Chapter 6
The Black-Scholes
Option Pricing
Model
1
© 2002 South-Western Publishing
Introduction
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The Black-Scholes option pricing model
(BSOPM) has been one of the most
important developments in finance in the
last 30 years
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basic valuation model for a European Call option
on a stock that pays no dividends
Has provided a good understanding of what
options should sell for
Has made options more attractive to individual
and institutional investors
The Model
C  SN (d1 )  Ke RT N (d 2 )
where
2 
S 
T
ln     R 
2 
K 
d1 
 T
and
d 2  d1   T
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The Model (cont’d)

Variable definitions:
S
K
e
R
T
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=
=
=
=
=
=
current stock price
option strike price
base of natural logarithms
riskless interest rate
time until option expiration
standard deviation (sigma) of returns on
the underlying security
natural logarithm
ln =
N(d1) and
N(d2) = cumulative standard normal distribution
functions
Development and Assumptions
of the Model

Derivation from:
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Physics
Mathematical short cuts
Arbitrage arguments
Fischer Black and Myron Scholes utilized
the physics heat transfer equation to
develop the BSOPM
Determinants of the Option
Premium
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Strike price
Time until expiration
Stock price
Volatility
Dividends
Risk-free interest rate
Strike Price

The lower the strike price for a given stock,
the more the call option should be worth
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the strike price does not change (not a risk
factor) so this concept is meaningful only in
considering how much more or less a call
with a different strike price would be worth.
......this is the only variable that does not
change over the life of the option!
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Because a call option lets you buy at a
predetermined striking price
Time Until Expiration
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The longer the time until expiration, the
more the option is worth
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The option premium increases for more distant
expirations for puts and calls
Time value decay or the decrease in value of a
call option as time elapses is measured by the
option’s ‘theta’
Theta is a measure of the sensitivity of a call
option price to the time remaining until
expiration
Stock Price

The higher the stock price, the more a given
call option is worth
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A call option holder benefits from a rise in the
stock price
the relationship between the stock price and the
call price is typically expressed as a single value
and is referred to as the ‘delta’
the delta is closer to one when the stock price is
high relative to the strike price and closer to
zero when the stock price is low relative to the
strike price
Volatility

The greater the price volatility (expected)
the more the option is worth
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call option prices are quite sensitive to a small
change in expected stock price volatility
volatility is the annualized standard deviation of
returns anticipated in the underlying asset over
the remaining term of the option
The volatility estimate ‘Vega’ cannot be directly
observed and must be estimated
Volatility

Expected volatility or the ‘vega’ is also related
to:
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the time to expiration factor - the longer the life
span of the option the greater the vega will be
which then means that it will have a greater
impact on the price of the option for a given
change in the volatility
the position of the option ie. At the money/in/out
of the money .....impact of volatility on the call
option price is greatest for options that are ‘at
the money’
Implied Volatility
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Instead of solving for the call premium,
assume the market-determined call
premium is correct
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Then solve for the volatility that makes the
equation hold
This value is called the implied volatility
Historical Versus Implied
Volatility
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
The volatility from a past series of prices is
historical volatility
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Implied volatility gives an estimate of what
the market thinks about likely volatility in
the future
Historical Versus Implied
Volatility (cont’d)
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Strong and Dickinson (1994) find
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Clear evidence of a relationship between the
standard deviation of returns over the past
month and the current level of implied volatility
That the current level of implied volatility
contains both an ex post component based on
actual past volatility and an ex ante component
based on the market’s forecast of future
variance
Dividends
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Black -Scholes model assumes zero dividends
Real World - A company that pays a large dividend
will have a smaller option premium than a company
with a lower dividend, everything else being equal
– The stock price falls on the ex-dividend date
– the model can be modified such that the option
premium in the model is calculated by
subtracting the present value of the dividends in
question from the stock price.
....so in effect, the stock price is adjusted down, which
all other things being equal, will bring down the value
of the call option
Risk-Free Interest Rate

The higher the risk-free interest rate, the higher the
option premium, everything else being equal
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A higher “discount rate” means that the call premium
must rise for the put/call parity equation to hold
BSOM model assumes no change in the interest
rate over the life of the option
Changes in RF interest rates have less of an impact
on the call price vs other variables we have looked
at.