Fi8000 Valuation of Financial Assets Fall Semester 2009 Dr. Isabel Tkatch Assistant Professor of Finance

The Value of a Call Option Assumptions: 1. Two Call options – European and American 2. The underlying asset is a stock that pays no dividends before expiration 3. The stock is traded 4. A risk free bond is traded Arbitrage restriction: C(European) = C(American) Hint: compare the payoff from immediate exercise to the lower bound of the European call option price. If CEu < CAm then you can make arbitrage profits, but the strategy is dynamic and involves transactions in the present and in a future date t < T.

Example There are two call options on the same stock (that pays no dividends), one is American and one is European. Both have the same expiration date (a year from now, T = 2) and exercise price (X = 100) but the American option costs more than the European (CEu= 5 < 6 = CAm). Assume that the buyer of the American call option considers to exercise after 6 months (t = 1). Show that if the semi-annual interest rate is rf = 5% then there is an opportunity to make arbitrage profits.

If the American Call is not Exercised before Expiration Time: → t = 0 t = 2 =T Strategy: ↓ State: → ST < X = 100 ST > X = 100 Buy Eu Call (date t=0) -CEU= -5 (ST - X) Sell Am Call CAM= 6 -(ST - X) Total CF CAM -CEU = 6 - 5 > 0 = 0

If the American Call is Exercised before Expiration on date t < T Time: → t = 0 t = 1 t = 2 =T Strategy: ↓ State: → St < X =100 St > X=100 ST < X =100 ST > X=100 Buy Eu Call (date t=0) -CEU= -5 (ST - X) Sell Am Call CAM= 6 -(St - X) Sell Stock (date t=1) St -ST Buy Bond -X FV(X) Total CF CAM -CEU = 6 - 5 > 0 = 0 = FV(X) - ST > X - ST > 0 = FV(X) - X > 0

Application Say only a European option is traded in the market and on date t=1 you really wish you could exercise since the price is much lower than the strike (say S1=150). Describe a strategy that will be as good as (if not better than) exercising a call option before expiration. Intuition – short stock and long bond will generate at least the same payoff as exercising an American call option. We can use this strategy to “exercise” the European call option. Note: if you own an American call option, the same intuition implies that you should guarantee the profit rather than exercise. The payoff of adding short stock and long bond to your American call is better than immediate exercise.

“Exercise” a European Call before Expiration Time: → t = -1 t = 0 t = 1 =T Strategy: ↓ St > X 150 > 100 ST < X 80 < 100 ST > X 130 > 100 170 > 100 Buy Eu Call (date t=-1) -CEU= -5 (ST - X)= 130-100 170-100 Sell Stock (date t=0) St=150 -ST=-80 -ST=-130 -ST=-170 Buy Bond -X=-100 FV(X)=105 Total CF -CEU=(- 5) St-X= 150-100= 50 = FV(X) - ST = 105 - 80 = 25 > 0 = FV(X) - X = 105 - 100 = 5 > 0

Application Explained The decision whether to “exercise” early or not depends on the trader’s risk preferences. A trader may decide that $50 are enough and not risk walking away with less (if the price drops below the current level, 150) or nothing (if the price drops below the strike, 100) on expiration date. Before expiration (date t=0), the payoff from exercising an American option is the same as that generated by the strategy in the previous slide, but the option expires. On the other hand, if you lock the profits (“exercise”: short stock + long bond) rather than exercise (use the American call option to buy stock and make the difference between the market price and the strike), you will get an additional positive payoff on expiration date t=T=1.