Session 2: Options I

1. Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006

2. Outline • Call and put options • The law of one price • Put-call parity • Binomial valuation

3. Options, Options Everywhere! • Compensation—employee stock options • Investment/hedging—exchange traded and OTC options on stocks, indexes, bonds, currencies, commodities, etc., exotics • Embedded options—callable bonds, convertible bonds, convertible preferred stock, mortgage-backed securities • Equity and debt as options on the firm • Real options—projects as options

4. Example..

5. Options The right, but not the obligation to buy (call) or sell (put) an asset at a fixed price on or before a given date. Terminology: Strike/Exercise Price Expiration Date American/European In-/At-/Out-of-the-Money

6. An Equity Call Option • Notation: C(S,E,t) • Definition: the right to purchase one share of stock (S), at the exercise price (E), at or before expiration (t periods to expiration).

7. Where Do Options Come From? • Publicly-traded equity options are not issued by the corresponding companies • An options transaction is simply a transaction between 2 individuals (the buyer, who is long the option, and the writer, who is short the option) • Exercising the option has no effect on the company (on shares outstanding or cash flow), only on the counterparty

8. Numerical example • Call option • Put option

9. Option Values at Expiration • At expiration date T, the underlying (stock) has market price ST • A call option with exercise price E has intrinsic value (“payoff to holder”) • A put option with exercise price E has intrinsic value (“payoff to holder”)

10. Long Call Payoff E ST Call Option Payoffs Short Call Payoff E ST

11. Long Put Payoff E ST Put Option Payoffs Short Put Payoff E E ST E

12. Stock Payoff ST Other Relevant Payoffs Risk-Free Zero Coupon Bond Maturity T, Face Amount E Payoff E ST

13. The Law of One Price • If 2 securities/portfolios have the same payoff then they must have the same price • Why? Otherwise it would be possible to make an arbitrage profit • Sell the expensive portfolio, buy the cheap portfolio • The payoffs in the future cancel, but the strategy generates a positive cash flow today (a money machine)

14. Payoff Payoff Call +Bond Stock + Put E E Payoff Payoff E E ST ST E E ST ST Put-Call Parity = =

15. Put-Call Parity Payoffs: Stock + Put = Call + Bond Prices: Stock + Put = Call + Bond Stock = Call – Put + Bond S = C – P + PV(E)

16. Introduction to binomial trees

17. What is an Option Worth? Binomial Valuation Consider a world in which the stock can take on only 2 possible values at the expiration date of the option. In this world, the option payoff will also have 2 possible values. This payoff can be replicated by a portfolio of stock and risk-free bonds. Consequently, the value of the option must be the value of the replicating portfolio.

18. Payoffs Stock Bond (rF=2%) Call (E=105) 137 102 32 100 100 C 73 102 0 1-year call option, S=100, E=105, rF=2% (annual) 1 step per year Can the call option payoffs be replicated?

19. Payoff (½)137 - (1.02) 35.78 = 32 Cost (1/2)100 - 35.78 = 14.22 Payoff (½)73 - (1.02) 35.78 = 0 Replicating Strategy Buy ½ share of stock, borrow $35.78 (at the risk-free rate). The value of the option is $14.22!

20. Solving for the Replicating Strategy The call option is equivalent to a levered position in the stock (i.e., a position in the stock financed by borrowing). 137 H - 1.02 B = 32 73 H - 1.02 B = 0 • H (delta) = ½ = (C+ - C-)/(S+ - S-) B = (S+ H - C+ )/(1+ rF) = 35.78 Note: the value is (apparently) independent of probabilities and preferences!

21. 156.25 51.25 125 100 100 0 80 64 Multi-Period Replication Stock Call (E=105) C+ C- 0 1-year call option, S=100, E=105, rF=1% (semi-annual) 2 steps per year

22. 51.25 0 Solving Backwards • Start at the end of the tree with each 1-step binomial model and solve for the call value 1 period before the end • Solution: H = 0.911, B = 90.21  C+ = 23.68 • C- = 0 (obviously?!) 156.25 rF = 1% 125 C+ 100

23. 23.68 0 The Answer • Use these call values to solve the first 1-step binomial model • Solution: H = 0.526, B = 41.68  C = 10.94 • The multi-period replicating strategy has no intermediate cash flows 125 rF = 1% 100 80

24. Building The Tree S++ S+ = uS S+ S- = dS S+- S S++ = uuS S-- = ddS S- S-- S+- = S-+ = duS = S

25. 156.25 125 100 100 80 64 The Tree! u =1.25, d = 0.8

26. Binomial Replication • The idea of binomial valuation via replication is incredibly general. • If you can write down a binomial asset value tree, then any (derivative) asset whose payoffs can be written on this tree can be valued by replicating the payoffs using the original asset and a risk-free, zero-coupon bond.

27. An American Put Option What is the value of a 1-year put option with exercise price 105 on a stock with current price 100? The option can only be exercised now, in 6 months time, or at expiration.  = 31.5573% rF = 1% (per 6-month period)

28. 156.25 0 125 100 100 5 80 64 Multi-Period Replication Stock Put (E=105) P+ P- 41

29. 0 5 156.25 100 5 41 80 125 P+ P- 100 64 Solving Backwards rF = 1% H = -0.089, B = -13.75  P+ = 2.64 rF = 1% H = -1, B = -103.96  P- = 23.96 25!! ------- The put is worth more dead (exercised) than alive!

30. 2.64 25.00 The Answer 125 rF = 1% 100 80 H = -0.497, B = -64.11  P = 14.42

31. Assignments • Reading • RWJ: Chapters 8.1, 8.4, 22.12, 23.2, 23.4 • Problems: 22.11, 22.20, 22.23, 23.3, 23.4, 23.5 • Problem sets • Problem Set 1 due in 1 week