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
Option Values at Expiration At expiration date T, the underlying (stock) has market price S T 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 Payoff STST E Long Call Payoff STST E Short Call
Put Option Payoffs Payoff STST E Long Put Payoff STST E Short Put E E
Other Relevant Payoffs Payoff STST Stock Payoff STST Risk-Free Zero Coupon Bond Maturity T, Face Amount E E
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 Stock + Put Payoff STST E STST E E = STST E Call +Bond Payoff STST E E =
Put-Call Parity Payoffs: Stock + Put = Call + Bond Prices: Stock + Put = Call + Bond Stock = Call – Put + Bond S = C – P + PV(E)
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 100 137 73 Bond (r F =2%) 100 102 Call (E=105) C 32 0 1-year call option, S=100, E=105, r F =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 Payoff ( ½ )73 - (1.02) 35.78 = 0 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+ r F ) = 35.78 Note: the value is (apparently) independent of probabilities and preferences!
Multi-Period Replication Stock 100 80 125 100 156.25 64 Call (E=105) 0 51.25 0 C+C+ C-C- 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 Solution: H = 0.911, B = 90.21 C + = 23.68 C - = 0 (obviously?!) 125 100 156.25 0 51.25 r F = 1% C+C+
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 100 80 125 0 23.68 r F = 1%
Building The Tree S S+S+ S-S- S -- S +- S ++ S + = uS S - = dS S ++ = uuS S -- = ddS S +- = S -+ = duS = S
The Tree! u =1.25, d = 0.8 100 80 125 100 156.25 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% r F = 1% (per 6-month period)
Solving Backwards 125 100 156.25 r F = 1% 5 0 P+P+ H = -0.089, B = -13.75 P + = 2.64 80 64 100 41 5 P-P- r F = 1% H = -1, B = -103.96 P - = 23.96 25!! ------- The put is worth more dead (exercised) than alive!
The Answer 100 80 125 25.00 2.64 r F = 1% H = -0.497, B = -64.11 P = 14.42
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