2. 2
Outline
Introduction
The Black-Scholes option pricing model
Calculating Black-Scholes prices from
historical data
Implied volatility
Using Black-Scholes to solve for the put
premium
Problems using the Black-Scholes model
3. 3
Introduction
The Black-Scholes option pricing model
(BSOPM) has been one of the most
important 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
4. 4
The Black-Scholes Option
Pricing Model
The model
Development and assumptions of the model
Determinants of the option premium
Assumptions of the Black-Scholes model
Intuition into the Black-Scholes model
6. 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
σ = standard deviation (sigma) of returns on
the underlying security
ln = natural logarithm
N(d1) and
N(d2) = cumulative standard normal distribution
functions
7. 7
Development and Assumptions
of the Model
Derivation from:
– Physics
– Mathematical short cuts
– Arbitrage arguments
Fischer Black and Myron Scholes utilized
the physics heat transfer equation to
develop the BSOPM
8. 8
Determinants of the Option
Premium
Striking price
Time until expiration
Stock price
Volatility
Dividends
Risk-free interest rate
9. 9
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
10. 10
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
11. 11
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
12. 12
Volatility
The greater the price volatility, the more the
option is worth
– The volatility estimate sigma cannot be directly
observed and must be estimated
– Volatility plays a major role in determining time
value
13. 13
Dividends
A company that pays a large dividend will
have a smaller option premium than a
company 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
14. 14
Risk-Free Interest Rate
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
15. 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
16. 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 a
dividend yield, the model will predict the
same call premium
– Robert Merton developed a simple extension to
the BSOPM to account for the payment of
dividends
17. 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
Ke
d
SN
e
C RT
dT
σ
σ
σ
−
=
+
−
+
=
−
= −
−
*
1
*
2
2
*
1
*
2
*
1
*
and
2
ln
where
)
(
)
(
18. 18
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
19. 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
20. 20
No Transaction Costs
There are no commissions and bid-ask
spreads
– Not true
– Causes slightly different actual option prices for
different market participants
21. 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 are
unknown or dynamic
22. 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
23. 23
Intuition Into the Black-Scholes
Model
The valuation equation has two parts
– 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)
24. 24
Intuition Into the Black-Scholes
Model (cont’d)
)
( 1
d
SN
C = )
( 2
d
N
Ke RT
−
−
Cash Inflow Cash Outflow
25. 25
Intuition Into the Black-Scholes
Model (cont’d)
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
26. 26
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
27. 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 pays
no dividends.
We need the interest rate and the stock volatility to
value the call.
28. 28
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%.
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.
29. 29
Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using the BSOPM:
2032
.
1589
.
5671
.
1589
.
0
2
5671
.
0610
.
70
75
.
70
ln
2
ln
2
2
1
=
+
+
=
+
+
=
T
T
R
K
S
d
σ
σ
30. 30
Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using the BSOPM (cont’d):
0229
.
2261
.
2032
.
1
2
−
=
−
=
−
= T
d
d σ
31. 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
32. 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
(.
2
1
=
−
=
−
=
−
−
e
d
N
Ke
d
SN
C RT
33. 33
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.
34. 34
Implied Volatility
Introduction
Calculating implied volatility
An implied volatility heuristic
Historical versus implied volatility
Pricing in volatility units
Volatility smiles
35. 35
Introduction
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
37. 37
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.
38. 38
An Implied Volatility Heuristic
For an exactly at-the-money call, the correct
value of implied volatility is:
T
R
K
T
P
C
)
1
/(
/
2
)
(
5
.
0
implied
+
+
=
π
σ
39. 39
Historical Versus Implied
Volatility
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
40. 40
Historical Versus Implied
Volatility (cont’d)
Strong and Dickinson (1994) find
– 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
41. 41
Pricing in Volatility Units
You cannot directly compare the dollar cost
of two different options because
– Options have different degrees of “moneyness”
– A more distant expiration means more time
value
– The levels of the stock prices are different
42. 42
Volatility Smiles
Volatility smiles are in contradiction to the
BSOPM, which assumes constant volatility
across all strike prices
– When you plot implied volatility against striking
prices, the resulting graph often looks like a
smile
44. 44
Using Black-Scholes to Solve
for the Put Premium
Can combine the BSOPM with put/call
parity:
)
(
)
( 1
2 d
SN
d
N
Ke
P RT
−
−
−
= −
45. 45
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
