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15-Option Markets

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Options Options are contracts. There are two sides to the contract

Long Side (option holder): Pays a premium upfront Gets to “call the shots” in the future

Short Side (option writer): Receives a premium upfront Must agree to go along with the decision of the “long

side”

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Payoffs vs. Profit Options are contracts with two transactions

Transaction 1: the premium is exchanged

Transaction 2: payoff at maturity

Profit: Sum of both transactions.

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Call Options The option to buy an asset at a pre-specified price called the “strike”

Notation: T = the time at which the option matures ST = the price of the asset at time T X = the strike price

Long position Pays a premium upfront Can buy the asset in the future at the strike price Payoff: Max(ST -X,0)

Short position: sell the option Receives a premium upfront Must sell the asset for X Payoff: -Max(ST -X,0)

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Put Options The option to sell an asset at a pre-specified price called the “strike”

Notation: T = the time at which the option matures ST = the price of the asset in the future X = the strike price

Long position Pays a premium upfront Can sell the asset in the future at the strike price Payoff: Max(X-ST ,0)

Short position: sell the option Receives a premium upfront Must buy the asset for X Payoff: -Max(X-ST ,0)

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Example Call option on Apache stock with strike=100.

“Premium” or “option price” = $8 At maturity what is total profit to the long side if

S=110? S=95? S=115?

What is the total profit to the short side?

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Example: If S=110 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.

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.

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Example Put option on Apache stock with strike=100.

Price = $8 At maturity what is total profit to the long side get if

S=110? S=95? S=115?

What is the total profit to the short side?

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Example: If S=95 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 5-8=-3, since the long side paid 8 to buy the option.

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.

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Relation Between S and X For calls:

S = X: Option is at-the-money. S > X: Option is in-the-money. S < X: Option is out-of-the-money.

For puts: S = X: Option is at-the-money. S > X: Option is out-of-the-money. S < X: Option is in-the-money.

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Option Contracts 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: Guarantees contract performance Members (brokers) post margins with the OCC Brokers require investor clients to post margins OCC is “middle man” for exercising options

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

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Options on Futures Contracts Call

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.

Short party: collects the premium and agrees to take the short side of the futures contract if the long party decides to do so.

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Options on Futures Contracts Put

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.

Short party: collects the premium and agrees to take the long side side of the futures contract if the long party decides to do so.

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Options on Futures vs. Other Options Call options:

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: When a put option on a bond is exercised, the long party

immediately sells the bond at the strike price.

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.

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Why Options on Futures? Futures contracts are often more liquid than

the underlying debt instrument.

Market participants would rather have the option based on the more liquid instrument. Greater price accuracy. Efficient hedging of the futures position.

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Option Profit Diagrams Long Call: Max(S-X,0) – premium

Short Call: -Max(S-X,0) + premium

X450

S

X450

Assume:

Option is exercised at the maturity date.

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Option Profit Diagrams Long Put: Max(X-S,0) – premium

Short Put: -Max(X-S,0) + premium

X450

S

X450

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Hedging Interest Rate Risk Short treasury futures contracts Buy puts on treasury futures

Put Profit Futures Contract Profit

Underlying T-Bond Price

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Example #1 A bank estimates that if rates increase by 50bp its equity will drop by

$4m (30%).

Treasury Futures Matures: 1 year Underlying asset: T-bond

FV: 100,000 Maturity: 10 years YTM: 10% Coupon: 0% PV=38,554

Futures Price = $38,554*(1.10) = $42,410 Assuming market participants can borrow and lend at 10%

Put option on bond Premium (price) $900 Expires: 1 year Strike = 42,410

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Example #1 If in 1 year, rates do increase by 50bp

Underlying asset: T-bond FV: 100,000 Maturity: 9 years YTM: 10.5% Coupon: 0% PV=40,714

Exercise put option Buy bond for 40,714 in market Sell it for 42,410 Payoff = 1,696

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Example #1 If manager buys 2,000 contracts

Bank pays 2,000*900=$1.8 million upfront

If rates do increase (e.g. 50bp) Exercises option, gets 2000*1696 = $3.39M

If rates decrease (e.g. 50bp) Do nothing Lose the $1.8 million premium

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Example #1 If manager instead went short 2000 futures contracts Pay nothing upfront

If rates do increase by 50bp Honor futures contract Gets 2000*1696 = $3.39M

If rates decrease by 50bp 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.

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Example #2 A client needs 5 million euros in six months.

Current dollar-euro fx-rate: $1.30 Asks bank to help hedge position.

Possibilities: Buy the euros now

Maybe not enough capital now Buy a call option on the Euros Go long a futures contract

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Example #2 Futures contract:

Note: there is some benefit to holding Euros, so futures price does not equal S0(1+rf)T To learn more take BusM 411

We’ll assume F0 = 1.31 Matures in 6 months

Call Contract: Strike= 1.31

Assume each futures/ call contract is for 100,000 Euros Go long 50 contracts Futures: essentially free Call option: must pay premium. Assume it’s $25,000/contract

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Example #2 Suppose dollar-euro fx-rate jumps to $1.40

Futures contract Honor position, buy Euros for 1.31100,00050=$6.55 million

Call Contract Pay premium upfront ($1.25 million) Exercise, buy Euros for 6.55 million

Suppose euro ex-rate jumps to 0.80 Futures contract

Honor position, buy Euros for 6.55 million Call Contract

Pay premium upfront ($1.25 million) Walk away from the contract at expiration Buy 5 million Euros for [5 million*(0.80)] = 4 million dollars

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Factors that Affect Option Prices (Premiums)

Maturity to Time

Price AssetUnderlying of Volatility

Price Strike

Price AssetUnderlying

Premiums Call Put

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Example Which option has the higher strike price? Call #1

Vol = 0.19 S0= 110 Time to maturity = 1 month price = $5

Call #2 Vol=0.18 S0= 110 Time to maturity = 1 month price = $7

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Example Call #1 must have the higher strike.

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.

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Preview to Pricing Options Recall how we priced bonds:

Price=PV(Future Cashflows)

Different approach to pricing derivatives: No arbitrage pricing

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Preview to Pricing Options Simple example of no arbitrage pricing:

Asset A with known price: $3

Consider a derivative contract on A: Payoff is 2*(payoff of A) What is price of derivative contract?

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Preview to Pricing Options Suppose price of derivative is > 6

Then do the following: Short derivative: get $6 Use to buy 2 shares of A Initially, you get $$$ since price(D)>6

In future: Liability: 2*payoff of A But you own 2 shares of A. Use these to payoff the liability. Done – keep money earned when positions were initiated.