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© 2004 South-Western Publishing1

Chapter 6

The Black-ScholesOption PricingModel



3

Introduction

The Black-Scholes option pricing model

(BSOPM) has been one of the mostimportant developments in finance in thelast 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 themodel

Determinants of the option premium

Assumptions of the Black-Scholes model

Intuition into the Black-Scholes model



5

The Model

T d d

T

T R K

S

d

d N Ked SN C RT

12

2

1

21

and

2ln

where

)()(



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

= standard deviation (sigma) of returns onthe underlying security

ln = natural logarithm

N(d1) and

N(d2) = cumulative standard normal distribution

functions



Development and Assumptions

of the Model

Derivation from:

–

Physics – Mathematical short cuts

– Arbitrage arguments

Fischer Black and Myron Scholes utilizedthe physics heat transfer equation todevelop the BSOPM



Determinants of the Option

Premium

Striking price

Time until expiration Stock price

Volatility

Dividends Risk-free interest rate



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



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



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



Volatility

The greater the price volatility, the more the

option is worth – The volatility estimate s igma cannot be directly

observed and must be estimated

– Volatility plays a major role in determining time

value



Dividends

A company that pays a large dividend will

have a smaller option premium than acompany with a lower dividend, everythingelse being equal

– Listed options do not adjust for cash dividends

– The stock price falls on the ex-dividend date



Risk-Free Interest Rate

The higher the risk-free interest rate, the

higher the option premium, everything elsebeing 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 adividend yield, the model will predict thesame call premium

– Robert Merton developed a simple extension to

the BSOPM to account for the payment ofdividends



17

The Stock Pays no Dividends

During the Option’s Life (cont’d)

The Robert Miller Option Pricing Model

T d d

T

T d R K

S

d

d N Ked SN eC RT dT

*

1

*

2

2

*

1

*

2

*

1

*

and

2ln

where

)()(



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 fordifferent 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 areunknown 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” weightedexpected stock price (an inflow)

– One gives the time-value of money adjustedexpected payment at exercise (an outflow)



24


Model (cont’d)

)( 1d SN C )( 2d N Ke RT

Cash Inflow Cash Outflow



25


Model (cont’d)

The value of a call option is the differencebetween the expected benefit fromacquiring the stock outright and paying theexercise price on expiration day



26

Calculating Black-Scholes

Prices from Historical Data

To calculate the theoretical value of a calloption 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



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 paysno dividends.

We need the interest rate and the stock volatility to

value the call.



28



Valuing a Microsoft Call Example (cont’d)

Consulting the “Money Rate” section of the Wall

Street Jou rnal , we find a T-bill rate with about 58days to maturity to be 6.10%.

To determine the volatility of returns, we need totake the logarithm of returns and determine theirvolatility. Assume we find the annual standard

deviation of MSFT returns to be 0.5671.



29




Using the BSOPM:

2032.1589.5671.

1589.02

5671.0610.

70

75.70ln

2ln

2

2

1

T

T R K

S

d



30




Using the BSOPM (cont’d):

0229.2261.2032.

12

T d d



31




Using normal probability tables, we find:

4909.)0029.(5805.)2032(.

N N



32




The value of the MSFT OCT 70 call is:

04.7$

)4909(.70)5805(.75.70

)()(

)1589)(.0610(.

21

e

d N Ked SN C RT



33




The call actually sold for $4.88.

The only thing that could be wrong in ourcalculation is the volatility estimate. This isbecause we need the volatility estimate over theoption’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 callpremium is correct

– Then solve for the volatility that makes theequation hold

–

This value is called the imp l ied volat il i ty



38

An Implied Volatility Heuristic

For an exactly at-the-money call, the correctvalue of implied volatility is:

T

R K

T P C

)1/(

/2)(5.0

implied



40

Historical Versus Implied

Volatility (cont’d)

Strong and Dickinson (1994) find

– Clear evidence of a relation between thestandard deviation of returns over the pastmonth and the current level of implied volatility

– That the current level of implied volatilitycontains both an ex post component based on

actual past volatility and an ex ante componentbased on the market’s forecast of future

variance



41

Pricing in Volatility Units

You cannot directly compare the dollar costof two different options because

– Options have different degrees of “moneyness”

– A more distant expiration means more timevalue

–

The levels of the stock prices are different



42

Volatility Smiles

Volat i li ty sm iles are in contradiction to theBSOPM, which assumes constant volatilityacross all strike prices

– When you plot implied volatility against strikingprices, the resulting graph often looks like asmile



43

Volatility Smiles (cont’d) Volatility Smile

Microsoft 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

I m p l i e d

V o l a t i l i t y ( % )

Current StockPrice



44

Using Black-Scholes to Solve

for the Put Premium

Can combine the BSOPM with put/callparity:

)()( 12 d SN d N Ke P RT



Problems Using the Black-

Scholes Model

Does not work well with options that aredeep-in-the-money or substantially out-of-the-money

Produces biased values for very low orvery 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

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