1

Determination of Forward and

Futures Prices

Chapter 5

2

The participantsHEDGERS:

OPEN FUTURES POSITIONS IN ORDER TO ELIMINATE SPOT PRICE

RISK.

SPECULATORS:OPEN RISKY FUTURES POSITIONS

FOR EXPECTED PROFITS.

ARBITRAGERS:OPEN SIMULTANEOUS FUTURES AND

SPOT POSITIONS IN ORDER TO MAKE ARBITRAGE PROFITS.

3

Supply and demand for forwards and futures will determine their market prices.

BUTThe forwards and futures markets

are NOT independent of the spot market.

If spot and futures prices do not maintain A SPECIFIC relationship, dictated by economic rationale, then, arbitragers will enter these markets. Their activities will tend to force the prices to realign.

4

A static model of price formation in the futures markets.

In the following slides we analyze the

Demand and Supply of futures by:Long and Short Hedgers

Long and Short SpeculatorsAnd then,

the Arbitrageurs activities in the spot

and futures markets.

5

Demand for LONG futures positions by long HEDGERS

Long hedgers want to hedge all of their risk exposure if the settlement price is less than or equal to the expected future spot price.

c

b

a

Od0 Quantity of long positions

Long hedgers want to hedge a decreasing amount of their risk exposure as the premium of the settlement price over the expected future spot price increases.

Ft (k)

Expt [St+k]

6

Supply of SHORT futures positions by short HEDGERS.

Short hedgers want to hedge a decreasing amount of their risk exposure as the discount of the settlement price below the expected future spot price increases.

f

e

d

QS0 Quantity of short positions

Short hedgers want to hedge all of their risk exposure if the settlement price is greater than or equal to the expected future spot price.

Ft (k)

Expt [St + k]

7

Equilibrium in a futures market with a preponderance of long hedgers.

D

S

D

Qd0 Quantity of

positions

Ft (k)

Expt [St + k]

S

Ft (k)e

Supply schedule

Demand schedule

Premium

QS

8

Equilibrium in a futures market with a preponderance of short hedgers.

S

D

Qd0 Quantity of positions

Ft (k)

Expt [St + k]

S

Ft (k)e

Supply schedule

Demand scheduleDiscount

D

QS

9

Demand for long positions in futures contracts by speculators.

0 Quantity of long positions

Ft (k)

Expt [St + k]

Speculators will not demand any long positions if the settlement price exceeds the expected future spot price.

Speculators demand more long positions the greater the discount of the settlement price below the expected future spot price.

c

b

a

10

Supply of short positions in futures contracts by speculators.

0 Quantity of short positions

Ft (k)

Expt [St + k]

Speculators supply more short positions the greater the premium of the settlement price over the expected future spot price

Speculators will not supply any short positions if the settlement price is below the the expected future spot pricef

e

d

11

Equilibrium in a futures market with speculators and a preponderance of short

hedgers.

S

D

Qd QE Qs0 Quantity of positions

Ft (k)

Expt [St + k]

S

Ft (k)e

Increased supply from speculators

Discount

D

Increased demand from speculators

12

Equilibrium in a futures market with speculators and a preponderance of long

hedgers.

S

D

0 Quantity of positions

Ft (k)

Expt [St + k]

S

Ft (k)e

Increased supply from speculators

Premium

D

QE

Increased demand from speculators

13Equilibrium in the spot market

0Quantity of the asset

Ft (k); St

Ft (k)e

Premium

QE

Spot demand

Excess supply of the asset when the spot market price is St

}

Spot supply

Expt [St + k]

14Equilibrium in the futures market

0Net quantity of long positions held by hedgers and speculators

Ft (k)

Expt [St + k]

Ft (k)e

Premium

Q

}Excess demand for long positions by hedgers and speculators when the settlement price is Ft (k)e

Schedule of excess demand by hedgers and speculators

15

• Arbitrage: A market situation whereby an

investor can make a profit with: no equity and no risk.

• Efficiency: A market is said to be efficient if

prices are such that there exist no arbitrage opportunities.Alternatively, a market is said to be inefficient if prices present arbitrage opportunities for investors in this market.

16

ARBITRAGE WITH FUTURES:

Arbitragers trade in both, the futures and the spot markets simultaneously. Then, they wait until delivery time and close their positions in both markets.Note: their profit is guaranteed when open their positions.

17

ARBITRAGE IN PERFECT MARKETS

CASH -AND-CARRY

DATE SPOT MARKET FUTURES MARKET

NOW 1. BORROW CAPITAL. 3. SHORT FUTURES.

2. BUY THE ASSET IN THE

SPOT MARKET AND CARRY

IT TO DELIVERY.

DELIVERY 1. REPAY THE LOAN 3. DELIVER THE STORED

COMMODITY TO CLOSE THE SHORT FUTURES POSITION

18

ARBITRAGE IN PERFECT MARKETS

REVERSE CASH -AND-CARRY

DATE SPOT MARKET FUTURES MARKET

NOW 1. SHORT SELL ASSET 3. LONG FUTURES

2. INVEST THE PROCEEDS

IN GOV. BOND

DELIVERY: 2. REDEEM THE BOND 3. TAKE DELIVERY ASSET TO CLOSE THE LONG

FUTURES POSITION

1. CLOSE THE SPOT SHORT POSITION

19

Notation

St = Spot price today. (Or S0).

Ft,T = Futures or forward price today for delivery at T. ( or F0,T).

T = Time until delivery date.

r = Risk-free interest rate.

20

ARBITRAGE IN PERFECT MARKETS(P103)

CASH -AND-CARRY

DATE SPOT MARKET FUTURES MARKET

NOW 1. BORROW CAPITAL: S0 3. SHORT FUTURES

t=0 2. BUY THE ASSET IN F0,T

THE SPOT MARKET AND CARRY IT TO DELIVERY

DELIVERY 1. REPAY THE LOAN 3. DELIVER THE STORED

T COMMODITY TO CLOSE THE SHORT FUTURES

POSITION

S0erT F0,T

21

ARBITRAGE IN PERFECT MARKETS

REVERSE CASH -AND-CARRY

DATE SPOT MARKET FUTURES MARKET

NOW 1. SHORT SELL ASSET: S0 3. LONG FUTURES

t=0 2. INVEST THE PROCEEDS F0,T

IN GOV. BOND

DELIVERY: 2. REDEEM THE BOND 3. TAKE DELIVERY T ASSET TO CLOSE THE LONG FUTURES POSITION

1. CLOSE THE SPOT SHORT POSITION

S0erT F0,T

22

Conclusion (p.103):When an Investment Asset

Provides NO INCOME and the only carrying cost is the interest

F0,T = S0erT

23

When an Investment Asset Provides a Known Dollar Income

(p.105)

F0,T = (S0 – I )erT

where I is the present value of the income

24

When an Investment Asset Provides a Known annual Yield,

q. (P.107)

F0,T = S0e(r–q )T

where q is the average yield during the lifeof the contract (expressed with continuouscompounding)

25

Valuing a Forward Contract(Page 107)

For the sake of comparison: K = Ft,T is the forward price today ,t , for

delivery at T.

At a later date, j, F0 = Fj,T.

ƒj , is the forward value at any time j; t ≤ j ≤ T.

Date: t j T

Ft,T Fj,T

26

Valuing a Forward Contract(p.108)

Again:• Suppose that, Ft,T is forward price

today ,t , for delivery at T and Fj,T is the forward price at date j, for delivery at T.

• At j, t ≤ j ≤ T, the value of a long forward contract, ƒj[L], is

fj[L] = (Fj,T – Ft,T )e–r(T-j)

27

Valuing a Forward Contract(p.108)

• At j, t ≤ j ≤ T, the value of a short forward contract fj[SH] is

fj[SH] = (Ft,T – Fj,T )e–r(T-j)

28

Forward vs Futures Prices• Forward and futures prices are

usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different:

• A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price

• A strong negative correlation implies the reverse

29

Stock Index (P. 111)

• Can be viewed as an investment asset paying a dividend yield

• The futures price and spot price relationship is therefore

F0,T = S0e(r–q )T where q is the dividend yield on the

portfolio represented by the index

30

Stock Index (continued)

• For the formula to be true it is important that the index represent an investment asset

• In other words, changes in the index must correspond to changes in the value of a tradable portfolio

• The Nikkei index viewed as a dollar number does not represent an investment asset

31

Stock Index ArbitrageWhen F0,T > S0e(r-q)T

an arbitrageur buys the stocks underlyingthe index and sells futures.

When F0,T < S0e(r-q)T

an arbitrageur buys futures and shorts orsells the stocks underlying the index.

32

Index Arbitrage • Index arbitrage involves

simultaneous trades in futures and many different stocks

• Very often a computer is used to generate the trades

• Occasionally (e.g., on Black Monday) simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0,T and S0 does not hold

33

• A foreign currency is analogous to a security providing a dividend yield

• The continuous dividend yield is the foreign risk-free interest rate

• It follows that if rf is the foreign risk-free interest rate

Futures and Forwards on Currencies (P113)

)Tf

r(reSF 00,T

34tTT

rr

rr

)/FC(FCSS

)/FC(FCFF

FORf

DOM

21t0

21t,T0,T

The same parameters used in my slides are noted as follows:

35

THE INTEREST RATES PARITY

If financial flows are unrestricted, the SPOT and FORWARD exchange rates and the INTEREST rates in any two

countries must satisfy the Interest Rates Parity:

1.2536%rr

1.9972e1.94775

.eS(USD/GBP) = F(USD/GBP)

UKUS

)(2)r(r

t)- )(Tr - (r

UKUS

UKUS

36

In the following derivations of theTheoretical Interest Rate Parity

and the practical Interest Rate Parityin the real world we denote:

DC = The Domestic currency.FC = The Foreign currency.DOM = domestic.FOR = foreign. Q = Amount borrowed domestically.P = Amount borrowed abroad.

37

NO ARBITRAGE: CASH-AND-CARRYTIME CASH FUTURES

t (1) BORROW Q. rDOM (4) SHORT FOREIGN CURRENCY

(2) BUY FOREIGN CURRENCY FORWARD Ft,T(DC/FC)

[Q]/S(DC/FC) = [Q]S(FC/DC)] AMOUNT:

(3) INVEST IN BONDS

DENOMINATED IN THE

FOREIGN CURRENCY rFOR

T (3) REDEEM THE BONDS EARN (4) DELIVER THE CURRENCY TO

CLOSE THE SHORT POSITION

(1) PAY BACK THE LOAN RECEIVE:

IN THE ABSENCE OF ARBITRAGE:

t)-(TrFOR)e[Q]S(FC/DC

t)-(TrFOR)e[Q]S(FC/DC

t)-(TrFOR]S(FC/DC)eF(DC/FC)[Qt)-(TrDOM[Q]e

t)-(Trt)(Tr FORD ]S(FC/DC)eF(DC/FC)[Q [Q]e

t)-)(Tr - (rtt,T

FORDOM(DC/FC)eS (DC/FC)F

38

NO ARBITRAGE: REVERSE CASH – AND - CARRYTIME CASH FUTURES

t (1) BORROW [P] . rFOR (4) LONG FOREIGN CURRENCY (2) BUY DOLLARS FORWARD Ft,T(DC/FC)

[P]S(DC/FC) AMOUNT IN DOLLARS:

(3) INVEST IN T-BILLS

FOR RDOM

T REDEEM THE T-BILLS EARN TAKE DELIVERY TO CLOSE

THE LONG POSITION

PAY BACK THE LOAN RECEIVE

IN THE ABSENCE OF ARBITRAGE:

t)-(TRDOM)e[P]S(DC/FC

t)-(TrDOM)e[P]S(DC/FC

F(DC/FC)

)e[P]S(DC/FC t)-T(rDOMt)-(TrFOR[P]e

t)-(TrFOR[P]eF(DC/FC)

)e[P]S(DC/FC t)-T(rDOM

t)-T)(r(rtt,T

FORDOM(DC/FC)eS (DC/FC)F

39

t)- )(Tr - (rtt,T

FORDOM(DC/FC)eS = (DC/FC)F

FROM THE CASH-AND-CARRY STRATEGY:

(DC/FC)Ft,T

FROM THE REVERSE CASH-AND-CARRY STRATEGY:

t)-)(Tr - (rt

FORDOM(DC/FC)eS (DC/FC)Ft,T

THE ONLY WAY THE TWO INEQUALITIES HOLD SIMULTANEOUSLY IS BY BEING AN EQUALITY:

t)-)(Tr - (rt

FORDOM(DC/FC)eS

40

Example:The six-months rates in the USA and

the EC are 4% and 7%, respectively. The current spot exchange rate is:S(USD/EUR) = USD1.49/EUR.

The no arbitrage six-months forward rate is:

F(USD/EUR) = 1.49e –[.04 - .07](.5) F(USD/EUR) = USD1.5125185/EUR

If the Forward market rate is other than the above, arbitrage is possible.

41

ARBITRAGE IN THE REAL WORLD

TRANSACTION COSTS

DIFFERENT BORROWING AND LENDING RATES

MARGINS REQUIREMENTS

RESTRICTED SHORT SALES AN USE OF PROCEEDS

STORAGE LIMITATIONS

* BID - ASK SPREADS

** MARKING - TO - MARKET

* BID - THE HIGHEST PRICE ANY ONE IS WILLING TO BUY AT NOW

ASK - THE LOWEST PRICE ANY ONE IS WILLING TO SELL AT NOW.

** MARKING - TO - MARKET: YOU MAY BE FORCED TO CLOSE YOUR POSITION BEFORE ITS MATURITY.

42

FOR THE CASH - AND - CARRY:

BORROW AT THE BORROWING RATE: rB

BUY SPOT FOR: SASK

SELL FUTURES AT THE BID PRICE: F(BID).

PAY TRANSACTION COSTS ON:

BORROWING

BUYING SPOT

SELLING FUTURES

PAY CARRYING COST

PAY MARGINS

43

THE REVERSE CASH - AND - CARRY

SELL SHORT IN THE SPOT FOR: SBID.

INVEST THE FACTION OF THE PROCEEDS ALLOWED BY LAW: f; 0 ≦ f ≦ 1.

LEND MONEY (INVEST) AT THE LENDING RATE:rL

LONG FUTURES AT THE ASK PRICE: F(ASK).

PAY TRANSACTION COST ON:

SHORT SELLING SPOT

LENDING

BUYING FUTURES

PAY MARGIN

44

With these market realities, a new no-arbitrage condition emerges:

BL < FBID < FASK < BU

As long as the futures price fluctuates between the bounds there is no possibility to make arbitrage profits

BU

BL

BU

BL

time

F

45

Example 1:S0,BID (1 - c)[1 + f(rBID )] < F0, T < S0,ASK (1 + c)(1 + rASK)

c is the % of the price which is a transaction cost.Here, we assume that the futures trades for one price.In order to understand the LHS of the inequality, remember that in the USA the rule is that you may invest only a fraction, f, of the proceeds from a short sale. So, in the reverse cash and carry, the arbitrager sells the asset short at the bid price. Then (1-f)S0,BID cannot be invested. Only fS0,BID is invested. Thus, the inequality becomes:

F0,T (1-f)(1-c)S0,BID + fS0,BID(1-c)(1+rBID)

F0,T S0,BID(1-c)(1 + frBID)

46

S0,BID(1-c)[1+f(rBID )]< F0,T< S0,ASK(1+c)(1+rASK)

S0,ASK = $90.50 / bbl

S0,BID = $90.25 / bbl

rASK = 12 %

rBID = 8 % c = 3 %

$90.25(.97)[1+f(.08)]<F0,T< $90.50(1.03)(1.12)

$87.5425 + f($7.0034) < F0,T < $104.4008

47

EXAMPLE 1. $87.5425 + f($7.0034) < F0,T < $104.4008

THE CASH-AND-CARRY costs: $104.4008/bbl. THE REVERSE CASH-AND-CARRY costs:87.5425+ f($7.0034). IF f=0.5 the lower bound of the futures becomes: $91.0042. In the real market, f = 1, for some large arbitrage firms and thus, for these firms the lower bound is: $94.5459.

48

Example 2: THE INTEREST RATES PARITY

In the real markets the forward exchange rate fluctuates within a band of rates without presenting

arbitrage opportunities.Only when the market forward exchange rate diverges from this band of rates

arbitrage exists.

Given are:

Bid and Ask domestic and foreign spot rates; forward rates and interest rates.

49

NO ARBITRAGE: CASH - AND - CARRYTIME CASH FUTURES

t (1) BORROW [Q]. rD,ASK (4) SHORT FOREIGN CURRENCY FORWARD

(2) BUY FOREIGN CURRENCY

[Q]/SASK(DC/FC) FBID (DC/FC)

(3) INVEST IN BONDS

DENOMINATED IN THE

FOREIGN CURRENCY rF,BID

T REDEEM THE BONDS DELIVER THE CURRENCY TO CLOSE THE SHORT POSITION

EARN:

PAY BACK THE LOAN RECEIVE:

IN THE ABSENCE OF ARBITRAGE:

t)-(TrASK

BIDF,(DC/FC)}e{[Q]/S

t)-(TrASKBID

BIDF,(DC/FC)e/S(DC/FC)[Q]Ft)-(Tr ASKD,[Q]e

t)-(TrASKBID

t)(Tr BIDF,ASKD, (DC/FC)e/S(DC/FC)[Q]F [Q]e

t)-)(Tr - (rASKBID

BIDF,ASKD,(DC/FC)eS (DC/FC)F

t)-(TrASK

BIDF,(DC/FC)e[Q]/S

50

NO ARBITRAGE:

REVERSE CASH - AND - CARRYTIME CASH FUTURES

t (1) BORROW [P] . rF,ASK (4) LONG FOREIGN CURRENCY FORWARD FOR FASK(DC/FC)

(2) EXCHANGE FOR

[P]SBID (DC/FC)

(3) INVEST IN T-BILLS

FOR rD,BID

T REDEEM THE T-BILLS EARN TAKE DELIVERY TO CLOSE THE LONG POSITION

RECEIVE in foreign currency, the amount:

PAY BACK THE LOAN

IN THE ABSENCE OF ARBITRAGE:

t)-(TrBID

BIDD,(DC/FC)e[P]S

(DC/FC)F

(DC/FC)e[P]S

ASK

t)-T(rBID

BIDD,

t)-(Tr ASKF,[P]e

t)-T)(r(rBIDASK

ASKF,BIDD,(DC/FC)eS (DC/FC)F

t)-(TrBID

BIDD,(DC/FC)e[P]S

t)-(Tr ASKF,[P]e (DC/FC)F

(DC/FC)e[P]S

ASK

t)-T(rBID

BIDD,

51

t)-T)(r(rBIDASK

ASKF,BIDD,(DC/FC)eS (DC/FC)F (2)

t)-)(Tr - (rASK

BIDF,ASKD,(DC/FC)eS

From Cash and Carry:

(DC/FC)F (1) BID

From reverse cash and Carry

Notice that:

RHS(1) > RHS(2)

Define: RHS(1) BU RHS(2) BL

(3) And FASK(DC/FC) > FBID(DC/FC)

52

BU

BL

FASKFASK(DC/FC) > FBID(DC/FC).

Arbitrage exists only if both ask and bid futures prices are above BU,

or both are below BL.

FBID

LASK B(DC/FC)F

UBID B(DC/FC)F

F($/D)

BU

BL

53

A numerical example:

Given the following exchange rates:

Spot Forward Interest rates

S(USD/NZ) F(USD/NZ) r(NZ) r(US)

ASK 0.4438 0.4480 6.000% 10.8125%

BID 0.4428 0.4450 5.875% 10.6875%

Clearly, F(ask) > F(bid). (USD0.4480NZ > USD0.4450/NZ)

We will now check whether or not there exists an opportunity for arbitrage profits. This will require comparing these

forward exchange rates to: BU and BL

54

t)-T)(r(rBIDASK

ASKNZ,BIDUS,(USD/NZ)eS (USD/NZ)F

t)-)(Tr - (rASK

BIDNZ,ASKUS,(USD/NZ)eS Inequality (1):

(USD/NZ)FBID

0.4450 < (0.4438)e(0.108125 – 0.05875)/12 = 0.4456 = BU

0.4480 > (0.4428)e(0.106875 – 0.06000)/12 = 0.4445 = BL

No arbitrage.

Lets see the graph

Inequality (2):

55

BU

BL

Clearly: FASK($/FC) > FBID($/FC).

An example of arbitrage:

FASK = 0.4480

FBID = 0.4465

FBID = 0.4450

LASK B4445.0 (USD/NZ)F

UBID B0.4456(USD/NZ)F

FFASK = 0.4480

0.4445

0.4456