© 2002 South-Western Publishing 1

Chapter 5

Option Pricing

2

Outline

IntroductionA brief history of options pricingArbitrage and option pricing Intuition into Black-Scholes

3

Introduction

Option pricing developments are among the most important in the field of finance during the last 30 years

The backbone of option pricing is the Black-Scholes model

4

Introduction (cont’d)

The Black-Scholes model:

tdd

t

trKS

d

dNKedSNC rt

12

2

1

21

and

2ln

where

)()(

5

A Brief History of Options Pricing: The Early Work

Charles Castelli wrote The Theory of Options in Stocks and Shares (1877)– Explained the hedging and speculation aspects

of options

Louis Bachelier wrote Theorie de la Speculation (1900)– The first research that sought to value derivative

assets

6

The Middle Years

Rebirth of option pricing in the 1950s and 1960s– Paul Samuelson wrote Brownian Motion in the

Stock Market (1955)– Richard Kruizenga wrote Put and Call Options:

A Theoretical and Market Analysis (1956)

James Boness wrote A Theory and Measurement of Stock Option Value (1962)

7

The Present

The Black-Scholes option pricing model (BSOPM) was developed in 1973

– An improved version of the Boness model– Most other option pricing models are modest

variations of the BSOPM

8

Basic Option Pricing Models

The theory of put/call parity The binomial option pricing model Binomial put pricing Binomial pricing with asymmetric branches The effect of time The effect of volatility

9

Arbitrage and Option Pricing

Finance is sometimes called “the study of arbitrage”– Arbitrage is the existence of a riskless profit

Finance theory does not say that arbitrage will never appear– Arbitrage opportunities will be short-lived

Option pricing techniques are based on basic arbitrage principles

Efficient market - equivalent assets should sell for the same price – ‘the law of one price’

10

The Theory of Put/Call Parity

Covered call and short put Covered call and long put No arbitrage relationships Variable definitions The put/call parity relationship

11

Put-Call Parity Theory

For a given underlying asset, the following factors form an interrelated complex:– Call price– Put price– Stock price and– Interest rate - risk free rate– time to expiration

.....the price of European style puts and calls on the same stock with the identical strike price and expiration dates have a special relationship

12

Put – Call Parity Theory

‘When European options are at the money and the stock pays no dividends, relative call prices should exceed relative put prices by an amount approximately equal to the riskless rate of interest for the option term times the stock price.’

….a theory about how to price options relative to other options

13

Put-Call Parity theory - Covered Call and Short Put

The profit/loss diagram for a covered call and for a short put are essentially equal

Covered call Short put

14

Put Call Parity theory - Covered Call and Long Put

A riskless position results if you combine a covered call and a long put

Covered call Long put Riskless position

+ =

....stock price at option expiration has no impact on the profit/loss position - ‘riskless’

15

Covered Call and Long Put

Riskless investments should earn the riskless rate of interest

If an investor can own a stock, write a call and buy a put and make a profit, arbitrage is present....and will be quickly taken advantage of.

16

Variable Definitions

C = call premium

P = put premium

S0 = current stock price

S1 = stock price at option expiration

K = option striking price

r = riskless interest rate

t = time until option expiration

17

No Arbitrage Relationships - State of Equilibrium

The covered call and long put position has the following characteristics:

– One cash inflow from writing the call (C)– Two cash outflows from paying for the put (P) and

paying interest on the bank loan (S)(r)– The principal of the loan (S) comes in but is

immediately spent to buy the stock– The interest on the bank loan is paid in the future

18

No Arbitrage Relationships

If there is no arbitrage and assuming European at the money options, then:

t

t

t

r

SrPC

r

SrPC

r

SrPCSS

)1(

0)1(

0)1(

19

No Arbitrage Relationships (cont’d)

If there is no arbitrage, then:

– The call premium should exceed the put premium by about the riskless rate of interest for the option term times the strike price

– The difference will be greater as: The stock price increases Interest rates increase The time to expiration increases

rr

r

S

PC

t

)1(

20

The Put/Call Parity Relationship

We now know how the call prices, put prices, the stock price, and the riskless interest rate are related

What about when the strike price is different from the stock price ie. In/out of the money options ?

tr

KSPC

)1(0

21

The Put-Call Parity Relationship

tr

KSPC

)1(0

…where the strike price is less than the stock price S0

(in the money call) the call price will reflect this intrinsic Value

22

The Put/Call Parity Relationship (cont’d)

Equilibrium Stock Price Example

You have the following information: Call price = $3.5 Put price = $1 Striking price = $75 Riskless interest rate = 5% Time until option expiration = 32 days

If there are no arbitrage opportunities, what is the equilibrium stock price?

23

The Put/Call Parity Relationship (cont’d)

Equilibrium Stock Price Example (cont’d)

Using the put/call parity relationship to solve for the stock price:

18.77$

)05.1(

00.75$00.1$50.3$

)1(

36532

0

tr

KPCS

24

Put-Call Parity – another look

Another way to illustrate the theory Instead of the covered call and long put with

borrowing – an equivalent situation is:

– A share of stock plus a put is equivalent to a call plus an investment in risk free bonds

25

The Binomial Option Pricing Model

A theory about establishing the option price from the factors that influence it:

– Another simplified pricing model - used to illustrate various influences on option prices:

Stock price Strike price Interest rates Stock price volatility

– The simplifying assumption is: a single period binomial model – there are only two possible outcomes and one time period

– The stock can go up or down in price over the period

26

Binomial Pricing Model

Assume the following:

– U.S. government securities yield 10% next year– Stock XYZ currently sells for $75 per share– There are no transaction costs or taxes– There are two possible stock prices in one year

….we want to construct a riskless portfolio of stock and options – one where the future value of the portfolio is independent of the future value of the stock.

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The Binomial Option Pricing Model (cont’d)

Possible states of the world:

$75

$50

$100

Stock PriceToday

Stock PriceOne Year Later

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The Binomial Option Pricing Model (cont’d)

A call option on XYZ stock is available that gives its owner the right to purchase XYZ stock in one year for $75– If the stock price is $100, the option will be

worth $25– If the stock price is $50, the option will be worth

$0

What should be the price of this option?

29

The Binomial Option Pricing Model (cont’d)

We can construct a portfolio of stock and options such that the portfolio has the same value regardless of the stock price after one year (no risk)

– Buy the stock and write N call options

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The Binomial Option Pricing Model (cont’d)

Possible portfolio values:

$75 – (N)($C)

$50

$100 - $25N

Total InvestmentToday

Total InvestmentOne Year Later

31

The Binomial Option Pricing Model (cont’d)

We can solve for N such that the portfolio value in one year must be $50:

2

50$25$100$

N

N

32

The Binomial Option Pricing Model (cont’d)

If we buy one share of stock today and write two calls, we know the portfolio will be worth $50 in one year

– The future value is known and riskless and must earn the riskless rate of interest (10%)

Discounting at 10% for one year the portfolio must be worth $45.45 today

33

The Binomial Option Pricing Model (cont’d)

Assuming no arbitrage exists:

The option must sell for $14.77 otherwise there would exist an arbitrage situation:

– If the option is selling for more than $14.77 sellers would step in or if the price was less than $14.77 buyers of the option would find it attractive

77.14$

45.45$275$

C

C

34

The Binomial Option Pricing Model (cont’d)

The option value is independent of the probabilities associated with the future stock price and hence:

The price of an option is independent of the expected return on the stock!

The price of an option tells you nothing about the future path of the stock price!

35

Binomial Put Pricing

Priced analogously to calls

You can combine puts with stock so that the future value of the portfolio is known– Assume a value of $100

36

Binomial Option Pricing Model

Establish a riskless or hedge portfolio where no arbitrage opportunity exists - state of equilibrium

37

Binomial Put Pricing (cont’d)

Possible portfolio values:

$75 + 2($P)

$50 + N($75 - $50)

$100

Today One Year Later

38

Binomial Put Pricing

Solve for N – how many put options to purchase – $50 + N(25) = $100– 25 N = $50– N = 2

39

Binomial Put Pricing (cont’d)

A portfolio composed of one share of stock and two puts will grow risklessly to $100 after one year (discounted to $90.91 in today’s dollars)

95.7$

91.90$275$

P

P

40

Back to Put-Call Parity

Let’s reconcile: Using binomial theory we arrived at a call

price of $14.77 and a put price of $7.95 Does this reconcile with the put-call parity

theory?

41

Binomial Pricing With Asymmetric Branches

The size of the up movement does not have to be equal to the size of the decline– E.g., the stock will either rise by $25 or fall by

$15

The logic remains the same:– First, determine the number of options

– Second, solve for the option price

42

The Effect of Time

More time until expiration means a higher option value – ….remember an option can have time value and

or intrinsic value

43

The Effect of Volatility

Higher volatility means a higher option price for both call and put options– E.g. high tech vs utility type stocks

….work through the same call option example with smaller range of possible outcomes (lower volatility)

44

Black-Scholes Model - the next evolution

Pricing logic remains the same – riskless investments should earn riskless rates of return

Input factors are the same – Stock price– Strike price– Time until expiration– Interest rates– Stock volatility

Using developing computer power in the early 70’s, the model moves to continuous time calculus – outcomes now not limited to only two, many different time intervals - in theory there are an infinite number of future outcomes

45

Continuous Time and Multiple Periods

Future security prices are not limited to only two values

– There are theoretically an infinite number of future states of the world

Requires continuous time calculus (BSOPM)

The pricing logic remains:

– A riskless investment should earn the riskless rate of interest