2Outline Call and put options The law of one price Put-call parity Binomial valuation
3Options, Options Everywhere! Compensation—employee stock optionsInvestment/hedging—exchange traded and OTC options on stocks, indexes, bonds, currencies, commodities, etc., exoticsEmbedded options—callable bonds, convertible bonds, convertible preferred stock, mortgage-backed securitiesEquity and debt as options on the firmReal options—projects as optionsWhy so much stuff on options?Bottom line—options is something every well-educated finance student should understand!
5OptionsThe 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 PriceExpiration DateAmerican/EuropeanIn-/At-/Out-of-the-Money
6An 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
7Where Do Options Come From? Publicly-traded equity options are not issued by the corresponding companiesAn 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
9Option Values at Expiration At expiration date T, the underlying (stock) has market price STA 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”)
10Call 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!
12Other Relevant Payoffs Risk-Free Zero Coupon BondMaturity T, Face Amount EPayoffSTStockPayoffEST
13The Law of One PriceIf 2 securities/portfolios have the same payoff then they must have the same priceWhy? Otherwise it would be possible to make an arbitrage profitSell the expensive portfolio, buy the cheap portfolioThe payoffs in the future cancel, but the strategy generates a positive cash flow today (a money machine)This should be familiar from FFM!
14Put-Call Parity Stock + Put = Call +Bond = Payoff ST E Payoff ST E Stock+put S<E S+(E-S)=ES>E S+0=SCall+bond S<E 0+E=ES>E (S-E)+E=SPayoffSTE=
15Put-Call Parity Payoffs: Stock + Put = Call + Bond Prices: Stock = Call – Put + BondS = C – P + PV(E)Value of zero coupon bond is just PV of face amount
17What 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.
18Payoffs 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 yearCan the call option payoffs be replicated?
19Replicating StrategyBuy ½ share of stock, borrow $35.78 (at the risk-free rate).Cost(1/2) = 14.22Payoff(½)137 - (1.02) = 32(½)73 - (1.02) = 0Beware the rounding!!The value of the option is $14.22!
20Solving 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 B = 3273 H B = 0H (delta) = ½ = (C+ - C-)/(S+ - S-)B = (S+ H - C+ )/(1+ rF) = 35.78Note: 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 priceWhy 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/ )+0.5(0/ )=0.5(125%)+0.5(-100%)=12.5%
21Multi-Period Replication 10080125156.2564Stock51.25Call (E=105)C+C-1-year call option, S=100, E=105, rF=1% (semi-annual)2 steps per year
22Solving BackwardsStart at the end of the tree with each 1-step binomial model and solve for the call value 1 period before the endSolution: H = 0.911, B = C+ = 23.68C- = 0 (obviously?!)156.2551.25rF = 1%125C+100
23The AnswerUse these call values to solve the first 1-step binomial modelSolution: H = 0.526, B = C = 10.94The multi-period replicating strategy has no intermediate cash flows12523.68100rF = 1%80Spreadsheet available (but make sure you can do the calculations by hand)
24Building The Tree S++ S+ = uS S+ S- = dS S S+- S++ = uuS S-- = ddS S- S+- = S-+ = duS = SYou do not need to know how to do this!
26Binomial ReplicationThe 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.
27An American Put OptionWhat 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. = % rF = 1% (per 6-month period)