CHAPTER 8Financial Options and Their Valuation

What are options?Why are they needed?Call and Put OptionsOption Pricing Models

Note: spreadsheet for Ch 16 due in next class

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What is an option? A contract that 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.

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

What is the single most importantcharacteristic of an option?

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Why are they needed?

To manage risk To help you with employee stock

options that you might receive To help with capital structure

decisions especially when convertible securities are used

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Call Option

An option to buy a specified number of shares of a security within some future period.

Exercise (or strike) price – the price stated in the option contract at which the security can be bought.

Option price – the market price of the option contract.

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Call Option…contd… Expiration date – the date the option

matures. Exercise value – the value of an

option if it were exercised today (Current stock price - Strike price).

Covered option – an option written against stock held in an investor’s portfolio.

Naked (uncovered) option – an option written without the stock to back it up.

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Example: Call option

Suppose you owned 100 shares of GCC’s stock which are selling at $53.5 per share on January 9, 2004. You sell person B the option to BUY these shares at $55 per share at any time until May 14, 2004.Buyer of option

Strike or exercise price

Covered option

‘Writer’ of option

American option

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Example…contd Who is the ‘writer’ of the option? Who is the buyer? Why would he buy this option? What is the strike or exercise price? Is it covered or a naked option? What type of option is this? American or

European? What is the exercise value?

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Further terminology In-the-money call – a call

option whose exercise price is less than the current price of the underlying stock.

Out-of-the-money call – a call option whose exercise price exceeds the current stock price.

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Option example A call option with an exercise price of

$25, has the following values at these prices:

Stock price Market price of call option

$25 $3.00 30 7.50 35 12.00 40 16.50 45 21.00 50 25.50Observe how the market price goes up as

the stock price exceeds the exercise price

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Determining option exercise value and option premium

Stock Strike Exercise Option Option

price price value pricepremium

$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 0.50Observe how the market price becomes closer to

the exercise value as the stock price rises. The option premium becomes smaller and smaller

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Why does the option premium decrease?

The premium of the option price over the exercise value declines as the stock price increases.

This is because there is virtually no chance that the stock will be out-of-the-money at expiration if the stock price is presently very high

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Call premium diagram

5 10 15 20 25 30 35 40 45 50

Stock

Price

Option value

30

25

20

15

10

5

Market price

Exercise value

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Observations: Call Premium diagram Market value of the option is zero

when the stock price is zero Market price of the option is

always greater than or equal to the exercise value

Market value of the option is greater than zero even when the option is out-of-the-money

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Observations: Call Premium diagram…contd

Value of the option is steadily increasing as the stock price is increasing

Options have considerable upside potential but limited downside risk. The most you lose is the price you paid for the option

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Options magnify both returns and losses You buy STI’s stock at $30 and the price

goes up to $40. You would have a 33% return on the stock.

Instead you bought a call option and the price goes up from $7.5 to $16.5 per option: a return of 120%!

If the stock price goes down to $25,then you would lose 17%, and if you had bought the option you would lose 60% (from $7.50 to $3.0)

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Some factors that affect the value of a Call option Market price vs. Strike price: the higher

the market price, the higher will be the option’s value

Level of strike price: the higher the strike price, the lower will be the option’s value

Length of option: the longer the option period the higher is the option price

Stock price volatility: the more volatile the stock the higher will be the option price. Chances of making a profit increase whilst the downside potential is limited

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Put Option Put option – an option to sell a

specified number of shares of a security within some future period.

Exercise (or strike) price – the price stated in the option contract at which the security can be sold.

Option price – the market price of the option contract.

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Put Option…contd Expiration date – the date the

option matures. Exercise value – the value of an

option if it were exercised today (Current stock price - Strike price).

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Example: Put Option

Suppose you have bought a put option to sell GCC’s shares at a price of $50 per share for 100 shares over the next 4 months

Buyer of optionStrike price

Draw exercise value versus stock price

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Example…contd..

Who is the buyer of the option? Why would you buy this option? What is the strike or exercise

price? What is the exercise value if the

stock price fell to $45?

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Put Option: Exercise Value

Exercise Value = ($50-$45) x 100 shares = $500

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Option Pricing Models

We find the price of an option using: Binomial Approach OR Black-Scholes Option Pricing Model

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Binomial Approach All option pricing models are based on

the concept of a riskless hedge E.g., suppose an investor (call her Hedger)

buys some shares of stock and simultaneously writes a call option on the stock.

Another example of a hedge: Buy shares of a sock and also buy a put option

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What are the assumptions of the Black-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.

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

Security buyers may borrow any fraction of the purchase price at the short-term, risk-free rate.

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What are the assumptions of the Black-Scholes Option Pricing Model?

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

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Which equations must be solved to find the Black-Scholes option price?

)][N(d Xe- )]P[N(d V

tσ - d dtσ

t] 2

[k ln(P/X) d

2

tk-

1

12

2

RF

1

RF

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Use the B-S OPM to find the option value of a call option with P = $27, X = $25, kRF = 6%, t = 0.5 years, and σ2 = 0.11.

0.6327 0.1327 0.5000 N(0.3391) )N(d0.7168 0.2168 0.5000 N(0.5736) )N(d

textbook the in 5- ATable From

0.3391 .7071)(0.3317)(0 - 0.5736 d

0.5736 .7071)(0.3317)(0

(0.5) )]20.11 [(0.06 )ln($27/$25

d

2

1

2

1

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Solving for option value

$4.0036 V

[0.6327]$25e - ]$27[0.7168 V

)][N(d Xe- )]P[N(d V )(0.06)(0.5-

2t-k

1RF

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How do the factors of the B-S OPM affect a call option’s value?

As the factor increases … Option value …

Current stock price IncreasesExercise price DecreasesTime to expiration IncreasesStock return variance Increases

Valuation of Put options We can use the price of the call

option to value a put option. We create 2 portfolios:

Portfolio 1 = Buy one put option and buy one share

Portfolio 2 = Buy one call option and keep cash equal to the exercise price

The payoffs of both portfolios will be the same

Portfolio payoffs

P < X P >= X

Put X -- P 0Stock P PPortfolio 1: X P

Stock Price at Expiration

Call 0 P -- XCash X XPortfolio 2: X P

Put-Call Parity

The two portfolios have identical payoffs

They must have identical values Bingo! We have created a put-call

parityPut option + Stock = Call option + PV of exercise

priceRearranging,Put option = Call option – Price + PV of exercise

price

Put option = V – P + Xe –rRFt