option vauation ii-2 ppt

7/25/2019 Option Vauation II-2 ppt

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Fi8000Fi8000

Option Valuation IIOption Valuation II

Milind ShrikhandeMilind Shrikhande


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Valuation of OptionsValuation of Options

Arbitrage Restrictions on the Values ofArbitrage Restrictions on the Values of

OptionsOptions

Quantitative Pricing ModelsQuantitative Pricing Models

Binomial modelBinomial model

A formula in the simple caseA formula in the simple caseAn algorithm in the generalAn algorithm in the general

BlackScholes model !a formula"BlackScholes model !a formula"


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Binomial Option Pricing ModelBinomial Option Pricing Model

AssumptionsAssum

ptions##

A single periodA single period $%o dates# time t&' and time t&( !e)piration"$%o dates# time t&' and time t&( !e)piration"

$he future !time (" stock price has onl* t%o$he future !time (" stock price has onl* t%o

possible valuespossible values $he price can go up or do%n$he price can go up or do%n

$he perfect market assumptions$he perfect market assumptions +o transactions costs, borro%ing and lending at the+o transactions costs, borro%ing and lending at the

risk free interest rate, no ta)es-risk free interest rate, no ta)es-


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ExampleExample$he stock price$he stock price Assume S& ./',Assume S& ./',

u& ('0 and d& !10"u& ('0 and d& !10"

S

Su=S!"u#

Sd=S!"d#

S=$%0

Su=$%%

Sd=$&8'%


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ExampleExample$he call option price$he call option price Assume 2& ./',Assume 2& ./',

$& ( *ear !$& ( *ear !e)piratione)piration""

(

(u= Max)Su*+,0-

(d= Max)Sd*+,0-

(

(u= $% = Max)%%*%0,0-

(d= $0

= Max)&8'%*%0,0-


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ExampleExample$he bond price$he bond price Assume r& 30Assume r& 30

!

!"r#

!"r#

$!

$!'0.

$!'0.


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/eplicating Portfolio/eplicating Portfolio

At time t&', %e can create a portfolio ofAt time t&', %e can create a portfolio of NNshares ofshares ofthe stock and an investment ofthe stock and an investment of BBdollars in the riskdollars in the riskfree bond4 $he pa*off of the portfolio %ill replicatefree bond4 $he pa*off of the portfolio %ill replicatethe t&( pa*offs of the call option#the t&( pa*offs of the call option#

NN5.// 65.// 6 BB5.(4'3 & ./5.(4'3 & ./

NN5.784/ 65.784/ 6 BB5.(4'3 & .'5.(4'3 & .'

Obviousl*, this portfolio should also have the sameObviousl*, this portfolio should also have the same

price as the call option at t&'#price as the call option at t&'#

NN5./' 65./' 6 BB5.( & 95.( & 9

:e get:e get NN&'4;3


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1ifferent /eplication 1ifferent /eplication

$he price of .( in the$he price of .( in the

>up? state#>up? state#$he price of .( in the$he price of .( in the

>do%n? state#>do%n? state#

2u

$!

$0

2d

$0

$!


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/eplicating Portfolios 3sing t4e/eplicating Portfolios 3sing t4e

State PricesState Prices:e can replicate the t&( pa*offs of the stock and:e can replicate the t&( pa*offs of the stock andthe bond using the state prices#the bond using the state prices#

qquu5.// 65.// 6 qqdd5.784/ & ./'5.784/ & ./'

qquu5.(4'3 65.(4'3 6 qqdd5.(4'3 & .(5.(4'3 & .(

Obviousl*, once %e solve for the t%o state pricesObviousl*, once %e solve for the t%o state prices%e can price an* other asset in that econom*4 @n%e can price an* other asset in that econom*4 @nparticular %e can price the call option#particular %e can price the call option#

qquu5./ 65./ 6 qqdd5.' & 95.' & 9

:e get:e get qquu&'43/1(,&'43/1(, qqdd&'4=


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ExampleExample$he put option price$he put option price Assume 2& ./',Assume 2& ./',

$& ( *ear !$& ( *ear !e)piratione)piration""

P

Pu= Max)+*Su,0-

Pd= Max)+*Sd,0-

P

Pu= $0 = Max)%0*%%,0-

Pd= $!'%

= Max)%0*&8'%,0-


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/eplicating Portfolios 3sing t4e/eplicating Portfolios 3sing t4e

State PricesState Prices:e can replicate the t&( pa*offs of the stock and:e can replicate the t&( pa*offs of the stock andthe bond using the state prices#the bond using the state prices#

qquu5.// 65.// 6 qqdd5.784/ & ./'5.784/ & ./'

qquu5.(4'3 65.(4'3 6 qqdd5.(4'3 & .(5.(4'3 & .(

But the assets are e)actl* the same and so are theBut the assets are e)actl* the same and so are thestate prices4 $he put option price is#state prices4 $he put option price is#

qquu5.' 65.' 6 qqdd5.(4/ & P5.(4/ & P

:e get:e get qquu&'43/1(,&'43/1(, qqdd&'4=


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56o Period Example56o Period Example

Assume that the current stock price is ./',Assume that the current stock price is ./',

and it can either go up ('0 or do%n 10 inand it can either go up ('0 or do%n 10 in

each period4each period4

$he one period riskfree interest rate is$he one period riskfree interest rate is

304304

:hat is the price of a uropean call option:hat is the price of a uropean call optionon that stock, %ith an e)ercise price of ./'on that stock, %ith an e)ercise price of ./'

and e)piration in t%o periodsand e)piration in t%o periods


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54e Stoc7 Price54e Stoc7 Price

S& ./', u& ('0 and d& !10"S& ./', u& ('0 and d& !10"

S=$%0

Su=$%%

Sd=$&8'%

Suu=$.0'%

Sud=Sdu=$%'%

Sdd=$&9'0%


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54e Bond Price54e Bond Price

r& 30 !for each period"r& 30 !for each period"

$!

$!'0.

$!'0.

$!'!:.

$!'!:.

$!'!:.


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54e (all Option Price54e (all Option Price

2& ./' and $& = periods2& ./' and $& = periods

(

(u

(d

(uu=Max).0'%*%0,0-=$!0'%

(ud=Max)%'%*%0,0-=$'%

(dd=Max)&9'0%*%0,0-=$0


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State Prices in t4e 56o Period 5reeState Prices in t4e 56o Period 5ree

:e can replicate the t&( pa*offs of the stock and the bond:e can replicate the t&( pa*offs of the stock and the bondusing the state prices#using the state prices#

qquu5.// 65.// 6 qqdd5.784/ & ./'5.784/ & ./'

qquu

5.(4'3 65.(4'3 6 qqdd

5.(4'3 & .(5.(4'3 & .(+ote that if u, d and r are the same, our solution for the+ote that if u, d and r are the same, our solution for thestate prices %ill not change !regardless of the price levelsstate prices %ill not change !regardless of the price levelsof the stock and the bond"#of the stock and the bond"#

qquu5S5!(6u" 65S5!(6u" 6 qqdd5S 5!(6d" & S5S 5!(6d" & S

qquu 5!(6r"5!(6r"tt66 qqdd 5!(6r"5!(6r"tt& !(6r"& !(6r"!t("!t("

$herefore, %e can use the same stateprices in ever* part$herefore, %e can use the same stateprices in ever* partof the tree4of the tree4


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54e (all Option Price54e (all Option Price

CCuu& '43/1( and C& '43/1( and Cdd& '4=


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:hat is the price of a uropean put option on:hat is the price of a uropean put option onthat stock, %ith an e)ercise price of ./' andthat stock, %ith an e)ercise price of ./' ande)piration in t%o periodse)piration in t%o periods

:hat is the price of an American call option on:hat is the price of an American call option onthat stock, %ith an e)ercise price of ./' andthat stock, %ith an e)ercise price of ./' ande)piration in t%o periodse)piration in t%o periods

:hat is the price of an American put option on:hat is the price of an American put option onthat stock, %ith an e)ercise price of ./' andthat stock, %ith an e)ercise price of ./' ande)piration in t%o periodse)piration in t%o periods


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uropean put option use the tree or theuropean put option use the tree or the

putcall parit*putcall parit*

:hat is the price of an American call:hat is the price of an American call

option if there are no dividends -option if there are no dividends -

American put option D use the treeAmerican put option D use the tree


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54e European Put Option Price54e European Put Option Price

CCuu& '43/1( and C& '43/1( and Cdd& '4=


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54e merican Put Option Price54e merican Put Option Price

CCuu& '43/1( and C& '43/1( and Cdd& '4=


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merican Put Optionmerican Put Option

+ote that at time t&( the option bu*er %ill+ote that at time t&( the option bu*er %ill

decide %hether to e)ercise the option ordecide %hether to e)ercise the option or

keep it till e)piration4keep it till e)piration4

@f the pa*off from immediate e)ercise is@f the pa*off from immediate e)ercise is

higher than the option value the optimalhigher than the option value the optimal

strateg* is to e)ercise#strateg* is to e)ercise#@f Ma)E 2S@f Ma)E 2Suu,' F G P,' F G Puu!uropean" &G )ercise!uropean" &G )ercise


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54e merican Put Option Price54e merican Put Option Price

CCuu& '43/1( and C& '43/1( and Cdd& '4=


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1eterminants of t4e Values1eterminants of t4e Values

of (all and Put Optionsof (all and Put OptionsVaria>leVaria>le ( ? (all Value( ? (all Value P ? Put ValueP ? Put Value

SSDD stock pricestock price @ncrease@ncrease HecreaseHecrease

++DD e)ercise pricee)ercise price HecreaseHecrease @ncrease@ncrease@@DD stock price volatilit*stock price volatilit* @ncrease@ncrease @ncrease@ncrease

55DD time to e)pirationtime to e)piration @ncrease@ncrease @ncrease@ncrease

rrDD riskfree interest rateriskfree interest rate @ncrease@ncrease HecreaseHecrease

1iA1iADD dividend pa*outsdividend pa*outs HecreaseHecrease @ncrease@ncrease


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Blac7*Sc4oles ModelBlac7*Sc4oles Model

Heveloped around (


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9 D call premium9 D call premium

S D stock priceS D stock price

2 D e)ercise price2 D e)ercise price

$ D time to e)piration$ D time to e)piration

r D the interest rater D the interest rate

IID std of stock returnsD std of stock returns

ln!J" D natural log of Jln!J" D natural log of J

eer$r$

D e)pEr$F & !=4;(81"D e)pEr$F & !=4;(81"r$r$

+!J" D standard normal+!J" D standard normal

cumulative probabilit*cumulative probabilit*

1 2

2

1

2 1

( ) ( )

1ln

2,

rTC S N d Xe N d

Where

ST r

Xd

T

d d T

=

+ + =

=


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54e 0,!# 1istri>ution54e 0,!# 1istri>ution

z=0

pdf(z)

C

C#


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Blac7*Sc4oles exampleBlac7*Sc4oles example

9 D 9 D

S D .7;4/'S D .7;4/'

2 D ./'2 D ./'

$ D '4=/ *ears$ D '4=/ *ears

r D '4'/r D '4'/ !/0 annual rate!/0 annual rate

compoundedcompounded

continuousl*"continuousl*"IID '41' !or 1'0"D '41' !or 1'0"

1 2

2

1

2 1

( ) ( )

1ln

2,

rTC S N d Xe N d

Where

ST r

Xd

T

d d T

=

+ + =

=


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Blac7*Sc4oles ExampleBlac7*Sc4oles Example

9 D 9 D

S D .7;4/'S D .7;4/'

2 D ./'2 D ./'

$ D '4=/ *ears$ D '4=/ *ears




1

1

2

2

( 0.1836)

( ) 0.4272

( 0.3336)

( ) 0.3693

$2.0526

d

N d

d

N d

and

C

= =

= =

=


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9ontinuous time and therefore continuous9ontinuous time and therefore continuous

compoundingcompounding

+!d" D loosel* speaking, +!d" is the >risk+!d" D loosel* speaking, +!d" is the >risk

adKusted? probabilit* that the call optionadKusted? probabilit* that the call option

%ill e)pire in the mone* !check the pricing%ill e)pire in the mone* !check the pricing

for the e)treme cases# ' and ("for the e)treme cases# ' and ("

ln!SL2" D appro)imatel*, a percentageln!SL2" D appro)imatel*, a percentage

measure of option >mone*ness?measure of option >mone*ness?


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P D Put premiumP D Put premium

S D stock priceS D stock price

2 D e)ercise price2 D e)ercise price

$ D time to e)piration$ D time to e)piration

r D the interest rater D the interest rate

IID std of stock returnsD std of stock returns

ln!J" D natural log of Jln!J" D natural log of J

eer$r$

D e)pEr$F & !=4;(81"D e)pEr$F & !=4;(81"r$r$

+!J" D standard normal+!J" D standard normal

cumulativecumulative

probabilit*probabilit*

[ ] [ ]2 1

2

1

2 1

1 ( ) 1 ( )

1ln2

,

rTP Xe N d S N d

Where

S T rX

dT

d d T

=

+ + =

=


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Blac7*Sc4oles ExampleBlac7*Sc4oles Example

P D P D

S D .7;4/'S D .7;4/'

2 D ./'2 D ./'

$ D '4=/ *ears$ D '4=/ *ears




1

1

2

2

( 0.1836)

( ) 0.4272

( 0.3336)

( ) 0.3693

$3.9315

d

N d

d

N d

and

P

= =

= =

=


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Stoc7 /eturn VolatilitDStoc7 /eturn VolatilitD

One approachOne approach##9alculate an estimate of the volatilit* using the9alculate an estimate of the volatilit* using the

historical stock returns and plug it in the optionhistorical stock returns and plug it in the option

formula to get pricingformula to get pricing

( )2

1

1

1( )

1

ln

n

t

t

tt

t

Est return average returnn

Where

Sreturn

S

=

=

=


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Stoc7 /eturn VolatilitDStoc7 /eturn VolatilitD

Another approachAnother approach##9alculate the stock return volatilit* implied b*9alculate the stock return volatilit* implied b*

the option price observed in the marketthe option price observed in the market

!a trial and error algorithm"!a trial and error algorithm"

1 2

2

1 2 1

( ) ( )

1ln

2 and

rTC S N d Xe N d

Where

ST r

Xd d d T

T

=

+ + = =


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Implied VolatilitD * exampleImplied VolatilitD * example

9 D .=4/9 D .=4/

S D .7;4/'S D .7;4/'

2 D ./'2 D ./'

$ D '4=/ *ears$ D '4=/ *ears

r D '4'/r D '4'/!/0 annual rate!/0 annual rate


continuousl*"continuousl*"IID D

1 2

2

1

2 1

( ) ( )

1ln

2,

rTC S N d Xe N d

Where

ST r

Xd

T

d d T

=

+ + =

=


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9 D 9 D

S D .7;4/'S D .7;4/'

2 D ./'2 D ./'

$ D '4=/ *ears$ D '4=/ *ears




1

1

2

2

( 0.1836)

( ) 0.4272

( 0.3336)

( ) 0.3693

$2.0526

d

N d

d

N d

and

C

= =

= =

=


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9 D .=4/ G .=4'/=39 D .=4/ G .=4'/=3

S D .7;4/'S D .7;4/'

2 D ./'2 D ./'

$ D '4=/ *ears$ D '4=/ *ears



continuousl*"continuousl*"IIN or G '41N or G '41

1 2

2

1

2 1

( ) ( )

1ln

2,

rTC S N d Xe N d

Where

ST r

Xd

T

d d T

=

+ + =

=


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9 D 9 D

S D .7;4/'S D .7;4/'

2 D ./'2 D ./'

$ D '4=/ *ears$ D '4=/ *ears




1

1

2

2

( 0.0940)

( ) 0.4626

( 0.2940)

( ) 0.3844

$2.9911

d

N d

d

N d

and

C

= =

= =

=


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9 D 9 D

S D .7;4/'S D .7;4/'

2 D ./'2 D ./'

$ D '4=/ *ears$ D '4=/ *ears



continuousl*"continuousl*"IID '41/ !or 1/0"D '41/ !or 1/0"

1

1

2

2

( 0.1342)

( ) 0.4466

( 0.3092)

( ) 0.3786

$2.5205

d

N d

d

N d

and

C

= =

= =

=


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pplication Portfolio Insurancepplication Portfolio Insurance

Options can be used to guarantee minimumOptions can be used to guarantee minimum

returns from an investment in stocks4returns from an investment in stocks4

Purchasing portfolio insurance !protective putPurchasing portfolio insurance !protective putstrateg*"#strateg*"#

ong one stockong one stock

Bu* a put option on one stockBu* a put option on one stock@f no put option e)ists, use a stock and@f no put option e)ists, use a stock and

a bond to replicate the put option pa*offs4a bond to replicate the put option pa*offs4


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Portfolio Insurance ExamplePortfolio Insurance Example

ou decide to invest in one share of eneral Pills !P"ou decide to invest in one share of eneral Pills !P"stock, %hich is currentl* traded for ./34 $he stock pa*s nostock, %hich is currentl* traded for ./34 $he stock pa*s nodividends4dividends4

ou %orr* that the stockTs price ma* decline and decide toou %orr* that the stockTs price ma* decline and decide to

purchase a uropean put option on Ps stock4 $he putpurchase a uropean put option on Ps stock4 $he putallo%s *ou to sell the stock at the end of one *ear for ./'4allo%s *ou to sell the stock at the end of one *ear for ./'4

@f the std of the stock price is@f the std of the stock price is II&'41 !1'0" and rf&'4'8 !80&'41 !1'0" and rf&'4'8 !80compounded continuousl*", %hat is the price of the putcompounded continuousl*", %hat is the price of the putoptionoption

:hat is the 9U from *our strateg* at time t&':hat is the 9U from *our strateg* at time t&':hat is the 9U at time t&( as a function of 'NS:hat is the 9U at time t&( as a function of 'NS$$N(''N(''


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:hat if there is no put option on the stock that:hat if there is no put option on the stock that*ou %ish to insure se the BWS formula to*ou %ish to insure se the BWS formula toreplicate the protective put strateg*4replicate the protective put strateg*4

:hat is *our insurance strateg*:hat is *our insurance strateg*:hat is the 9U from *our strateg* at time t&':hat is the 9U from *our strateg* at time t&'

Suppose that one %eek later, the price of theSuppose that one %eek later, the price of thestock increased to .3', %hat is the value of thestock increased to .3', %hat is the value of the

stocks and bonds in *our portfoliostocks and bonds in *our portfolioXo% should *ou rebalance the portfolio to keepXo% should *ou rebalance the portfolio to keepthe insurancethe insurance


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$he total time t&'$he total time t&' (F(Fof the protective putof the protective put

!insured portfolio" is#!insured portfolio" is#

9U9U''& S& S''PP'' & S& S''2e2er$r$Y(+!dY(+!d=="Z6S"Z6S''Y(+!dY(+!d(("Z"Z

& S& S''Y+!dY+!d(("Z"Z2e2er$r$Y(+!dY(+!d=="Z"Z

And the proportion invested in the stock is#And the proportion invested in the stock is#

%%stockstock&S&S''Y+!dY+!d(("Z"ZLESLES''Y+!dY+!d(("Z"Z2e2er$r$Y(+!dY(+!d=="ZF"ZF


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$he proportion invested in the stock is#$he proportion invested in the stock is#

%%stockstock & S& S''Y+!dY+!d(("Z"ZLESLES''Y+!dY+!d(("Z"Z62e62er$r$Y(+!dY(+!d=="ZF"ZF

Or, if %e remember the original !protectiveOr, if %e remember the original !protectiveput" strateg*#put" strateg*#

%%stockstock & S& S''Y+!dY+!d(("Z"ZLESLES'' 6 P6 P''FF

Uinall*, the proportion invested in the bond is#Uinall*, the proportion invested in the bond is#%%bondbond& (%& (%stockstock


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Sa* *ou invest .(,''' in the portfolio toda* !t&'"Sa* *ou invest .(,''' in the portfolio toda* !t&'"

$ime t & '#$ime t & '#

%%stockstock& /35'4;83/L!/36=418" & ;/47/0& /35'4;83/L!/36=418" & ;/47/0

Stock value & '4;/7/[.(,''' & .;/74/Stock value & '4;/7/[.(,''' & .;/74/Bond value & '4=7// [.(,''' & .=7/4/Bond value & '4=7// [.(,''' & .=7/4/

nd of %eek ( !%e assumed that the stock price increasednd of %eek ( !%e assumed that the stock price increasedto .3'"#to .3'"#

Stock value & !.3'L./3"5.;/74/ & .8'841


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+o% *ou have a .(,'/74=; portfolio+o% *ou have a .(,'/74=; portfolio

$ime t & ( !beginning of %eek ="#$ime t & ( !beginning of %eek ="#

%%stockstock& 3'5'487;3L!3'6(431" & 8=4/10& 3'5'487;3L!3'6(431" & 8=4/10

Stock value & '48=/1[.(,'/74=; & .8;'4'3Stock value & '48=/1[.(,'/74=; & .8;'4'3Bond value & '4(;7; [.(,'/74=; & .(874=(Bond value & '4(;7; [.(,'/74=; & .(874=(

@4e4 *ou should rebalance *our portfolio !increase the@4e4 *ou should rebalance *our portfolio !increase theproportion of stocks to 8=4/10 and decrease the proportionproportion of stocks to 8=4/10 and decrease the proportion

of bonds to (;47;0"4of bonds to (;47;0"4:h* should %e rebalance the portfolio Should %e:h* should %e rebalance the portfolio Should %erebalance the portfolio if %e use the protective put strateg*rebalance the portfolio if %e use the protective put strateg*%ith a real put option%ith a real put option


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Practice Pro>lemsPractice Pro>lems

B\M 9h4 =(# ;(', (;,(8B\M 9h4 =(# ;(', (;,(8

Practice set# 137=4Practice set# 137=4

option vauation ii-2 ppt

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