FUTURES

FUTURES

DefinitionDefinition

• Futures are marketable forward contracts.

• Forward Contracts are agreements to buy or sell a specified asset (commodities, indices, debt securities, currencies, etc.) at an agreed-upon price (f) for purchase or delivery on a specified date (delivery date: T).

• Futures are marketable forward contracts.

• Forward Contracts are agreements to buy or sell a specified asset (commodities, indices, debt securities, currencies, etc.) at an agreed-upon price (f) for purchase or delivery on a specified date (delivery date: T).

Futures ExchangesFutures Exchanges

• Futures are traded on organized exchanges:– CBOT

– CME

– NYFE

• The exchanges provide marketability:

• Listings– Standardization

– Position Traders

– Clearinghouse

• Futures are traded on organized exchanges:– CBOT

– CME

– NYFE

• The exchanges provide marketability:

• Listings– Standardization

– Position Traders

– Clearinghouse

Futures PositionsFutures Positions

• Long Position: Agreement to buy.

• Short Position: Agreement to sell.

• Long Hedge: Taking a long position in futures to protect against a price increase.

• Short Hedge: Taking a short position in a futures to protect against a price decrease.

• Long Position: Agreement to buy.

• Short Position: Agreement to sell.

• Long Hedge: Taking a long position in futures to protect against a price increase.

• Short Hedge: Taking a short position in a futures to protect against a price decrease.

ClearinghouseClearinghouse

Like the OCC, the futures clearinghouse guarantees each contract (both long and short positions) and acts as an intermediary, breaking up each contract after it has been established.

Like the OCC, the futures clearinghouse guarantees each contract (both long and short positions) and acts as an intermediary, breaking up each contract after it has been established.

ExampleExample

• Suppose A buys a September Wheat Futures contract (5,000 bu.) from B for fo = $2.50/bu.– A is long; B is Short

• After the contract is established, the CH steps in and breaks up the contract.

• Suppose A buys a September Wheat Futures contract (5,000 bu.) from B for fo = $2.50/bu.– A is long; B is Short

• After the contract is established, the CH steps in and breaks up the contract.

CH RecordsCH Records

• A agrees to buy at $2.50.

• B agrees to sell at $2.50.



Example Continued

• Suppose the price of wheat increases, causing the September futures price to increase to ft = $3.00.

• Suppose A decides to close by going short.

• New Contract: A agrees to sell September Wheat futures at $3.00 to C.

– A is short; C is long.

• After the contract is established, the CH breaks it up.

CH Records



• A agrees to sell at $3.00.

• C agrees to buy at $3.00.

CH owes A$0.50

At Expiration

• In the absence of arbitrage, the price on an expiring futures contract must be equal to the spot price.

f ST T

Example Continued

• At the September expiration, suppose the spot price of wheat is at $3.50/bu.

• B is short and needs to close by going long (B is not a farmer).

• C is long and needs to close by going short (C does not need 5000 bu of wheat).– New Contract: B agrees to buy September Wheat (that is

expiring) from C for $3.50.• CH breaks up the contract.

CH RecordsCH Records


• C agrees to buy at $3.00.

• B agrees to buy at $3.50.

• C agrees to sell at $3.50.

B owes CH $0.50

CH owes C $.50

Long Futures Hedge

• Take long position in futures to protect against an increase in the spot price.

• EXAMPLE:– OJ distributor plans to buy 15,000 lbs of frozen OJ in September.

To protect against an increase in the spot price of OJ, the distributor goes long in one OJ futures contract (size = 15,000 lbs) at fo = $0.96/lb.

– At delivery, the distributor buys OJ on the spot market at the spot price and closes the futures position by going short in the expiring futures at a futures price equal to the spot price.

Cost at T

S fT T $ 0 . 9 0 $ 0 . 9 6 $ 1 . 0 0

O J C o s t = S ( 1 5 , 0 0 0 ) 1 3 5 0 0 1 4 4 0 0 1 5 0 0 0

F Tf [ $ . ]9 6 1 5 0 0 0 - 9 0 0 0 6 0 0

C o s t = R o w 2 - R o w 3 1 4 4 0 0 1 4 4 0 0 1 4 4 0 0

Short Futures Hedge

• Take short position in futures to protect against a decrease in the spot price.

• EXAMPLE:– Wheat farmer plans to sell 5000 bu. of wheat in September. To

protect against a decrease in the spot price, the farmer goes short in a September wheat futures at fo = $2.40

– At delivery, the farmer sells wheat on the spot market at the spot price and closes the futures position by going long in the expiring futures at a futures price equal to the spot price.

Revenue at T

S fT T $ 2 . 0 0 $ 2 . 4 0 $ 3 . 0 0

O J R e v . = S ( 5 0 0 0 ) 1 0 0 0 0 1 2 0 0 0 1 5 0 0 0

F Tf [ $ 2 . ]4 0 5 0 0 0 2 0 0 0 0 - 3 0 0 0

R e v = R o w 2 + R o w 3 1 2 0 0 0 1 2 0 0 0 1 2 0 0 0

Hedging Risk

Quantity Risk

Quality Risk

Timing Risk

Speculative Positions

• Pure Outright Position: – Long Position (Bullish)

– Short Position (bearish)

• Spread– Intracommodity Spread: long and short in futures on the

same underlying asset but with different expirations.– Intercommodity Spread: Long and short in futures with

different underlying assets but the same expiration.

Initial Margin Requirements

• Initial Margin: Cash or RF securities that must be deposited with the broker to secure the position. Initial margin (Mo) is equal to a porportion (m) times the contract value.

• Example: September wheat contract at fo = $2.40 (long or short) with m = .10:

M0 10 40 . [($2. )(5000)] $1200

Maintenance Margin Requirements

• Maintenance Margin: Keep the equity value of the commodity account (Eq) equal to a proportion (90% to 100%) of initial margin.

Eq M position Value 0 ( .. )

Example• September wheat prices increase from $2.40 to

$2.42. With a 100% maintenance margin requirement, a long position would be overmargined and a short position would be undermargined:

LONG Eq

Eq Overm ined

SHORT Eq

Eq Underm ined

: $1200 ($2. . )(5000)

$1300 arg

: $1200 ($2. $2. )(5000)

$1100 arg

42 2 40

40 42

Undermargined Positions

• If an account is undermargined, the investor must deposit additional funds to satisfy the maintenance margin requirement. If the investor does not do this, then she will receive a margin call from the broker instructing her that her account will be closed unless she deposits the requisite funds.

• When the equity value of the account meets the maintenance margin requirement, the account is said to be marked to market.

Other Points

• Equity accounts are adjusted daily.

• Futures Funds are often set up where the funds of investors are used to buy RF securities which the fund uses to satisfy the margin requirements for the futures. Such funds can be viewed as overmargined futures positions.

Futures Pricing

• Basis (B):

• Carrying Cost Model: Equilibrium futures price is equal to the net cost of carrying the underlying asset to expiration. This relation is governed by arbitrage.

B f S

B Normal

B Inverted

t t t

t

t

0

0

Pricing Futures on PDB

• Let So = spot price of PDB with maturity of 91 days + T; Rf = RF rate or repo rate with maturity of T; fo = price of PDB futures expiring at T.

f S R fT

0 0 1 ( )

Example

• Price on spot PDB maturing in 161 days is So = 97.5844; 70-day RF rate is 6.38%.

• Equilibrium price of PDB futures with expiration of 70 days (or T= 70/365):

f070 36597 5844 10638 98 74875 . ( . ) ./

Arbitrage

• Overpriced:• If the market price of PDB futures is at 99, an arbitrageur

could earn a riskless profit of 99-.98.74875 = 0.25125 (times $1M) by:– Borrowing $97.5844 at Rf = 6.38% , then buying 161-day SPOT

PDB at So = 97.5844;

– taking short position in a PDB futures expiring in 70 days at fm = 99.

• At T, the arbitrageur would sell the spot PDB on the futures (it would now have a maturity of 91 days) and pay off her loan.

Arbitrage

• Underpriced:• If the market price of PDB futures is at 98, an investor

holding 161-day spot PDB could earn a riskless profit of 98.74875-98 = 0.74875 (times $1M) by:– Selling the PDBs for $97.5844, then investing the proceeds in RF

security for 91 days at Rf = 6.38%;

– taking long position in a PDB futures expiring in 70 days at fm = 98.

• At T, the arbitrageur would buy the spot PDB on the futures (it would now have a maturity of 91 days) for 98 and receive 98.74875 from her investment.

Pricing Futures on Stock Portfolio

• Carrying Cost Model:

f S R D

where

D value of dividends at T

Example S R D

ice on a futures on stock portfolio with T is

f

fT

T

T

f T

0 0

0

025

1

8%, 50

25

108 50 41

( )

:

.

: $150, $1. .

Pr .

$150( . ) $1. $151..

Pricing Commodity Futures


f S R kT TRC

where

k Storage ts per unit of com ity per period

TRC transportation t

fT

0 0 1

( )

:

cos mod .

cos

Example: Pricing Commodity Futures

• In June, the spot price of a bushel of wheat is $2.00, the annual storage cost is $0.30/Bu, Rf = 8%, and transportation cost of transporting wheat from the destination point on the futures contract to a grain elevator is $0.01/bu. The equilibrium price of a September wheat futures (T = .25) is $2.124/bu:

f02500 108 30 25 01 124 $2. ( . ) ($0. )(. ) $0. $2..

Pricing Relation Between Futures with Different Expirations


f f R k T T TRC DT T TT T

T2 1 12 1

21 2 1 ( ) ( )

Financial Futures• Stock Index Futures

• Foreign Currency Futures

• Debt Securities

Stock Index Futures

• Types:– SP 500 (CME, Multiplier = 500)– MMI (CBT, Multiplier = 250)– SP OTC (CME, Multiplier = 500)

• Cash Settlement Feature• Multiplier • Use: Speculation, hedging, and portfolio

management.

Hedging Portfolio Future Value

Example:• Portfolio manager plans to liquidate a $50M portfolio in

September. The portfolio is well-diversified with a beta of 1.25. The current S&P 500 is at 1250 and there is a September S&P 500 futures index trading at fo = 1250. (Note futures and spot prices are usually not equal.)

• Hedging Strategy: Go short in 100 September index futures contracts:

nV

f

Mf

0

0

125

1250100

( . )$50

( )(500)

Hedged Value at TST 1.25g V g MT( . )501125 f 3+2

1000 -.25 $37.50M $12.5M$50M

1125 -.125 $43.75M 6.25M50M

1250 0 $50.00M 0 50M

1375 .125 $56.25M -6.25M50M

1500 .25 $62.5M -12.5M50M

gST

1250

1250 f TS ( )(500)[ ]100 1250

Portfolio Uses• Speculating on Unsystematic Risk

• Market Timing

• Dynamic Portfolio Insurance

nV

ff TR 0

00[ ]

nV

ff

S

0

0

Pricing Stock Index Futures

• Let So = spot price of stock index (S&P 500); Rf = RF rate or repo rate with maturity of T; D = dividend per share on portfolio underlying the index which can be estimated from a proxy portfolio; fo = price of index future expiring at T.

f S R DfT

0 0 1 ( )

Proxy Portfolio

• Stock Index futures are often priced in terms of a proxy portfolio. A Proxy portfolio is a portfolio which is highly correlated with the index (could be 30-stock portfolio or a MF). This portfolio can be viewed as equivalent to holding hypothetical shares in the index.

• For example, if the S&P 500 is at 1200, a $10M well-diversified portfolio with a beta of 1 and expected dividends at T worth $250,000 could be viewed as owning 8333.333 hypothetical index shares that are selling at $1200 per share and paying a dividend per share of $30.

Example

• Spot index (S&P 500) is at 1200 and RF rate is 8% for RF securities maturing in 180 days.

• Using the proxy portfolio, the equilibrium price S&P 500 futures with expiration of 180 days (or T= .5 per year:

07.121730$)08.1(1200 5.0 f

Index Arbitrage

• Overpriced:

• If the futures were priced at fm = 1245, an arbitrageur could earn a riskless profit by going long in the proxy portfolio and short in the futures:– Borrow $10M and buy portfolio.– Go short in 8333.333/500 = 16.6667 futures.

CF at TC l o s i n gP o s i t i o n s

g = - . 1 0S = 1 0 8 0

g = + . 1 0S = 1 3 2 0

D e b t :$ 1 0 ( . ) .M 1 0 8 5

- $ 1 0 . 3 9 2 M - $ 1 0 . 3 9 2 M

P o r t f o l i o : $ 1 0 M ( 1 + g )

$ 9 M $ 1 1 M

F u t u r e s :( . ) ( 5 0 0 ) [ ]1 6 6 6 6 7 1 2 4 5 f T

$ 1 . 3 7 5 M - $ . 6 2 5 M

D i v i d e n d s $ . 2 3 3 M $ . 2 3 3 M

C F $ . 1 9 1 3 M $ . 1 9 1 3 M

Foreign Currency Futures

• Traded on the IMM.

• Futures on major currencies:– DM (125,000)– BP (25,000)– FF (250,000)

• Use: Hedging and speculation.

Pricing Currency Futures

• Carrying cost for currency futures is the interest rate parity model discussed in many international text:

f ER

R

where

R Foreign RF Rate

US

F

T

F

0 0

1

1

:

.. ..

f E f0

Pricing Currency Futures

R R E FC T year

E FC FC

us F

f

4%, 6%, 40 1

40104

106392

0 $0. / ,

$0. /.

.$0. /

Pricing Currency Futures• Covered Interest Arbitrage:

E FC

Borrow at

Convert to FC FC

Invest in Foreign urity at

Enter forward contract to sell FC at E FC

At T

FC FC

fM

fM

$0. / :

$40,

: ($40, )( . / $) $100,

sec

$. / .

:

($. / ) $42,

$40, ( . ) $41,

$800

40

000 4%

000 2 5 000

6%

106000 40

106000 40 400

000 1104 600

IRPT and Cutoff Exchange Rate• Use the IRPT to determine the cutoff expected

exchange rate for determining whether to invest in domestic or foreign RF security.

if E E E Invest in foreign

if E E E Indifferent

if E E E Invest in domestic

Assume risk neutral market

T f

T f

T f

( )

( )

( )

IRPT and Cutoff Exchange Rate• Example:

R R E FC T year

E FC FC

us F

f

4%, 6%, 40 1

40104

106392452

0 $0. / ,

$0. /.

.$0. /

E E FC

convert to FC invest at for year

At T FC FC convert to FC FC

Rate

T( ) $0. /

$1 .

: . . . $: . ($0. / ) $1.

$1.

$1.

41

2 5 6%

2 5 106 2 65 2 65 41 0865

08651 8 65%

b g

IRPT and Cutoff Exchange Rate

• Use Ef from IRPT as curtoff rate:

E E ER

R

If we assume risk aversion then add RP

E E ER RP

R

TC us

T

FT

TC us

T

FT

( )( )

( )

, :

( )( )

( )

0

0

1

1

1

1

Cross Exchange Rate Relation

• Cross Rates:Given

DM or FC

FF or FF

Then

DM FF

DM FF

DMFF

DM FF

If DM FF Arbitrage

:

. $1 $. /

$1 $. /

:

. $1

.

.

.

/ .

2 5 40

4 25

2 5 4

2 5 4

2 5

4625

625

Cross Exchange Rate Relation

• Triangular Arbitrage:

Given DM FF

buy FF convert to DM DM FF FF

convert DM to DM DM

. /

$1 . (. / )( )

. $: ( . )($0. / ) $1.

7

4 2 8 7 4

2 8 2 8 40 12

Speculation

• Expect Exchange rate to decrease -- appreciation of the dollar.

R R E FC T year

E FC FC

Expect E E FC

us F

f

T

4%, 6%, 40 1

40104

106392

38

0 $0. / ,

$0. /.

.$0. /

: ( ) $. /

Speculate by going short in

forward contract or futures

at E FCf $. /392

Money Market

Borrow FC

Convert to

At T

Investment

Buy FC to pay debt FC FC

:

,

$40,

:

: $40, ( . ) $41,

: ( )($. / ) $40,

,

100 000

000

000 104 600

106000 38 280

1 320

Hedging Example

• Expecting a receipt of 625,000 DM in September.

• September DM futures is trading at fo = $0.40/DM.

• Hedging Strategy: Go short in 5 September DM futures:– nf = 625000DM/125000DM

Hedged Dollar Revenue at T

ET E DMT( )625000 f C2+C3

$.35/DM $218,750 $31,250$250,000

$.40/DM $250,000 0 $250,000

$.45/DM $281,250 -31,250$250,000 f TDM DM E (5)( , )[$. / ]125 000 40

Hedging Example• Hedging with money market:R R E E FC T yearus F f 6%, 6%, 40 10 $0. / ,

BorrowDM

DM

Convert to and invest at

At T

receipt of DM

pay debt DM

Investment

625 000

106589 622 64

849 6%.

625 000

625 000

859 106 000

,

., .

$: $235,

:

: ,

: ,

: $235, ( . ) $250,

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