15 -Option Markets 1 Options l l

15 -Option Markets 1

Options l l Options are contracts. There are two sides to the contract l Long Side (option holder): l l l Pays a premium upfront Gets to “call the shots” in the future Short Side (option writer): l l Receives a premium upfront Must agree to go along with the decision of the “long side” 2

Payoffs vs. Profit l Options are contracts with two transactions l Transaction 1: the premium is exchanged l Transaction 2: payoff at maturity l Profit: Sum of both transactions. 3

Call Options l The option to buy an asset at a pre-specified price called the “strike” l Notation: l T = the time at which the option matures l ST = the price of the asset at time T X = the strike price l l Long position l Pays a premium upfront l Can buy the asset in the future at the strike price l Payoff: Max(ST -X, 0) l Short position: sell the option l Receives a premium upfront l Must sell the asset for X l Payoff: -Max(ST -X, 0) 4

Put Options l The option to sell an asset at a pre-specified price called the “strike” l Notation: l T = the time at which the option matures l ST = the price of the asset in the future X = the strike price l l Long position l Pays a premium upfront l Can sell the asset in the future at the strike price l Payoff: Max(X-ST , 0) l Short position: sell the option l Receives a premium upfront l Must buy the asset for X l Payoff: -Max(X-ST , 0) 5

Example l Call option on Apache stock with strike=100. l l At maturity what is total profit to the long side if l l “Premium” or “option price” = $8 S=110? S=95? S=115? What is the total profit to the short side? 6

Example: If S=110 l In this case, it makes sense to exercise the option. The long side can buy Apache for 100 (strike) and sell it for 110 (market value) on the maturity date. The option payoff is therefore 10. The profit to the long side is 10 -8=2, since the long side paid 8 to buy the option. l The short side suffers a negative payoff. She must buy Apache for 110 (market value) and sell it for 100 (strike). The option payoff is therefore -10. Total profit is -2, since the short side received 8 upfront to take the short side. 7

Example l Put option on Apache stock with strike=100. l l At maturity what is total profit to the long side get if l l Price = $8 S=110? S=95? S=115? What is the total profit to the short side? 8

Example: If S=95 l In this case, it makes sense to exercise the option. The long side can buy Apache for 95 (market value) sell it for 100 (strike) on the maturity date. The option payoff is therefore 5. The profit to the long side is 58=-3, since the long side paid 8 to buy the option. l The short side suffers a negative payoff. She must buy Apache for 100 (strike) when it is only worth 95 (market value). The option payoff is therefore -5. Total profit is 3, since the short side received 8 upfront to take the short side. 9

Relation Between S and X l For calls: l l S = X: Option is at-the-money. S > X: Option is in-the-money. S < X: Option is out-of-the-money. For puts: l l l S = X: Option is at-the-money. S > X: Option is out-of-the-money. S < X: Option is in-the-money. 10

Option Contracts l l l European option: can only be exercised on the expiration date. American option: can be exercised on any day prior to and including the expiration date. Options Clearing Corporation: l l Guarantees contract performance Members (brokers) post margins with the OCC Brokers require investor clients to post margins OCC is “middle man” for exercising options 11

Underlying Assets l l l l Stocks Indices Futures Contracts Foreign Currencies Swaps (Swaptions) Beef Lumber 12

Options on Futures Contracts l Call l Long party: pays a premium to “put on” the long side of a futures contract with a specified futures price. Is not obligated to do so, however. l Short party: collects the premium and agrees to take the short side of the futures contract if the long party decides to do so. 13

Options on Futures Contracts l Put l Long party: pays a premium to “put on” the short side of a futures contract with a specified futures price. Is not obligated to do so, however. l Short party: collects the premium and agrees to take the long side of the futures contract if the long party decides to do so. 14

Options on Futures vs. Other Options l Call options: l l l When a call option on a bond is exercised, the long party immediately pays the strike and buys the bond. When a call option on a bond futures contract is exercised, the long party obligates herself to buy the bond in the future on some specific date for the futures (strike) price Put options: l When a put option on a bond is exercised, the long party immediately sells the bond at the strike price. l When a put option on a bond futures contract is exercised, the long party obligates herself to sell the bond in the future on some specific date for the futures (strike) price. 15

Why Options on Futures? l Futures contracts are often more liquid than the underlying debt instrument. l Market participants would rather have the option based on the more liquid instrument. l l Greater price accuracy. Efficient hedging of the futures position. 16

Option Profit Diagrams l Long Call: Max(S-X, 0) – premium Assume: Option is exercised at the maturity date. X 450 l S Short Call: -Max(S-X, 0) + premium X 450 17

Option Profit Diagrams l Long Put: Max(X-S, 0) – premium X 450 l S Short Put: -Max(X-S, 0) + premium 450 X 18

Hedging Interest Rate Risk l l Short treasury futures contracts Buy puts on treasury futures Put Profit Futures Contract Profit Underlying T-Bond Price 19

Example #1 l A bank estimates that if rates increase by 50 bp its equity will drop by $4 m (30%). l Treasury Futures l l Matures: 1 year Underlying asset: T-bond l l l Futures Price = $38, 554*(1. 10) = $42, 410 l l FV: 100, 000 Maturity: 10 years YTM: 10% Coupon: 0% PV=38, 554 Assuming market participants can borrow and lend at 10% Put option on bond l l l Premium (price) $900 Expires: 1 year Strike = 42, 410 20

Example #1 l If in 1 year, rates do increase by 50 bp l Underlying asset: T-bond l FV: 100, 000 l Maturity: 9 years l YTM: 10. 5% l Coupon: 0% l PV=40, 714 l Exercise put option l Buy bond for 40, 714 in market l Sell it for 42, 410 l Payoff = 1, 696 21

Example #1 l If manager buys 2, 000 contracts l l If rates do increase (e. g. 50 bp) l l Bank pays 2, 000*900=$1. 8 million upfront Exercises option, gets 2000*1696 = $3. 39 M If rates decrease (e. g. 50 bp) l l Do nothing Lose the $1. 8 million premium 22

Example #1 l If manager instead went short 2000 futures contracts Pay nothing upfront l If rates do increase by 50 bp l l Honor futures contract Gets 2000*1696 = $3. 39 M If rates decrease by 50 bp l l l PV of bonds would be 44185 Honor futures contract For each contract lose (42410 -44185)=1, 775 Total loss: 2000*1775 = 3. 6 million Would be offset by increase in bank equity. 23

Example #2 l A client needs 5 million euros in six months. l l l Current dollar-euro fx-rate: $1. 30 Asks bank to help hedge position. Possibilities: l Buy the euros now l l l Maybe not enough capital now Buy a call option on the Euros Go long a futures contract 24

Example #2 l Futures contract: l l Call Contract: l l Note: there is some benefit to holding Euros, so futures price does not equal S 0(1+rf)T l To learn more take Bus. M 411 We’ll assume F 0 = 1. 31 Matures in 6 months Strike= 1. 31 Assume each futures/ call contract is for 100, 000 Euros l l l Go long 50 contracts Futures: essentially free 2 Call option: must pay premium. Assume it’s $25, 000/contract 5

Example #2 l Suppose dollar-euro fx-rate jumps to $1. 40 l l l Futures contract l Honor position, buy Euros for 1. 31´ 100, 000´ 50=$6. 55 million Call Contract l Pay premium upfront ($1. 25 million) l Exercise, buy Euros for 6. 55 million Suppose euro ex-rate jumps to 0. 80 l l Futures contract l Honor position, buy Euros for 6. 55 million Call Contract l Pay premium upfront ($1. 25 million) l Walk away from the contract at expiration l Buy 5 million Euros for [5 million*(0. 80)] = 4 million dollars 26

Factors that Affect Option Prices (Premiums) Premiums Call Put 27

Example l l l Which option has the higher strike price? Call #1 l Vol = 0. 19 l S 0= 110 l Time to maturity = 1 month l price = $5 Call #2 l Vol=0. 18 l S 0= 110 l Time to maturity = 1 month l price = $7 28

Example l Call #1 must have the higher strike. l The volatility of option #1 is higher, while every other factor which impacts option prices is the same. If the strikes of the two options were the same, then the price of call #2 would be lower (because of the lower volatility). But the price of call #2 is higher. Something besides volatility must be causing the price of call #2 to be higher. The only possibility is the lower strike price. 29

Preview to Pricing Options l Recall how we priced bonds: Price=PV(Future Cashflows) l Different approach to pricing derivatives: l No arbitrage pricing 30

Preview to Pricing Options l Simple example of no arbitrage pricing: l Asset A with known price: $3 l Consider a derivative contract on A: l l Payoff is 2*(payoff of A) What is price of derivative contract? 31

Preview to Pricing Options l Suppose price of derivative is > 6 l Then do the following: l Short derivative: get $6 l Use to buy 2 shares of A l Initially, you get $$$ since price(D)>6 l In future: l Liability: 2*payoff of A l But you own 2 shares of A. l Use these to payoff the liability. l Done – keep money earned when positions were initiated. 32