© 2004 South-Western Publishing 1 Chapter 5 Option Pricing.

© 2004 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

A Brief History of Options Pricing: 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

A Brief History of Options Pricing: 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

Arbitrage and Option Pricing

Introduction Free lunches The theory of put/call parity The binomial option pricing model Put pricing in the presence of call options Binomial put pricing Binomial pricing with asymmetric branches The effect of time

9

Arbitrage and Option Pricing (cont’d)

The effect of volatility Multiperiod binomial option pricing Option pricing with continuous

compounding Risk neutrality and implied branch

probabilities Extension to two periods

10

Arbitrage and Option Pricing (cont’d)

Recombining binomial trees Binomial pricing with lognormal returns Multiperiod binomial put pricing Exploiting arbitrage American versus European option pricing European put pricing and time value

11

Introduction

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

12

Free Lunches

The apparent mispricing may be so small that it is not worth the effort– E.g., pennies on the sidewalk

Arbitrage opportunities may be out of reach because of an impediment– E.g., trade restrictions

13

Free Lunches (cont’d)

A University Example

A few years ago, a bookstore at a university was having a sale and offered a particular book title for $10.00. Another bookstore at the same university had a buy-back offer for the same book for $10.50.

14

Free Lunches (cont’d)

Modern option pricing techniques are based on arbitrage principles– In a well-functioning marketplace, equivalent

assets should sell for the same price (law of one price)

– Put/call parity

15

The Theory of Put/Call Parity

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

16

Introduction

For a given underlying asset, the following factors form an interrelated complex:– Call price– Put price– Stock price and– Interest rate

17

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

18

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

+ =

19

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

20

No Arbitrage Relationships

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 (Sr)– 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

21

No Arbitrage Relationships (cont’d)

If there is no arbitrage, then:

)1(

0)1(

0)1(

r

SrPC

r

SrPC

r

SrPCSS

22

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

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

rr

r

S

PC

)1(

23

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

24

The Put/Call Parity Relationship

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

tr

KSPC

)1(0

25

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?

26


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

27


To understand why the law of one price must hold, consider the following information:

C = $4.75P = $3

S0 = $50K = $50R = 6.00%t = 6 months

28


Based on the provided information, the put value should be:

P = $4.75 - $50 + $50/(1.06)0.5 = $3.31

– The actual call price ($4.75) is too high or the put price ($3) is too low

29


To exploit the arbitrage, arbitrageurs would:– Write 1 call @ $4.75– Buy 1 put @ $3– Buy a share of stock at $50– Borrow $48.56 at 6.00% for 6 months

These actions result in a profit of $0.31 at option expiration irrespective of the stock price at option expiration

30


Stock Price at Option Expiration

$0 $50 $100

From call 4.75 4.75 (45.25)

From put 47.00 (3.00) (3.00)

From loan (1.44) (1.44) (1.44)

From stock (50.00) 0.00 50.00

Total $0.31 $0.31 $0.31

31

The Binomial Option 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

32

The Binomial Option Pricing Model (cont’d)

Possible states of the world:

$75

$50

$100

Today One Year Later

33


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?

34


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– Buy the stock and write N call options

35


Possible portfolio values:

$75 – (N)($C)

$50

$100 - $25N


36


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

2

50$25$100$

N

N

37


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%) The portfolio must be worth $45.45 today

38


Assuming no arbitrage exists:

The option must sell for $14.77!

77.14$

45.45$275$

C

C

39


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

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

40

Put Pricing in the Presence of Call Options

In an arbitrage-free world, the put option cannot also sell for $14.77; If it did, an astute arbitrageur would:

Buy a 75 call Write a 75 put Sell the stock short Invest $68.18 in T-bills

These actions result in a cash flow of $6.82 today and a cash flow of $0 at option expiration

41

Put Pricing in the Presence of Call Options

Activity Cash Flow Today

Portfolio Value at Option Expiration

Price = $100 Price = $50

Buy 75 call -$14.77 $25 0

Write 75 put +14.77 0 -$25

Sell stock short +75.00 -100 -50

Invest $68.18 in T-bills

-68.18 75.00 75.00

Total $6.82 $0.00 $0.00

42

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

43

Binomial Put Pricing (cont’d)

Possible portfolio values:

$75

$50 + N($75 - $50)

$100


44

Binomial Put Pricing (cont’d)

A portfolio composed of one share of stock and two puts will grow risklessly to $100 after one year

95.7$

91.90$275$

P

P

45

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

46

The Effect of Time

More time until expiration means a higher option value

47

The Effect of Volatility

Higher volatility means a higher option price for both call and put options– Explains why options on Internet stocks have a

higher premium than those for retail firms

48

Multiperiod Binomial Option Pricing

In reality, prices change in the marketplace minute by minute and option values change accordingly

The logic of binomial pricing can be easily extended to a multiperiod setting using the recursive methods of solving for the option value

49

Option Pricing With Continuous Compounding

Continuous compounding is an assumption of the Black-Scholes model

Using continuous compounding to revalue the call option from the previous example:

88.14$

00.50$))(275($ 10.

C

eC

50

Risk Neutrality and Implied Branch Probabilities

Risk neutrality is an assumption of the Black-Scholes model

For binomial pricing, this implies that the option premium contains an implied probability of the stock rising

51

Risk Neutrality and Implied Branch Probabilities (cont’d)

Assume the following:– An investor is risk-neutral– He can invest funds risk free over one year at a

continuously compounded rate of 10%– The stock either rises by 33.33% or falls 33.33% in

one year

After one year, one dollar will be worth $1.00 x e.10 = $1.1052 for an effective annual return of 10.52%

52


A risk-neutral investor would be indifferent between investing in the riskless rate and investing in the stock if it also had an expected return of 10.52%

We can determine the branch probabilities that make the stock have a return of 10.52%

53


Define the following:– U = 1 + percentage increase if the stock

goes up– D = 1 – percentage decrease if the stock

goes down– Pup = probability that the stock goes up– Pdown = probability that the stock goes down– ert = continuously compounded interest rate

factor

54


The average stock return is the weighted average of the two possible price movements:

%22.346578.01

%78.656667.03333.1

6667.01052.1

)(

down

up

up

rt

up

rtdownup

P

P

P

DU

DeP

eDPUP

55


If the stock goes up, the call will have an intrinsic value of $100 - $75 = $25

If the stock goes down, the call will be worthless

The expected value of the call in one year is:

45.16$)0$3422.0()25$6578.0(

56


Discounted back to today, the value of the call today is:

88.14$1052.1/45.16$

57

Extension to Two Periods

Assume two periods, each one year long, with the stock either rising or falling by 33.33% in each period

What is the equilibrium value of a two-year European call shown on the next slide?

58

Extension to Two Periods (cont’d)

$75

$50

$100


$133.33 (UU)

$66.67 (UD = DU)

$33.33 (DD)

Two Years Later

59


The option only winds up in the money when the stock advances twice (UU)– There is a 65.78% probability that the call is

worth $58.33 and a 34.22% probability that the call is worthless

72.34$1052.1/37.38$

37.38$)0$3422.0()33.58$6578.0(

60


There is a 65.78% probability that the call is worth $34.72 in one year and a 34.22% probability that the call is worthless in one year– The expected value of the call in one year is:

66.20$1052.1/84.22$

84.22$)0$3422.0()72.34$6578.0(

61


$20.66

$0

$34.72


$58.33 (UU)

$0 (UD = DU)

$0 (DD)

Two Years Later

62

Recombining Binomial Trees

If trees are recombining, this means that the up-down path and the down-up path both lead to the same point, but not necessarily the starting point

To return to the initial price, the size of the up jump must be the reciprocal of the size of the down jump

63

Binomial Pricing with Lognormal Returns

Black-Scholes assumes that security prices follow a lognormal distribution– With lognormal returns, the size of the

upward movement U equals:

– The probability of an up movement is:

te

DU

DeP

t

up

64

Multiperiod Binomial Put Pricing

To solve for the value of a put using binomial logic, just change the terminal intrinsic values and work backward just as with call pricing

The branch probabilities do not change

65

Exploiting Arbitrage

Arbitrage Example

Binomial pricing results in a call price of $28.11 and a put price of $2.23. The interest rate is 10%, the stock price is $75, and the striking price of the call and the put is $60. The expiration date is in two years.

What actions could an arbitrageur take to make a riskless profit if the call is actually selling for $29.00?

66

Exploiting Arbitrage (cont’d)

Arbitrage Example (cont’d)

Since the call is overvalued, and arbitrageur would want to write the call, buy the put, buy the stock, and borrow the present value of the striking price, resulting in the following cash flow today:

Write 1 call $29.00Buy 1 put ($2.23)Buy 1 share ($75.00)Borrow $60e-(.10)(2) $49.12

$0.89

67

Exploiting Arbitrage (cont’d)

Arbitrage Example (cont’d)

The value of the portfolio in two years will be worthless, regardless of the path the stock takes over the two-year period.

68

American Versus European Option Pricing

With an American option, the intrinsic value is a sure thing

With a European option, the intrinsic value is currently unattainable and may disappear before you can get at it

An American option should be worth more than a European option

69

European Put Pricing and Time Value

With a European put, the longer the option’s life, the longer you must wait to see sales proceeds

More time means greater potential dispersion in underlying asset values, and this pushes up the put value

A European put’s value with respect to time until expiration is indeterminate

70

European Put Pricing and Time Value (cont’d)

Often, an out-of-the-money put will increase in value with more time

Often, an in-the-money put decreases in value for more distant expirations

71

Intuition Into Black-Scholes

Continuous time and multiple periods

72

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 risk less investment should earn the riskless rate

of interest

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© 2004 South-Western Publishing 1 Chapter 5 Option Pricing.

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