The Black-Scholes-Merton The Black-Scholes-Merton ModelModel

Chapter 13

1

B-S-M model is used to determine the option price of any underlying stock. They believed that stock follow a random walk.

Proportional changes in the stock price(relative return) in a short period of time is normally distributed and stock price at any future time is lognormal distributed.

2

The Black-Scholes-Merton ModelThe Black-Scholes-Merton Model

Stock return have log normal Stock return have log normal distributiondistribution

Determine the price change over an period not at the end of period.

Stock price follow random walk, next movement of price is completely independent of past prices.

For longer horizon, there can be upward and downward movement but not hold for very small period like daily or hourly periods. pure random

Returns are measured by price relative; ratio of the price at two successive time intervals rather than absolute value. To measure it, we need to assume continuous compounding and we take natural log.

Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008 3

Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008 4

The Stock Price AssumptionThe Stock Price Assumption

Consider a stock whose price is SIn a short period of time of length t,

the return on the stock is normally distributed:

where is expected return and is volatility

ttS

S

2,

Stock return have log normal Stock return have log normal distributiondistribution

We consider the distribution of the logarithm of the return relative rather than the distribution of stock price?

We know stock price cant not be negative. Minimum value is 0 and loss cant exceed 100%. So assuming stock price normally distributed will consider negative value also which is simply wrong.

A log normal distribution fits the description as it consider only positive value. So as an alternative we may imagine the distribution of log return. If log return are ND, then distribution of the stock price will be log normal.

5

6

The Lognormal PropertyThe Lognormal Property

It follows from this assumption that

Since the logarithm of ST is normal, ST is lognormally distributed

,2

lnln

or

,2

lnln

22

0

22

0

TTSS

TTSS

T

T

Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008 7

The Lognormal DistributionThe Lognormal Distribution

E S S e

S S e e

TT

TT T

( )

( ) ( )

0

02 2 2

1

var

Assumptions of the Black and Assumptions of the Black and Scholes Model:Scholes Model:

The stock pays no dividends during the option's life European exercise terms are used means option can

only be exercised on the expiration date Markets are efficient suggests that people cannot

consistently predict the direction of the market or an individual stock. 

Interest rates remain constant and known Returns are lognormally distributed. this assumption

suggests, returns on the underlying stock are normally distributed, which is reasonable for most assets that offer options.

9

Continuously Compounded Return Continuously Compounded Return (Equations 13.6 and 13.7), page 279) (Equations 13.6 and 13.7), page 279)

If x is the continuously compounded return.

10

,2

ln1

=

22

0

0

Tx

S

S

Tx

eSS

T

xTT

11

The Concepts Underlying Black-The Concepts Underlying Black-ScholesScholes

The option price and the stock price depend on the same underlying source of uncertainty

We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty

The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate

12

The Black-Scholes FormulasThe Black-Scholes Formulas(See pages 291-293)(See pages 291-293)

TdT

TrKSd

T

TrKSd

dNSdNeKp

dNeKdNScrT

rT

10

2

01

102

210

)2/2()/ln(

)2/2()/ln( where

)( )(

)( )(

13

The N(x) FunctionThe N(x) Function

N(d1) It is the delta of the option which represents the fraction of stock bought for each call written. Fraction of stock owned. It is the probability that the expected value in risk neutral world, using the risk-free interest rate, of the expected asset price at expiration St at time T.

N(d2) is the strike price times the probability that the strike price will be paid in a risk neutral world. It is the probability of call becoming ITM that is spot price exceeding the strike price. The expected value of cash outflow is )( X 2dNeK rT

14

Properties of Black-Scholes FormulaProperties of Black-Scholes Formula

As S0 becomes very large c tends to

S0 – Ke-rT and p tends to zero

As S0 becomes very small c tends to zero and p tends to Ke-rT – S0

When volatility approaches zero, d1 and d2 tends to infinite and standard normal become 1

15

Implied VolatilityImplied Volatility

The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price. This is the volatility implied by an option price observed in the market.

Implied volatility shows the market’s opinion of the stock’s potential moves, but it doesn’t forecast direction. If the implied volatility is high, the market thinks the stock has potential for large price swings in either direction,  Implied volatility often tends to be higher for out-the-money (OTM) and in-the-money (ITM) options compared to at-the-money.

Composite implied volatility for the stock is calculated by taking a suitable weighted average of the individual implied volatility of various options

Since most option trading volume usually occurs in at-the-money (ATM) options, these are the contracts generally used to calculate IV. Once we know the price of the ATM options, we can use an options pricing model and a little algebra to solve for the implied volatility.

Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008 16

Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008 17

18

DividendsDividends

The “dividend” should be the expected reduction in the stock price and call option price expected

European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into Black-Scholes

The stock’s price goes down by an amount reflecting the dividend per share. The effect of this is to reduce the value of calls and to increase the value of puts.

19

DividendsDividends

Adjusting stock price with the dividend would result in . q is dividend yield (compounding basis)

TdT

TqrKSd

T

TqrKSd

dNSdNeKp

dNeKdNScrT

rT

10

2

01

1qt-

02

21-qt

0

)2/2()/ln(

)2/2()/ln( where

)( e)(

)( )( e

-qt0eS

20

American CallsAmerican Calls

An American call on a non-dividend-paying stock should never be exercised early

An American call on a dividend-paying stock should only ever be exercised immediately prior to an ex-dividend date

Suppose dividend dates are at times t1, t2, …tn. Early exercise is sometimes optimal at time ti if the dividend at that time is greater than ]1[ )( 1 ii ttreK