mod4-optn-pricing.pptx

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Module 4: Option Pricing

1



Put-Call Parity

•

Both portfolios are worth the same onmaturity – expiration of options

• Both are uropean options – hence can

not !e exercised !efore expiry date• "ence their #alue must !e e$ual today

• %hus:

• %his is called Put-Call Parity& 't pro#idesa !asis for deri#ing the price of a callfrom a put ( #ice #ersa) when stri*e

price ( expiry date are same

0S p Kec rT +=+

−

+



Contents

• Our focus will !e on ,toc* Optionsonly !ecause that is the typical

model applied for understandingoptions

• "owe#er the principles are

applica!le withwithout smallmodi.cations) if re$uired) to othertypes of underlying assets also

/



Contents

• Binomial model of OptionPricing

• Blac*-,choles model of OptionPricing

•

0elated issues

4



Option Pricing – Binomial Model foruropean Options

•

Based on the concept of !inomial trees• !inomial tree is a diagrammatic

representation of the #arious paths

that the price of the underlying stoc*may follow during the lifetime of anoption

• 2nderlying assumption: ,toc* pricefollows a random wal*

• 'n each time inter#al there is a certain

pro!a!ility that the stoc* price will 5



Binomial Model

• 6o ar!itrage argument: 't is an

assumption that is made in order toarri#e at the e$uili!rium price of theoption& s long as there are ar!itrage

opportunities there will !e dise$uili!riumin the options mar*et ( price will !eunsta!le&

• 7ogic: portfolio in the lines of a co#eredcall is set up with an in#estment hori8one$ual to the maturity of the option& 'tconsists of a long position in some shares

( a short position in 1 call option& %he 9



Binomial Model: 7ogic

•

%he components of the portfolio areli*ely to #ary in #alue o#er time&"owe#er it is constructed in a way soas to *eep the o#erall #alue of theportfolio constant at maturity&

• Because the portfolio #alue at maturity

is sta!le it is ris*-free& "ence it mustearn a return e$ual to the ris*-freerate&

•

On this !asis the #alue of the portfolio;



Binomial Model: 7ogic

•

%he #alue of the option is deri#ed fromthe cost of setting up the portfolio atthe inception&

•

Portfolio consists of a long position in <no& of shares ( a short position in 1uropean call option&

•

%he stoc* price can ha#e two possi!leoutcomes at maturity: it can mo#e upor down&

•

=hen the portfolio is designed to !e >



Binomial Model: ssumptions

•

6o ar!itrage assumption: 'mpact onpricing

• Perfect ( competiti#e mar*ets – 6o

transaction costs) 6o margins) 6otaxes) short sales are allowed) fractionsof securities can !e traded) !orrowing( lending rates are same: 'mplies freetrading in all mar*ets

• 0is* free interest rate) si8e of uptic* (

si8e of downtic* are *nown in e#ery @



Binomial Model: 6o ssumptions On

•

6o assumption made on theactual pro!a!ilities on theuptic*downtic*

• 6o assumption made onexpected returns on the stoc*

• 6o assumption made on thein#estorsA degree of ris*

a#ersion 1



Binomial Model: 6otations

• , : ,pot price at time

• f : Current price of call option3uropean

• % : %ime to expiration

• u : Multiple !y which the stoc* pricechanges during % 3u 1 so thatending price D ,u

• d : Multiple !y which the stoc* pricechanges during % 3d E 1 so thatending price D ,

d11



Binomial Model

•

7et < !e the no& of shares in which longposition is ta*en

• 'f the stoc* price increases at maturity

then portfolio #alue D <,u - f u • 'f the stoc* price decreases at maturity

then portfolio #alue D <,d - f d

• %he portfolio will !e said to !e ris*-freewhen its #alue at maturity is same inall conditions& %hus: <,u - f u D <,d - f d

O0 1&

d S uS

f f d u

00 −

−=∆

1+



Binomial Model

• %he portfolio is ris*-free& "ence it mustearn ris*-free rate of interest in order toa#oid ar!itrage opportunities& Gi#en the#alue of portfolio at maturity its present

#alue will !e• Cost of setting up the portfolio at

inception:

• 'n a no ar!itrage situation the cost ofsetting up must !e e$ual to the present

#alue of the #alue at maturity& %hus:

rT

u e f uS −−∆= )( 0

f S −∆0

rT

u

rT

rT

u

e f ueS f

e f uS f S

−−

−

+−∆=

−∆=−∆

)1(

OR )(

0

00

1/



Binomial Model

=hen the expression for < is su!stitutedwe get:

+&

/&

$uations + ( / are applica!le for optionpricing when stoc* price follows one-period!inomial model&

•

%he underlying principle is called 0is*-

rT

u

rT

rT

u

e f ueS f

e f uS f S

−−

−

+−∆=

−∆=−∆

)1(

OR )(

0

00

d u

d e pwhere

f p pf e f

rT

d u

rT

−

−=

−+= −

:

])1([

14



Binomial Model: 0is* 6eutral Haluation

•

'n#estors are assumed to !e ris*neutral – they are indiFerent towardsris*? so they re$uire 6O

compensation for ris*• "ence expected return on all

securities is the ris*-free rate of

return• 'n a ris* neutral world in#estors are

indiFerent to the actual pro!a!ilities

of the stoc* price mo#ing updown –15




• %he portfolio is designed to !e ris*-

free – the no& of shares is such thatthe #alue of the portfolio remains thesame whether the stoc* mo#es up ordown

• %he terms p ( 31 – p in e$uations +

( / are not the true pro!a!ilities ofthe up ( down mo#ements of thestoc*

•

0ather they may !e interpreted to !e19




• %hus the option price today is the

expected payoF from the option infuture) discounted !ac* at the ris*-free rate

• %he terms p and 31 – p wheninterpreted to !e the pro!a!ilities ofthe stoc* price mo#ing updown in aris* neutral world are called ris*neutral pro!a!ilities

•

't can !e shown: with ris* neutral 1;



Binomial Model: ,ome 'mportant spects

• position in an option is ris*ier than the

position in the underlying asset• "ence the discount rate to e#aluate its

payoFs must !e greater than the expected

return on the underlying asset in the realworld

• "owe#er without *nowing the options #aluenow the discount rate for option cannot !e

*nown ( without *nowing the discount rateits #alue now cannot !e calculated

• Haluing the option in a ris* neutral

framewor* is easy !ecause discount rate for1@



%wo ,tep Binomial Model

• %here are two consecuti#e periods

of time during which the shareprice can consecuti#ely change (so can the option price

• %he option price at time can !earri#ed at in a stepwise manner

along the !inomial tree using thesame procedure ( formulae usedin case of one period !inomial

model +1



Multi-,tep Binomial Model

• s the no& of stepsperiods is

increased we get scenarios that tendto !e closer to reality

• "owe#er such scenarios are too

complicated to !e sol#ed manually• "ence in real life the !inomial model

can !e applied only !y software

• %he result will !e more accurate thanthe one period !inomial model

++



Multi-,tep Binomial Model• 'n !oth +-step ( multi-step !inomial models

the expiry period of the option is !ro*endown into + or more su!-periods

• ,o in the last nodes of the tree there aremultiple possi!le stoc* prices ( optionpayoFs ( their pro!a!ilities – Kointpro!a!ilities of the up ( down mo#ements ofstoc* price through #arious paths

• ,o the expression of the expected payoF ofthe option at the end of the last time step isexpanded !ecause of the #arious possi!ilities

+/



Binomial Model: Other Hariants

a& Options on stoc*s paying

continuous di#idends

!& Options on stoc* indices

c& Options on currenciesd& Options on futures

+4




a& Options on stoc*s paying

continuous di#idend yield 3$:

• %he !inomial principle still

remains the same• 'n a ris* neutral world the total

return to the stoc*holders fromdi#idends ( capital gains will!e D 0is*-free rate 3r which is

compounded continuously+5




a& Options on stoc*s paying continuous

di#idend yield 3$:• %he !inomial pricing model for options

on stoc*s not paying any di#idends uses

ris*- free rate which is e$ui#alent to thecapital gains yield in a ris* neutral world

• =ith 8ero di#idends) total stoc* return

D capital gains yield D ris*-free rate• %he rele#ant discount rate is the capital

gains yield in 8ero di#idends case+9

i i l d l h i




a& Options on stoc*s paying continuous

di#idend yield 3$:• =ith *nown di#idend yield the capital

gains yield will !e D r – $

• Because the %otal yield D Li#idend yield Capital Gains yield D 0is*-free rate 3r

• s di#idends reduce the stoc* price it

will grow at the rate of: r – $

• "ence su!stitute r with r – $ in theoriginal !inomial pricing model

+;

Bi i l M d l O h H i




!& Options on stoc* indices:

• ,toc* indices are assumed to pro#idea *nown cumulati#e di#idend yield $3from the stoc*s comprising theindex

• "ence the !inomial pricing of options

on stoc* indices is similar in principleto the #aluation of an option on astoc* paying *nown di#idend yield

+>

Bi i l M d l O h H i




c& Options on Currencies:

• foreign currency is an asset pro#idinga yield at the ris*-free rate of interest3rf in the foreign country

• "ence the appropriate discount rate isthe domestic ris*-free interest rate 3rminus foreign ris*-free interest rate: r – rf

• %he model is similar to that of stoc*swith *nown di#idend yields ( stoc*indices

+@

Bi i l M d l Oth H i t




d& Options on Nutures:

• %here is no cost of entering into along or short position in a futures

• 'n a ris* neutral world the expectedgrowth rate of the price of a futurescontract should !e 8ero

•

,o the expected futures price at theend of the time inter#al should !ethat at time 8ero: pNu 31 – pNd D

N /

Bi i l M d l Oth H i t




d& Options on Nutures:

• pNu 31 – pNd D N

• %hus the ris* neutral pro!a!ility

of a up mo#ement in the futuresprice 3p will !e:

d u

d p

−

−=1

/1

Binomial Model Critical ppraisal



Binomial Model: Critical ppraisal

• %he principal merit of the !inomial

model is its exi!ility• 't can !e used to e#aluate a wide

#ariety of options including mericanoptions

• MaKor limitation: if we increase the

no& of time steps then the model willre$uire a #ery large no& of inputswhich can !e handled only !y a

computer/+



Nrom Binomial %o Blac*-,choles Model 3B,M

• Binomial model is a discrete time model• 't allows for a time inter#al 3t !etween

price mo#ements

• 'f t tends to 8ero then in the limit twotypes of distri!utions are possi!le

a& 6ormal – if price changes tend to 8ero

as t tends to 8ero!& Poisson – if price changes remain large

as t tends to 8ero 3allows for sudden

Kumps /4

N Bi i l % Bl * , h l M d l 3B,M




• B,M explicitly assumes that the price

process is continuous• 't uses normal distri!ution as the

limiting distri!ution as t tends to 8ero

• ,toc* prices cannot assume –#e#alues !ecause of limited lia!ility ofshareholders – hence cannot !e

normally distri!uted• %he distri!ution of natural log of stoc*

prices is assumed to !e normal in B,M/5

N Bi i l % Bl * , h l M d l 3B,M




• B,M is a special case of the Binomial

model• =e reach the B,M from the !inomial if we

reduce the time inter#al to extremely

short inter#als such that in !etween time 3!eginning ( time % 3expiry there arein.nite no& of time inter#als) each one isextremely short

• %his would happen if trading is acontinuous phenomenon – for somemar*ets this assumption is close to reality

/9

Blac* ,choles Model 3B,M



Blac*-,choles Model 3B,M

• ,toc* prices cannot !e less than

3explained !efore ( stoc* returnscannot !e less than –1I

• "ence the closely approximatingdistri!ution for !oth stoc* prices (stoc* returns is the lognormaldistri!ution

• B,M is originally applica!le touropean call options on stoc*s not

paying di#idends/;

B,M ssumptions



B,M: ssumptions• 6o di#idends paid on the stoc*

• 6o transaction costs• 0is*-free interest rate is *nown ( is

constant during the life of option

•,hort-selling of stoc* is allowed

• Call can !e exercised only on expiry

• ,toc* prices change randomly ( trading

ta*es place continuously• ,toc* prices ( returns at any point of time

are !est explained !y lognormal distri!ution

/>

B,M ssumptions



B,M: ssumptions

• Most of the assumptions of the BM

are present in the B,M• %wo additional assumptions:

•

%rading is continuous – thishappens when mar*ets are alwaysopen

• %he stoc* price !eha#iour o#ertime follows a stochastic process

called Geometric Brownian Motion/@

B,M: ssumptions



B,M: ssumptions• %here are two critical assumptions: 1&

#olution of stoc* prices o#er time followsGeometric Brownian Motion? +& t any pointof time the stoc* price ( stoc* return arelognormally distri!uted

• %he 1st assumption tells how the parametersof the lognormal distri!ution in the +nd assumption will change o#er time

• 'f stoc* returns are lognormally distri!utedthen ln(St / St-1) will !e normally distri!uted&

ln(St / St-1) is the continuously compounded

return during 1 time inter#al 3t-1 to t4

B,M: 0easona!leness of 7ognormality



B,M: 0easona!leness of 7ognormality

• 'f returns are lognormally distri!uted

then lowest possi!le return in anyperiod is -100% 3whereas if returns arenormally distri!uted there is some

pro!a!ility that returns will !e less than-1I

• 7ognormal return distri!ution is s*ewed

to the right !ecause while the lowestreturn is -1I) there is no limit on thehighest return - hence right s*ewed

41

7ognormal Property of ,toc* Prices



& 7ognormal Property of ,toc* Prices

• xpected return 3 p&a& on stoc* (

#olatility of returns 3Q p&a& are *nown• 2nderlying assumption: Percentage

3Proportionate changes in stoc* price in

a short time period are normallydistri!uted& %his is related to theGeometric Brownian Motion&

•

Lue to 7ognormal property of stoc*prices ln3, % is normally distri!uted with

mean ( s&d& expressed in terms of , )

( Q 4+





• 2nderlying assumption: Percentage

3Proportionate changes in stoc* pricein a short time period are normallydistri!uted: 1&

• ln3, % is normally distri!uted withmean ( s&d& expressed in terms of , )

( Q

+& due tolognormality

or /&

),( 2 t t S

S ∆∆≈

∆σ µ φ

],)2

[(lnln 22

0 T T S S T

σ σ µ φ −≈−

],)

2

([lnln 22

0 T T S S T σ σ

µ φ −+≈

4/





• Calculate the @5I con.dence

inter#al of the prices of a stoc*after 9 months) gi#en the

following data:• Current stoc* price: 4)

• xpected return: 19I pa)

• Holatility: +I pa

44

B Listri!ution of Continuously



B& Listri!ution of ContinuouslyCompounded 0ate of 0eturn

• =hat is the Pro!a!ility distri!ution ofcontinuously compounded rate of return parealised !etween time ( time % J 3Based on7ognormal Property

• LiFerent from !ecause is &M& of the annualreturns whereas continuously compounded

return is !ased on the G&M& of returns o#er #erysmall inter#als of time !etween ( %

• %he a!o#e distri!ution is normal with mean (s&d& expressed in terms of ) Q ( %

45

B Listri!ution of 0ate of 0eturn



B& Listri!ution of 0ate of 0eturn

Nrom relationship + earlier it can !e said:

%hus as time % increases the standardde#iation of continuously compoundedreturn pa !etween times ( % declines

0

0 ln

1

or S

S

T xeS S

T xT

T ==

),2

(

22

T x σ σ µ φ −≈

49

B Listri!ution of 0ate of 0eturn



B& Listri!ution of 0ate of 0eturn

• Calculate the @5I con.dence

inter#al of a#erage rate of return3GM return realised on a

continuously compounded !asison a stoc* o#er a period of /years gi#en the following data:

• xpected return: 1;I pa• Holatility: +I pa

4;

C Holatility



C& Holatility

• Holatility 3Q in stoc* returns 3( hence

in stoc* price is measured annually• Q can !e estimated from the #olatility

of the continuously compounded daily

returns i&e& #olatility in log of dailyprice relati#es: ln3,i ,i-1

• 'f s D s&d of daily log price relati#es

then: stimated annual #olatility:• ,tandard error of estimate:n2

∧

σ

Γ

=

Λ sσ

4>

C Holatility



C& Holatility

• Holatility 3Q in stoc* returns 3( hence

in stoc* price is measured annually• Rear is measured in trading days not in

calendar days& "ence 1 year D +5+

trading days• 'f #olatility is estimated on a daily !asis:

• 7ife of an option 3% in years :

252dayg per tradinVolatility p.a.Volatility ×=

252

expiryoptiontilldaystradingof No.=T

4@

C Holatility



C& Holatility

• g: %he s&d of continuously

compounded daily returns is 1&+19I&stimate the #olatility of annualreturns and the standard error of

estimate? the s&d is estimated with +data points of daily returns& ssume 1year D +5+ trading days&

5

L Basics of the B,M



L& Basics of the B,M

• %he same source of uncertainty aFects the

option price ( the stoc* price• %he #alue of a stoc* option expressed in terms

of the #alue of the underlying stoc* does notdepend on the expected return on the stoc*

• portfolio can !e formed out of the option (the stoc* such that the uncertainty is eliminated

• %he portfolio !ecomes ris*-less for a #ery small

inter#al of time only• ,o the portfolio has to !e re!alanced #ery

fre$uently in order to *eep it ris*-free in e#eryinstant of time

51

%h Bl * , h l M t N l



%he Blac*-,choles-Merton Normulae

T d T

T r K S d

T

T r K S d

d N S d N e K p

d N e K d N S c

rT

rT

σ−=σ

σ−+=

σ

σ++=

−−−=

−=

−

−

10

2

01

102

210

)22()ln(

)22()ln(

)()(

)()(

where

5+

Blac*-,choles Model: 6umerical



Blac*-,choles Model: 6umerical

•

Sue: Calculate the price of a /-month uropean call option on anon-di#idend paying stoc*& %he

stri*e price is '60 +5 when thecurrent price is '60 +5& %he ris*-free interest rate is 1I p&a& and

#olatility is /I p&a& =hat will !ethe price of a put optionJ

5/

B,M: dKustment for Li#idends



B,M: dKustment for Li#idends

•

Tnown di#idends• Tnown di#idend yield

54

B,M: dKustment for Tnown Li#idends




• Payment of di#idends reduces thestoc* price

• Li#idend payments tend to ma*e calloptions less #alua!le ( put options

more #alua!le• ,o su!tract the present #alue of the

expected di#idends from current price

!efore inputting the same in the model• ,o in the B,M replace the factor , !y:

, – PH 3Li#idends55





nother way of understanding this:

• stoc* price may !e considered to !ethe sum of two components: 3a a

0is*less component and 3! a 0is*ycomponent

• %he ris*less component corresponds

with the *nown di#idends• =hen there are no di#idends the

current stoc* price is only the function

of the ris*y component59





• =hen there are *nown di#idends the ris*less

component is the present #alue of all di#idendsoccurring during the life of the option

• By the time the option expires all di#idends

would ha#e !een paid – hence there will !e noris*less component

• ,o stoc* price after payment of di#idends onlyreects the ris*y component

• 'n present #alue terms: the present #alue ofris*y component should !e e$ual to stoc* priceat time minus present #alue of di#idends

5;





• Sue: Calculate the price of auropean Call option on a stoc*which has ex-di#idend dates in +-

monthsA and 5-monthsA time& achtime the di#idend will !e '60 &5& %he current stoc* price is '60 4)

stri*e price is '60 4 ( #olatility is/I p&a& %he life of the option is 9months ( ris*-free interest rate is

@I&5>

B,M: dKustment for Tnown Li#idend Rield




• 7imitation associated with the pre#iousapproach is that if the option expirationperiod is long then it is unrealistic to saythat di#idends are *nown

• more realistic assumption is thatdi#idend yield 3y D di#idends currentmar*et price of the asset is *nown (

shall remain constant during the life ofthe option

• 0eplace , !y ,e-y% in the B, e$uations5@





• %he current stoc* price is multipliedwith a discount factor calculated onthe !asis of the di#idend yield to

account for the expected drop in#alue from di#idend payments

• %he interest rate is oFset !y the

di#idend yield !ecause the carryingcost of the stoc* decreases due tothe di#idend yield

9

B,M: Modi.ed for Tnown Li#idend Rield



B,M: Modi.ed for Tnown Li#idend Rield

T d T

T yr K S d

T

T yr K S d

d N eS d N e K p

d N e K d N eS c

yT rT

rT yT

σ

σ

σ

σ

σ

−=−−+

=

+−+=

−−−=

−=

−−

−−

10

2

01

102

210

)22()ln(

)22()ln( !"ere

)()(

)()(

91

B,M: Tnown Li#idend Rield



B,M: Tnown Li#idend Rield

•

Calculate the price of a uropeanCall option on a stoc* whichpro#ides a continuous di#idend

yield of +&5I& %he current stoc*price is '60 5) stri*e price is '605 ( #olatility is 4I p&a& %he life

of the option is 9 months ( ris*-free interest rate is @I&

mod4-optn-pricing.pptx

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