© 2004 south-western publishing 1 chapter 6 the black-scholes option pricing model
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© 2004 South-Western Publishing 1
Chapter 6
The Black-Scholes Option Pricing Model
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
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
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
The Model
Tdd
T
TRKS
d
dNKedSNC RT
12
2
1
21
and
2ln
where
)()(
6
The Model (cont’d)
Variable definitions:S = current stock priceK = option strike pricee = base of natural logarithmsR = riskless interest rateT = time until option expiration = standard deviation (sigma) of returns on
the underlying securityln = natural logarithm
N(d1) and
N(d2) = 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
8
Determinants of the Option Premium
Striking price Time until expiration Stock price Volatility Dividends Risk-free interest rate
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
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
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
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
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
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
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
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
The Stock Pays no Dividends During the Option’s Life (cont’d)
The Robert Miller Option Pricing Model
Tdd
T
TdRKS
d
dNKedSNeC RTdT
*1
*2
2
*1
*2
*1
*
and
2ln
where
)()(
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
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
No Transaction Costs
There are no commissions and bid-ask spreads– Not true– Causes slightly different actual option prices for
different market participants
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
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
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
Intuition Into the Black-Scholes Model (cont’d)
)( 1dSNC )( 2dNKe RT
Cash Inflow 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
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
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 $70.75 on August 23 (58 days before option expiration). Microsoft pays no dividends.
We need the interest rate and the stock volatility to value the call.
28
Calculating Black-Scholes Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Consulting the “Money Rate” section of the Wall Street Journal, we find a T-bill rate with about 58 days to maturity to be 6.10%.
To determine the volatility of returns, we need to take the logarithm of returns and determine their volatility. Assume we find the annual standard deviation of MSFT returns to be 0.5671.
29
Calculating Black-Scholes Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using the BSOPM:
2032.1589.5671.
1589.02
5671.0610.
7075.70
ln
2ln
2
2
1
T
TRKS
d
30
Calculating Black-Scholes Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using the BSOPM (cont’d):
0229.2261.2032.12
Tdd
31
Calculating Black-Scholes Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using normal probability tables, we find:
4909.)0029.(
5805.)2032(.
N
N
32
Calculating Black-Scholes Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
The value of the MSFT OCT 70 call is:
04.7$
)4909(.70)5805(.75.70
)()()1589)(.0610(.
21
e
dNKedSNC RT
33
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
Implied Volatility
Introduction Calculating implied volatility An implied volatility heuristic Historical versus implied volatility Pricing in volatility units Volatility smiles
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
Calculating Implied Volatility
Sigma cannot be conveniently isolated in the BSOPM– 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.
38
An Implied Volatility Heuristic
For an exactly at-the-money call, the correct value of implied volatility is:
TRK
TPC
)1/(
/2)(5.0implied
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
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
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
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
Volatility Smiles (cont’d)
Volatility SmileMicrosoft 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
Imp
lie
d V
ola
tili
ty (
%)
Current Stock Price
44
Using Black-Scholes to Solve for the Put Premium
Can combine the BSOPM with put/call parity:
)()( 12 dSNdNKeP RT
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