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
2
3. Introduction
The
Black-Scholes option pricing model
(BSOPM) has been one of the most
important developments in finance in the
last 50 years
–
–
3
Has provided a good understanding of what
options should sell for
Has made options more attractive to individual
and institutional investors
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
4
5. The Model
C = SN (d1 ) − Ke − RT N (d 2 )
where
σ2
S
T
ln + R +
2
K
d1 =
σ T
and
d 2 = d1 − σ T
5
6. The Model (cont’d)
Variable
S
K
e
R
T
σ
=
=
=
=
=
=
ln =
definitions:
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(d1) and
N(d2) =
6
cumulative standard normal distribution
functions
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
7
8. Determinants of the Option
Premium
Striking
price
Time until expiration
Stock price
Volatility
Dividends
Risk-free interest rate
8
9. Striking Price
The
lower the striking price for a given
stock, the more the option should be worth
–
9
Because a call option lets you buy at a
predetermined striking price
10. Time Until Expiration
The
longer the time until expiration, the
more the option is worth
–
10
The option premium increases for more distant
expirations for puts and calls
11. Stock Price
The
higher the stock price, the more a given
call option is worth
–
11
A call option holder benefits from a rise in the
stock price
12. Volatility
The
greater the price volatility, the more the
option is worth
–
–
12
The volatility estimate sigma cannot be directly
observed and must be estimated
Volatility plays a major role in determining time
value
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
–
–
13
Listed options do not adjust for cash dividends
The stock price falls on the ex-dividend date
14. Risk-Free Interest Rate
The
higher the risk-free interest rate, the
higher the option premium, everything else
being equal
–
14
A higher “discount rate” means that the call
premium must rise for the put/call parity
equation to hold
15. Assumptions of the BlackScholes 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
15
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
–
16
Robert Merton developed a simple extension to
the BSOPM to account for the payment of
dividends
17. The Stock Pays no Dividends
During the Option’s Life (cont’d)
The Robert Miller Option Pricing Model
*
C * = e − dT SN (d1* ) − Ke − RT N (d 2 )
where
σ2
S
T
ln + R − d +
2
K
d1* =
σ T
and
17
*
d 2 = d1* − σ T
18. European Exercise Style
A
European option can only be exercised
on the expiration date
–
–
18
American options are more valuable than
European options
Few options are exercised early due to time
value
19. Markets Are Efficient
The
BSOPM assumes informational
efficiency
–
–
19
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. No Transaction Costs
There
are no commissions and bid-ask
spreads
–
–
20
Not true
Causes slightly different actual option prices for
different market participants
21. Interest Rates Remain Constant
There
–
–
21
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. Prices Are Lognormally
Distributed
The
logarithms of the underlying security
prices are normally distributed
–
22
A reasonable assumption for most assets on
which options are available
23. Intuition Into the Black-Scholes
Model
The
–
–
23
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. Intuition Into the Black-Scholes
Model (cont’d)
C = SN (d1 )
Cash Inflow
24
− Ke
− RT
N (d 2 )
Cash Outflow
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
25
26. Calculating Black-Scholes
Prices from Historical Data
To
calculate the theoretical value of a call
option using the BSOPM, we need:
–
–
–
–
–
26
The stock price
The option striking price
The time until expiration
The riskless interest rate
The volatility of the stock
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.
27
We need the interest rate and the stock volatility to
value the call.
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%.
28
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. Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using the BSOPM:
σ2
S
T
ln + R +
2
K
d1 =
σ T
.56712
70.75
0.1589
ln
+ .0610 +
2
70
=
= .2032
.5671 .1589
29
30. Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using the BSOPM (cont’d):
d 2 = d1 − σ T
= .2032 − .2261 = −.0229
30
31. 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
31
32. 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
32
− (.0610 )(.1589 )
(.4909)
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.
33
34. Implied Volatility
Introduction
Calculating
implied volatility
An implied volatility heuristic
Historical versus implied volatility
Pricing in volatility units
Volatility smiles
34
35. Introduction
Instead
of solving for the call premium,
assume the market-determined call
premium is correct
–
–
35
Then solve for the volatility that makes the
equation hold
This value is called the implied volatility
36. Calculating Implied Volatility
Sigma
cannot be conveniently isolated in
the BSOPM
–
36
We must solve for sigma using trial and error
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.
37
38. An Implied Volatility Heuristic
For
an exactly at-the-money call, the correct
value of implied volatility is:
σ implied
38
0.5(C + P) 2π / T
=
T
K /(1 + R )
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
39
40. Historical Versus Implied
Volatility (cont’d)
Strong
–
–
40
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. Pricing in Volatility Units
You
cannot directly compare the dollar cost
of two different options because
–
–
–
41
Options have different degrees of “moneyness”
A more distant expiration means more time
value
The levels of the stock prices are different
42. Volatility Smiles
Volatility
smiles are in contradiction to the
BSOPM, which assumes constant volatility
across all strike prices
–
42
When you plot implied volatility against striking
prices, the resulting graph often looks like a
smile
44. Using Black-Scholes to Solve
for the Put Premium
Can
combine the BSOPM with put/call
parity:
P = Ke
44
− RT
N (− d 2 ) − SN (− d1 )
45. 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
–
Increases as the time until expiration increases
May
yield unreasonable values when an
option has only a few days of life remaining
45
