1 課程十一：期貨與遠期契約 futures and forwards 本講義僅供上課教學之用。

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課程十一：期貨與遠期契約 Futures and Forwards

本講義僅供上課教學之用。

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何謂期貨• Futures and forward contracts are similar to options in

that they specify purchase or sale of some underlying security at some future date.

• However a future contract means an obligation of both sides.

• It is a commitment rather than an investment.• Example: no. 2 hard winter wheat or no. 1 soft red

wheat delivered at an approved warehouse by December 31, 1998.

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Basics of Futures ContractsLong position – commits to purchase the commodity.Short position – commits to deliver.The initial investment is zero however some margin is required.At maturity: Profit to long = Spot pr. at maturity – Original futures pr. Profit to short = Original futures pr. –Spot pr. at maturity零和遊戲，一個願打，一個願挨。 (speculator & hedger)

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期貨與遠期契約的比較

• The later cash flow is mark-to-market for a future contract and is concentrated in one point for the forward contract.

• Futures are standardized and do not specify the counterside.

• Futures are traded in exchanges.

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Futures and Forwards

Forward – the original version. It is very specific and not transferable.A and B write an agreement to deliver a precisely specified product and no side can leave the agreement.This type of contract is good for hedgers and not attractive for speculators.

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Futures and Forwards

Speculators would like to have more flexibility and an ability to sell the contract before maturity.In exchange they provide a high liquidity and price stability, small bid-ask spread and a good default protection.Future contract can be sold, the default risk is very small, the actual delivery is rarely used.

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Futures Markets的商品分類• Currencies

– all major currencies, including cross rate• Agricultural

– corn, wheat, meat, coffee, sugar, lumber, rice• Metals and Energy

– copper, gold, silver, oil, gas, aluminum• Interest Rates Futures

– eurodollars, T-bonds, LIBOR, Municipal, Fed funds• Equity Futures

– S&P 500, NYSE index, OTC, FT-SE, Toronto

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Marking to MarketExample: initial margin on corn is 10%1 contract is 5,000 bushels, price of one bushel is 2.2775,

so you have to post the initial margin = $1,138.75 = 0.1*2.2755*5000

If the futures price goes from 2.2775 to 2.2975the clearinghouse credits the margin account of the long

position for 5000 bushels x 2 cents or $100 per contract.

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Marking to Market

Your balance

time

Initialmargin

Maint.margin

margin call

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Convergence Property期貨價格特性

• The futures price and the spot price must converge at maturity.

• Otherwise there will be an arbitrage based on actual delivery.

• Sometimes delivery is costly!• Cash delivery: sometimes is allowed, sometimes

is the only way to deliver.

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Spot-Futures Parity Theorem

• Create a riskless position involving a futures contract and the spot position.

• Buy one stock for S and take a short position in it. • The only difference is from dividends.• Thus F + D – S is riskless.• The amount of money invested is S.

S

SDFrf

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Spot-Futures Parity Theorem Cost-of-carry relationship

)1()1( drSDrSF ff

Tf drSF )1(

For contract maturing in T periods

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Relationship for Spreads

)(12

2

1

12

2

1

)1)(()(

)1()(

)1()(

TTf

Tf

Tf

drTFTF

drSTF

drSTF

This is a rough approximation based on anassumption that there is a single source ofrisk and all contracts are perfectly correlated.

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Basis Risk and Hedging基差風險The basis is the difference between the futures price and the spot price. (At maturity it approaches zero).This risk is important if the futures position is not held till maturity and is liquidated in advance.Spread position is when an investor is long a futures with one term and short with another.

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Futures versus Expected Spot預期理論

Expectation hypothesis states F0=E(PT)

this hypothesis is be true if all market participants were risk neutral.

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Futures versus Expected Spot

Normal Backwardation (Keynes), commodities are used by hedgers to reduce risk. In order to induce speculators to take the opposite positions, the producers must offer a higher return. Thus speculators enter the long side and have the expected profit of

E(PT) – F0 > 0

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Futures versus Expected SpotContango is similar to the normal backwardation,

but the natural hedgers are the purchasers of a commodity, rather than suppliers. Since speculators must be paid for taking risk, the opposite relation holds:

E(PT)–F0 < 0

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Contango

Futures versus Expected Spot

Expectation Hypothesis

Normal Backwardation

Futureprice

delivery time

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期貨交易• Stock index futures• Program trading• Index arbitrage• Foreign exchange futures• Interest rates futures• Swaps and futures

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Stock Index Futures股價指數期貨

Settled in cash only (to avoid transaction cost)Example:an S&P contract with futures price 450If the final value is 455, the profit is500(455-450) = $2,500Broad market indexes are highly correlated

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Stock Index Correlation

DJIA MMI SP500 VL NYSE

DJIA 1.000 .9779 .9774 .8880 .9750

MMI 1.000 .9497 .8104 .9403

SP500 1.000 .9137 .9972

VL 1.000 .9337

NYSE 1.000

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Synthetic Stock Index• Let’s invest $40M in stock market for 1 month.• One can use futures instead of buying separate st

ocks (saving on transaction fees).• Assume that the index is 400. The 1m futures pri

ce is 404, the T-bill rate is 1% (for a month).• We enter 200 futures contracts and invest the $40

M in T-bills.

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Synthetic Stock IndexWe enter 200 futures contracts and invest the $40M in T-

bills. In 1m from now the index is say 410. Then we have from

the T-bills,40M*0.01 = $400,000and from the futures:200*500*(410-404)=$600,000Total of $1M, as if we had all the money in the stock index.

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Index Arbitrage指數套利• Is very simple theoretically, however in practice it would

be very difficult to buy quickly hundreds of stocks in right proportions.

• As a result the program trading is used.• Designated order turnaround (DOT) – a system that

enables traders to send coordinated orders to the exchange via computer.

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Foreign Exchange Markets

• Useful for international companies trying to protect against changes in exchange rates.

• The forward market is informal. A network of banks and brokers (very big!). Not standardized, no marking to market.

• Currency futures are traded on CME, LIFFE, etc. This market is smaller, but liquid and open for everybody.

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Interest Rate

• October 79. IBM issues new debt of $1B. Underwriting syndicate led by Salomon Brothers.

• The underwriters bought the issue from IBM at 9.62% for 7-year notes and 9.41% for 25 year bonds (about 4 bp above treasuries).

• On October 4, the underwriters started selling the issue. However they were able to place only about 70% of the issue.

• About $250–$300M left with investment banks.

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Interest RateSaturday, October 6, the FED raised IR by 1%.The bonds have fallen by 5% of their value.Salomon lost about $3.5M.

However Salomon hedged by shorting about $100M in GNMA and Treasury futures. This hedge worked excellent in this case offsetting the loss on the bond issue.所以，利用期貨來避利率風險

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Commodity Futures商品期貨Storage costs, carrying cost, insurance – C.

CF at 0 CF at TStrategy 1Buy asset – P0 PT – CBorrow P0 P0 – P0(1+rf)Strategy 2futures 0 -F0 +PT

Total 0 F0 = P0(1+rf)+ C

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Commodity Futures

Using the no-arbitrage principle we getF0 – P0(1+rf) – C = 0

F0 = P0(1+rf) + Cdenote c = C/P0 (percentage r

ate of carrying cost)

F0 = P0(1+rf + c)

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Commodity FuturesF0 = P0(1+rf + c)

More generally c can be any type of costs involved in holding a security.

This equation is derived assuming that an actual storage takes place if it is not reasonable (economically) the equation should NOT be used.

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Agricultural Commodity

Price pattern

1st harvest 2nd harvest time

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Agricultural Futures

1. Find the expected spot price.2. Calculate the present value of the commodity

(using CAPM for discount factor).3. Calculate the present value of the futures price.4. Both present values should be equal.

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Agricultural Futures1. Find the expected spot price.

$1.45 per pound of orange juice in 6m2. Calculate the present value of the commodity (using

CAPM for discount factor). = 0.117 gives r = 5.49%, PV = 1.45/1.05490.5

3. Calculate the present value of the futures price.F0/1.0450.5

4. Both present values should be equal.F0/1.0450.5 = 1.45/1.05490.5

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General Rule

Equate the present value of the future payment of F0 and the present value of the commodity to be received.

TT

Tf k

PE

r

F

110

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General Rule

T

fT

TT

Tf

k

rPEF

k

PE

r

F

1

1

11

0

0