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Black-Sholes Model

Prof. Rafi Eldor

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BSM Model

Prof Fisher BLACK and Prof Myron SCHOLES

published in 1973 a paper, that won a Nobel

prize, on the valuation of an European call

option. Their paper was based on an arbitrage

model. At the same time, Prof Robert MERTON

worked and finally published the same

valuation formula. He also won a Nobel prize in

1997.

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BS Assumptions

1.Perfect market

2.Constant risk free rate

3.Constant standard deviation

4.No dividends

5.The dynamics of the stock price is a

GEOMETRIC BROWNIAN MOTION

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World of Certainty

Value of a all option that would be in the money:

rTrT XeSXTSeC

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BS Call Value

)2(*)1(*)( dNXedNSXC rt

t

trXSnd

)5.0()/(11

2

According to BS model, value of Euro call option is:

tdd 12

Where N(d1) and N(d2) are the values of d1 and d2,

taking from the standard normal distribution tables.

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Graphical Exposition

The graphical exposition of the value of a call

option according to BS model:

Value of a call

BS

Value

Intrinsic

value

Value of

the asset

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Black-Scholes example

Black-Scholes example :

%20

%20

2466.0365/90

200

205

r

T

X

S

The last 3 values are in annual terms

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Example (continued)

N200205 )2466.0(2.0

21

eddNC

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Useful terms

37.9

0014.0

78.0

21.17

62.14

5

h

C

XeS

XS

rT

Intrinsic value

Downside limit

BS value

Option delta

Option gamma

Option omega

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Graphical exposition of a Call

BS Call option value C(200)

SBC /

Underlyin

g asset

(Value at

expiration)

5

205200

14.62

C=17.21

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BS value of a PUT option:

)1(*)2(*)( dNSdNXeXP rt

Value at expiration

BS value

BS PUT value

Value of the

underlying

asset

Graphical exposition of a PUT

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Parameters Impact

CALLPUT

S

X

r

t?

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Implied Volatility

Inserting the data of the 5 parameters would

give rise to the BS option value. On the other

hand, if we use the market value for the

option and solving for the value of the

volatility, we receive the value that the

market participants assume regarding

volatility. This number is called implied

volatility.

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The Option Greeks

We can get the derivatives with regard to

each of the parameters. Those derivatives

are called: The Option Greeks.

DerivativeParameterSimbol

Gamma

DeltaUnderlying assetS

Exercise price X

RhoRate of interestr

VegaStandard deviation

ThetaTime to expirationt