Ch 13 Hull Fundamentals 8 the d

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Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. ull !"13

Valuing Stock Options:

The Black-Scholes-MertonModel

Chapter 13

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Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. ull!"13

The Black-Scholes-Merton

Random Walk Assumption

Consider a stock whose price is S In a short period of time of length tthe

return on the stock (S/S) is assumed tobe normal with mean tand standarddeviation

is expected return and is volatility

2

t


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The Lognormal Propert

hese assumptions imply ln STis normally

distributed with mean!

and standard deviation!

"ecause the logarithm of STis normal# STis

lognormally distributed

3

TS )2/(ln 20 +

T


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The Lognormal Propert

continued

where [m,v$ is a normal distribution withmean mand variance v

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[ ]

[ ]TTS

S

TTSS

T

T

22

0

22

0

,)2(ln

,)2(lnln

+

or


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The Lognormal !istri"ution

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E S S e

S S e e

T

T

T

T T

( )

( ) ( )

=

= 0

0

2 2 2 1

var


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The #$pected Return

he expected value of the stock price attime Tis S0e

T

he return in a short periodt is t"ut the expected return on the stock

with continuous compounding is 2/2his reflects the difference between

arithmetic and geometric means

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Mutual %und Returns &See BusinessSnapshot '()' on page (*(+

%uppose that returns in successive yearsare 1&'# '# 3'# *' and &'

he arithmetic mean of the returns is 1+'he returned that would actually be

earned over the five years (the geometric

mean) is 1,+'


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The Volatilit

he volatility is the standard deviation of thecontinuously compounded rate of return in 1year

he standard deviation of the return in timetis

If a stock price is -& and its volatility is &'

per year what is the standard deviation ofthe price change in one day.

8

t


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,ature o Volatilit

olatility is usually much greater when themarket is open (i,e, the asset is trading)

than when it is closed0or this reason time is usually measured

in trading days2 not calendar days whenoptions are valued

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#stimating Volatilit rom .istorical !ata &page (*/-(*0+

1, ake observations S0, S1, . . . , Sn at intervalsof years (e,g, for weekly data 1/&)

,

Calculate the continuously compoundedreturn in each interval as!

3, Calculate the standard deviation#s# of the ui4s

+, he historical volatility estimate is!


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The 1oncepts 2nderling Black-

Scholes

he option price and the stock price dependon the same underlying source of uncertainty

5e can form a portfolio consisting of the

stock and the option which eliminates thissource of uncertainty

he portfolio is instantaneously riskless and

must instantaneously earn the risk*free rate

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The Black-Scholes %ormulas&See page (*3+

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TdT

TrKSd

T

TrKSd

dNSdNeKp

dNeKdNSc

rT

rT

=

+=

++=

=

=

10

2

01

102

210

)2/2()/ln(

)2/2()/ln(

)()(

)()(

where


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The ,&$+ %unction

N(x) is the probability that a normally distributedvariable with a mean of 6ero and a standard deviationof 1 is less thanx

%ee tables at the end of the book


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Properties o Black-Scholes %ormula

7s S0becomes very large ctends to S0#

Ke-rTandptends to 6ero

7s S0becomes very small ctends to 6eroandptends toKe-rT#S0

5hat happens as becomes very large.

5hat happens as Tbecomes very large.


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Risk-,eutral Valuation

he variable does not appear in the "lack*%choles e8uation

he e8uation is independent of all variablesaffected by risk preference

his is consistent with the risk*neutralvaluation principle

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Appling Risk-,eutral Valuation

1, 7ssume that the expectedreturn from an asset is therisk*free rate

, Calculate the expected payofffrom the derivative

3, 9iscount at the risk*free rate

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Valuing a %or4ard 1ontract 4ith

Risk-,eutral Valuation

:ayoff is ST#K

;xpected payoff in a risk*neutral world isS0e

rT#K

:resent value of expected payoff is

e-rT[S0erT#K]=S0#Ke-rT

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5mplied Volatilit

he implied volatility of an option is thevolatility for which the "lack*%choles pricee8uals the market price

he is a one*to*one correspondencebetween prices and implied volatilities

raders and brokers often 8uote impliedvolatilities rather than dollar prices

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The V56 5nde$ o S7P 8** 5mplied

Volatilit9 an) ;**/ to Sept) ;*';


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!i


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American 1alls

7n 7merican call on a non*dividend*payingstock should never be exercised early

7n 7merican call on a dividend*paying stockshould only ever be exercised immediatelyprior to an ex*dividend date

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Black=s Appro$imation or !ealing 4ith

!iust before the final ex*dividenddate

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Ch 13 Hull Fundamentals 8 the d