Black Scholes Model.pdf

BSM

1. © 2004 South-Western Publishing 1
Chapter 6
The Black-Scholes
Option Pricing
Model

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

5. 5
The Model
Tdd
T
TR
K
S
d
dNKedSNC
RT
s
s
s
-=
÷
÷
ø
ö
ç
ç
è
æ
++÷
ø
ö
ç
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æ
=
-=
-
12
2
1
21
and
2
ln
where
)()(

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
s= standard deviation (sigma) of returns on
the underlying security
ln= natural logarithm
N(d
1
) and
N(d
2
) =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
Tdd
T
TdR
K
S
d
dNKedSNeC
RTdT
s
s
s
-=
÷
÷
ø
ö
ç
ç
è
æ
+-+÷
ø
ö
ç
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æ
=
-=
--
*
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)
)(
1dSNC= )(
2dNKe
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
=
÷
÷
ø
ö
ç
ç
è
æ
++÷
ø
ö
ç
è
æ
=
÷
÷
ø
ö
ç
ç
è
æ
++÷
ø
ö
ç
è
æ
=
T
TR
K
S
d
s
s

30. 30
Calculating Black-Scholes
Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using the BSOPM (cont’d):
0229.2261.2032.
12
-=-=
-= Tdd s

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(.
21
=
-=
-=
-
-
e
dNKedSNC
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

36. 36
Calculating Implied Volatility
Sigma cannot be conveniently isolated in
the BSOPM
–We must solve for sigma using trial and error

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
RK
TPC
)1/(
/2)(5.0
implied
+
+
=
p
s

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

43. 43
Volatility Smiles (cont’d)
Volatility Smile
Microsoft August 2000
0
10
20
30
40
50
60
404550556065707580859095100105
Striking Price
I
m
p
l
i
e
d

V
o
l
a
t
i
l
i
t
y

(
%
)
Current Stock
Price

44. 44
Using Black-Scholes to Solve
for the Put Premium
Can combine the BSOPM with put/call
parity:
)()(
12 dSNdNKeP
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