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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

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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

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The Model

Tdd

T

TRK

S

d

dNKedSNCRT

12

2

1

21

and

2ln

where

)()(

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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 distribution

functions

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Development and Assumptions

of the Model

Derivation from:

Physics Mathematical short cuts

Arbitrage arguments

Fischer Black and Myron Scholes utilizedthe physics heat transfer equation to

develop the BSOPM

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Determinants of the Option

Premium

Striking price

Time until expiration Stock price

Volatility

Dividends Risk-free interest rate

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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

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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

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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

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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

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Dividends

A company that pays a large dividend will

have a smaller option premium than acompany 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

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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

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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

Prices are lognormally distributed

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The Stock Pays no Dividends

During the Options Life

If you apply the BSOPM to two securities,

one with no dividends and the other with adividend yield, the model will predict the

same call premium

Robert Merton developed a simple extension to

the BSOPM to account for the payment ofdividends

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The Stock Pays no Dividends

During the Options Life (contd)

The Robert Miller Option Pricing Model

Tdd

T

TdRK

S

d

dNKedSNeCRTdT

*

1

*

2

2

*

1

*

2

*

1

*

and

2ln

where

)()(

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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

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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

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No Transaction Costs

There are no commissions and bid-ask

spreads Not true

Causes slightly different actual option prices for

different market participants

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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

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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

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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 adjusted

expected payment at exercise (an outflow)

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Intuition Into the Black-Scholes

Model (contd)

)( 1dSNC )( 2dNKeRT

Cash Inflow Cash Outflow

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Intuition Into the Black-Scholes

Model (contd)

The value of a call option is the difference

between the expected benefit fromacquiring the stock outright and paying the

exercise price on expiration day

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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

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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