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
• Straddle
  • 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.