© 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