C15.0008 Corporate Finance Topics Summer 2006 Session 2: Options I C15.0008 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 bonds, convertible preferred stock, mortgage-backed securities Equity and debt as options on the firm Real options—projects as options Why so much stuff on options? Bottom line—options is something every well-educated finance student should understand!

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). eBay Jan o6 call exercise price of $45 C(39, 45, 1/3 year) Currently out-of-the-money

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 ST E Payoff E ST This is just the payoff at expiration—it does not include the price/premium!

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

Other Relevant Payoffs Risk-Free Zero Coupon Bond Maturity T, Face Amount E Payoff ST Stock Payoff E 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) This should be familiar from FFM!

Put-Call Parity Stock + Put = Call +Bond = Payoff ST E Payoff ST E Stock+put S<E S+(E-S)=E S>E S+0=S Call+bond S<E 0+E=E S>E (S-E)+E=S Payoff ST E =

Put-Call Parity Payoffs: Stock + Put = Call + Bond Prices: Stock = Call – Put + Bond S = C – P + PV(E) Value of zero coupon bond is just PV of face amount

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

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 (½)73 - (1.02) 35.78 = 0 Beware the rounding!! The value of the option is $14.22!

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! Spreadsheet available (but make sure you can do the calculations by hand) All the information about probabilities and preferences (discount rates) is contained in the current stock price Why not use DCF? What is the correct discount rate for the call option? Equal to stock, greater than, less than? Calculate expected/required return on the stock (assuming prob. of 0.5) E[r(S)]=0.5(137/100-1)+0.5(73/100-1)=0.5(37%)+0.5(-27%)=5% Calculate expected/required return on the call (assuming prob. of 0.5) E[r(C)]=0.5(32/14.22-1)+0.5(0/14.22-1)=0.5(125%)+0.5(-100%)=12.5%

Multi-Period Replication 100 80 125 156.25 64 Stock 51.25 Call (E=105) C+ C- 1-year call option, S=100, E=105, rF=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 Solution: H = 0.911, B = 90.21  C+ = 23.68 C- = 0 (obviously?!) 156.25 51.25 rF = 1% 125 C+ 100

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 23.68 100 rF = 1% 80 Spreadsheet available (but make sure you can do the calculations by hand)

Building The Tree S++ S+ = uS S+ S- = dS S S+- S++ = uuS S-- = ddS S- S+- = S-+ = duS = S You do not need to know how to do this!

The Tree! u =1.25, d = 0.8 100 80 125 156.25 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% rF = 1% (per 6-month period)

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

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

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

Assignments Reading Problem sets 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 Next class: Black-Scholes, intro to real options (decision trees)