Lecture 15 - Options I

16. Institutional details Options traded on many exchanges in the US and around the world: At least __ different countries have exchanges covering assets from around the world. Underlying assets: (7) 2011 trading volume: $706 billion. Most option trading in US takes place on ____

(BATS, BOX, CBOE, CME, GEMINI, ISE, MIAX, NASDAQ, NYSE, PHLX) 21 every major index; over 3,000 US companies' stock; hundreds of ETFs; futures for commodities, indices, and interest rates; exchange rates; and many more. CBOE

9. Put option A put option is At expiration, spot price is ST . If you were long a put option with strike K , would you exercise that option?

A put option is the right to sell an asset at the strike price. Right, but not obligation, to sell underlying asset at a pre-specified price. If ST < K ? Yes, can buy on open market for ST and sell to put counter-party for $K , generating profit K − ST If ST ≥ K ? No, better to sell to open market for $ST than to put counter-party for $K .

25. Leveraged speculation If you have really good information or a strong opinion about a stock, it's often better to buy options instead of buying/shorting the underlying. (Buy calls if you have good/bad news, puts if you have good/bad news.) Why?

Buy calls if you have good news, puts if you have bad news Because it is a cheap source of leverage: For a small investment, call/put option gives you a big exposure to the underlying asset. For example, a typical at-the-money 3 month call option will lose 100% of its value if the stock goes down 0.1%, but will double in value if it goes up 10%, quadruple if it goes up 20%. Very risky, but can be very profitable if you are right.

4. Call option premium What is the total long call profit? What is the total short call profit? Why is this?

Call buyer must pay call seller at time t (trade date, not expiration date) for the right to buy at time T: If you long the call (so purchase the call), you must pay Ct now in order to have the option of buying at time T and price K. ---This means that the maximum you will get at time T is St - K (in other words, if St is greater than K, then you can exercise the option and buy at K and sell at the higher St price). The minimum you will get is 0 by not exercising the option. ---This means that overall the profit will be max(St - K, 0) - Ct In you short the call (so sell the call option without owning the underlying stock), you will get Ct now with the potential to have to purchase the stock for St and only receive K if the short call buyer exercises the option. ---This means the maximum you will have to pay at time T is St - K (the call buyer will pay you K for the stock). The minimum you will have to pay at time T is 0 if the call purchaser does not exercise the option. --- This means the overall profit will be Ct - max(St - K,0)

15. Terminology Call option is: Out of the money if In the money if At the money if Put option is: Out of the money if In the money if At the money if

Call option is: Out of the money if S < K (call is useless for now). In the money if S > K . At the money if S = K . Put option is: Out of the money if S > K (put is useless for now). In the money if S < K . At the money if S = K.

21. Hedging Can use options to hedge against some market risk. For example, pension indexed to market? Can buy Short a company's stock but worried the market may go up? Can buy

Can buy market puts to put a "floor" on your pension value. Can buy market calls.

19. Hedging Can limit the downside of a long stock position by... This works because if stock value... Like an ______ for the stock position.

Can limit the downside of a long stock position by buying a put (protective put). if stock value falls below K , put covers any additional losses. (Put worth K − S , stock worth S , total is K ). insurance policy

23. Hedging Can also hedge against industry or company risks outside your portfolio: (2 examples)

Generally not allowed to buy put options for a company you work at. Can buy puts for suppliers, industry peers, and other companies whose stock returns are correlated to your company.

18. Purposes of using options: 4 main purposes of using options

Hedging. Leveraged speculation in the underlying asset. Trading on volatility. Other strategies.

2. Call option at expiration At expiration, spot price is ST . If you were long a call option with strike K , would you exercise that option?

If ST ≤ K ? No, better to pay ST on open market than K using call option. If ST > K ? Yes, can buy using call option for K and sell for ST , getting profit of ST − K.

10. What are the possible long put payoffs? What are the possible short put payoffs? Long (trader with right to sell) never ____ at expiration, Short (trader who sold right to sell) never ____ at expiration

Long (trader with right to sell) never loses at expiration, Short (trader who sold right to sell) never wins.

3. Call option at expiration What are the payoffs for a long call? What are the payoffs for a short call? Long call (buying the call) never receives a ______ payoff at expiration. Short call (selling the call) never receives a ______ payoff at expiration.

Long never receives a negative payoff at expiration Short never receives a positive payoff at expiration.

38. Protective put What is a protective put? You are guaranteed a payoff at least equal to ______ Example: suppose the strike price is K . Your portfolio consists of one share of stock and one put option. The payoff of your total portfolio at option expiration date will be: image

Protective put = Purchase stock and also long a put option on the stock. at least equal to the put option's exercise price.

11. Put option premium What is the total profit for a long put? What is the total profit for a short put? Why is this?

Put buyer must pay put seller at time t (trade date, not expiration date) for the right to sell at time T : If you long the put you purchase at time t for Pt the right to sell at time T at price K. ---The maximum you will make is K - St if K > St. The minimum you will make is 0 if you don't exer K < St ---The profit will then be max(K - St, 0) - Pt If you short the put you sell at time t for Pt the right right for someone else to sell to you the stock at time T for price K. ---The maximum you will have to pay is K - St if K > St. The minimum you will pay is 0 if K < St (so the put buyer won't exercise the put). ---Therefore the profit will be Pt - max(K-St, 0)

20. Hedging Similarly, can limit the downside of a short stock position by ____

Similarly, can limit the downside of a short stock position by buying a call.

32. Other strategies Cont Long Strangle: Long Butterfly:

Strangle: a long strangle consists of a long put with strike price X1 and a long call with strike price X2 > X1, both on the same underlying and having the same maturity date. ---If price of underlying, ST, at time T is less than X1 and X2 , you will exercise the long put and receive X1 - ST. So in total you will get X1 - ST - premium for put - premium for call. ---If price of underlying, ST, at time T is greater than X1 but less than X2, you will not exercise either options. You will just pay the two premiums. ---If price of underlying, ST, at time T is greater than both X1 and X2, then you will exercise the call option. You will receive X2 - ST - premium for put - premium for call. Butterfly: a long butterfly consists of a long call with strike price X1, a long call with strike price X2 > X1, and two short calls with strike price (X1 + X2)/2 all on the same underlying and having the same maturity date. ---If the underlying price, ST, is less than X1 at time T, none of the calls will be exercised. You will just pay the premium for the two longs and receive the premium for the two shorts. ---If the underlying price, ST, is between X1 and X2 and less than (X1 + X2)/2, then you will make X1 - ST and the other calls won't be exercised. So in total you will receive X1 - ST + diff between premium received for two shorts and premium paid for two longs. ---If the underlying price, ST, is between X1 and X2 and more than X1 and X2, then you will make X1 - ST and the two shorts will be exercised, so you will receive (lose) (ST - (X1+X2)/2)*2. So in total you will receive X1 - ST - ((ST - (X1+X2)/2)*2) + diff between premium received for two shorts and premium paid for two longs. ---If the underlying price, ST, is above X2, then you will make X1 - ST and X2 - ST and the two shorts will be exercised, so you will receive (lose) (ST - (X1+X2)/2)*2. So in total you will receive X1 - ST + X2 - ST - ((ST - (X1+X2)/2)*2) + diff between premium received for two shorts and premium paid for two longs.

44. Put-call parity Many uses and rearrangements of this equation: (2) Valid for ______ options Other extensions available for ____

Tells you how to make a put if you only have call options. Can estimate future dividends Div if you have both Ct and Pt . Valid for European options Other extensions available for American options, if there are short-sale costs, etc.

46. Put-call parity: Example Suppose there is an European call option and an European put option on the same underlying stock, both with strike price $100 and time to maturity of 1 year. The risk-free rate is 2% (continuously compounded APR), and the current price of the underlying stock is $95. (b) Suppose the price of the put option is currently $5.5. Is there an arbitrage opportunity? If yes, how would you implement it? Specify the positions and cash flows.

We know that the protective put position is undervalued compared to the calls-plus-bills position. So we short the calls-plus-bills position and long the protective put position

22. Protective Put Protective put graph (3) Protective put payoffs when St < K and when St > K

You purchase stock at time t and purchase put P at time t. The put allows you to sell the stock at time T for X. If the price of the stock at time T, ST, is greater than X, then you won't exercise the put and will get ST cash inflow If the price of the stock at time T is less than X, then you will exercise the put and sell for X. So the puts payoff is X - ST. You will also sell the stock and receive ST, so total cash inflow is X.

28. Trading on volatility Possible reasons to "buy volatility" using straddles: (2)

You think markets will be very volatile over the next month but don't know which direction; You are worried that volatility in oil markets will be bad for your portfolio and you want to hedge against this risk.

30. Graph of bullish spread

You won't have any payoff until ST > X1, and then you won't break even until ST = X1 + (premium received for short - premium paid for long).

39. Protective put Puts can be used as insurance against ______ Protective puts lock in _____ The cost of the insurance is _____ Options can be used for ______, not just for speculation.

against stock price declines. lock in a minimum portfolio value. is the put premium. for risk management

40. Call-plus-bills strategy A call-plus-bills portfolio can also provide _______ Consider a strategy of buying a call option and, in addition, buying Treasury bills with face value equal to the strike price of the call, and with the maturity date equal to the expiration date of the option. The payoff of this portfolio at expiration date:

also provide limited downside risk with unlimited upside potential.

24. Hedging Companies themselves can use options on _____ (2) to hedge against their operating risk. Sometimes better than using forwards because

commodities and currencies because the only real concern is the downside risk.

36. Basic price bounds for call option Call option can never have a ______ value Call option will be worth CT = max(ST − K, 0) at expiration, which implies _____. Equivalent to holding a call option, you can _______. Getting the share today costs St , meaning price of call today must be _____ Combining the two bounds above, we get:

have a negative value, Ct ≥ 0. This is because if the strike price (price you can buy the stock at thanks to call option) is more than the stock price on the date of the call maturity, you can just choose not to exercise the call option. which implies CT ≤ ST you can buy a stock today and hold it till the expiration date; must be Ct ≤ St 0 ≤ Ct ≤ St

37. Basic price bounds for put option Put option can never have a ________ value Put option worth PT = max(K − ST , 0) at expiration, which implies ______ Equivalently, you can _________. Present value of K is _____, meaning price of put today must be ____. Combining the two bounds above, we get:

have a negative value: Pt ≥ 0. This is because if the strike price (price you can sell the stock at thanks to put option) is less than the stock price on the date of the put maturity, you can just choose not to exercise the put option. which implies PT ≤ K . you can deposit certain amount of dollars at risk-free rate till the expiration date is Ke^[−r (T−t)]; must be Pt ≤ Ke^[−r (T−t)] 0 ≤ Pt ≤ Ke^[−r (T−t)]

29. Other strategies Bullish spread: Bearish spread:

long a call option with strike price X1, short another call option on the same underlying, having the same maturity date, but with strike price X2 > X1. By selling a call option, the investor receives a premium, which partially offsets the price they paid for the first call. This is a way to profit from stock price increases while offsetting the premium on the long call with a short call. ---If at T the underlying price, ST, is below X1, you will not exercise your long call and the purchaser of your shorted call will not exercise their call. You will make the difference between the premium received for the short and the premium you paid for the long ---If at T the underlying price. ST, is above X1 but below X2, then you will exercise your long call. You will make ST - X1. The purchaser of your short call will not exercise their call. So you will make ST - X1 + the difference between the premium received for the short and the premium you paid for the long. ---If at T the underlying price, ST, is above both X1 and X2, then you AND the short purchaser will exercise the call option. You will make the difference between X2 and X1, X2 - X1 + the difference between the premium received for the short and the premium you paid for the long. As a result, the gains earned from buying with the first call option are capped at the strike price of the sold option. Bearish spread: short a put option with strike price X1, long another put option on the same underlying, having the same maturity date, but with strike price X2 > X1. By selling a put option, you agree to buy the underlying at time T at price X1 and receive a premium. By longing a put option, you pay a premium in order to have the option of selling the underlying for price X2 at time T. ---If the price, ST, of the underlying is under X1 at time T, then both you and the purchaser of the shorted put will exercise the option. You will receive X2 - ST for the long put. You will also receive lose X1 - ST fir the short. So in total you will receive X2 - X1 + the difference between the premium received for the short and the premium paid for the long. ---If the price, ST, of the underlying is under X2 but above X1, you will exercise your long but the short purchaser will not exercise their put option. You will receive X2 - ST for the long put. So in total you will receive X2 - ST + the difference between the premium received for the short and the premium paid for the long. ---If the price, ST, of the underlying is above X2 and above X1, neither you nor he purchaser of the short will exercise the option. You will receive the difference between the premium received for the short and the premium paid for the long.

12. Option premium Pt is the

payment option buyer makes to option seller at trade date t.

5. Option premium Ct is...

payment option buyer makes to option seller at trade date t.

6. Call option payoff and profit Suppose today you buy a call option on a stock with a strike price $100. The call option costs $14 and expires in 1 year. The payoff of the call option in 1 year is Your profit will be graph this (image) At what price (of stock in 1 year) do you break even?

payoff of call option in one year is max(ST − $100, 0). Your profit will be max(ST − $100, 0)−$14. Profits do not become positive until the stock price at expiration exceeds $114: ST − 100 = $14 ⇒ ST = $114.

17. Institutional details (Cont'd) Unit of trade: one stock option "contract," typically pertains to ____ Most options are American meaning you can_______, rather than European meaning you can ______. Indices are "cash settled":

pertains to 100 underlying shares. Most options are American meaning you can exercise any time up to expiration, rather than European meaning you can only exercise at exactly expiration date. Indices are "cash settled": since you can't directly buy/sell the underlying, option seller pays option buyer a cash amount equal to max(ST − K, 0) for calls, max(K − ST , 0) for puts.

41. Put-call parity The payoff to the protected put strategy is identical to ______ The cost of the protected put strategy: The cost of the call-plus-bills strategy: No arbitrage means that these two strategies _______ How can this no arbitrage relationship be shown? (image)

the payoff to the call-plus-bills strategy. The cost of the protected put strategy: St + Pt The cost of the call-plus-bills strategy: Ct + Ke−r (T−t) these two strategies should cost you same amount of dollar today to establish

1. Call Options A call option gives its holder the right to ______ Maturity date T: Strike price K: Spot price St :

the right to purchase an asset for a pre-specified price. ---Right, but not obligation, to buy underlying asset at a pre-specified price (you can choose not to buy). Maturity date T: date at which call option buyer has right to buy the asset from the call option seller. Strike price K: purchase price at which call option buyer may buy the underlying from the call option seller at maturity. Spot price St : market price for the underlying asset at time t.

26. Trading on volatility Using options markets, can bet on ____ of returns as opposed to directions of returns. Use a long straddle: Payoff of longstraddle is ________, premium is ______ If you buy a long straddle, you win if ________

volatility Use a long straddle: combination of a long put and a long call with the same strike price and same expiration date. Payoff of longstraddle is max(ST − K, K − ST ), premium is Ct + Pt . If you buy a long straddle, you win if there is a big move in either direction. Hence, it can be thought of as buying volatility.