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© 2004 South-Western Publishing1
Chapter 6
The Black-ScholesOption PricingModel
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3
Introduction
The Black-Scholes option pricing model
(BSOPM) has been one of the mostimportant developments in finance in thelast 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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4
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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5
The Model
T d d
T
T R K
S
d
d N Ked SN C RT
12
2
1
21
and
2ln
where
)()(
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6
The Model (cont’d)
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 todevelop 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, everythingelse 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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15
Assumptions of the Black-
Scholes Model
The stock pays no dividends during the
option’s life European exercise style
Markets are efficient
No transaction costs
Interest rates remain constant
Prices are lognormally distributed
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16
The Stock Pays no Dividends
During the Option’s Life
If you apply the BSOPM to two securities,
one with no dividends and the other with adividend yield, the model will predict thesame call premium
– Robert Merton developed a simple extension to
the BSOPM to account for the payment ofdividends
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17
The Stock Pays no Dividends
During the Option’s Life (cont’d)
The Robert Miller Option Pricing Model
T d d
T
T d R K
S
d
d N Ked SN eC RT dT
*
1
*
2
2
*
1
*
2
*
1
*
and
2ln
where
)()(
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19
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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20
No Transaction Costs
There are no commissions and bid-ask
spreads – Not true
– Causes slightly different actual option prices fordifferent market participants
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21
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 areunknown or dynamic
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22
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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23
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 adjustedexpected payment at exercise (an outflow)
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24
Intuition Into the Black-Scholes
Model (cont’d)
)( 1d SN C )( 2d N Ke RT
Cash Inflow Cash Outflow
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25
Intuition Into the Black-Scholes
Model (cont’d)
The value of a call option is the differencebetween the expected benefit fromacquiring the stock outright and paying theexercise price on expiration day
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26
Calculating Black-Scholes
Prices from Historical Data
To calculate the theoretical value of a calloption 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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27
Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example
We would like to value a MSFT OCT 70 call in the
year 2000. Microsoft closed at $70.75 on August 23
(58 days before option expiration). Microsoft paysno dividends.
We need the interest rate and the stock volatility to
value the call.
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Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Consulting the “Money Rate” section of the Wall
Street Jou rnal , we find a T-bill rate with about 58days to maturity to be 6.10%.
To determine the volatility of returns, we need totake the logarithm of returns and determine theirvolatility. Assume we find the annual standard
deviation of MSFT returns to be 0.5671.
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Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using the BSOPM:
2032.1589.5671.
1589.02
5671.0610.
70
75.70ln
2ln
2
2
1
T
T R K
S
d
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30
Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using the BSOPM (cont’d):
0229.2261.2032.
12
T d d
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31
Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using normal probability tables, we find:
4909.)0029.(5805.)2032(.
N N
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32
Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
The value of the MSFT OCT 70 call is:
04.7$
)4909(.70)5805(.75.70
)()(
)1589)(.0610(.
21
e
d N Ked SN C RT
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Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
The call actually sold for $4.88.
The only thing that could be wrong in ourcalculation is the volatility estimate. This isbecause we need the volatility estimate over theoption’s life, which we cannot observe.
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Implied Volatility
Introduction
Calculating implied volatility An implied volatility heuristic
Historical versus implied volatility
Pricing in volatility units
Volatility smiles
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Introduction
Instead of solving for the call premium,assume the market-determined callpremium is correct
– Then solve for the volatility that makes theequation hold
–
This value is called the imp l ied volat il i ty
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An Implied Volatility Heuristic
For an exactly at-the-money call, the correctvalue of implied volatility is:
T
R K
T P C
)1/(
/2)(5.0
implied
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Historical Versus Implied
Volatility (cont’d)
Strong and Dickinson (1994) find
– Clear evidence of a relation between thestandard deviation of returns over the pastmonth and the current level of implied volatility
– That the current level of implied volatilitycontains both an ex post component based on
actual past volatility and an ex ante componentbased on the market’s forecast of future
variance
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Pricing in Volatility Units
You cannot directly compare the dollar costof two different options because
– Options have different degrees of “moneyness”
– A more distant expiration means more timevalue
–
The levels of the stock prices are different
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Volatility Smiles
Volat i li ty sm iles are in contradiction to theBSOPM, which assumes constant volatilityacross all strike prices
– When you plot implied volatility against strikingprices, the resulting graph often looks like asmile
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Volatility Smiles (cont’d) Volatility Smile
Microsoft August 2000
0
10
20
30
40
50
60
40 45 50 55 60 65 70 75 80 85 90 95 100 105
Striking Price
I m p l i e d
V o l a t i l i t y ( % )
Current StockPrice
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Using Black-Scholes to Solve
for the Put Premium
Can combine the BSOPM with put/callparity:
)()( 12 d SN d N Ke P RT
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Problems Using the Black-
Scholes Model
Does not work well with options that aredeep-in-the-money or substantially out-of-the-money
Produces biased values for very low orvery 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