Futures and Options

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Forward contracts are useful, but only up to a point. They don’t eliminate the risk of default among the parties involved in the trade. For example, merchants might default on the forward agreements if they found the same product cheaper elsewhere, leaving farmers with the goods and no buyers. Conversely, farmers could also default if prices went up dramatically before the forward contract delivery date, and they could sell to someone else at a much higher price. Therefore, a standardized contract was required to address this issue.

A legally binding, standardized agreement to buy or sell a standardized commodity, specifying quantity and quality at a set price on a future date. A great advantage of standardized contracts is that they are easy to trade.
As a result, the contracts usually changed hands many times before their specified delivery dates. Many people who never intended to make or take delivery of a commodity began to actively engage in buying and selling futures contracts.

Why? They were ―speculating‖ — taking a chance that as market conditions changed they would be able to buy or sell the contracts at a profit.

The ability to eliminate a ―position‖ on a contract by buying or selling it before the delivery date — called ―offsetting‖ — is a key feature of futures trading.

Promise to pay $20,000 Buyer Promise to deliver 5, 000 bushels Seller

Promise to pay $20,000

Promise to pay $20,000

Buyer

Clearinghouse

Seller

Promise to deliver 5, 000 bushels

Promise to deliver 5, 000 bushels

25

Basis Calculation Example Spot (Cash) Price August Futures Price Basis $42 $47 -5 (In market lingo, the basis is "$5 under August".)

30

Table 3.2 Gold Prices and the Basis
(July 11) Contract CASH JUL (this year) AUG OCT DEC FEB (next year) APR JUN AUG OCT DEC Prices 353.70 354.10 355.60 359.80 364.20 368.70 373.00 377.50 381.90 386.70 391.50 The Basis -.40 -1.90 -6.10 -10.50 -15.00 -19.30 -23.80 -28.20 -33.00 -37.80

32

Prices
Cash Basis Futures

Present

Time Maturity

May Cash Futures Basis $6.60 $6.70 -$0.10

August $7.20 $7.30 -$0.10

Net $0.60/bu $0.60/bu +$0.00 ( unchanged)

May
Cash Futures Basis $6.60 $6.70 -$0.10

August
$7.20 $7.27 -$0.07

Net
$0.60/bu (loss) $0.57/bu (gain) +$0.03 (Stengthened by $0.03)

May
Cash Futures Basis $6.60 $6.70 -$0.10

August
$6.20 $6.33 -$0.13

Net
-$0.40/bu (loss in cash) -$0.37/bu (gain) -$0.03 (Weakened by $0.03)

Short basis

Strengthen Weakening Is exposed ing of basis of basis to Long hedger Short Hedger likes likes Short basis risk Long basis risk Long basis

Basis

Basis

39

Delivery or cash settlement
• Most commodity futures contracts are written for completion of the futures contract through the physical delivery of a particular good. • Most financial futures contracts allow completion through cash settlement. In cash settlement, traders make payments at the expiration of the contract to settle any gains or losses, instead of making physical delivery.

Offset or reversing trade
• If you previously sold a futures contract, you can close out your position by purchasing an identical futures contract. The exchange will cancel out your two positions.

Exchange-for-physicals (EFP) or ex-pit transaction
• Two traders agree to a simultaneous exchange of a cash commodity and futures contracts based on that cash commodity.

Suppose B is long in futures in wheat and today the price of the futures is $3.95 and next day, the buyer B finds that people are paying $4.15 per bushel for wheat. If B believes that the price of wheat will not go any higher, then B might sell a wheat futures contract for $4.15 to someone else.

In this situation, B has made a reversing trade.

Since B is involved in two wheat contracts, one as a seller and one as a buyer, B is obligated to deliver 5000 bushels to clearing house and clearing house in turn is required to deliver it back to B. The moment B offsets his positions, clearing house will immediately cancel both of them, and B will be able to withdraw $2000 from his account.

An Exchange-for-Physicals Transaction
Before the EFP Trader A Long 1 wheat futures Wants to acquire actual wheat Trader A Agrees with Trader B to purchase wheat and cancel futures Receives wheat; pays Trader B Reports EFP to exchange; exchange adjusts books to show that Trader A is out of the market Trader B Short 1 wheat futures Owns wheat and wishes to sell EFP Transaction Trader B Agrees with Trader A to sell wheat and cancel futures Delivers wheat; receives payment from Trader A Reports EFP to exchange; exchange adjusts books to show that Trader B is out of the market

Futures game

The procedures that protect clearinghouse from potential losses due to noncompliance of the buyer or seller are:
• Impose initial margin requirements on both buyers and sellers • Mark to market the accounts of buyers and sellers every day • Impose daily maintenance margin requirements on both buyers and sellers.

A performance bond is a deposit to cover losses you may incur on a futures contract as it is marked-to-market. A maintenance performance bond is a minimum amount of money (a lesser amount than the initial performance bond) that must be maintained on deposit in your account. A performance bond call is a demand for an additional deposit to bring your account up to the initial performance bond level.

In stock trading, margin refers to a partial deposit you put up with your broker to purchase securities, while borrowing the remaining amount (typically half) from the broker (expecting to pay interest).

In futures, this ―down payment‖ is actually a good faith deposit you pay to indicate that you will be able to ensure fulfillment of the contract.

Futures contracts require an initial performance bond in an amount determined by the exchange itself. This amount is roughly 5% to 15% of the total purchase price of the futures contract. This margin covers only a part of the protection against the total loss in the case of default.
Therefore, the use of marking to market coupled with a maintenance margin requirement provides the requisite amount of additional protection.

Day

Price of wheat

Event

Amount

Equity in account

If maintenance margin were not required 1 2 3 4 4 4.10 3.95 4.15 Deposit initial margin Mark to market Mark to market Mark to market 1000 500 -750 1000 1000 1500 750 1750

With required maintenance margin 1 2 4 4.10 Deposit initial margin Mark to market Buyer withdraws cash 3 3.95 Mark to market Buyer deposits cash 4 4.15 Mark to market 1000 500 -500 -750 750 1000 1000 1500 1000 250 1000 2000

Reversing trade and withdrawal of cash

-2000

0

At the end of the trading day your position is marked-to-themarket. That is, the clearing house will settle your account on a cash basis. Money will be added to your performance bond balance if your position has made a profit that day. If you’ve sustained a loss that day, money is deducted from your performance bond account.

This rebalancing occurs each day after the close of trading.

If your position has lost money and the balance in the performance bond account has fallen below the maintenance level, a performance bond call will be issued.

That means you have to put in more money to bring the account up to the initial performance bond level.

How Trading Affects Open Interest
Time t=0 t=1 t=2 t=3 Action Trading opens for the popular widget contract. Trader A buys and Trader B sells 1 widget contract. Trader C buys and Trader D sells 3 widget contracts. Trader A sells and Trader D buys 1 widget contract. (Trader A has offset 1 contract and is out of the market. Trader D has offset 1 contract and is now short 2 contracts.) Trader C sells and Trader E buys 1 widget contract. Trader B C D E All Traders Long Position 2 2 1 3 3 Open Interest 0 1 4 3

t=4 Ending Positions

3 Short Position 1

Spot Price $420 $410 $400 $390 $380

Spot Market Gain or Loss % Return $2,000 $1,000 0 -$1,000 -$2,000 5% 2.5% 0% -2.5% -5%

Futures Market Gain or Loss % Return $2,000 $1,000 0 -$1,000 -$2,000 100% 50% 0% -50% -100%

Futures Pricing

There are three main theories of future pricing
• The expectations hypothesis • Normal backwardation • A full carrying charge market

Hypothesis: The futures price for a commodity is what the marketplace expects the cash price to be when the delivery month arrives.

The expectation hypothesis is a good predictor because it provides an important source of information about what the future price is likely to be. It works like a price discovery mechanism.

Normally, the futures price exceeds the spot price; this market is called contango. If the futures price is less than the spot price, this is called backwardation, or an inverted market.

A hedger (for example, a farmer) who is selling a futures contract is trying to lock in the price of the commodity in future. i.e. the hedger is trying to reduce the risk, but this risk has to be borne by somebody i.e. speculators. Now question is if the future price equals the spot price + storage costs + other costs exactly, what the speculator will earn by bearing the risk?

Therefore, the speculator will agree to that future price where he expects that the spot price on the delivery date will be higher than futures price. E(ST)>F0

This is called normal backwardation.

A hedger (for example, a manufacturer) who is buying a futures contract is trying to lock in the price of the commodity in future. i.e. the hedger is trying to reduce the risk, but this risk has to be borne by somebody i.e. speculators. Now question is if the future price equals the spot price + storage costs + other costs exactly, what the speculator will earn by bearing the risk?

Therefore, the speculator will agree to that future price where he expects that the spot price on the delivery date will be less than futures price. E(ST)<F0

This is called normal contango.

A full carrying charge market occurs when futures prices reflect the cost of storing and financing (borrowing) the commodity until the delivery month. In the world of certainty, the futures price is equal to the current spot price plus the carrying charges until the delivery month.

To the extent that markets adhere to the following equations markets are said to be at ―full carry‖:

F 0 , t  S 0 (1  C 0 , t ) F 0 , d  F 0 , n (1  Cn , d )
If the futures price is higher than that specified by above equations, the market is said to be above full carry. If the futures price is below that specified by the above equations, the market is said to be below full carry.

To determine if a market is at full carry, consider the following example:

Suppose that:
September Gold December Gold Bankers Rate $410.20 $417.90 7.8%

Step 1: compute the annualized percentage difference between two futures contracts.
AD 

(F ) F
0, d 0. N

12 M 1

Where
• AD = Annualized percentage difference • M = Number of months between the maturity of the futures contracts.

12 $417.90 3 AD  1 $410.20

(

)

AD  0.0772
Step 2: compare the annualized difference to the interest rate in the market. The gold market is almost always at full carry. Other markets can diverge substantially from full carry.

A spread is the difference in price between two futures contracts on the same commodity for two different maturity dates:

Spread  F 0, t  k  F 0, t
F0,t = The current futures price for delivery of the product at time t.
• This might be the price of a futures contract on wheat for delivery in 3 months.

F0,t+k = The current futures price for delivery of the product at time t +k.
• This might be the price of a futures contract for wheat for delivery in 6 months.

Spread relationships are important to speculators.

We know that there is a relationship between the price of the commodity in the cash market and price of that commodity in the futures market. The futures market price should reflect the storage cost of that commodity unto that future date plus the cash price of that commodity today and any other costs. If futures price is more than this price (= cash price + storage cost + other costs) then there is a possibility of arbitrage. One will purchase the commodity today, store it and at the same time short a futures contract to deliver it on the futures date.

Since there is a difference in prices, there is a scope for arbitrage.

The common way to value a futures contract is by using the Cost-of-Carry Model. The Cost-of-Carry Model says that the futures price should depend upon two things:
• The current spot price. • The cost of carrying or storing the underlying good from now until the futures contract matures.

Assumptions:
• There are no transaction costs or margin requirements. • There are no restrictions on short selling. • Investors can borrow and lend at the same rate of interest.

Suppose you buy the corn now for the current cash price of S0 per bushel and store it until you have to deliver it at date T, the forward price you would be willing to commit would have to be high enough to cover
• The present cost of the corn and • The cost of storing the corn until contract maturity

These storage costs involve
• Commission paid to the warehouse for storing • Cost of financing the initial purchase • LESS cash flows received by owing the asset.

F0,T = S0 + SC0,T

= S0 + (PC0, T + i 0, T – D0, T)

The Cost-of-Carry Model can be expressed as:

F 0 , t  S 0(1  C 0 , t )
S0 F0,t = = the current spot price the current futures price for delivery of the product at time t.

C0,t = the percentage cost required to store (or carry) the commodity from today until time t. The cost of carrying or storing includes:
• Storage costs • Insurance costs • Transportation costs • Financing costs

Cash-and-carry arbitrage
• When futures are overpriced

Reverse cash-and-carry arbitrage
• When futures are underpriced

A cash-and-carry arbitrage occurs when a trader borrows money, buys the goods today for cash and carries the goods to the expiration of the futures contract. Then, delivers the commodity against a futures contract and pays off the loan. Any profit from this strategy would be an arbitrage profit.
0 1

1. Borrow money 2. Sell futures contract 3. Buy commodity

4. Deliver the commodity against the futures contract 5. Recover money & payoff loan

The futures price must be greater than or equal to the spot price of the commodity plus the carrying charges necessary to carry the spot commodity forward to delivery.
F 0 , t  S 0(1  C 0 , t )
0 1

1. Borrow $400 2. Buy 1 oz gold 3. Sell futures contract

4. Deliver gold against futures contract 5. Repay loan

Cash-and-Carry Gold Arbitrage Transactions
Prices for the Analysis: Spot price of gold Future price of gold (for delivery in one year) Interest rate Transaction t=0 Borrow $400 for one year at 10%. Buy 1 ounce of gold in the spot market for $400. Sell a futures contract for $450 for delivery of one ounce in one year. Remove the gold from storage. Deliver the ounce of gold against the futures contract. Repay loan, including interest. $400 $450 10% Cash Flow +$400 - 400 0 Total Cash Flow t=1 $0 $0 +450 -440 Total Cash Flow +$10

A reverse cash-and-carry arbitrage occurs when a trader sells short a physical asset. The trader purchases a futures contract, which will be used to honor the short sale commitment. Then the trader lends the proceeds at an established rate of interest. In the future, the trader accepts delivery against the futures contract and uses the commodity received to cover the short position. Any profit from this strategy would be an arbitrage profit.
0 1

1. Sell short the commodity 2. Lend money received from short sale 3. Buy futures contract

4. Accept delivery from futures contract 5. Use commodity received to cover the short sale

The futures price must be equal to or less than the spot price of the commodity plus the carrying charges necessary to carry the spot commodity forward to delivery.
0

F 0 , t  S 0(1  C 0 , t ) 1
4. Collect proceeds from loan 5. Accept delivery on futures contract 6. Use gold from futures contract to repay the short sale

1. Sell short 1 oz. gold 2. Lend $420 at 10% interest 3. Buy a futures contract

Reverse Cash-and-Carry Gold Arbitrage Transactions
Prices for the Analysis Spot price of gold Future price of gold (for delivery in one year) Interest rate Transaction t=0 Sell 1 ounce of gold short. Lend the $420 for one year at 10%. Buy 1 ounce of gold futures for delivery in 1 year. Collect proceeds from the loan ($420 x 1.1). Accept delivery on the futures contract. Use gold from futures delivery to repay short sale. $420 $450 10% Cash Flow +$420 - 420 0 Total Cash Flow t=1 $0

+$462 -450 0 Total Cash Flow +$12

Transactions for Arbitrage Strategies
Market Debt Physical Futures Cash-and-Carry Borrow funds Buy asset and store; deliver against futures Sell futures Reverse Cash-and-Carry Lend short sale proceeds Sell asset short; secure proceeds from short sale Buy futures; accept delivery; return physical asset to honor short sale commitment

Since the futures price must be either greater than or equal to the spot price plus the cost of carrying the commodity forward by rule #1.
And the futures price must be less than or equal to the spot price plus the cost of carrying the commodity forward by rule #2. The only way that these two rules can reconciled so there is no arbitrage opportunity is by the cost of carry rule #3. Rule #3: the futures price must be equal to the spot price plus the cost of carrying the commodity forward to the delivery date of the futures contract.

F 0 , t  S 0(1  C 0 , t )

If prices were not to conform to cost of carry rule #3, a cash-and carry arbitrage profit could be earned. Recall that we have assumed away transaction costs, margin requirements, and restrictions against short selling.

As we have just seen, there must be a relationship between the futures price and the spot price on the same commodity. Similarly, there must be a relationship between the futures prices on the same commodity with differing times to maturity. The following rules address these relationships: Cost-of-Carry Rule 4 Cost-of-Carry Rule 5 Cost-of-Carry Rule 6

The distant futures price must be greater than or equal to the nearby futures price plus the cost of carrying the commodity from the nearby delivery date to the distant delivery date.

F 0, d  F 0, n(1  Cn , d )
F0,d = the futures price at t=0 for the distant delivery contract maturing at t=d.

Fo,n = the futures price at t=0 for the nearby delivery contract maturing at t=n.

Cn,d = the percentage cost of carrying the good from t=n to t=d.

If prices were not to conform to cost of carry rule # 4, a cash-and-carry arbitrage profit could be earned.

0

1

2

1. Buy futures contract w/exp in 1 yrs. 2. Sell futures contract w/exp in 2 years 3. Contract to borrow $400 from yr 1-2

4. Borrow $400 5. Take delivery on 1 yr to exp futures contract. 6. Place the gold in storage for one yr.

7. Remove gold from storage 8. Deliver gold against 2 yr. futures contract 9. Pay back loan

0

1

2

1. Buy futures contract w/exp in 1 yrs. 2. Sell futures contract w/exp in 2 years 3. Contract to borrow $400 from yr 1-2

4. Borrow $400 5. Take delivery on 1 yr to exp futures contract. 6. Place the gold in storage for one yr.

7. Remove gold from storage 8. Deliver gold against 2 yr. futures contract 9. Pay back loan

Gold Forward Cash-and-Carry Arbitrage
Prices for the Analysis Futures price for gold expiring in 1 year Futures price for gold expiring in 2 years Interest rate (to cover from year 1 to year 2) Transaction t=0 Buy the futures expiring in 1 year. Sell the futures expiring in 2 years. Contract to borrow $400 at 10% for year 1 to year 2. Total Cash Flow t=1 Borrow $400 for 1 year at 10% as contracted at t = 0. Take delivery on the futures contract. Begin to store gold for one year. $400 $450 10% Cash Flow +$0 0 0 $0

+$400 - 400 0 Total Cash Flow $0 +$450 - 440 Total Cash Flow + $10

t=2

Deliver gold to honor futures contract. Repay loan ($400 x 1.1)

The nearby futures price plus the cost of carrying the commodity from the nearby delivery date to the distant delivery date cannot exceed the distant futures price. Or alternatively, the distant futures price must be less than or equal to the nearby futures price plus the cost of carrying the commodity from the nearby futures date to the distant futures date.

F0,d  F0,n 1 Cn,d 
If prices were not to conform to cost of carry rule # 5, a reverse cashand-carry arbitrage profit could be earned.

0

1

2

1. Sell futures contract w/exp in 1 yrs. 2. Buy futures contract w/exp in 2 years 3. Contract to lend $400 from yr 1-2

4. Borrow 1 oz. gold 5. Deliver gold on 1 yr to exp futures contract. 6. Invest proceeds from delivery for one yr.

7. Accept delivery on exp 2 yr futures contract 8. Repay 1 oz. borrowed gold. 9. Collect $400 loan

Gold Forward Reverse Cash-and-Carry Arbitrage
Prices for the Analysis: Futures price for gold expiring in 1 year Futures price for gold expiring in 2 years Interest rate (to cover from year 1 to year 2) Transaction t=0 Sell the futures expiring in one year. Buy the futures expiring in two years. Contract to lend $440 at 10% from year 1 to year 2. Total Cash Flow t=1 Borrow 1 ounce of gold for one year. Deliver gold against the expiring futures. Invest proceeds from delivery for one year. Total Cash Flow t=2 Accept delivery on expiring futures. Repay 1 ounce of borrowed gold. Collect on loan of $440 made at t = 1. $440 $450 10% Cash Flow +$0 0 0 $0 $0 + 440 - 440 $0

- $450 0 + 484 Total Cash Flow + $34

Since the distant futures price must be either greater than or equal to the nearby futures price plus the cost of carrying the commodity from the nearby delivery date to the distant delivery date by rule #4. And the nearby futures price plus the cost of carrying the commodity from the nearby delivery date to the distant delivery date can not exceed the distant futures price by rule #5.

The only way that rules 4 and 5 can be reconciled so there is no arbitrage opportunity is by cost of carry rule #6.

The distant futures price must equal the nearby futures price plus the cost of carrying the commodity from the nearby to the distant delivery date.

F 0, d  F 0, n(1  Cn , d )
If prices were not to conform to cost of carry rule #6, a cash-and-carry arbitrage profit or reverse cash-and-carry arbitrage profit could be earned. Recall that we have assumed away transaction costs, margin requirements, and restrictions against short selling.

Since the futures or forwards don’t require front-end from either the long or short transaction; therefore, the contract’s initial market value is usually zero.

Portfolio A

Composition

Value at 0

Value at t Vt (0,T)

Value at T ST – F (0,T)

Long forward 0 contract established at t at price of F(0,T) Long position in asset and loan of F (0,T)(1+r)-(T-t) established at t Long position in security and short one forward contract N/A

B

St – F(0,T)(1+r)-(T-t) St – F (0, T) e-r(T-t) , for a continuous compounding case

ST – F (0,T)

Long position payment is S0

To close the forward contract, investor will sell the underlying for F (0,T)

In other words, investor is investing S0, which will grow @ risk-free rate of interest to F(0,T) in time T. i.e. S0 will grow by (1+r)T to S0 (1+r)T . Therefore, F(0,T) = S0 (1+r)T.

Value of a contract at start

Value of contrac t at expirat ion VT (O, T) = ST – F(0,T)

Price at expirat ion

Price of a contract any time

Value of contract prior to expiration

Price of the contract at start

Value of contract that pays cash dividend

Price of contract that pays cash dividend

F V0 (0, T) O =0 R W A R D

For a known income Vt (O, T) = St – DtT – F(0,T)(1+r)-(T-t) Vt(0,T) = St – F(0,T)(1+r)-(T-t) Vt(0,T) = St – F(0, T)e-r(T-t) , for a continuous compounding case Vt (O, T) = St – DtT – F(0,T)e-r(T-t) For a continuous compounding case F(0,T) = S0(1+r)T F(0, T) = S0erT, for a continuous compoundi ng case For a known dividend yield Vt (O, T) = St e - δc (T-t) – F(0,T) e –
rc (T-t)

f0(T) = S0(1+r)T DT f0(T) = S0erT – DT for a continuous compounding case OR f0(T) = (S0 – D0)(1+r)T f0(T) = (S0 – D0) erT f0(T) = S0 e(rc δc)T

F(T,T) = ST fT(T) = ST F U T U R E S v0 (T) = 0 vt(T) = ft(T) – ft1(T)

f t(T) = F (0,T) = S0 (1+r)T S0erT, for a continuous compoundin g case

For a continuous compounding case

vt(T) = ft(T) – ft-1(T)

If storage costs are considered f0(T) = (S0 + s) erT f0(T) = S0 e(rc + sc)T

Value of a futures contract

Value of a contract at start

Value of contract at expiration

Value of contract prior to expiration

Value of contract that pays cash dividend

Price at expirati on

Price of a contract

Price of contract that pays cash dividend

F O R W A R D

V0 (0, T)= 0

VT(O, T) = ST – F(0,T)

Vt(0,T) = St – F(0,T)(1+r)-(T-t)

Vt (O, T) = St – DtT – F(0,T)(1+r)-(T-t) Vt (O, T) = St e - δc (T-t) – F(0,T) e –rc (T-t) For a continuous compounding case

F(T,T) = ST

F(0,T) = S0(1+r)T

Vt(0,T) = St – F(0, T)e-r(T-t) , for a continuous compounding case

F(0, T) = S0erT, for a continuous compoundin g case
fT(T) = ST f t(T) = F (0,T) = S0 (1+r)T f0(T) = S0(1+r)T - DT f0(T) = S0erT – DT for a continuous compounding case OR f0(T) = (S0 – D0)(1+r)T f0(T) = (S0 – D0) erT f0(T) = S0 e(rc - δc)T

F U T U R E S

v0 (T) = 0

vt(T) = ft(T) – ft-1(T)

vt(T) = ft(T) – ft-1(T)

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