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)

- gives its holder the right to purchase the
- 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

- gives its holder the right to sell an asset for a specified price (exercise or
- 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

- An

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.