Chapter 6
The Black-Scholes
Option Pricing
Model
1
© 2004 South-Western Publishing
Transition from discrete to
continuous time T→lim0
2
Natural Logarithm and e

3
Continuous compounding/discounting
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
4
The Model (cont’d)

Variable definitions:
S
K
e
R
T

5
=
=
=
=
=
=
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:
–
–
–

6
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





7
Striking price
Time until expiration
Stock price
Volatility
Risk-free interest rate
Striking Price

8
The lower the striking price for a given
stock, the more the option should be worth
Time Until Expiration

The longer the time until expiration, the
more the option is worth
–
9
The option premium increases for more distant
expirations for puts and calls
Stock Price
Pricing based on discounted cash flows; Payoff
occurs at expiration = S-K; best bet on S
The role of two counterparties in the trade
10
Volatility

The greater the price volatility, the more the
option is worth
–
–
11
The volatility estimate sigma cannot be directly
observed and must be estimated
Volatility plays a major role in determining time
value
Risk-Free Interest Rate

The higher the risk-free interest rate, the
higher the option premium, everything else
being equal
–
12
A higher “discount rate” means that the call
premium must rise for the put/call parity
equation to hold
Assumptions of the BlackScholes Model






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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
European Exercise Style

A European option can only be exercised
on the expiration date
–
–
14
American options are more valuable than
European options
Few options are exercised early due to time
value
Markets Are Efficient

The BSOPM assumes informational
efficiency
–
–
15
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
No Transaction Costs

There are no commissions and bid-ask
spreads
–
–
16
Not true
Causes slightly different actual option prices for
different market participants
Interest Rates Remain Constant

There is no real “riskfree” interest rate
–
–
17
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
Prices Are Lognormally
Distributed
–
18
the log-normal distribution is the probability
distribution of any random variable whose
logarithm is normally distributed
Intuition Into the Black-Scholes
Model

The valuation equation has two parts
–
–
19
One gives a “pseudo-probability” weighted expected
stock price (an inflow)
One gives the time-value of money adjusted expected
payment at exercise (an outflow)
Time Value based on volatility
Example

20
Option at the money and 0 risk free rate
Time Value based on volatility
Example
21
Time Value based on volatility
Example
22
23
24
25
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Intrinsic Value plus Time Value

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The value of a call option is the difference
between the expected benefit from
acquiring the stock outright and paying the
exercise price on expiration day
Calculating Black-Scholes
Prices from Historical Data

To calculate the theoretical value of a call
option using the BSOPM, we need:
–
–
–
–
–
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The stock price
The option striking price
The time until expiration
The riskless interest rate
The volatility of the stock
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 pays
no dividends.
29
We need the interest rate and the stock volatility to
value the call.
Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Consulting the “Money Rate” section of the Wall
Street Journal, we find a T-bill rate with about 58
days to maturity to be 6.10%.
30
To determine the volatility of returns, we need to
take the logarithm of returns and determine their
volatility. Assume we find the annual standard
deviation of MSFT returns to be 0.5671.
Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using the BSOPM:
2 
S 
T
ln     R 
2 
K 
d1 
 T
.56712 
 70.75  
0.1589
ln 
   .0610 
2 
 70  

 .2032
.5671 .1589
31
Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using the BSOPM (cont’d):
d 2  d1   T
 .2032  .2261  .0229
32
Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using normal probability tables, we find:
N (.2032)  .5805
N (.0029)  .4909
33
Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
The value of the MSFT OCT 70 call is:
C  SN (d1 )  Ke
 RT
N (d 2 )
 70.75(.5805)  70e
 $7.04
34
 (.0610)(.1589)
(.4909)
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 our
calculation is the volatility estimate. This is
because we need the volatility estimate over the
option’s life, which we cannot observe.
35
Implied Volatility

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Introduction
Calculating implied volatility
An implied volatility heuristic
Historical versus implied volatility
Pricing in volatility units
Volatility smiles
Introduction

Instead of solving for the call premium,
assume the market-determined call
premium is correct
–
–
37
Then solve for the volatility that makes the
equation hold
This value is called the implied volatility
Calculating Implied Volatility

Sigma cannot be conveniently isolated in
the BSOPM
–
38
We must solve for sigma using trial and error
Calculating Implied Volatility
(cont’d)
Valuing a Microsoft Call Example (cont’d)
The implied volatility for the MSFT OCT 70 call is
35.75%, which is much lower than the 57% value
calculated from the monthly returns over the last
two years.
39
An Implied Volatility Heuristic

For an exactly at-the-money call, the correct
value of implied volatility is:
 implied
40
0.5(C  P) 2 / T

T
K /(1  R)
Historical Versus Implied
Volatility
41

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
Historical Versus Implied
Volatility (cont’d)

Strong and Dickinson (1994) find
–
–
42
Clear evidence of a relation 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
Pricing in Volatility Units

You cannot directly compare the dollar cost
of two different options because
–
–
–
43
Options have different degrees of “moneyness”
A more distant expiration means more time
value
The levels of the stock prices are different
Volatility Smiles

Volatility smiles are in contradiction to the
BSOPM, which assumes constant volatility
across all strike prices
–
44
When you plot implied volatility against striking
prices, the resulting graph often looks like a
smile
Volatility Smiles (cont’d)
Volatility Smile
Microsoft August 2000
60
Current Stock
Price
Implied Volatility (%)
50
40
30
20
10
0
40
45
45
50
55
60
65
70
75
80
Striking Price
85
90
95
100
105
Using Black-Scholes to Solve
for the Put Premium

Can combine the BSOPM with put/call
parity:
P  Ke
46
 RT
N (d 2 )  SN (d1 )
Problems Using the BlackScholes Model


Does not work well with options that are
deep-in-the-money or substantially out-ofthe-money
Produces biased values for very low or
very high volatility stocks
–

47
Increases as the time until expiration increases
May yield unreasonable values when an
option has only a few days of life remaining