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The Black-Scholes
Option Pricing
Model
© 2002 South-Western Publishing

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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
–
–
– 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

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C

SN ( d
1
)

Ke

RT
N ( d
2
) where d
1
 ln



S
K








R
T


2
2


 T and d
2
 d
1
 
T

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 Variable definitions:
S =
K = e =
R =
T =

= ln = 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
N(d
1
) and
N(d
2
) = cumulative standard normal distribution functions

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 Derivation from:
– Physics
–
–
Mathematical short cuts
Arbitrage arguments
 Fischer Black and Myron Scholes utilized the physics heat transfer equation to develop the BSOPM

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 Strike price
 Time until expiration
 Stock price
 Volatility
 Dividends
 Risk-free interest rate

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 The lower the strike price for a given stock, the more the call option should be worth
– Because a call option lets you buy at a predetermined striking price
 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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 The longer the time until expiration, the more the option is worth
– 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’
– Theata is a measure of the sensitivity of a call option price to the time remaining until expiration

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 The higher the stock price, the more a given call option is worth
– 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

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 The greater the price volatility (expected) the more the option is worth
– 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

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
Expected volatility or the ‘vega’ is also related to:
– 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’

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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

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 The higher the risk-free interest rate, the higher the option premium, everything else being equal
–
A higher “discount rate” means that the call premium must rise for the put/call parity equation to hold

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 Instead of solving for the call premium, assume the market-determined call premium is correct
– Then solve for the volatility that makes the equation hold
– This value is called the implied volatility

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 The volatility from a past series of prices is historical volatility
 Implied volatility gives an estimate of what the market thinks about likely volatility in the future

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 Strong and Dickinson (1994) find
–
–
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