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Introduction
The Black-Scholes option pricing model
(BSOPM) has been one of the mostimportant developments in finance in the
last 50 years
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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The Black-Scholes Option
Pricing Model
The model
Development and assumptions of themodel
Determinants of the option premium
Assumptions of the Black-Scholes model
Intuition into the Black-Scholes model
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The Model
Tdd
T
TRK
S
d
dNKedSNCRT
12
2
1
21
and
2ln
where
)()(
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The Model (contd)
Variable definitions:S = current stock price
K = option strike price
e = base of natural logarithms
R = riskless interest rate
T = time until option expiration
= standard deviation (sigma) of returns onthe underlying security
ln = natural logarithm
N(d1) and
N(d2) = cumulative standard normal distribution
functions
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Development and Assumptions
of the Model
Derivation from:
Physics Mathematical short cuts
Arbitrage arguments
Fischer Black and Myron Scholes utilizedthe physics heat transfer equation to
develop the BSOPM
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Determinants of the Option
Premium
Striking price
Time until expiration Stock price
Volatility
Dividends Risk-free interest rate
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Striking Price
The lower the striking price for a given
stock, the more the option should be worth Because a call option lets you buy at a
predetermined striking price
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Time Until Expiration
The longer the time until expiration, the
more the option is worth The option premium increases for more distant
expirations for puts and calls
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Stock Price
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
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Volatility
The greater the price volatility, the more the
option is worth The volatility estimate s igma cannot be directly
observed and must be estimated
Volatility plays a major role in determining time
value
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Dividends
A company that pays a large dividend will
have a smaller option premium than acompany with a lower dividend, everything
else being equal
Listed options do not adjust for cash dividends
The stock price falls on the ex-dividend date
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Risk-Free Interest Rate
The higher the risk-free interest rate, the
higher the option premium, everything elsebeing equal
A higher discount rate means that the call
premium must rise for the put/call parity
equation to hold
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Assumptions of the Black-
Scholes Model
The stock pays no dividends during the
options life European exercise style
Markets are efficient
No transaction costs
Interest rates remain constant
Prices are lognormally distributed
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The Stock Pays no Dividends
During the Options Life
If you apply the BSOPM to two securities,
one with no dividends and the other with adividend yield, the model will predict the
same call premium
Robert Merton developed a simple extension to
the BSOPM to account for the payment ofdividends
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The Stock Pays no Dividends
During the Options Life (contd)
The Robert Miller Option Pricing Model
Tdd
T
TdRK
S
d
dNKedSNeCRTdT
*
1
*
2
2
*
1
*
2
*
1
*
and
2ln
where
)()(
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European Exercise Style
A European option can only be exercised
on the expiration date American options are more valuable than
European options
Few options are exercised early due to time
value
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Markets Are Efficient
The BSOPM assumes informational
efficiency People cannot predict the direction of the
market or of an individual stock
Put/call parity implies that you and everyone
else will agree on the option premium,regardless of whether you are bullish or bearish
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No Transaction Costs
There are no commissions and bid-ask
spreads Not true
Causes slightly different actual option prices for
different market participants
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Interest Rates Remain Constant
There is no real riskfree interest rate
Often the 30-day T-bill rate is used Must look for ways to value options when the
parameters of the traditional BSOPM are
unknown or dynamic
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Prices Are Lognormally
Distributed
The logarithms of the underlying security
prices are normally distributed A reasonable assumption for most assets on
which options are available
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Intuition Into the Black-Scholes
Model
The valuation equation has two parts
One gives a pseudo-probability weightedexpected stock price (an inflow)
One gives the time-value of money adjusted
expected payment at exercise (an outflow)
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Intuition Into the Black-Scholes
Model (contd)
)( 1dSNC )( 2dNKeRT
Cash Inflow Cash Outflow
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Intuition Into the Black-Scholes
Model (contd)
The value of a call option is the difference
between the expected benefit fromacquiring the stock outright and paying the
exercise price on expiration day
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Calculating Black-Scholes
Prices from Historical Data
To calculate the theoretical value of a call
option using the BSOPM, we need: The stock price
The option striking price
The time until expiration
The riskless interest rate The volatility of the stock
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Problems Using the Black-
Scholes Model
Does not work well with options that are
deep-in-the-money or substantially out-of-
the-money
Produces biased values for very low or
very high volatility stocks
Increases as the time until expiration increases
May yield unreasonable values when an
option has only a few days of life remaining
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