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

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
C

SN
(
d
1 ) 
Ke

RT N
(
d
2 ) where
d
1  ln   
S K
      
R

T
  2 2   
T
and
d
2 
d
1  
T

The Model (cont’d) 6

Variable definitions: S K e R = T =
 =
= = = ln = 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(d 1 ) and N(d 2 ) = 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

The Stock Pays no Dividends During the Option’s Life (cont’d) 17 The Robert Miller Option Pricing Model
C
* 
e

dT SN
(
d
1 * ) 
Ke

RT N
(
d
2 * ) where
d
1 *  ln
S K
  
R

d
  2 2  
T

T
and
d
2 * 
d
1 *  
T

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)
C 
SN
(
d
1 ) 
Ke

RT N
(
d
2 )
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

Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example 27 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.

Calculating Black-Scholes Prices from Historical Data 28 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:
d
1  ln  ln
S K
70   
R
  2 2 
T
 
T
70 .
75    .
0610  .
5671 2 2   0 .
1589  .
2032 .
5671 .
1589

30 Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (cont’d) Using the BSOPM (cont’d):
d
2 
d
1    .
2032
T
 .
2261   .
0229

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

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
(
d
1 ) 
Ke

RT N
(
d
2 )  70 .
75 (.
5805 )  70
e
 (.
0610 )(.
1589 ) (.
4909 )  $ 7 .
04

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:
 implied  0 .
5 (
C K

P
) /( 1  2
R
)
T
 /
T

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 Smile Microsoft August 2000
60 50 40 Current Stock Price 30 20 10 0 40 45 50 55 60 65 70 75
Striking Price
80 85 90 95 100 105

44 Using Black-Scholes to Solve for the Put Premium

Can combine the BSOPM with put/call parity:
P

Ke

RT N
( 
d
2 ) 
SN
( 
d
1 )

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