The document outlines the Black-Scholes model for European call and put options, originally developed by Fischer Black and Myron Scholes in 1973, with contributions from Robert Merton. It details the assumptions necessary for the model, the valuation formulas for options, and incorporates graphical expositions. Additionally, it discusses the concept of implied volatility and the Option Greeks used to analyze the sensitivity of option prices to various parameters.

1. 1
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Black-Sholes Model
Prof. Rafi Eldor

2. 2
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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.

3. 3
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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

4. 4
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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
trXSn
d

 )5.0()/(1
1
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.

6. 6
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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

7. 7
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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)
SB
C /
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
Delta
Underlying assetS
Exercise priceX
RhoRate of interestr
Vega
Standard deviation
Theta
Time to expirationt





