1. FUTURES AND OPTIONS Chapter 16
Futures and Options Relations
Futures Option Contracts

2. Put-Call-Futures Parity
Conversion:
Long in futures at fo
Long in put
Short in call
At expiration the value of the position will be X-fo regardless of the price of the underlying asset.

3. Put-Call-Futures Parity

4. Put-Call-Futures Parity
Note: If the carrying-Cost Model holds and the futures and option expire at the same time, then put-call-futures parity and put-call parity are the same.
Proof:

5. BOPM Defined in Terms of Futures Contracts
The replicating portfolio underlying the BOPM can be defined in terms of futures positions instead of the spot
Consider the example for the single-period BOPM for currency options presented in Chapter 15:
u = 1.1, d = .95, Rus = .05, RF = .03, X = $1.50, Eo = $1.50, and Co = $0.066.
Suppose there is a futures contract on the currency that expires in one period and assume that the carrying-cost model (IRPT) holds.

6. BOPM Defined in Terms of Futures Contracts

7. BOPM Defined in Terms of Futures Contracts
Replicating Portfolio:
Go long in Ho futures contracts and borrow Bo dollars.

8. BOPM Defined in Terms of Futures Contracts
Solve for Ho and Bo where:
Solution:

9. BOPM Defined in Terms of Futures Contracts
Equilibrium Price
The same price obtained with a replicating portfolio using the spot position.

10. BOPM Defined in Terms of Futures Contracts
If the call is mispriced, then the arbitrage can be defined in terms of the futures position. For example, if the market price of the currency call were $0.075, an arbitrageur would sell the call at $0.75, go long in Ho = currency futures at Ef = $1.529, and invest $0.066 in a risk-free security. This would yield an initial CF of .009 and no liabilities at T (see Table ).
This is a much simpler arbitrage strategy than the one using a spot position.

11. Futures Options
Futures options give the holder the right to take a futures position:
Futures Call Option gives the holder the right to go long. When the holder exercises, she obtains a long position in the futures at the current price, ft, and the assigned writer takes the short position and pays the holder ft - X.
Futures Put Option gives the holder the right to go short. When the holder exercises, she obtains a short position at the current futures price, ft, and the assigned writer takes the long position and pays the put holder X - ft.
Futures options on Treasuries, stock indices, currency, and commodities.

12. Futures Options Call on S&P 500 Futures: X = 1250
C = 10, Multiplier = 500
Futures and options futures have same expiration.

13. Futures Options Put on SP 500 Futures X = 1250 P =10, multiplier = 500
Futures and options futures have same expiration.

14. Put-Call Parity
Put-call parity for futures options is formed with a conversion: Long in futures at fo, long in put, and Short in call.
At expiration the value of the position will be X-fo regardless of the price of the underlying futures.
If the futures option, spot option, and futures expire at the same time and the carrying-cost model holds, then put-call-futures, put-call spot and put-call on futures option are the same.

15. BOPM for Futures Option
BOPM for a futures option is the same as the BOPM for a spot if the futures and option expire at the same time and if the carrying cost model holds.
If the futures and futures option do not expire at the same time, then the BOPM for futures option will differ.

16. BOPM for Futures Option

17. BOPM for Futures Options
Replicating Portfolio:
Go long in Ho futures contract and borrow Bo dollars.

18. BOPM for Futures Options
Solve for Ho and Bo where:
Solution:

19. BOPM for Futures Options
Equilibrium Price

20. Black Model for Futures Options
Equilibrium Price
Black Model includes fo instead of So and there is no interest rate.
If the carrying-cost model holds and the futures and futures option expire at the same time, then the Black futures option model is the same as the B-S OPM for spot.