Properties of Stock Options Chapter 9 Properties of Stock Options

Notation C : American Call option price c : European 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 cont comp c : European call option price p : European put option price S0 : Stock price today X : Strike price T : Life of option : Volatility of stock price

Effect of Variables on Option Pricing X T  r D c p C P – – + + – – + + + + ? + + + + – – + + – – + +

Upper & Lower Bound for Option Prices We derive the Upper & Lower bounds for option prices. If the option prices is above the upper & lower bound , there are profitable arbitraging opportunities An upper & lower bound for the value of a call-option The value of an American or European call-option does not exceed the value of the stock lower bound for a European call-option on non dividend paying stock is S - X e-rT An upper & lower bound for the value of a put-option The value of American or European put option can never exceed X. The value of a European put-option is less than or equal to X e- rT. A lower bound for a European put-option is X e-rT - S

Calls: An Arbitrage Opportunity? Suppose that c = 3 S0 = 20 T = 1 r = 10% X = 18 D = 0 Is there an arbitrage opportunity?

Lower Bound for European Call Option Prices; No Dividends c  max(S0 –Xe –rT, 0)

Puts: An Arbitrage Opportunity? Suppose that Is there an arbitrage opportunity? p = 1 S0 = 37 T = 0.5 r =5% X = 40 D = 0

Lower Bound for European Put Prices; No Dividends p  max(Xe -rT–S0, 0)

Early Exercise Usually there is some chance that an American option will be exercised early An exception is an American call on a non-dividend paying stock. This should never be exercised early

Early Exercise on a Non dividend Paying stock Call Option- not optimal to exercise early If its hold till expiry it protect from falling stock price Time Value is more Put Option- may be optimal to exercise early if it is sufficiently deep in the money Receiving now is always worth more than in future That’s why American put option value is always worth more than European

Put-Call Parity To replicate the gain/loss characteristics of a long stock position, one would purchase a call and write a put simultaneously. The call and put would have the same strike price and the same expiration. By taking these two combined positions (Long call and short put), we can replicate a third one (Long stock).  Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008

Put-Call Parity Put–Call parity defines a relationship between the price of a European call option and European put option, both with the identical strike price and expiry, namely that a portfolio of long a call option and short a put option is equivalent to (and hence has the same value as) a single future contract at this strike price and expiry. This is because if the price at expiry is above the strike price, the call will be exercised, while if it is below, the put will be exercised, and thus in either case one unit of the asset will be purchased for the strike price, exactly as in a future contract. A protective put (holding the stock and buying a put) will deliver the exact payoff as a long call (buying one call and investing the present value (PV) of the exercise price). Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008

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 They must therefore 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 ?

The Impact of Dividends on Lower Bounds to Option Prices