1. CONTINGENT CLAIMS èAny risky security has a payoff that is contingent on the “state of the world” èe.g., equity and debt in our asset substitution and underinvestment problem examples èSome securities make a payoff in a range of states or in one state, but zero otherwise èe.g., lottery tickets, life insurance, my World Series tickets, options

2. OPTION: A Contingent Claim

3. CALL OPTION PAYOFF

4. PUT OPTION PAYOFF

5. PROPERTIES OF CALL OPTION PRICES

6. IMPLICATIONS FOR CALL OPTION PRICES (nondividend-paying stocks) ÊThe price of a European call option is at least the maximum of zero or the stock price minus the present value of the exercise price ËAn American call option will never be exercised early ÌAmerican call option must have same value as equivalent European call option

7. OPTION TIME PREMIUM: The Value of Waiting

8. TIME PREMIUM CHARACTERISTICS èSince the payoff on an option is asymmetric, the ability to wait is valuable: èOption time premium is generally positive èTime premium decreases as an option becomes further in-the-money or out-of-the-money èTime premium is greatest when the option is exactly at the money

9. OPTIONS AND RISK èWhat happens if risk of underlying asset increases? èGreater chance of a larger payoff èDownside payoffs are limited to zero èImplication: Option values increase with risk of underlying asset

10. ÍPUT-CALL PARITY (nondividend-paying stocks) Buy stock, buy put, sell call (option exercise price = X) Stock Price Call Exer? Put Exer? Wealth S > X yes no X S < X no yes X S = X ?? X  S 0 + P 0 - C 0 = X/(1+r f ) T

11. OPTIONS ARE EVERYWHERE Explicit Traded options Executive stock options Call provisions on bonds Convertible bonds Warrants Hidden Capital budgeting: (postponement, abandonment) Tax timing Common stock and risky debt

12. OPTION ELEMENTS OF EQUITY AND RISKY DEBT èEquity is like a call option on firm assets èEquityholders have “sold” assets to bondholders èBy paying off the debt obligation (exercise price) equityholders can buy back assets èRisky debt contains an embedded put èEquityholders can put the assets to the bondholders and cancel bondholders’ claim

13. MARKET VALUE BALANCE SHEET

14. SIMPLE CASE: TWO POSSIBLE FUTURE STATES OF THE WORLD Stock price now is 100/1.05 = 95.24 Stock pays no dividend Next year, stock price goes up ( by factor u =1.68) to 160 or down (by factor d =.63) to 60

15. PRICING A EUROPEAN CALL OPTION Suppose current stock price is $95.24, and we know that, one year from now, stock price will be either $60 or $160 Consider two investment strategies: 1. Buy ten call options on stock with exercise price of $150. Cost = ? 2. Buy one share of stock and borrow 60/1.05 at 5% int. rate. Total cost = 95.24 - 57.14 = 38.1

16. PRICING A EUROPEAN CALL OPTION Payoff from two strategies after one year: StrategyIf S = 60If S = 160 1 0 10x10 = 100 2 60 - 60 = 0 160 - 60 = 100 èStrategy 2 is a replicating strategy è Since both strategies have the same payoff, they must have the same cost èValue of call = $38.1/10 = 3.81

17. BLACK-SCHOLES MODEL C = call price S = stock price X = exercise price r f = risk-free rate T = time to expiration  = std. dev. stock price changes N( ) = cumulative std. Normal probabilities

18. PUT OPTION PRICING We can use put-call parity to derive a Black-Scholes put option pricing model

19. IMPORTANT ASSUMPTIONS èNondividend-paying stock èConstant interest rate èStochastic process governing stock price movements stays constant