.
7.2 T ans o med Black-Scholes Equa ion
He e, we ecall om he second chap e whe e he analy ical solu ion o Linea Black-
Scholes Equa ion was gi en. Le V(S, ) be he op ion p ice , hen
∂V
∂ +1
2σ2∂2V
∂S2+ S ∂V
∂S − V = 0.(7.1)
Ha ing he ini ial and bounda y condi ions which a e:
V(0, τ)=0,
V(S, τ)≈S, as S → ∞,
V(S, T) = max(S−K, 0).
74
He e, we ha e Kis he s ike p ice, σis he ola ili y, Tis he expi y ime and is
he isk ee in e es a e. Then we apply changes o he a iables:
S=Kex,
=T−τ
(1/2)σ2,
V=K (x, ).
Which gi es us he
∂
∂τ =∂2
∂x2+ (k−1)∂
∂x −k ,
whe e k=2
σ2. Now The ini ial condi ion is:
(x, 0) = max(ex−1,0).
7.3 F ac ional De i a i e
The mos common ac ional De i a i es a e:
Capu o: The ac ional de i a i e o g(x) is de ined as [58]:
Dβg(x) = Kn−βDng(x) = 1
Γ(n−β)Zx
0
(x− )n−β−1gn( )d .
o n−1< β ≤n,n∈N,x > 0.
Mi ag-Le le Func ion: This unc ion Mβ(x) is de ined as [54]:
Mβ(x) = ∞
X
j=0
xj
Γ(βj + 1),
whe e β > 0 and x∈C.
7.4 F ac ional Black-Scholes Equa ion o Time
He e , we asses he ac ional Black-Scholes Equa ion o ime [40]:
∂β
∂τβ=∂2
∂x2+ (k−1)∂
∂x −k , 0< β ≤1.(7.2)
I has he ini ial condi ion (x, 0) = ex−1, x > 0. Adomian Decomposi ion Me hod
is used in he equa ion (7.2) o ge :
∂
∂τ =∂1−β
∂τ1−β∂2
∂x2+ (k−1)∂
∂x −k .
75
We a e going o in eg a e on he bo h sides wi h espec o τ o ge :
(x, τ) = (x, 0) + Zτ
0∂1−β
∂s1−β∂2 (x, s)
∂x2+ (k−1)∂ (x, s)
∂x −k (x, s)ds.
We deno e (x, 0) = 0. Then
(x, τ) = Zτ
0∂1−β
∂s1−β∂2 0(x, s)
∂x2+ (k−1)∂ 0(x, s)
∂x −k 0(x, s)ds.
The epe i i e me hod yields o m > 1
m(x, τ) = Zτ
0∂1−β
∂s1−β∂2 m−1(x, s)
∂x2+ (k−1)∂ m−1(x, s)
∂x −k m−1(x, s)ds.
(7.3)
The e o e om Adomian Decomposi ion Me hod, we ha e he solu ion o (7.2)
by
(x, τ) = ∞
X
m=0
m(x, τ).
We a e going o use he ini ial condi ion:
(x, 0) = ex−1, x > 0.
So,
0=ex−1.
F om (7.3) , we e alua e he es o he e ms o he solu ion.
1(x, τ) = Zτ
0∂1−β
∂s1−β∂2 0
∂x2+ (k−1)∂ 0
∂x −k 0ds.
=−ex−kτβ
Γ(β+ 1) + (ex−1) −kτβ
Γ((β+ 1)
2(x, τ) = Zτ
0∂1−β
∂s1−β∂2 1(x, 0)
∂x2+ (k−1)∂ 1(x, 0)
∂x −k 1(x, 0)ds.
=−ex(−kτβ)2
Γ(2β+ 1) + (ex−1) (−kτβ)2
Γ((2β+ 1)
3(x, τ) = Zτ
0∂1−β
∂s1−β∂2 2(x, 0)
∂x2+ (k−1)∂ 2(x, 0)
∂x −k 2(x, 0)ds.
=−ex(−kτβ)3
Γ(3β+ 1) + (ex−1) (−kτβ)3
Γ((3β+ 1).
76
So, he solu ion o he equa ion (7.2) by he Adomian Decomposi ion Me hod,
(x, τ) = ∞
X
m=0
m(x, τ)
=ex(−kτβ
Γ(β+ 1) −(−kτβ)2
Γ(2β+ 1) +(−kτβ)3
(Γ3β+ 1) −···)
+ (ex−1)(1 −−kτβ
Γ(β+ 1) +(−kτβ)2
Γ(2β+ 1) +···)
=ex(1 −Mβ(−kτβ)) + (ex−1)Mβ(−kτβ)
=ex−Mβ(−kτβ).
whe e Mβ(x) is called he Mi ag-Le le unc ion o one pa ame e . (x, τ) = ex−
Mβ(−kτβ) is he exac solu ion o (7.2). And Mi ag-Le le unc ion has he p ope y
[57, 46] when β→1, he solu ion (7.2) ag ees wi h he exac solu ion o he Black-
Schole PDE.
7.5 F ac ional Black-Scholes Equa ion o Space
He e, we conside he ac ional Black-Scholes PDE o space [77] . In his cas, We
ha e ac ional de i a i e in space. This is in he sense o Riemann-Liou ille. The
Le y densi y o a Log S able is gi en below:
CLS(x) = (Bq |x|−1−βx < 0,
Bpx−1−βx < 0.
He e B > 0, p, q ∈[−1,1], p +q= 1 and 0 < β ≤2. We eplace p= 0 and q= 1.
Then his p ocedu e con e s in o he well known Fini e Momen Log S able Me hod.
The ollowing ac ional pa ial di e en ial equa ion is gi en by (Fo Eu opean Op ion
P ice):
∂V (x, τ)
∂τ + +1
2σβsec (βπ
2)∂V (x, τ)
∂x −1
2σβsec (βπ
2)∂βV(x, τ)
∂+xβ= V (x, τ) (7.4)
He e ∂βV(x,τ)
∂+xβis called he le Riemann-Liou ille ac ional de i a i e , is isk-less
in e es a e and σis ola ili y. Now , whene e β→2, (7.4) ma ches wi h Black-
Scholes PDE.
We sol e his e y equa ion using Adomian Decomposi ion Me hod. A i s ,we se
V(x, 0) = ex−1.
77
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