## Extra Innings 1: Options Contracts

Option Contracts

• Call option
• gives its holder the right to purchase the underlying asset for a specified price (exercise or strike price) on or before some specified expiration date
• the holder (long leg) must buy this option for a premium from a seller (short leg)
• if the option is exercised, the short leg must deliver the stock to the option holder (the seller of the option may have to buy the stock on the open market)
• Put option
• gives its holder the right to sell an asset for a specified price (exercise or strike price) on or before some specified expiration date
• the holder (long leg) must buy this option for a premium from a seller (short leg)
• if the option is exercised, the seller must buy the stock from the option holder
• Moneyness
• in the money (ITM): the option would produce profits for its holder if exercised
• out of the money (OTM): to exercise the option would be unprofitable
• at the money (ATM): the strike and the spot price are equal
• American vs. European Options
• An American Option allows its holder to exercise on or before expiration
• A European Option allows its holder to only exercise at expiration

Payoffs to Option Contracts

• Call Option
• the option holder will only exercise the option if the spot price exceeds the strike, otherwise he will let the option expire
• thus, the payoff to the long leg is $max(S_T-X,0)$
• Note that the writer of a call option does not have limited liability
• Put Option
• the option holder will only exercise the option if the spot price is less than the strike; otherwise he will let the option expire
• thus, the payoff to the long leg is $max(X-S_T,0)$
• Does the holder of a short put have limited liability?
• Would you ever exercise an American option before expiration?

Option Strategies

• Protective Put
• combines a long put with a long stock position
• long put provides insurance against downside risk of long stock position
• pays $X$ below the strike and $S_T$ above the strike
• Covered Call
• combines a long stock position with a short call
• long stock position provides insurance when long call is ITM
• pays $S_T$ below the strike and $X$ above the strike
• can be long or short
• combines a long (short) put with a long (short) call
• provides a play on volatility
• long straddle pays $X-S_T$ below the strike and $S_T-X$ above the strike

Put-Call Parity

• Some of our option strategies have payoffs that mimic those of other strategies
• For example, consider a long call coupled with a short put
• this strategy pays $S_T-X$ in all states of the world
• this same payoff can be achieved by going long in the underlying stock and short a bond that pays the strike price at expiration
• since the payoffs of these two positions are the same, they must be priced the same
• this fact implies that $C_0-P_0 = PV(S_T-X) = S_0 - X/(1+r)^T$

Pricing Options Contracts using Binomial Trees

• The trick is to replicate the option payoff.
• Example
• Suppose a stock sells at $\$$100, and the price will either double to \$$200 or fall in half to$\$$50 by year-end. • Consider a call option with a strike price of \$$125 and a time to expiration of one year.
• If the interest rate is 8 percent, what is the value of the call?
• We can mimic the payoff to the call option by borrowing $\$$23.15 and going long in .5 shares of the stock. • The initial outlay on the mimicking portfolio is \$$50-$\$$23.15 = \$$26.85, which must be the value of the call.
• In practice, we need to do two things:
• One find the appropriate hedge ratio, $H$, that equates the payoff of $HS^+ - C^+ = HS^- - C^-$ in both states. (There are alternative hedge portfolios that will work as long as they are risk-free.)
• Find the present value of the hedged portfolio and solve for $C_0$.
• In general, $H = \frac{C^+ -C^-}{S^+-S^-}$ and $C_0 = HS_0-\frac{HS^- -C^-}{(1+r_f)^T} = HS_0-\frac{HS^+ -C^+}{(1+r_f)^T}$.
• If we can price a single tree, we can price multiple trees.