Properties ofStock Option Prices

Chapter 10, 7th editionChapter 9, pre 7th edition

Assets Underlying Options

• Stocks• Foreign Currency

• Stock Indices

• Futures

Notation

• c : European call option price

• p : European put option price

• S0 : Stock price currently

• K : Strike price• T : Life of option • : Volatility of stock

price

• C : American Call option price

• P : American Put option price

• ST :Stock price at option maturity

• D : Present value of dividends during option’s life

• r : Risk-free rate for maturity T with continuous compounding

Factors affecting Stock Option Prices

• S0 : Current stock price• K : Strike price• T : Life of option • : Volatility of stock price• r : Risk-free rate for maturity T with

continuous compounding• D : Present value of dividends during

option’s life

Payoff Patterns from OptionsWhat is the Option Position in Each Case?

K = Strike price, ST = Price of asset at maturity

Payoff Payoff

ST STK

K

Payoff Payoff

ST STK

K

Profit Patterns from Options

Profit

STK

Profit

ST

K

Profit

ST

K

Profit

STK

Long Call Short Call

Long Put Short Put

S0 : Current stock price

• Call option payoff: amount by which stock price exceeds strike price (K)

• Call options therefore become more valuable as stock price ___________

• Put option payoff: amount by which strike price (K) exceeds stock price (S0 )

• Put options therefore become more valuable as stock price ____________

K: Strike or Exercise Price

• Call options: a lower strike price results in a more valuable call option

• Put options: a higher strike price results in a more valuable put option

T: Time to expiration

• Intrinsic value and time value of an option• Both put and call options become more valuable

as time to expiration __________• Does this apply to both American and European?

• European options can only be exercised at expiration thus dividend payments could have a negative effect on the holder of the option (specifically the holder of European calls)

: Volatility of stock price

• Volatility: uncertainty of stock price movements• As volatility increases, the chance that the stock

will do well or poorly increases (Bombardier, Abitibi)

• Does the value of a call option increase with increased volatility?

• Does the value of a put option decrease with decreased volatility?

r: Risk free interest rate

• 30-day T-bill rate as an example • CAPM and discounted cash flows• As ‘r’ increases, value of future cash flows decreases

causing the stock price to drop• How does this affect a call option?• How does this affect a put option?• As ‘r’ increases, and all other variables, including the

stock price, are held constant, the value of a call option increases while the value of a put option decreases

D: Cash dividends

• Dividends reduce the stock price on the ex-dividend date (what is the ex-dividend date)

• Is this good news for the holder of call options?• Is this bad news for the holder of put options?

•Declaration date– This is the date on which the board of directors announces to shareholders and the market as a whole that the company will pay a dividend. •Ex-date or Ex-dividend date– On (or after) this date the security trades without its dividend. If you buy a dividend paying stock one day before the ex-dividend you will still get the dividend, but if you buy on the ex-dividend date, you won't get the dividend. Conversely, if you want to sell a stock and still receive a dividend that has been declared you need to sell on (or after) the ex-dividend day. The ex-date is the second business day before the date of record. •Date of record– This is the date on which the company looks at its records to see who the shareholders of the company are. An investor must be listed as a holder of record to ensure the right of a dividend payout. •Date of payment (payable date) – This is the date the company mails out the dividend to the holder of record. This date is generally a week or more after the date of record so that the company has sufficient time to ensure that it accurately pays all those who are entitled.

                                                               

Effect of Increasing Each Variable on Option Pricing

c p C PVariable

S0

KTr*D

+ + –+

? ? + ++ + + +

-+ -+

–– – +

– + – +•As ‘r’ increases, and all other variables, including the stock price, are held constant, the value of a call option increases while the value of a put option decreases•In general, however, an increase in interest rates leads to a drop in stock prices which decreases the value of a call option and increases the value of a put option

American vs European Options

An American option is worth at least as much as the corresponding European option

C cP p

Upper Bound for Call Option Prices; No Dividends

• Call option (European or American) gives the holder the right to buy one share of a stock for a certain price

c S0

C S0

• Otherwise buy stock instead of option

Lower Bound for Call Option Prices; No Dividends

c S0 –Ke -rT

Upper and Lower bounds of Options Prices

If an option price is above the upper bound and below the lower bound, there are profitable opportunities for arbitrage.

Calls: An Arbitrage Opportunity?

• Suppose that

c = 3 S0 = 20 T = 1 r = 10% K = 18 D = 0

• Is there an arbitrage opportunity?

Formal Argument for Lower Bound Call Option Price

– Portfolio A: European call on a stock + PV of the strike price in cash (c + Ke-rT)

– Portfolio B: one share (SO)

Upper Bound for Put Option Prices; No Dividends

• Put option (European or American) gives the holder the right to sell one share of a stock for a certain price (K)

p K

P K

Lower Bound for Put Prices; No Dividends

p Ke-rT–S0

Puts: An Arbitrage Opportunity?

• Suppose that

p = 1 S0 = 37 T = 0.5 r =5%

K = 40 D = 0

• Is there an arbitrage opportunity?

Formal Argument for Lower Bound Put Option Price

– Portfolio C: European put on the stock + the stock (p + SO)

– Portfolio D: PV of the strike price in cash, Ke-rT

Put-Call Parity; No Dividends

• Consider the following 2 portfolios:– Portfolio A: European call on a stock + PV of the

strike price in cash– Portfolio C: European put on the stock + the stock

• Both are worth MAX(ST , K ) at the maturity of the options

• European, thus, they must be worth the same today– This means that

c + Ke -rT = p + S0

Arbitrage Opportunities• Suppose that

c = 3 S0 = 31

T = 0.25 r = 10%

K =30 D = 0

• What are the arbitrage possibilities when

p = 2.25 ? p = 1 ?

Option Positions

– Long call = max (St – K, 0)– Short call = min (K – St, 0)– Long put = max (K – St, 0)– Short put = min (St – K, 0)

Early Exercise of American calls• Never optimal to exercise an American call

option early on non-dividend stock (does this guideline apply to European options also?)

• Example: stock price is $50 and strike price is $40 with 1 month to expiration; investor plans to hold the stock

• Reasons not to exercise early a call option:– $40 could be invested for 1 month and paid at

expiration (delay paying the strike price)– Stock price could drop before the end of month

(Holding the call provides insurance against stock price falling below strike price )

Early Exercise of American puts

• Always optimal to exercise an American put option early (does this guideline apply to European options also?)

• Strike price is $25 and stock price is $0.02• May be optimal to forgo the insurance and

exercise early to realize the strike price immediately (invest the money).

Extension of Put-Call Parity

• European options; D > 0

c + D + Ke -rT = p + S0

D = present value of all dividends during the life of the option

Questions5th and 6th edition: 9.2, 9.3, 9.7, 9.11, 9.12, 9.14, 9.15

7th edition: 10.2, 10.3, 10.7, 10.11, 10.12, 10.14, 10.15