Option Types
- A call is an option to buy
- A put is an option to sell
- A European option can only be exercised at the end of its life
- An American option can be exercised at any time
- Assets underlying exchange-traded options:
- stocks, ETFs, Foreign Currencies, Stock Indices, Futures
Market Structure
Market Structure for OTC Options
Market Structure in Option Exchanges and Clearinghouses
Payoffs from Options
Payoffs from long positions:
- Call:
- Put:
Trading Strategies
Options & Underlying Assets
Combinations of Options
Bull
- Long call at & Short call at , where
- (or) Long put at & Short put at , where
Bear
- Short put at & Long put at where
- (or) short call at & Long call at where
Box
- Long a call & Short a put at + Long a put and Short a call at , where
A Box is a combination of a bull call spread and a bear put spread. If all options are European, a box spread is worth the present value of the difference between the strike prices. If they are American, this is not necessarily so.
Butterfly
- Long calls at & + Short 2 calls at , where
- (or) Long puts at & + Short 2 puts at , where
Straddle
- Long a call and a put at the same and expiry
Strips and Straps
- Strip: Buy a call and 2 (or more) puts at the same and expiry
- Strap: Buy 2 (or more) calls and a put at the same and expiry
Strangle
- Long call at and put at , where and at the same expiry
Calendar
- Short a short term call & Long a longer term call at the same
- (or) Short a short term put & Long a longer term put at the same
Put-Call Parity
Consider two portfolios
- A: European call on a stock + zero-coupon bond that pays at time
- C: European put on the stock + the stock
Payoffs
γ
€ | γ
€ | ||
Portfolio A | Call option
Zero-coupon bond
Total | ||
Portfolio C | Put option
Share
Total |
Both are worth at the maturity of the options. Therefore,
If there are dividends
where is the present value of dividends. Or,
where
Factors Affecting Options Prices
Notations:
- European call option price
- American call option price
- European put option price
- American put option price
- Stock price today
- Stock price at option maturity
- Strike price
- Present value of dividends during optionβs life
- Life of option
- Volatility of stock price
- Risk-free rate for maturity with continuous compounding
Variable | ||||
+ | - | + | - | |
- | + | - | + | |
? | ? | + | + | |
+ | + | + | + | |
+ | - | + | - | |
- | + | - | + |
Options Prices Bounds
Upper Bounds
An American or European call option cannot be worth more than the underlying share:
An American or European put option cannot be worth more than the exercise price:
A European put option cannot be exercised before time . Therefore, it cannot be worth more than the PV of
Lower Bounds
European Call
Consider 2 portfolios:
- A: 1 European call option + Cash equal
- B: 1 Share
No arbitrage, thus
European Put
Consider 2 portfolios:
- C: 1 European put option + 1 Share
- D: Cash equal
No arbitrage, thus
Early Exercise of American Option
American options may be exercised before maturity. However, it is never optimal to exercise an American call option on a non-dividend paying stock.
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