Physical principles in option pricing: analysis and valuation based on the Black-Scholes model

G adu amaie ako lana / T abajo in de g ado
Fisikako g adua / G ado en F´ısica
Physical P inciples in Op ion P icing
Analysis and alua ion based on he Black-Scholes model
Egilea/Au o :
I˜nigo Jos´e P´e ez G´amiz
Zuzenda ia/Di ec o :
Hegoi Manzano Mo o
Leioa, 2022eko ekaina en 24a / Leioa, 24 de junio de 2022
Con en s
1 In oduc ion 2
2 Objec i es 3
3 Basic inancial concep s 4
3.1 Wha isanop ion?............................... 4
3.2 Vanillaop ions ................................. 4
3.2.1 Callop ions............................... 5
3.2.2 Pu op ions ............................... 6
3.3 In e es a e................................... 6
3.4 A bi age .................................... 7
3.5 Di idends .................................... 7
3.6 Po olio..................................... 7
3.7 Long posi ion, sho posi ion . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 B ownian mo ion 8
4.1 De ini ion .................................... 8
4.2 Di usionequa ion................................ 8
4.3 Wiene p ocess ................................. 10
5 Asse p icing 10
5.1 Randomwalkmodel .............................. 10
5.2 I ˆo’sLemma................................... 11
5.3 Geome ic B ownian mo ion . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6 The Black-Scholes model 15
7 The Black-Scholes o mula o Eu opean op ions 17
7.1 De i a ion.................................... 17
7.2 Nume icalexample ............................... 21
8 Ame ican op ions 22
8.1 Gene alconcep s ................................ 22
8.2 Ame icanpu op ions.............................. 23
8.3 Solu ion o he ee bounda y p oblem . . . . . . . . . . . . . . . . . . . . 25
8.4 Fini e-di e ence o mula ion: The C ank-Nicolson me hod . . . . . . . . . 27
8.5 Ma ix o mula ion ............................... 30
8.6 TheLUme hod................................. 31
8.7 Py honcode................................... 34
8.8 Nume icalexample ............................... 36
9 Conclusions 38
1
1 In oduc ion
Nowadays he wo ld o inance is inc easingly based on ma hema ical and physical models.
Tha is why many inancial companies hi e ma hema icians and physicis s o build and
analyse hose models. The b anch o inance ha deals wi h hese asks is known as
quan i a i e inance.
I is said ha quan i a i e inance was bo n in 1900 when Louis de Bachelie published
his Ph.D. hesis The heo y o specula ion. He in oduced he concep o B ownian mo ion
o app oxima e asse p ices andom pa h. The asse could be, o ins ance, a s ock.
F om ha momen , many o he heo ies ha e been de eloped in an a emp o p edic
he beha iou o inancial ma ke s and ins umen s. In 1973 Fische Black and My on
Scholes in oduced he Black-Scholes model o p ice de i a i es wi hin hei publica ion
The p icing o op ions and co po a e liabili ies. They we e awa ded in 1997 wi h a Nobel
P ize o his wo k. Al hough i had ce ain limi a ions, The Black-Scholes model became
a e e ence o many o he subsequen models. The cons an changes in ma ke s equi e
he e olu ion o models and he appea ance o new upda ed ones. The main ocus ield has
always been o manage he ola ili y and mos o he buil models a e based on s ochas ic
p ocesses [1].
This p ojec explo es an applica ion o physics o s udy inancial sys ems. In pa -
icula , we will ocus on he physical p inciples in op ion p icing. Op ions a e inancial
ins umen s whose p ice depends on he p ice o an unde lying asse . This means ha we
will ha e o model he beha iou o he unde lying asse o be able o model he p ice o
he op ion. Some concep s like B ownian mo ion and di usion p ocess can be ex apo-
la ed o hose asks as we will see. The e is mo e han one model o p ice op ions, bu we
will only analyse he Black-Scholes model as i can be ela ed o physical concep s. This
model makes ce ain assump ions ha a e no e y ealis ic in eal ma ke s, bu i gi es
a good quali a i e o e iew o how op ions a e p iced. In ac , many subsequen mo e
sophis ica ed models a e based on i .
Fi s o all, we will in oduce in Sec ion 3 some basic inancial no ions which a e
necessa y o unde s and he es o he p ojec . I is e y impo an o assimila e wha
is an op ion and how does i wo k.
In Sec ion 4 we will explain he physical concep s. This includes he de ini ion o
B ownian mo ion, i s ela ion wi h di usion p ocesses, inding a solu ion o he di usion
equa ion and making a ma hema ical o mula ion o he B ownian mo ion ( his is known
as a Wiene p ocess). These aspec s will be ex apola ed la e o inance h oughou he
p ojec .
Once he inancial and physical concep s a e clea , we will y o model he beha iou
o he unde lying asse p ice in Sec ion 5. We will assume a andom walk model o i .
This model is based on he de ini ion o Wiene p ocess made in Sec ion 4.3. We will ind
ou ha he asse p ice ollows a Geome ic B ownian mo ion and ha he p obabili y
densi y unc ion o i s loga i hm su e s a di usion p ocess. Howe e , we will p e iously
need o in oduce a ma hema ical concep called I ˆo’s Lemma.
Knowing he beha iou o he unde lying asse p ice, we will be able o build he
Black-Scholes model ha leads o he pa ial di e en ial equa ion used o p ice op ions.
This is done in Sec ion 6.
Ou nex s ep will be o ind a solu ion o he Black-Scholes equa ion o p ice Eu opean
op ions in Sec ion 7. This is known as he Black-Scholes Fo mula. We will ans o m he
Black-Scholes equa ion in o a di usion equa ion and use he solu ion ound in Sec ion
2
4.2. The p oblem o aluing he op ion will be ea ed as a di usion p ocess. We will also
analyse a nume ical example o he de i ed solu ion.
In Sec ion 8 we will s udy he case o p icing Ame ican op ions. We will see ha we
canno explici ly sol e he Black-Scholes equa ion and ha we ha e o deal wi h a ee
bounda y p oblem. We will ans o m he ee bounda y p oblem in o a linea comple-
men a i y p oblem o elimina e he dependence on he ee bounda y. A e ha , we will
w i e he p oblem using he ini e-di e ence o mula ion and he C ank-Nicolson scheme.
An algo i hm which includes he LU me hod will be buil o sol e he p oblem. Finally,
we will ansla e his algo i hm in o ou own Py hon code and un some simula ions.
To inish he p ojec , we will make some conclusions based on he analysed opics.
2 Objec i es
The main objec i e o he p ojec is o es ablish a connec ion be ween physics and pinance,
analysing he physical p inciples in Op ion P icing. Mo eo e , he p ojec s also has he
ollowing pa icula objec i es:
1. Lea n ha he asse p ice beha iou can be modelled wi h a B ownian mo ion and
isualize he di usion p ocess su e ed by he p obabili y densi y unc ion o i s
loga i hm.
2. De i e he Black-Scholes equa ion used o p ice op ions and sol e i as a di usion
p ocess o p ice Eu opean op ions.
3. Unde s and he ee bounda y p oblem o Ame ican op ions and sol e i using he
C ank-Nicolson o mula ion and an algo i hm based on he LU me hod.
4. W i e ou own Py hon code wi h he men ioned algo i hm and un a simula ion
wi h a nume ical example o analyse esul s.
5. Deepen in ma hema ical esolu ions, unde s anding he ma hema ical de elopmen s
beyond he le el o de ail gi en by he bibliog aphy.
3
3 Basic inancial concep s
Be o e we s a de eloping ou p ojec , we need o in oduce some basic inancial concep s
ha will help us o unde s and he es o he opics.
3.1 Wha is an op ion?
An op ion is a con ac be ween wo pa ies, he holde and he w i e , on an unde lying
asse . The holde pays a compensa ion ( he p emium) o he w i e o ha e he igh ,
bu no he obliga ion, o buy o sell he unde lying asse a an ag eed p ice ( he s ike
o exe cise p ice) by a speci ic da e (expi a ion da e o ma u i y) [2]. In case he
holde wan ed o exe cise his igh o buy o sell he unde lying asse , he w i e would
ha e he obliga ion o do he opposi e mo emen , ha is, sell o buy i . A he same ime,
he holde o he op ion can sell his igh o execu ion o a hi d pa y in exchange o a
p emium be o e he expi a ion da e. The unde lying asse is usually a s ock o a bond,
bu i could also be an index, an in e es a e o e en commodi ies. To cla i y concep s,
we a e going o explain hem one by one:
•The op ion is he con ac ha depends on he unde lying asse , which can be a
s ock, a bond,...
•The holde o he op ion is he one who has he igh o execu e he op ion o buy
o sell he unde lying asse . The holde can also sell ha igh o ano he pe son.
•The w i e o he op ion is he one who ecei es a p emium o w i ing he op ion
bu la e depends on wha he holde decides.
•The p emium is he alue ha has o be paid o acqui e he posi ion o holde ,
ha is, i is he p ice o he op ion. P icing an op ion means inding he p emium.
•The s ike o exe cise p ice is he ag eed amoun o money ha he holde
pays/ ecei es when execu ing he op ion o buy/sell he unde lying asse .
•The expi a ion da e o ma u i y is he da e by which he holde can exe cise he
op ion.
The e a e a ious ypes o op ions a ending o di e en pa ame e s. The i s main
di e en ia ion is made be ween call and pu op ions. Call op ions gi e he holde he
igh o buy he unde lying asse , whe eas pu op ions gi e he chance o sell i . The
simples ones a e he Eu opean call and pu op ions, whe e he holde can exe cise his
igh jus a expi a ion da e. The e a e also Ame ican call and pu op ions. In his case,
he igh o buy o sell he unde lying asse can be execu ed a any ime be o e expi a ion
da e. Eu opean and Ame ican op ions a e ypically called Vanilla op ions i hey ha e
no o he special condi ion o exe cise he op ion. Apa om Vanilla op ions, he e a e
o he ypes o op ions like Exo ic op ions, whe e he condi ions o execu ion a e di e en
[3]. Howe e , we a e no going o wo k wi h hem in his p ojec .
3.2 Vanilla op ions
Vanilla op ions a e he simples ype o op ions and a e usually aded on an exchange
(ma ke place whe e inancial ins umen s a e aded). They a e di ided in o call and pu
op ions.
4

3.2.1 Call op ions
As i has been explained be o e, call op ions gi e he holde he igh o buy he un-
de lying asse . Depending on whe he i is Ame ican o Eu opean, he pu chase can be
done be o e o jus a expi a ion da e. Call op ions a e execu ed when he p ice o he
unde lying asse is highe han he s ike p ice. This means we a e buying i cheape han
he ac ual p ice.
We a e now going o see an example o be e unde s and how do call op ions wo k.
Le us conside a holde A and a w i e B. They ag ee on an Eu opean call op ion o e a
s ock whose cu en p ice is 100 $. The expi a ion da e is 6 mon hs and he s ike p ice
is also 100 $. The holde pays 5 $ o he w i e as he p emium o he op ion. As i is an
Eu opean op ion, he e a e wo possible scena ios a expi a ion da e. I he s ock p ice
goes up o, o ins ance, 110 $, he holde would execu e his igh o buy i o 100 $. He
would ge a payo o 10 $ and a p o i o 5 $ aking in o conside a ion he 5 $ he had
paid as he p emium. I , on he con a y, he s ock alue goes down o 90 $, he holde
would no buy he s ock o 100 $. In ha case, he would lose he 5 $ ini ially paid o
he w i e . I he op ion we e Ame ican ins ead o Eu opean, i could be exe cised a any
ime be o e expi a ion da e, bu we will see la e ha i should only be done a ma u i y
o call op ions.
We can deduce and plo in Figu e 1.a a gene al exp ession o he payo and p o i o
an Eu opean call op ion a ma u i y om he poin o iew o he holde [3].
Payo : max{S(T)−K, 0},P o i : max{S(T)−K, 0}−C( i, T, K) (1)
whe e he 0 ep esen s he case o no execu ion and
•T is he expi a ion da e o ma u i y
•K is he exe cise o s ike p ice
•S(T) is he p ice o he unde lying asse a ma u i y
•C( i,T,K) is he p ice o he call op ion ( he p emium) when i is bough a ini ial
ime i
Rega ding Figu e 1.a, we can isualize ha he op ion has o be exe cised in some cases
despi e he p o i is nega i e. In hose si ua ions, exe cising he op ion means educing
he loss o money.
In ou example, we had ha K= 100 $, i=0, T= 0.5 yea , C( i, T, K) = 5 $. I we
conside he i s case in which he s ock p ice aised o 110 $, hen S(T) = 110 $ and so
he payo and p o i a e
Payo : max{110 −100,0}= 10 $, P o i : max{110 −100,0}−5 = 5 $ (2)
and, in he o he case, whe e he s ock p ice ell o 90 $,
Payo : max{90 −100,0}= 0 $, P o i : max{90 −100,0}−5 = −5 $ (3)
We can also de ine he payo unc ion in gene al a a ime o bo h Eu opean and
Ame ican call op ions om he poin o iew o he holde :
Payo : max{S( )−K, 0}(4)
The shape o his payo unc ion will be he same as he shape o he payo unc ion
in Figu e 1.a, bu o ano he S( ).
5
3.2.2 Pu op ions
Pu op ions g an he holde he igh o sell he unde lying asse . Once again, depend-
ing on i i is Ame ican o Eu opean, he sale can be done be o e o a ma u i y. In his
case, he holde would only sell he unde lying asse i i s p ice wen below he s ike
p ice.
We can also ga he an exp ession o he payo and p o i o an Eu opean pu op ion
a expi a ion da e om he poin o iew o he holde and plo hem in in Figu e 1.b [3].
Payo : max{K−S(T),0},P o i : max{K−S(T),0}−P( i, T, K) (5)
The a iables K,T, i,S(T) a e he same as in he case o he call op ion and
P( i, T, K) is he p ice o he pu op ion ( he p emium) when i is bough a ini ial ime
i. Fo Eu opean and Ame ican pu op ions, he gene al payo unc ion a a ime om
he poin o iew o he holde would be:
Payo : max{K−S( ),0}(6)
Once mo e, he shape o his payo unc ion will be he same as he shape o he
payo unc ion in Figu e 1.b, bu o ano he S( ).
(a) Call op ion (b) Pu op ion
Figu e 1: Payo unc ion and p o i unc ion o Eu opean op ion a ime T om he
poin o iew o he holde .
3.3 In e es a e
When bo owing money o deposi ing i in a bank, he e is a cha ge o bene i o he
ope a ion a he end. The pa ame e ha measu es his change o money is he in e es
a e. Fo ins ance, i an indi idual in oduces an amoun o money X( 0) in he bank a
ime 0, he money a ime conside ing a con inuous and cons an in e es a e will be
[4]:
X( ) = X( 0)e ( − 0)(7)
This ela ion can also be exp essed in di e en ial o m:
dX
X= d (8)
6
The opposi e calcula ion is known as he p esen alue:
X( 0) = X( )e− ( − 0)(9)
When an in es men has ze o isk, he in e es a e is called isk- ee a e.
3.4 A bi age
The concep o a bi age e e s o he possibili y o making ins an aneous isk-less p o i
wi h an in es men . This is ob iously no a desi able si ua ion in inancial ma ke s. In
ac , mos o he inancial heo ies a e de eloped assuming he absence o a bi age.
I is some imes possible o make isk-less p o i in an in es men , o ins ance, deposi -
ing money in he bank a a isk- ee a e (8). Howe e , his is no a si ua ion o a bi age
since he p o i is no ins an aneous [4].
3.5 Di idends
Di idends a e ea nings ha someone ecei es o owning an asse . The ypical case is
he company ha dis ibu es some o i s ea nings as di idends be ween i s sha eholde s.
Di idends a e ecei ed a a speci ic ime o pe iodically. In ou opic o s udy, di idends
will be gi en by he unde lying asse .
3.6 Po olio
Apo olio is a se o inancial ins umen s like op ions, s ocks, bonds o commodi ies,
which aims o p o ide bene i s. Po olios ypically end o di e si y in es men s in o de
o educe he isk o loss [5].
3.7 Long posi ion, sho posi ion
When alking abou op ions, a long posi ion e e s o he si ua ion o being he holde
o an op ion. By con as , a sho posi ion means ha he in es o sells he igh o
execu ion o he op ion.
In he case o s ocks, he long posi ion is o buy he s ock, bu he sho posi ion,
which is usually known as sho selling, consis s on selling a s ock ha he in es o does
no own. The objec i e o sho selling is o bene i om a all in he p ice o he s ock.
The in es o bo ows a s ock and immedia ely sells i o ano he pe son and, a e some
ime, pays he lende he p ice o he s ock a ha ime. I he p ice o he s ock has
allen, he in es o ob ains p o i [6].
7
4 B ownian mo ion
Once we ha e he main inancial concep s, we a e going o explain now se e al physical
no ions ha will be used h oughou he p ojec . Fi s o all, we need o known wha
is a B ownian mo ion and how does i go e n a di usion p ocess. We will also ind a
solu ion o he di usion equa ion. Mo eo e , we a e going o in oduce he ma hema ical
o mula ion o he B ownian mo ion, known as he Wiene p ocess.
4.1 De ini ion
The B ownian mo ion is he mo emen o pa icles in a luid, which can be a gas o
a liquid. Collisions wi h he molecules o he luid make his mo emen andom and
unp edic able.
I was he bo anis Robe B own he i s o obse e he mo emen in 1827 when
looking a pollen g ains suspended in wa e . Tha is why i is called B ownian mo ion.
He disco e ed minuscule pa icles andomly mo ing in a wa e d op. F om ha momen ,
many quali a i e hypo hesis we e p oposed by scien is s, bu i was no un il 1905 when
Albe Eins ein de eloped a quan i a i e model. He based his heo y in h ee main
p inciples [7]:
•The exis ence o he pa icles.
•The mo emen o he pa icles in a luid is due o he eno mous numbe o collisions
wi h he luid molecules.
•The mo emen o he molecules is so complex ha i can jus be p obabilis ically
desc ibed as a esul o many independen hi s.
As a consequence o he huge numbe o collisions pe uni ime ha su e s a pa icle,
Eins ein s udied he p oblem as a whole a he han indi idually o each pa icle. He
wo ked wi h a densi y unc ion o he pa icles u(x, τ) and disco e ed ha i sa is ied he
di usion equa ion [8]:
∂u
∂τ =D∂2u
∂x2(10)
whe e xis he posi ion a iable ha ollows a B ownian mo ion and τis he ime. This
equa ion sugges s us ha di usion p ocesses a e go e ned by he B ownian mo ion. The
densi y unc ion o he pa icles su e s a di usion as a esul o he B ownian mo ion
ollowed by he pa icles. The di usion p ocess can be seen as he mac oscopic mani es-
a ion o he mic oscopic B ownian mo ion o pa icles in a luid [9]. We a e now going o
analyse he di usion equa ion in de ail because i will be essen ial h oughou he p ojec .
4.2 Di usion equa ion
The di usion equa ion is a pa ial di e en ial equa ion wi h se e al applica ions in
physics, enginee ing and e en inance as we will see in his wo k. I is widely used in
physics o model he low o hea in a con inuous medium [4]. Tha is why i is some imes
known as he hea equa ion. The di usion equa ion gi es he ime e olu ion (τ) o he
p obabili y densi y unc ion uo a a iable x ha ollows a B ownian mo ion, as we ha e
ecen ly seen. We a e going o de i e a solu ion o he equa ion.
8
6 The Black-Scholes model
Once we ha e s udied he beha iou o he asse p ice, we a e going o apply he Black-
Scholes analysis ha leads o he pa ial di e en ial equa ion used o p ice op ions. The
Black-Scholes model was de eloped in 1973 by Fische Black and My on Scholes and
became a e e ence o many o he subsequen models. They ecei ed a Nobel P ize
in 1997 o hei wo k. Ne e heless, as we ha e said be o e, he model makes ce ain
assump ions ha a e no e y ealis ic in eal ma ke s, bu quali a i ely is a g ea i s
app oxima ion. Be o e in oducing i , we a e going o explain he assump ions [4]:
•The unde lying asse p ice ollows he Geome ic B ownian mo ion. This does no
mean ha he Black-Scholes analysis canno be applied wi h any o he model ha
is no he one o he andom walk p e iously explained.
•The isk- ee in e es a e and he ola ili y σa e known du ing he li e o he
op ion.
•The unde lying asse pays no di idends.
•I is possible o buy o sell a ac ional numbe o he unde lying asse and sho
selling is allowed.
•The e is no possibili y o a bi age.
•The e a e no ansac ion cos s o buying o selling an op ion o unde lying asse .
Le us suppose ha he p ice o an op ion V(S, ) (does no ma e i i is a call o pu
op ion) depends only on he unde lying asse p ice Sand ime . Applying I ˆo’s Lemma
(30), aking in o conside a ion ha X≡S,λ≡µS and β≡σS, we can w i e [4]:
dV =∂V
∂ d +∂V
∂S dS +1
2σ2S2∂2V
∂S2d (44)
I we in oduce ela ion (18) o dS in o (44), we each o he exp ession ha gi es
he andom walk ollowed by he op ion p ice V:
dV =∂V
∂ +µS ∂V
∂S +1
2σ2S2∂2V
∂S2d +σS ∂V
∂S dW (45)
Now we a e going o build a po olio made up o one op ion (long posi ion) and he
sho sell o ∆ sha es o he unde lying asse . The alue o he po olio, Π, in his case
is:
Π = V−∆S(46)
and he di e en ial o his alue will be:
dΠ = dV −∆dS (47)
Subs i u ing exp essions (18) and (45) in o (47), we ge ha he alue o he po olio
Π also ollows a andom walk:
dΠ = ∂V
∂ +µS ∂V
∂S +1
2σ2S2∂2V
∂S2−µ∆Sd +σS ∂V
∂S −∆dW (48)
15

Rega ding his exp ession, we can choose ∆ so ha we elimina e he andom compo-
nen con aining dW:
∆ = ∂V
∂S (49)
As we can see, in his case ∆ is he a e o change o he op ion p ice Vwi h espec
o he unde lying asse p ice S. The esul is ha he change in he alue o he po olio
is de e minis ic:
dΠ = ∂V
∂ +1
2σ2S2∂2V
∂S2d (50)
We a e now going o conside some a gumen s ela ed o a bi age. We suppose ha
he e a e no ansac ion cos s o buying o selling an op ion o unde lying asse , as said
be o e. Acco ding o ela ion (8), when in es ing an amoun Π a a isk- ee a e , he
e u n o he in es men a e a ime will be:
dΠ
Π= d (51)
As a esul , he change in Π is:
dΠ = Π d (52)
The igh -hand side o (50) has o be equal o he igh -hand side o (52) so ha he e
is no a bi age. I i was g ea e , anyone could bo ow an amoun Π a a isk- ee a e
and in es i in he po olio. The esul would be a iskless p o i because he inc emen
in he alue o he po olio ( igh -hand side o (50)) is g ea e han he amoun ha has
o be e u ned o bo owing ( igh -hand side o (52)). This means ha he e is a bi age.
I , in con as , he igh -hand side o (50) was smalle han he igh -hand side o (52),
i would be possible o sho sell he po olio (sell he op ion Vand buy ∆ sha es o S)
and in es ha Π amoun in a bank a a isk- ee a e . Once again, he e is iskless
p o i and, as a esul , a bi age.
As we ha e s a ed ha he e is no place o a bi age, we mus impose ha :
Π d =∂V
∂ +1
2σ2S2∂2V
∂S2d (53)
Di iding by d a bo h sides and in oducing exp essions (46) and (49) in o (53):
V−∂V
∂S S =∂V
∂ +1
2σ2S2∂2V
∂S2(54)
we each o:
∂V
∂ +1
2σ2S2∂2V
∂S2+ S ∂V
∂S − V = 0 (55)
This is known as he Black-Scholes pa ial di e en ial equa ion (PDE). I is a
linea , pa abolic and backwa d in ime equa ion. Any op ion whose p ice only depends
on ime and he unde lying asse p ice S, mus e i y his equa ion, as long as he
assump ions made so a a e e i ied. Consequen ly, sol ing he Black-Scholes equa ion
gi es way o p icing op ions.
16
I is in e es ing o ema k ha µ, he a e age a e o g ow h o he unde lying asse
p ice S, does no appea in he equa ion. The only pa ame e om he s ochas ic di e -
en ial equa ion (18) ha a ec s he p ice o he op ion is he ola ili y σ. This means
ha , al hough he e migh be a disc epancy be ween people in he es ima ion o µ, he
p ice o he op ion would be he same anyway [4].
7 The Black-Scholes o mula o Eu opean op ions
7.1 De i a ion
We ha e al eady ound a PDE o p ice op ions, so now i is ime o look o solu ions
ha sa is y i . The Black-Scholes o mula is a well-known exp ession ha gi es p ice
o Eu opean op ions. I can be ob ained sol ing he Black-Scholes equa ion (55) wi h
bounda y and inal condi ions.
We a e going o de i e he o mula o an Eu opean pu op ion P(S, ). Remembe
ha an Eu opean pu op ion g an ed i s holde he igh o sell he unde lying asse only
a expi a ion da e T. We s a om he Black-Scholes equa ion
∂P
∂ +1
2σ2S2∂2P
∂S2+ S ∂P
∂S − P = 0 (56)
The equa ion is backwa d in ime due o he opposi e sign o e m ∂P/∂ wi h espec
o he o he pa ial de i a i es ∂2P/∂S2and ∂P/∂S. Fo ha eason, we need a inal
condi ion o sol e he equa ion, his is, a condi ion a expi a ion da e T. Rega ding
exp ession (5) o he payo o an Eu opean pu op ion, we de e mine ha he op ion
p ice a Thas o be equal o he payo o a oid a bi age [4]:
P(S, T) = max{K−S, 0}(57)
We also equi e bounda y condi ions. Acco ding o he inal condi ion (57), when he
asse p ice is 0 a expi a ion da e T, he p ice o he op ion is K:P(0, T) = K. I we wan
o de e mine P(0, ), we jus ha e o calcula e he p esen alue (9) o P(0, T). Assuming
a cons an in e es a e , we ge ha he i s bounda y condi ion is:
P(0, ) = Ke− (T− )(58)
On he o he hand, when he asse p ice ends o ∞, he pu op ion is e y unlikely
o be exe cised, so i loses i s alue and he second bounda y condi ion is:
P(S, )−→ 0,as S−→ ∞ (59)
Once we ha e he inal and bounda y condi ions, we a e going o sol e he equa ion.
We a e going o make some ans o ma ions o y o con e he Black-Scholes equa ion
o he pu op ion (56) in o a di usion equa ion(10). The solu ions o he o wa d
di usion equa ion a e al eady known o us. Howe e , we ha e seen ha he Black-
Scholes is backwa d. Since we wan o con e i in o a o wa d di usion equa ion, we
in oduce a new ime a iable τ=T− . Le us now assume he ollowing ans o ma ions
[10]:
x= ln S
K+ −1
2σ2τ(60)
17
P(S, ) = p(x, τ) (61)
p(x, τ) = e− τ g(x, τ) (62)
Wi h his modi ica ions, we ha e ha
∂P
∂ =−∂p
∂τ +∂p
∂x
∂x
∂τ =−∂(e− τ g)
∂τ −∂(e− τ g)
∂x
∂x
∂τ
=−e− τ − g +∂g
∂τ + −1
2σ2∂g
∂x(63)
∂P
∂S =∂p
∂x
∂x
∂S =e− τ ∂g
∂x
1
S(64)
∂2P
∂S2=∂
∂S e− τ ∂g
∂x
1
S=∂
∂S e− τ ∂g
∂x1
S+e− τ ∂g
∂x −1
S2=
=∂
∂x e− τ ∂g
∂x∂x
∂S
1
S−e− τ ∂g
∂x
1
S2=e− τ
S2∂2g
∂x2−∂g
∂x(65)
and so equa ion (56) becomes
e− τ − g +∂g
∂τ + −1
2σ2∂g
∂x=1
2σ2S2e− τ
S2∂2g
∂x2−∂g
∂x+ S e− τ
S
∂g
∂x − e− τ g
(66)
Simpli ying we ge :
∂g
∂τ =σ2
2
∂2g
∂x2(67)
We ha e eached a di usion equa ion as we wan ed. We had p e iously ound ha
G een’s unc ion (15) is a undamen al solu ion o his equa ion, so, in his case, we can
w i e (we ake ˆ
G0= 1):
g(x, τ) = 1
√2πσ2τe−x2
2σ2τ(68)
Adding he ac o e− τ gi en by (62) o his esul , we ge G een’s unc ion o he
Black-Scholes equa ion [10]:
G(x, τ) = 1
√2πσ2τe−x2
2σ2τ− τ (69)
I we e e se he ans o ma ions made, i can be e i ied ha (69) sa is ies equa ion
(56). Howe e , i does no sa is y all he condi ions. Fo ins ance, he inal condi ion
(57), which u ns in o an ini ial condi ion wi h he change τ=T− , is no e i ied as
G(x, 0) −→ δ(x).
In Sec ion 4.2 we saw ha hanks o he linea i y o he di usion equa ion, he e
was ano he way o ob ain a solu ion using an ini ial dis ibu ion and G een’s unc ion.
18
We e e o exp ession (17). This means ha we can u n ou p oblem o aluing a pu
op ion in o a di usion p oblem whe e he inal condi ion (57) (ini ial condi ion in ime
amewo k τ) is he ini ial dis ibu ion. G een’s unc ion ac s as a p opaga o , backwa ds
in ime and o wa d in ime τ, o he ini ial dis ibu ion. This ini ial dis ibu ion has
he ollowing exp ession:
e τ max{K−S, 0}|τ=0 = max{K−S, 0}(70)
The i s ac o e τ comes om ans o ma ion (62) and he second one, which is
max{K−S, 0}, om he inal condi ion. I we in oduce a a iable x′= ln(S/K), which
co esponds o xwhen τ= 0, he ini ial dis ibu ion can be w i en as
max{K−S, 0}=Kmax{1−S
K,0}=Kmax{1−ex′,0}a τ= 0 (71)
Now ha we ha e he ini ial dis ibu ion, we can calcula e i s ime τe olu ion in e-
g a ing he con ibu ion o all poin s x′wi h hei G een’s unc ion as in (17). Each poin
x′ ep esen s a possible alue o Sa τ= 0 (a =T). G een’s unc ion G(x−x′, τ) ac s
as he p opaga o om x′ o x[10].
p(x, τ) = Z∞
−∞
Kmax{1−ex′,0}G(x−x′, τ)dx′(72)
Rega ding he ini ial dis ibu ion (71), we can see ha i can be di ided in o wo
pa s: K(1 −ex′) when x′<0 and 0 when x′>0. This means ha we can jus in eg a e
(72) om −∞ o 0.
p(x, τ) = Z0
−∞
K1−ex′G(x−x′, τ)dx′=Z0
−∞
K1−ex′1
√2πσ2τe−(x−x′)2
2σ2τ− τ dx′
=Ke− τ
√2πσ2τZ0
−∞
e−(x′−x)2
2σ2τdx′−Z0
−∞
e−(x′−x)2
2σ2τ+x′dx′
=Ke− τ
√2πσ2τ Z0
−∞
e−(x′−x)2
2σ2τdx′−Z0
−∞
e−[(x′−(x+σ2τ)]2+x2−(x+σ2τ)2
2σ2τdx′!
=Ke− τ
√2πσ2τZ0
−∞
e−(x′−x)2
2σ2τdx′−ex+σ2τ
2Z0
−∞
e−(x′−x−σ2τ)2
2σ2τdx′=I1+I2
(73)
In he second e m we ha e comple ed he squa e in he exponen . Be o e we con inue
in eg a ing, we need o in oduce he s anda d no mal cumula i e dis ibu ion unc ion:
N(y) = 1
√2πZy
−∞
e−1
2z2dz (74)
This unc ion ep esen s he p obabili y o a andom a iable Y, which has a no mal
dis ibu ion, o be less o equal o y. Once we ha e his exp ession, we a e going o use
i in (73).
I1=Ke− τ
√2πσ2τZ0
−∞
e−(x′−x)2
2σ2τdx′=Ke− τ
√2πZ−x/√σ2τ
−∞
e−z2
2dz =Ke− τ N−x
√σ2τ(75)
19
To sol e his e m, we ha e made a change o a iable z=x′−x
√σ2τ. Fo he second e m,
he ans o ma ion is z=x′−x−σ2τ
√σ2τ.
I2=−Ke− τ
√2πσ2τex+σ2τ
2Z0
−∞
e−(x′−x−σ2τ)2
2σ2τdx′=−Ke− τ ex+σ2τ
21
√2πZ(−x−σ2τ)/√σ2τ
−∞
e−z2
2dz
=−Ke− τ ex+σ2τ
2N−x+σ2τ
√σ2τ(76)
Joining bo h e ms, we each:
p(x, τ) = I1+I2=Ke− τ N−x
√σ2τ−Ke− τ ex+σ2τ
2N−x+σ2τ
√σ2τ(77)
I we now eplace xand τby hei o iginal exp essions, we ge ha [10]:
P(S, ) = Ke− (T− )N −ln S
K+ −1
2σ2(T− )
pσ2(T− )!−SN −ln S
K+ +1
2σ2(T− )
pσ2(T− )!
(78)
This is he well-known Black-Scholes o mula o a pu op ion. I gi es p ice o Eu-
opean pu op ions as a unc ion o he unde lying asse p ice and ime. Remembe ha
he o mula was de i ed conside ing cons an in e es a e and ola ili y σdu ing he
li e o he op ion. Fo Ame ican pu op ions his o mula is no alid because hey can
be exe cised a any ime be o e ma u i y Tand so, as we will see la e , he bounda y
condi ions a e no he same.
The Black-Scholes o mula o a call op ion can be ob ained ollowing he same p ocess,
bu wi h di e en bounda y and inal condi ions. The o mula in ha case is [10]:
C(S, ) = SN ln S
K+ +1
2σ2(T− )
pσ2(T− )!−Ke− (T− )N ln S
K+ −1
2σ2(T− )
pσ2(T− )!
(79)
and he inal and bounda y condi ions a e [4]
C(S, T) = max{S−K, 0}(80)
C(0, )=0 C(S, )∼Sas S−→ ∞ (81)
20

7.2 Nume ical example
We a e now going o use bo h Black-Scholes o mulas (78)(79) o a nume ical example.
We conside he ollowing pa ame e s:
•Risk- ee in e es a e =0.08 (8%)
•Vola ili y σ=0.3 √yea −1(30%)
•Expi a ion da e T= 0.75 yea (9 mon hs)
•S ike p ice K= 15$
We i s plo in Figu e 3.a he alue o he Eu opean call op ion Cas a unc ion o he
p ice o he unde lying asse S o a gi en ime = 0. The plo also con ains he alue o
he op ion a expi a ion =T, which acco ding o (80), is equal o he payo unc ion.
Rega ding he cu e o = 0, we can see ha when S ends o 0, he op ion p ice Calso
app oaches 0, and as Sinc eases conside ably, he alue o he op ion linea ly g ows wi h
Swi h uni a y slope. This means ha he bounda y condi ions (81) a e e i ied.
On he o he hand, we ep esen in Figu e 3.b he alue o he Eu opean pu op ion
Pas a unc ion o he p ice o he unde lying asse S o a gi en ime = 0. Once
again, he plo also con ains he alue o he op ion a expi a ion =T, which is equal
o he payo unc ion as he inal condi ion (57) en o ces. In he case o = 0, when he
unde lying asse p ice Sis 0, he alue o he op ion is sligh ly smalle ha he s ike
p ice K= 15$, pa icula ly Ke− (T− ). This alue is he same as he bounda y condi ion
(58). When Sinc eases signi ican ly, he op ion p ice P ends o 0, he same way as in
he o he bounda y condi ion (59).
As we ha e seen, in bo h cases he inal and bounda y condi ions a e sa is ied, so he
beha iou o he op ions alue was he expec ed.
(a) C(S) o = 0 and =T(b) P(S) o = 0 and =T.
Figu e 3: C(S) and P(S) o = 0 and =T.
21
8 Ame ican op ions
We ha e sol ed he Black-Scholes equa ion o Eu opean call and pu op ions, bu we s ill
do no ha e he way o p ice Ame ican op ions. We ha e seen ha we can no explici ly
sol e he equa ion o Ame ican op ions due o he possibili y o ea ly exe cise. I is ime
now o analyse he p oblem o p icing Ame ican op ions and ind a solu ion o i .
8.1 Gene al concep s
Ame ican op ions, unlike Eu opean op ions, can be exe cised a any ime be o e expi-
a ion da e. We a e soon going o see ha he possibili y o ea ly exe cise changes he
bounda y condi ions, so he Black-Scholes o mula does no apply o hose cases. In ac ,
he possibili y o ea ly exe cise gi es he holde mo e lexibili y, so we could expec he
alue o an Ame ican op ion o be highe han he alue o an Eu opean op ion. This can
be shown using a bi age a gumen s.
We a e going o analyse he case o a pu op ion. The e a e alues o he unde lying
asse p ice S o which he alue o an Eu opean pu op ion is less han he payo unc ion
max{K−S, 0}. As a esul , i we conside ha he p ice o an Ame ican pu op ion is he
same as he Eu opean pu op ion, we can buy an Ame ican op ion o Pand immedia ely
exe cise i o K, ob aining a p o i o K−S−Pwi hou isk. The e is a bi age. As an
example, we conside he case o S= 0. Acco ding o he payo unc ion max{K−S, 0},
he payo is K. I we emembe ha S= 0 was a bounda y condi ion (58) wi h op ion
alue Ke− (T− ), we can see ha he payo unc ion is g ea e han he alue o he op ion,
K > Ke− (T− ), so we a e in he si ua ion desc ibed be o e. This can be isualised in
Figu e 3.b. To a oid a bi age, we mus impose he ollowing condi ion o Ame ican pu
op ions [4]:
P(S, )≥max{K−S, 0}(82)
Simila ly, o Ame ican call op ions he condi ion would be:
C(S, )≥max{S−K, 0}(83)
A expi a ion da e, he p ice o Ame ican op ions has o be equal o he payo unc ion,
ha is, he p ice o he Ame ican op ions is equal o he p ice o he Eu opean op ions
gi en by he Black-Scholes o mula.
P(S, T) = max{K−S, 0}C(S, T) = max{S−K, 0}(84)
Ame ican call op ions a e special since hey should ne e be exe cised be o e ma u i y
unless hey pay di idends. Fo now, we ha e always conside ed op ions wi h no di idends
and we a e going o con inue doing he same. This is jus a ema k o cla i y ha Ame ican
call op ions a e jus exe cised a expi a ion da e. The eason o his is ha ea ly exe cise
equi es he immedia e paymen o he s ike p ice Kand i is mo e p o i able o keep
ha money, o ins ance in he bank, wi h i s isk- ee in e es a e un il ma u i y. This
can shown building wo po olios as i is explained in Re [13]:
•Po olio A: Ame ican call op ion C+ money in he bank Ke− (T− )
•Po olio B: one sha e S
22
I we exe cise he call op ion in po olio A a ea ly ime ex < T because S > K, he
payo will be S−K+Ke− (T− ex), which is smalle han Sin po olio B. I o he wise we
exe cise he op ion a T, he payo o A will be max(S−K, 0) + K= max(S, K), which
is always g ea e han o equal o Sin B. Consequen ly, non-di idend paying Ame ican
call op ions should only be exe cised a ma u i y and i s p ice is he same as Eu opean
call op ions.
We a e now going o ocus on he s udy o Ame ican pu op ions since o hem ea ly
exe cise is op imal in some cases.
8.2 Ame ican pu op ions
As ea ly exe cise is possible o Ame ican pu op ions, he e mus also be some alues o
S o which he exe cise o he op ion is op imal be o e ma u i y. A each ime , he e a e
wo egions o S: one wi h he alues o S o which he op ion should be exe cised and
he o he wi h he alues o S o which he op ion should be hold. The poin ha ma ks
he bounda y be ween hose wo egions a ime is e e ed as he op imal exe cise
p ice and i is deno ed by S ( ). This op imal exe cise p ice is unknown o us. Tha is
why he p oblem o p icing an Ame ican pu op ion is called a ee bounda y p oblem.
We ha e o sol e a p oblem di ided in wo egions whe e we do no know whe e is he
bounda y. We will ha e ha e o ind ou his bounda y as pa o he solu ion.
F ee bounda y p oblems a e eally common in physics. Fo ins ance, he S e an p ob-
lem ha desc ibes he join e olu ion o a liquid and a solid phase is a ee bounda y
p oblem [14]. Ano he ypical example is he obs acle p oblem, which consis s in inding
he equilib ium con igu a ion o an elas ic memb ane whose bounda y is held ixed, and
which is cons ained o lie abo e a gi en obs acle [15].
We a e now going o o mula e he ee bounda y p oblem o Ame ican pu op ions.
Le us conside a pu op ion wi h alue P(S, ). As we ha e seen be o e, he alue o he
op ion e i ies ha [4]
P(S, )≥max{K−S, 0}(85)
and he inal condi ion is
P(S, T) = max{K−S, 0}(86)
An Ame ican pu op ion should be ea ly exe cised a a ime < T when he alue o S
is lowe han o equal o he op imal exe cise p ice S ( ). In hose cases, he p ice o he
op ion mus be P(S, ) = max{K−S, 0}. Con e sely, i S > S , he op ion should be hold
since i s p ice is P(S, )>max{K−S, 0}and i is mo e p o i able o sell he op ion han
o execu e i . When S > S , he op ion p ice ollows he Black-Scholes equa ion. The
combina ion o his wo ac s makes he Black-Scholes equa ion (56) become an inequali y.
We can now analyse how he Black-Scholes equa ion becomes an inequali y. We build
he same po olio as in (46), wi h he same alue o del a (49). Fo he pu op ion, i
would be:
Π = P−∂P
∂S S(87)
23
As ea ly exe cise is possible wi h Ame ican op ions, he a bi age a gumen used o
Eu opean op ions, whe e he e u n o he po olio had o be equal o he e u n o money
deposi ed in he bank, is no alid. This means ha exp ession (53) is no an equali y
anymo e. In his case, all we can say is ha he e u n o he po olio canno be g ea e
han he money in es ed in he bank:
Π d ≥∂P
∂ +1
2σ2S2∂2P
∂S2d (88)
I we in oduce he alue o he po olio (87) in o (88), we each o:
∂P
∂ +1
2σ2S2∂2P
∂S2+ S ∂P
∂S − P ≤0 (89)
We can see ha he Black-Scholes equa ion has become an inequali y o Ame ican pu
op ions. The exp ession is an equali y when he op ion should be hold and an inequali y
when he exe cise is op imal, so we can w i e ha o he egion 0 ≤S < S ( ):
P(S, ) = max{K−S, 0}=K−S(90)
∂P
∂ +1
2σ2S2∂2P
∂S2+ S ∂P
∂S − P < 0 (91)
and o he o he egion, S ( )< S < ∞:
P(S, )> K −S(92)
∂P
∂ +1
2σ2S2∂2P
∂S2+ S ∂P
∂S − P = 0 (93)
We now need o impose wo condi ions a he ee bounda y S ( ). We suppose ha
S ( ) is smalle han he s ike p ice Kso as o ha e a posi i e payo . The i s condi ion
comes om he con inui y o P(S, ) a S=S ( ):
P(S ( ), ) = max{K−S ( ),0}=K−S ( ) (94)
The second one, om he con inui y o he a e o change o del a (49) o P(S, ) a
S=S ( ):
∂P
∂S S=S
=−1 (95)
To unde s and he eason o his second condi ion, we a e going o conside he wo
o he possible scena ios: ∂P/∂S < −1 and ∂P/∂S > −1. In he i s case, an inc ease
in S om S ( ) implies a dec ease o P(S, ) below max{K−S, 0}. Acco ding o (85),
P(S, ) canno be smalle han max{K−S, 0}, so his si ua ion is no possible. In he
second case, a dec ease in S om S ( ) induces a mis alued inc emen o he op ion alue,
gi ing ise o possibili ies o a bi age. This is no a desi ed si ua ion. As a esul , he
only possible scena io is ha ∂P/∂S =−1 [4].
We ha e al eady o mula ed he ee bounda y p oblem o p ice Ame ican pu op ions.
We ha e he p oblem di ided in o wo egions and we ha e wo condi ions a he bounda y
o he egions. We also ha e a inal condi ion.
24
8.6 The LU me hod
The me hod we a e going o use o sol e he p oblem is known as he LU me hod.
Re [4] sol es he p oblem wi h he P ojec ed SOR me hod, no he LU. I uses he LU
me hod wi h he implici scheme o p ice Eu opean op ions. We a e going o apply he
same p ocedu e bu o Ame ican op ions wi h he C ank-Nicolson scheme. This me hod
is used o sol e sys ems o linea equa ions as (Aum+1 −bm) = 0. I is based on he LU
decomposi ion, which consis s on ac o izing a ma ix, in his case A, in o a p oduc o a
lowe iangula ma ix Land an uppe iangula ma ix U, so ha A=LU.
A=














1 + α−α
20. . . 0
−α
21 + α−α
2
.
.
.
0−α
2
......0
.
.
.......−α
2
0. . . 0−α
21 + α














=














1 0 0 . . . 0
lN−+1 1....
.
.
0.........0
.
.
.......0
0. . . 0lN+−21














·














yN−+1 zN−+1 0. . . 0
0yN−+2 ....
.
.
0.........0
.
.
.......zN+−2
0. . . 0 0 yN+−1














(147)
To de e mine ln,ynand znwe ha e o mul iply he ma ices
LU =
















yN−+1 zN−+1 0. . . 0
lN−+1yN−+1 lN−+1zN−+1 +yN−+2 ....
.
.
0.........0
.
.
.......zN+−2
0. . . 0lN+−2yN+−2lN+−2zN+−2+yN+−1
















(148)
and equal he esul o ma ix A. He e we ind ha
zn=−α
2ln=−α
2yn
o n=N−+ 1, . . . , N+−2 (149)
yN−+1 = 1 + α(150)
31

yn= (1 + α)−α2
4yn−1
o n=N−+ 2, . . . , N+−1 (151)
As we can see, we a e jus in e es ed in he alues o yn o n=N−+ 1, . . . , N+−1.
We can now di ide he p oblem (Aum+1 −bm) = 0 in o wo sub-p oblems:
Lwm=bmUum+1 =wm(152)
This is he same as doing LUum+1=bm, bu wi h an in e media e ec o wm.
Subs i u ing exp essions (149), (150) and (151) in o he ma ices Land U, we ha e ha
he sub-p oblems a e














1 0 0 . . . 0
−α
2yN−+1 1....
.
.
0.........0
.
.
.......0
0. . . 0−α
2yN+−21














·














wm
N−+1
.
.
.
wm
0
.
.
.
wm
N+−1














=














bm
N−+1
.
.
.
bm
0
.
.
.
bm
N+−1














(153)
and














yN−+1 −α
20. . . 0
0yN−+2 −α
2
.
.
.
0 0 ......0
.
.
.......−α
2
0. . . . . . 0yN+−1














·














um+1
N−+1
.
.
.
um+1
0
.
.
.
um+1
N+−1














=














wm
N−+1
.
.
.
wm
0
.
.
.
wm
N+−1














(154)
We begin wi h he i s sys em (153). We can di ec ly de i e he alue o wm
N−+1 and
hen inc easing n om N−+ 1, ob ain he e ms wm
nknowing he p e ious one wm
n−1.
wm
N−+1 =bm
N−+1 (155)
wm
n=bm
n+αwm
n−1
2yn−1
o n=N−+ 2, . . . , N+−1 (156)
In he second sys em (154), we di ec ly ge um+1
N+−1and hen dec easing n om N+−1,
we can ob ain he e ms um+1
nknowing um+1
n+1 .
um+1
N+−1=wm
N+−1
yN+−1
(157)
um+1
n=wm
n+α
2um+1
n+1
yn
o n=N+−2, . . . , N−+ 1 (158)
32
Wi h all hese exp essions, we can build he algo i hm o he LU me hod o sol e ou
linea sys em (Aum+1 =bm) in (145). We suppose ha Aand bma e al eady known.
The algo i hm has he ollowing s eps:
1. Find all he alues yns a ing om yN−+1 (150) and using (151).
2. Calcula e he ec o wms a ing wi h he componen wm
N−+1 (155) and hen using
(156) o ob ain he es .
3. Ob ain he ec o um+1 s a ing om um+1
N+−1(157) and calcula ing he o he com-
ponen s wi h (158).
This is he algo i hm ha sol es he linea sys em (Aum+1 =bm), bu his is jus
one pa o he esolu ion o he linea complemen a i y p oblem. The algo i hm o he
whole p oblem (145)(146) is he ollowing:
1. In he beginning, we ha e he ma ix Aand he ini ial (138) and bounda y condi-
ions (139). Wi h he ini ial condi ion, we can ob ain u0( i s ime s ep m= 0).
2. We calcula e b0and sol e he linea sys em Au1=b0using he LU me hod o
ob ain u1. The e ms u1
N−and u1
N+do no appea in he solu ion o he linea
sys em bu a e known om he bounda y condi ions. Acco ding o he exp ession
um+1 ≥ m+1, e e y componen o he ec o u1has o be g ea e han o equal
o he co esponden componen o he ec o 1. This means ha when we a e
calcula ing u1
nusing (157) and (158), we ha e o check whe he i is g ea e han o
equal o 1
n. In case i was smalle , we would ha e o o ce u1
n= 1
n.
3. Once we ha e u1, we calcula e b1 o be able o sol e he sys em Au2=b1and
ob ain u2. We ha e o check ha u2≥ 2 he same way as be o e and do he
necessa y eadjus men s.
4. The p ocess is epea ed un il we calcula e uM.
We ha e seen ha in some cases we ha e o o ce he alue o um
n o be equal o m
n.
This sugges s us ha we ha e o make some changes in exp essions (157) and (158):
um+1
N+−1= max wm
N+−1
yN+−1
, m+1
N+−1(159)
um+1
n= max wm
n+α
2um+1
n−1
yn
, m+1
n o n=N+−2, . . . , N−+ 1 (160)
We a e now going o build ou own Py hon code o he whole algo i hm o esolu ion.
The algo i hm gi es us he alues o uwi h espec o xand τ, bu we a e also going o
calcula e he alues o P(S, ) e e sing he ans o ma ions. Mo eo e , we a e going o
y o ind he op imal exe cise p ice x (τ) a e e y ime s ep and hen con e i in o
S ( ). To ob ain x a a ime s ep mδτ, we ha e o ind he alue o n o which um
n= m
n
and which e i ies ha o n′> n,um
n′> m
n′, excluding n=N−and n=N+.
33
8.7 Py hon code
We a e now going o w i e ou Py hon code o he algo i hm.
impo ma h
impo ma plo lib.pyplo as pl
class Ame icanPu :
de __ini __(sel , , sigma, T, K):
sel .__ =
sel .__sigma =sigma
sel .__T =T
sel .__K =K
sel .__q =2* /(sigma *sigma)
sel .__S =[]
sel .__ =[]
sel .__P =[[]]
sel .__S =[]
de ge _ (sel ):
e u n sel .__
de ge _S(sel ):
e u n sel .__S
de ge _P(sel ):
e u n sel .__P
de ge _S (sel ):
e u n sel .__S
de payo (sel , x, au): #Payo unc ion (x, au)
e u n ma h.exp(0.25 * (sel .__q + 1)*(sel .__q + 1)* au) *
max(ma h.exp(0.5 * (sel .__q - 1)*x) -ma h.exp(0.5 * (sel .__q + 1)*x), 0.0)
de om_u_ o_P(sel , u, x, au): #T ans o ma ion om u o P
P=[]
o iin ange(len(u)):
P.append(sel .__K *ma h.exp(-0.5 * (sel .__q - 1)*x[i]
- 0.25 * (sel .__q + 1)*(sel .__q + 1)* au) *u[i])
e u n P
@s a icme hod
de ind_y(alpha, N): #Me hod ha inds all he y_n
y=[1+alpha]
o iin ange(1, N):
y.append(1+alpha -alpha *alpha /(4*y[i - 1]))
e u n y
de ind_b(sel , u, x, au, alpha): #Me hod ha calcula es b^m
b=[]
o iin ange(1,len(u) - 1):
C=(1-alpha) *u[i] +0.5*alpha *(u[i + 1]+u[i - 1])
b.append(C)
b[0]+= 0.5 * alpha *sel .payo (x[0], au)
b[-1]+= 0.5 * alpha *sel .payo (x[-1], au)
e u n b
34
de LU(sel , b, y, x, au, alpha, leng h): #LU sol e
w=[b[0]]
check_s =T ue
o iin ange(1, leng h):
w.append(b[i] +0.5*alpha *w[i - 1]/y[i - 1])
u=[0 o kin ange(leng h)]
u[leng h - 1]=max(w[leng h - 1]/y[leng h - 1], sel .payo (x[leng h], au))
o jin ange(leng h - 2,-1,-1):
u[j] =max((w[j] +0.5*alpha *u[j + 1]) /y[j], sel .payo (x[j + 1], au))
i check_s and u[j] == sel .payo (x[j + 1], au):
x =x[j + 1]
check_s =False
sel .__S .append(sel .__K *ma h.exp(x ))
e u n u
de alues(sel , d au, dx, Nmin, Nplus): #Me hod ha gi es he alues o P(S, )
alpha =d au /(dx *dx)
N=Nplus -Nmin + 1
M=in (0.5 * sel .__sigma *sel .__sigma *sel .__T /d au)
sel .__P =[[0 o iin ange(N)] o jin ange(M)]
sel .__ =[sel .__T]
x=[]
u=[]
o lin ange(N):
x.append((Nmin +l) *dx)
sel .__S.append(sel .__K *ma h.exp(x[l]))
u.append(sel .payo (x[l], 0.0)) # Finding u^0
sel .__P[0][:] =sel . om_u_ o_P(u, x, 0.0)
y=sel . ind_y(alpha, len(u) - 2)
o iin ange(1, M): #Finding u^m om m=1 up o m=M
au =i*d au
sel .__ .append(sel .__T - au /(0.5 * sel .__sigma *sel .__sigma))
b=sel . ind_b(u, x, au, alpha)
u[1:N - 1]=sel .LU(b, y, x, au, alpha, len(u) - 2)
u[0]=sel .payo (x[0], au)
u[N - 1]=sel .payo (x[N - 1], au)
sel .__P[i][:] =sel . om_u_ o_P(u, x, au)
pass
i __name__ == '__main__':
ame ican_pu =Ame icanPu ( =, sigma=, T=, K=)
ame ican_pu . alues(d au=, dx=, Nmin=, Nplus=)
=ame ican_pu .ge _ ()
S=ame ican_pu .ge _S()
P=ame ican_pu .ge _P()
S =ame ican_pu .ge _S ()
To c ea e an objec ha ep esen s an Ame ican pu op ion in his code, we need o
se an in e es a e , a ola ili y σ, an expi a ion da e Tand a s ike p ice K. Once he
objec is c ea ed, we ha e o call he me hod alues i we wan o p ice he op ion. The
inpu s o his me hod a e he ime in e al d au, he xin e al dx and he minimum an
maximum numbe s o n, ha is, Nmin and Nplus. The me hod applies he algo i hm
o esolu ion and ills he lis s and Sand he ma ix Pwi h he co esponden alues. A
ow in Pma ix ep esen s a ime s ep and con ains he alues o he op ion o all alues
o xa ha ime s ep. The me hod also ills he lis S con aining he op imal exe cise
p ice a each ime s ep. The ans o ma ions om u(x, τ) o P(S, ) a e in e nally made.
35
8.8 Nume ical example
We a e now going o un he code wi h a nume ical example. We conside an Ame ican
pu op ion and he ollowing pa ame e s:
•Risk- ee in e es a e =0.03 (3%)
•Vola ili y σ=0.35 √yea −1(35%)
•Expi a ion da e T= 0.5 yea (6 mon hs)
•S ike p ice K= 10$
Fi s ly, we calcula e he alues o he op ion wi h d au = 0.0005, dx = 0.00125,
Nmin =−4000 and Nplus = 600. The p og am ills he lis s ,S,S and he ma ix P.
I we we e in e es ed in he alue o he op ion o some speci ic Sand , we could ake i
om P. The same o S as a unc ion o .
We a e now going o plo some esul s. On he one hand, we ep esen in 3D in Figu e
4 he alue o he op ion Pas a unc ion o he unde lying asse p ice Sand ime .
When he asse p ice S ends o 0, he op ion alue Pgoes o he s ike p ice K= 10$
he same way as he payo unc ion (max{K−S, 0}), and as Sinc eases signi ican ly,
Papp oaches 0. This means ha he bounda y condi ions (123) a e e i ied. The plo
also con ains he op imal exe cise bounda y, which is he ed line, o he gi en he alues
o he op ion. The alues o Pabo e he line a e cases o which he op ion would be
execu ed and he alues unde he line, o which i would be hold.
Figu e 4: 3D plo o P(S, ). The ed line ep esen s he op imal exe cise bounda y.
We a e also going o plo he 2D unc ion P(S) o wo di e en imes = 0 and
=T, as well as he op imal exe cise p ice unc ion S ( ). The esul ing plo s a e he
ones in Figu e 5. In he i s one, Figu e 5.a, we obse e ha o =T, he alue o P(S)
is equal o he payo unc ion (Figu e 1.b) as i has o be acco ding o he inal condi ion
(86). When = 0, he unc ion P(S) is g ea e han o equal o he payo unc ion, so
condi ion (85) is sa is ied. F om Figu es 4 and 5.a and he analysis o he bounda y and
inal condi ions we ha e made, we can in e ha he beha iou o P(S, ) is co ec .
36

Rega ding Figu e 5.b, we deduce ha when ime app oaches expi a ion da e, he op-
imal exe cise p ice inc eases and ends o he s ike p ice K= 10$. This is logical since
a ime =T,P(S, T) = max{K−S, 0}and exe cise should be done only i S < K.
The egion behind he line is he exe cise egion and he egion abo e he line is he
one o hold. As ime e ol es, we a e mo e p edisposed o execu e he op ion and sell
he asse despi e ob aining a smalle payo . The eason o his is ha , a ea ly imes,
he alue o he op ion is g ea e han he payo unc ion o mo e alues o S(i is i-
sualized in Figu e 5.a) and, in hose cases, i is be e o hold he op ion han o execu e i .
(a) P(S) o = 0 and =T(b) Op imal exe cise p ice unc ion S ( )
Figu e 5: Func ion P(S) o = 0 and =Tand op imal exe cise p ice unc ion S ( ).
37
9 Conclusions
In his wo k we ha e es ablished a connec ion be ween physics and inance. Conc e ely,
we ha e analysed some physical concep s in he alua ion o op ions.
Fi s o all, we ha e lea n some basic inancial concep s. I was c ucial o unde s and
wha is an op ion o he es o he p ojec . We ha e also in oduced some physical
no ions ela ed o B ownian mo ion. We ha e seen ha di usion p ocesses a e go e ned
by he B ownian mo ion and we ha e ound di e en solu ions o he di usion equa ion
based on G een’s unc ion. Fu he mo e, we ha e explained he ma hema ical o mula ion
o he B ownian mo ion, ha is, he Wiene p ocess.
Wi h he main inancial and physical concep s al eady assimila ed, we ha e analysed
he beha io o he unde lying asse p ice. We ha e seen ha i can modelled wi h a
andom walk based on a B ownian mo ion. We ha e ound ha in he andom walk
model, he asse p ice ollows a Geome ic B ownian mo ion and ha he p obabili y
densi y unc ion o i s loga i hm su e ed a di usion p ocess. To isualize his di usion
p ocess, we ha e plo ed some unc ions.
Ou nex s ep has been o pose he Black-Scholes model based on he andom walk
model o he unde lying asse . We ha e de i ed he Black-Scholes pa ial di e en ial
equa ion used o p ice op ions. Once we had he equa ion, we ha e looked o an explici
solu ion o Eu opean pu op ions. This was he Black-Scholes o mula. We ha e sol ed
he p oblem as a di usion p oblem wi h he payo unc ion playing he ole o he ini ial
dis ibu ion. Howe e , i s we ha e had o ans o m he Black-Scholes equa ion in o
a di usion equa ion. Once we had he Black-Scholes o mulas o bo h Eu opean call
and pu op ions, we ha e used hem wi h a nume ical example. We ha e ound ha he
beha iou o he op ions was he expec ed.
In he las sec ion, we ha e s udied he p oblem o p icing Ame ican op ions. We
ha e seen ha we canno ind an explici solu ion and ha o Ame ican call op ions,
ea ly exe cised is ne e ecommended. We ha e analysed he ee bounda y p oblem o
Ame ican pu op ions and ans o med i in o a linea complemen a i y p oblem o ob ain
a solu ion. Once mo e. we ha e wo ked wi h he Black-Scholes equa ion ans o med in o
a di usion equa ion. W i ing he p oblem as a linea complemen a i y one, elimina ed
he dependence on he ee bounda y and made esolu ion easie . We ha e used he
ini e di e ence o mula ion and he C ank-Nicolson scheme wi h ma ices o sol e he
p oblem. We ha e also buil an algo i hm ha uses he LU me hod o sol e linea sys ems
o equa ions and ansla ed i in o ou own Py hon code. The las s ep has been o un his
code wi h a nume ical example and analyse he esul s. We ha e seen ha he unc ion o
he op ion alue had he igh beha iou and we ha e also ob ained he op imal exe cise
p ice unc ion. The alue o he op imal exe cise p ice inc eased and con e ged o he
s ike p ice as ime app oached expi a ion da e.
O e all, we ha e ound ha he e is a ela ion be ween physics and inance. Pa icu-
la ly, we ha e ound ha he B ownian mo ion and di usion equa ion play an impo an
ole in he alua ion o op ions. The in luence o physics in inance is a ac and ha is
why many inancial lines o in es iga ion in ol e he use o physics.
38
Re e ences
[1] Mau o Cesa, A b ie his o y o quan i a i e inance, P obabili y, Unce -
ain y and Quan i a i e Risk, Sp inge (2017), DOI: h ps://doi.o g/10.1186/
s41546-017-0018-3
[2] Op ions: Calls and Pu s, Co po a e Finance Ins i u e, URL: h ps://
co po a e inanceins i u e.com/ esou ces/knowledge/ ading-in es ing/
op ions-calls-and-pu s/
[3] Jaksa C i anic, P icing Op ions wi h Ma hema ical Models, Cal ech,
Online cou se om Cou se a, URL:h ps://es.cou se a.o g/lea n/
p icing-op ions-wi h-ma hema ical-models
[4] Paul Wilmo , Sam Howison, Je Dewynne, The Ma hema ics o Financial De i a-
i es, Camb idge Uni e si y P ess (1995)
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