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Chapter 6

Financial Options

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Topics in Chapter

Financial Options Terminology Option Price Relationships Black-Scholes Option Pricing Model Put-Call Parity

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What is a financial option?

An option is a contract which gives its holder the right, but not the obligation, to buy (or sell) an asset at some predetermined price within a specified period of time.

Call Option: Right to buy at a prespecified price

Put Option: Right to sell at a prespecified price

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What is the single most importantcharacteristic of an option?

It does not obligate its owner to take any action. It merely gives the owner the right to buy or sell an asset.

That’s why you need to pay a price to have these options, they are not free.

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Option Terminology

Call option: An option to buy a specified number of shares of a security within some future period at a specific price.

Put option: An option to sell a specified number of shares of a security within some future period at a specific price.

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Option Terminology

Strike (or exercise) price: The price stated in the option contract at which the security can be bought or sold.

Call Payoff = max(0, Stock Price – Strike Price)

Put Payoff = max(0, Strike Price - Stock Price)

Expiration date: The last date the option can be exercised.

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Option Terminology (Continued)

Exercise value: The value of a call option if it were exercised today = Current stock price - Strike price. [Note: The exercise value is zero if the stock price is less than the strike price.]

Option price: The market price of the option contract.

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An Example: Union Pacific (UNP)

Closed at $64.69 on Nov. 12 Considerable volatility over past year All examples represent European options

http://finance.yahoo.com/q/bc?s=

UN

Phttp://finance.yahoo.com

/q/bc?s=U

NP

You are here!

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Payoff Diagrams

Underlying Underlying PricePrice

Option

Option

Payoff

Payoff

$0$0

$75$75 Underlying Underlying PricePrice

Option

Option

Payoff

Payoff

$0$0

$K$K

Payoff diagrams plot option holder’s payoff versus the price of the underlying security. K=Strike Price.

Call Option, K=$75Call Option, K=$75 Put Option, K=$55Put Option, K=$55

With a put, on the other hand, With a put, on the other hand, holder won’t exercise right to holder won’t exercise right to

sell at $55 if they can sell UNP sell at $55 if they can sell UNP for $65.for $65.

Why exercise call and pay Why exercise call and pay $75, when you buy UNP on $75, when you buy UNP on

the market for less than the market for less than $75?$75?

Once UNP is above $75, Once UNP is above $75, call holder will exercise call holder will exercise

option to buy UNP at $75.option to buy UNP at $75.

But say UNP drops to $45, the But say UNP drops to $45, the put holder will exercise to sell put holder will exercise to sell

at $55, a difference of $10.at $55, a difference of $10.

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Option Terminology (Continued) Time value: Option price minus the

exercise value. It is the additional value because the option has remaining time until it expires.

Covered option: A call option written against stock held in an investor’s portfolio.

Naked (uncovered) option: An option sold without the stock to back it up.

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Option Terminology (Continued)

In-the-money call: A call whose strike price is less than the current price of the underlying stock.

Out-of-the-money call: A call option whose strike price exceeds the current stock price.

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Terminology An option is said to be in the money if, were it

able to be exercised immediately, the payoff would be positive.

An option is at the money if the underlying is at the strike price.

An option is out of the money if the holder wouldn’t execute immediately, were they able to.

Underlying Underlying PricePrice

Option

Option

Payoff

Payoff

$0$0

$75$75 Underlying Underlying PricePrice

Option

Option

Payoff

Payoff

$0$0

$55$55

Call Option, K=$75Call Option, K=$75 Put Option, K=$55Put Option, K=$55

In the In the moneymoney

Out of the Out of the moneymoneyIn the In the

moneymoneyOut of the Out of the

moneymoney

At the At the moneymoney

At the At the moneymoney

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Options cost money!

Underlying Underlying PricePrice

Option

Option

Payoff

Payoff

$0$0

$75$75

Payoff diagrams suggest that holding options has no downside. Is this possibly true?

Call Option, K=$75Call Option, K=$75 Of course not. To buy a call or put option, investors must pay a premium, which must equal the value of option. This is reflected by changing the payoff diagram to a P&L daigram.

Option P

rofitO

ption Profit

Option Option PremiumPremium

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Option Terminology (Continued)

LEAPS: Long-term Equity AnticiPation Securities that are similar to conventional options except that they are long-term options with maturities of up to 2 ½ years.

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Consider the following data:

Strike price = $25.

Stock Price Call Option Price

$25 $3.00

30 7.50

35 12.00

40 16.50

45 21.00

50 25.50

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Exercise Value of Option

Price of stock (a)

Strike price (b)

Exercise valueof option (a)–(b)

$25.00 $25.00 $0.00

30.00 25.00 5.00

35.00 25.00 10.00

40.00 25.00 15.00

45.00 25.00 20.00

50.00 25.00 25.00

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Market Price of Option

Price of stock (a)

Strike price (b)

Exer.val. (c)

Mkt. Price of opt. (d)

$25.00 $25.00 $0.00 $3.00

30.00 25.00 5.00 7.50

35.00 25.00 10.00 12.00

40.00 25.00 15.00 16.50

45.00 25.00 20.00 21.00

50.00 25.00 25.00 25.50

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Time Value of Option

Price of stock (a)

Strike price (b)

Exer.Val. (c)

Mkt. P of

opt. (d)

Time value

(d) – (c)

$25.00 $25.00 $0.00 $3.00 $3.00

30.00 25.00 5.00 7.50 2.50

35.00 25.00 10.00 12.00 2.00

40.00 25.00 15.00 16.50 1.50

45.00 25.00 20.00 21.00 1.00

50.00 25.00 25.00 25.50 0.50

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5 10 15 20 25 30 35 40 Stock Price

Option value

30

25

20

15

10

5

Market price

Exercise value

Call Time Value Diagram

This area:

Time Value

Because of time value, options always less at a price higher than their exercise value

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Option Time Value Versus Exercise Value

The time value, which is the option price less its exercise value, declines as the stock price increases.

This is due to the declining degree of leverage provided by options as the underlying stock price increases, and the greater loss potential of options at higher option prices.

Binomial Option Pricing Model

Basic method for option pricing is through Binomial Models: assume stock can either go up or down.

Use riskless hedge argument Book’s very simple example

(homework 6.7) Useful formula: Hedge Ratio =

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Cu - Cd

Pu - Pd

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Assumptions of theBlack-Scholes Option Pricing Model?

The stock underlying the call option provides no dividends during the call option’s life.

There are no transactions costs for the sale/purchase of either the stock or the option.

RRF is known and constant during the option’s life.

(More...)

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Assumptions (Continued) Security buyers may borrow any

fraction of the purchase price at the short-term risk-free rate.

No penalty for short selling and sellers receive immediately full cash proceeds at today’s price.

Call option can be exercised only on its expiration date.

Security trading takes place in continuous time, and stock prices move randomly in continuous time.

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V = P[N(d1)] - Xe -rRFt[N(d2)]

d1 = t 0.5

d2 = d1 - t 0.5

ln(P/X) + [rRF + (2/2)]t

What are the three equations that make up the OPM?

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What is the value of the following call option according to the OPM?

Assume: P = $27 X = $25 rRF = 6% t = 0.5 years σ2 = 0.11

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d1 = {ln($27/$25) + [(0.06 + 0.11/2)](0.5)}

÷ {(0.3317)(0.7071)}

d1 = 0.5736.

d2 = d1 - (0.3317)(0.7071)

d2 = 0.5736 - 0.2345 = 0.3391.

First, find d1 and d2.

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Second, find N(d1) and N(d2)

N(d1) = N(0.5736) = 0.7168. N(d2) = N(0.3391) = 0.6327.

Note: Values obtained from Excel using NORMSDIST function. For example:

N(d1) = NORMSDIST(0.5736)

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Third, find value of option.

V = $27(0.7168) - $25e-(0.06)(0.5)

(0.6327) = $19.3536 - $25(0.97045)(0.6327) = $4.0036.

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What impact do the following parameters have on a call option’s value?

Current stock price: Call option value increases as the current stock price increases.

Strike price: As the exercise price increases, a call option’s value decreases.

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Impact on Call Value (Continued) Option period: As the expiration date is

lengthened, a call option’s value increases (more chance of becoming in the money.)

Risk-free rate: Call option’s value tends to increase as rRF increases (reduces the PV of the exercise price).

Stock return variance: Option value increases with variance of the underlying stock (more chance of becoming in the money).

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Summary of Pricing Factor Relationships

Impact onCall Value

Impact onPut Value

Underlying Price

Positive Negative

Strike Price Negative Positive

Time to Maturity

Positive Positive

Underlying Volatility

Positive Positive

Interest Rates Negative Negative

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Put Options

A put option gives its holder the right to sell a share of stock at a specified stock on or before a particular date.

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Put-Call Parity

Portfolio 1: Put option, Share of stock, P

Portfolio 2: Call option, V PV of exercise price, X

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Portfolio Payoffs at Expiration Date T for PT<X and PT≥X

PT<X PT≥X

Port. 1 Port. 2 Port. 1 Port. 2

Stock PT PT

Put X-PT 0

Call 0 PT-X

Cash X X

Total X X PT PT

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Put-Call Parity Relationship (Can do HW 6.4)

Portfolio payoffs are equal, so portfolio values also must be equal.

Put + Stock = Call + PV of Exercise Price

Put + P = V + Xe

-rRFt

Put = V – P + Xe

-rRFt