Bbbbbbbb Ssssssssssss Model

8/3/2019 Bbbbbbbb Ssssssssssss Model

1/18

Black- Scholes Model

1


2/18

2

Introduction

The Black-Scholes option pricing model

(BSOPM) has been one of the mostimportant 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


3/18

3

The Model

Tdd

T

TRK

S

d

dNKedSNCRT

=

++

=

=

12

2

1

21

and

2ln

where

)()(


4/18

4

The Model (contd)

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 distributionunctions


5/18

5

Development and Assumptions

of the Model

Derivation from:

Physics Mathematical short cuts

Arbitrage arguments


6/18

6

Determinants of the Option

Premium

Striking price

Time until expiration Stock price

Volatility

DividendsRisk-free interest rate


7/187

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


8/188

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


9/189

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


10/1810

Volatility

The greater the price volatility, the more the

option is worth.


11/1811

Dividends

A company that pays a large dividend will

have a smaller option premium than acompany with a lower dividend, everything

else being equal


12/1812

Risk-Free Interest Rate

The higher the risk-free interest rate, the

higher the option premium, everything elsebeing equal.


13/1813

Assumptions of the Black-

Scholes Model

The stock pays no dividends during the

options life European exercise style

Markets are efficient

No transaction costs

Interest rates remain constant


14/1814

European Exercise Style

A European option can only be exercised

on the expiration date American options are more valuable than

European options


15/1815

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 ofwhether you are bullish or bearish


16/18

16

No Transaction Costs

There are no commissions and bid-ask

spreads.


17/18

17

Interest Rates Remain Constant

There is no real riskfree interest rate


18/18

18

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

Bbbbbbbb Ssssssssssss Model

Documents