Session 2 Options I C 15 0008 Corporate

Session 2: Options I C 15. 0008 Corporate Finance Topics Summer 2006

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

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 preferred stock, mortgagebacked securities • Equity and debt as options on the firm • Real options—projects as options

Example. .

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

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).

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

Numerical example • Call option • Put option

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”)

Call Option Payoffs Long Call Short Call Payoff E ST

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

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

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)

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

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

Introduction to binomial trees

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.

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

Replicating Strategy Buy ½ share of stock, borrow $35. 78 (at the risk-free rate). Cost (1/2)100 - 35. 78 = 14. 22 Payoff (½)137 - (1. 02) 35. 78 = 32 The value of the option is $14. 22! Payoff (½)73 - (1. 02) 35. 78 = 0

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+ r. F) = 35. 78 Note: the value is (apparently) independent of probabilities and preferences!

Multi-Period Replication 156. 25 Stock 125 Call (E=105) C+ 100 51. 25 80 0 C- 64 0 1 -year call option, S=100, E=105, r. F=1% (semi-annual) 2 steps per year

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 156. 25 125 51. 25 C+ 100 r. F = 1% 0 • Solution: H = 0. 911, B = 90. 21 C+ = 23. 68 • C- = 0 (obviously? !)

The Answer • Use these call values to solve the first 1 -step binomial model 125 23. 68 r. F = 1% 100 80 0 • Solution: H = 0. 526, B = 41. 68 C = 10. 94 • The multi-period replicating strategy has no intermediate cash flows

Building The Tree S++ S+ S- = d. S S+- S S+ = u. S S++ = uu. S S-- = dd. S SS-- S+- = S-+ = du. S = S

The Tree! u =1. 25, d = 0. 8 156. 25 100 100 80 64

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.

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% r. F = 1% (per 6 -month period)

Multi-Period Replication 156. 25 Stock 125 Put (E=105) P+ 100 0 80 5 P- 64 41

Solving Backwards 156. 25 125 0 P+ r. F = 1% 100 5 H = -0. 089, B = -13. 75 P+ = 2. 64 100 80 5 P- 64 r. F = 1% 41 H = -1, B = -103. 96 P- = 23. 96 ------- 25!! The put is worth more dead (exercised) than alive!

The Answer 125 2. 64 r. F = 1% 100 80 25. 00 H = -0. 497, B = -64. 11 P = 14. 42

Assignments • Reading – RWJ: Chapters 8. 1, 8. 4, 22. 12, 23. 4 – Problems: 22. 11, 22. 20, 22. 23, 23. 4, 23. 5 • Problem sets – Problem Set 1 due in 1 week