0 determination of forward and futures prices chapter 3

1

Determination of Forward and Futures

Prices

Chapter 3

2

• 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.

3

SHORT SELLING STOCKSAn Investor may call a broker and ask to “sell a

particular stock short.”This means that the investor does not own shares

of the stock, but wishes to sell it anyway. The investor speculates that the stock’s share

price will fall and money will be made upon buying the shares back at a lower price. Alas, the investor does not own shares of the stock. The broker will lend the investor shares from the broker’s or a client’s account and sell it in the investor’s name. The investor’s obligation is to hand over the shares some time in the future, or upon the broker’s request.

4

SHORT SELLING STOCKS

Other conditions:The proceeds from the short sale cannot be

used by the short seller. Instead, they are deposited in an escrow account in the investor’s name until the investor makes good on the promise to bring the shares back.

Moreover, the investor must deposit an additional amount of at least 50% of the short sale’s proceeds in the escrow account.

This additional amount guarantees that there is enough capital to buy back the borrowed shares and hand them over back to the broker, in case the shares price increases.

5

SHORT SELLING STOCKSThere are more details associated with

short selling stocks. For example, if the stock pays dividend, the short seller must pay the dividend to the lender. Moreover, the short seller does not gain interest on the amount deposited in the escrow account, etc. We will use stock short sales in many of strategies associated with derivatives.

In terms of cash flows: St is the cash flow from selling

the stock short on date t.-ST is the cash flow from

purchasing the back on date T.

6

• Risk-Free Asset: is a security of investment whose return carries no risk. Thus, the return on this security is known and guaranteed in advance.

• Risk-Free Borrowing And LandingRisk-Free Borrowing And Landing: By purchasing the risk-free asset, investors lend their capital and earn the risk-free rate.By selling the risk-free asset, investors borrow capital at the risk-free rate.

7

• The One-Price Law:There exists only one risk-free rate in an efficient economy.

Continuous Compounding and Discounting:Calculating the future value of a series of cash flows or, the present value of the cash flows, respectively, in a continuous time framework.

8

Compounded InterestAny principal amount, P, invested at an

annual interest rate, r, compounded annually, for T years would grow to AT = P(1 + r)T.

If compounded Quarterly:

AT = P(1 +r/4)4T.

In general, with m compounding periods every year, the periodic rate becomes r/m and mT is the number of compounding periods. Thus, P grows to:

AT = P(1 +r/m)mT.

9

Monthly compounding becomes: AT = P(1 +r/12)12T

and daily compounding yields:AT = P(1 +r/365)365T

Eample: T =10 years; r =12%; P = $100.1. Simple annual compounding yields:

A10 = $100(1+ .12)10 = $310.582. Monthly compounding yields:

A10 = $100(1 + .12/12)120  = $330.033. Daily compounding yields:A10 = $100(1 + .12/365)3,650 = $331.94.

10

In the early 1970s, banks came up with the following economic reasoning: Since the bank has depositors’ money all the time, this money should be working for the depositor all the time!

This idea, of course, leads to the concept of continuous compounding.

mT

T mr1PA

Observe that continuous time means that the number of compounding periods every year, m, increases without limit. This implies that the length of every compounding time period goes to zero and thus, the periodic interest rate, r/m, becomes smaller and smaller.

11

This reasoning implies that we need to solve:

}mr1{PLimitA

mT

mT

.PeA

:years Tafter P of valuecompoundedly continuous for the expression the

yieldslimit thisofsolution The

}.r

m11{(P)LimitA

rTT

rT)rm(

mT

12

EXAMPLE, continued: First, recall that:

}x11{Limite

x

x

example: x e1 210 2.593742461,000 2.7169239310,000 2.71814592In the limit e = 2.718281828…

13

EXAMPLE, continued: Recall that inour example: T= 10 years and r = 12% and

P=$100. Thus, P=$100 invested at an annual rate of 12%. will grow to by the factor:

Compounding FactorSimple 3.105848208Quarterly 3.262037792Monthly

3.300386895Daily 3.319462164Continuously 3.320116923

14 rate.interest compoundedly continuous theisr where

,e

by it gmultiplyinby present for thediscountedly continuous becan

,CF flow,cash t period any timeFor .eAP

:is A of valuediscountedlycontinuous theT, andr ,AGiven

rt -

t

rT -T

T

T

15

Continuous Compounding

(Page 43)• In the limit as we compound more and more frequently we obtain continuously compounded interest rates.

• $100 grows to $100eRT when invested at a continuously compounded rate R for time T.

• $100 received at time T discounts to $100e-RT at time zero when the continuously compounded discount rate is R.

16

Conversion Formulas (Page 44)DefineRc : continuously compounded

rateRm: same rate with compounding

m times per year

1emR

mR1mlnR

/mRm

mc

c

17

FUTURES and SPOT PRICES:AN ECONOMICS MODEL of

DEMAND and SUPPLYSPECULATORS: WILL OPEN RISKY FUTURES

POSITIONS FOR EXPECTED PROFITS.

HEDGERS: WILL OPEN FUTURES POSITIONS IN ORDER TO ELIMINATE ALL PRICE RISK.

ARBITRAGERS: WILL OPEN SIMULTANEOUS FUTURES AND CASH POSITIONS IN ORDER TO MAKE ARBITRAGE PROFITS.

18

HEDGERS:HEDGERS TAKE FUTURES POSITIONS IN ORDER

TO ELIMINATE PRICE RISK.

THERE ARE TWO TYPES OF HEDEGESA LONG HEDGE

TAKE A LONG FUTURES POSITION IN ORDER TO LOCK IN THE PRICE OF AN ANTICIPATED

PURCHASE AT A FUTURE TIME

A SHORT HEDGE

TAKE A SHORT FUTURES POSITION IN ORDER TO LOCK IN THE SELLING PRICE OF AN

ANTICIPATED SALE AT A FUTURE TIME.

19

ARBITRAGE WITH FUTURES:

SPOT MARKET FUTURES MARKET

Contract to buy the product LONG futures

Contract to sell the product SHORT futures

20

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]

21

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]

22

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

23

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

24

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

25

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

26

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

27

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

28Equilibrium 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]

29Equilibrium in the futures market

0Net quantity of long positions held by hedgers and speculators

Ft (k)

Expt [St + k]

Ft (k)ePremium

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

30

ARBITRAGE IN PERFECT MARKETS

CASH -AND-CARRY

DATE SPOT MARKET FUTURES MARKETNOW 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

31

ARBITRAGE IN PERFECT MARKETS REVERSE CASH -AND-CARRY

DATE SPOT MARKET FUTURES MARKETNOW 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

32

Notation

S0: Spot price today. (Or St)F0,T: Futures or forward price today

for delivery at T. ( or Ft,T)T: Time until delivery dater: Risk-free interest rate for

delivery date.

33

Gold Example (From Chapter 1)

• For gold F0 = S0(1 + r )T (assuming no storage costs)• If r is compounded continuously

instead of annually F0 = S0erT

PROOF:

34

ARBITRAGE IN PERFECT MARKETS CASH -AND-CARRY

DATE SPOT MARKET FUTURES MARKETNOW 1. BORROW CAPITAL: S0 3. SHORT FUTURES 2. BUY THE ASSET IN F0,T

THE SPOT MARKET AND CARRY IT TO DELIVERY

DELIVERY 1. REPAY THE LOAN 3. DELIVER THE STORED COMMODITY TO CLOSE THE SHORT FUTURES POSITION

S0erT F0,T

35

ARBITRAGE IN PERFECT MARKETS REVERSE CASH -AND-CARRY

DATE SPOT MARKET FUTURES MARKET

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

2. INVEST THE PROCEEDS F0,T

IN GOV. BOND

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

THE LONG FUTURES POSITION

1. CLOSE THE SPOT SHORT POSITION

S0erT F0,T

36

Extension of the Gold Example(Page 46, equation 3.5)

• For any investment asset that provides no income and has no storage costs

F0 = S0erT

37

When an Investment Asset Provides a Known Dollar

Income (page 48, equation 3.6)

F0 = (S0 – I )erT

where I is the present value of the income

38

When an Investment Asset Provides a Known Yield (Page 49, equation 3.7)

F0 = S0e(r–q )T

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

39

Valuing a Forward ContractPage 50

• Suppose that K is delivery price in a forward

contract, F0,T is forward price today for delivery at T

• The value of a long forward contract, ƒ, is

ƒ = (F0,T – K )e–rT

• Similarly, the value of a short forward contract is

(K – F0,T )e–rT

40

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

41

Stock Index (Page 52)

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

• The futures price and spot price relationship is therefore

F0 = S0e(r–q )T

where q is the dividend yield on the

portfolio represented by the index

42

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

43

Index Arbitrage• When F0>S0e(r-q)T , an arbitrageur buys

the stocks underlying the index and sells futures.

• When F0<S0e(r-q)T , an arbitrageur buys futures and shorts or sells the stocks underlying the index.

44

Index Arbitrage (continued)

• 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

45

• 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 (Page 55-58)

)Tfr(reSF 0T0,

46

THE INTEREST RATES PARITYWherever financial flows are unrestricted, exchange rates,

the forward rates and the interest rates in any two countries must maintain a NO- ARBITRAGE relationship:

Interest Rates Parity.

./FC)eS(FC = /FC)F(FC t)- )(Tr - (rDOMDOM

FORDOM

47

NO ARBITRAGE: CASH-AND-CARRYTIME CASH FUTURES

t (1) BORROW $A. rDOM (4) SHORT FOREIGN CURRENCY

(2) BUY FOREIGN CURRENCY FORWARD Ft,T($/FC) A/S($/FC) [=AS(FC/$)] AMOUNT:

(3) INVEST IN BONDS

DENOMINATED IN THE

FOREIGN CURRENCY rFOR

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

EARN CLOSE THE SHORT POSITION

(1) PAY BACK THE LOAN RECEIVE:

IN THE ABSENCE OF ARBITRAGE:

t)-(TrFORAS(FC/$)e

t)-(TrFORAS(FC/$)et)-(TrFORFC/$)eF($/FC)AS(t)-(TrDOMAe

t)-(Trt)(Tr FORD S(FC/$)e F($/FC)AAe

t)-)(Tr - (rtTt,

FORDOM($/FC)eS ($/FC)F

48

NO ARBITRAGE:

REVERSE CASH – AND - CARRYTIME CASH FUTURES

t (1) BORROW FC A. rFOR (4) LONG FOREIGN CURRENCY (2) BUY DOLLARS FORWARD Ft,T($/FC)

AS($/FC) AMOUNT IN DOLLARS:

(3) INVEST IN T-BILLS

FOR RDOM

T REDEEM THE T-BILLS TAKE DELIVERY TO CLOSE

EARN THE LONG POSITION

PAY BACK THE LOAN RECEIVE

IN THE ABSENCE OF ARBITRAGE:

t)-(TR DOMAS($/FC)e

t)-(TrDOMAS($/FC)e

F($/FC)AS($/FC)e t)-T(rDOM

t)-(TrFORAet)-(TrFORAe F($/FC)

AS($/FC)e t)-T(rDOM

t)-T)(r(rtTt,

FORDOM($/FC)eS ($/FC)F

49

t)- )(Tr - (rtTt,

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

FROM THE CASH-AND-CARRY STRATEGY:

($/FC)F Tt,

FROM THE REVERSE CASH-AND-CARRY STRATEGY: t)-)(Tr - (r

tFORDOM($/FC)eS ($/FC)F Tt,

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

t)-)(Tr - (rt

FORDOM($/FC)eS

50

ON MAY 25 AN ARBITRAGER OBSERVES THE FOLLOWING MARKET PRICES:

S(USD/GBP) = 1.5640 <=> S(GBP/USD) = .6393

F(USD/GBP) = 1.5328 <=> F(GBP/USD) = .6524

RUS = 7.85% ; RGB = 12%

CASH AND CARRY

TIME CASH FUTURES

MAY 25 (1) BORROW USD100M AT 7. 85% SHORT GBP 68,477,215 FORWARD

FOR 209 DAYS FOR DEC. 20, FOR USD1.5328/GBP

(2) BUY GBP63,930,000

(3) INVEST THE GBP63,930,000

IN BRITISH BONDS

DEC 20 RECEIVE GBP68,477,215 DELIVER GBP68,477,215

FOR USD104,961,875.2

REPAY YOUR LOAN:

ARBITRAGE PROFIT: USD104,961,875.2 - USD104,597,484.3 = USD364,390.90

1.5273 = 1.5640e = F 365209.12) - (.0785

lTheoretica

7,484.3 USD104,59= 100Me 365209.0785

215GBP68,477, = e63,930,000 365209.12

51

Futures on Consumption Assets (Page 59)

F0 S0 e(r+u )T

where u is the storage cost per unit time as a percent of the asset value.

Alternatively,

F0 (S0+U )erT

where U is the present value of the storage costs.

52

The Cost of Carry (Page 60)

• The cost of carry, c, is the storage cost plus the interest costs less the income earned.

• For an investment asset F0 = S0ecT • For a consumption asset F0 = S0ecT

• The convenience yield on the consumption asset, y, is defined so that F0 = S0 e(c–y )T

53

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.

54

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:BORROWINGBUYING SPOTSELLING FUTURESPAY CARRYING COSTPAY MARGINS

55

THE REVERSE CASH - AND - CARRYSELL 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 LENDINGBUYING FUTURESPAY MARGIN

56

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

BL < F < 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

57

Example 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 the rule in the USA 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 while fS0,BID(1+rBID) is invested. Thus, the inequality becomes:

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

F0,T S0(1 + frBID)

58

EXAMPLE 1.

S0,BID (1 - T)[1 + f(rL )] < F0, t < S0,ASK (1 + T)(1 + rB)

S0,ASK = $20.50 / bbl S0,BID = $20.25 / bbl rASK = 12 % rBID = 8 % c = 3 %

$20.25(.97)[1+f(.08)]<F0,t< $20.50(1.03)(1.12)

$19.6425 + f($1.57) < F0,t < $23.6488

DEPENDING ON f, ANY FUTURES PRICE BETWEEN THE TWO LIMITS WILL LEAVE NO ARBITRAGE OPPORTUNITIES. THE CASH-AND-CARRY WILL COST $23.6488/bbl. THE REVERSE CASH-AND-CARRY WILL COST 19.6425 + f(1.62). IF f=0.5 THE LOWER BOUND IS $20.45. IN THE REAL MARKET, f = 1, FOR SOME LARGE ARBITAGE FIRMS AND THEIR LOWER BOUND IS $21.26. THUS, IT IS CLEAR THAT THERE ARE DIFFERENT ARBITRAGE BOUNDS APPLICABLE TO DIFFERENT INVESTORS. THE TIGHTER THE BOUNDS, THE GREATER ARE THE ARBITRAGE OPPORTUNITIES.

59

Example 2.: THE INTEREST RATES PARITY

In the real markets, buyers pay the ask price while sellers receive the bid price. Moreover, borrowers pay the ask interest rate while lenders only

receive the bid interest rate. Therefore, in the real markets, it is possible for the forward exchange rate to fluctuate within a band of

rates without presenting arbitrage opportunities.Only when the market forward exchange rate diverges from this band of rates arbitrage exists.

60

NO ARBITRAGE: CASH - AND - CARRYTIME CASH FUTURES

t (1) BORROW $A. rD,ASK (4) SHORT FOREIGN CURRENCY FORWARD

(2) BUY FOREIGN CURRENCY

A/SASK($/FC) FBID ($/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,($/FC)eA/S

t)-(TrBID

FOR$/FC)e($/FC)A/S(Ft)-(Tr ASKD,Ae

t)-(TrASKBID

t)(Tr BIDF,ASKD, ($/FC)e($/FC)A/S FAe

t)-)(Tr - (rASKBID

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

t)-(TrASK

BIDF,($/FC)eA/S

61

NO ARBITRAGE:

REVERSE CASH - AND - CARRYTIME CASH FUTURESt (1) BORROW FCA . rF,ASK (4) LONG FOREIGN CURRENCY FORWARD FOR FASK(USD/FC)

(2) EXCHANGE FOR ASBID (USD/FC)

(3) INVEST IN T-BILLS FOR rD,BID

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

EARN RECEIVE in foreign currency, the amount:

PAY BACK THE LOAN

IN THE ABSENCE OF ARBITRAGE:

t)-(TrBID

BIDD,($/FC)eAS

($/FC)F($/FC)eAS

ASK

t)-T(rBID

BIDD,

t)-(Tr ASKF,Ae

t)-T)(r(rBIDASK

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

t)-(TrBID

BIDD,($/FC)eAS

t)-(Tr ASKF,Ae ($/FC)F($/FC)eAS

ASK

t)-T(rBID

BIDD,

62

t)-T)(r(rBIDASK

ASKF,BIDD,($/D)eS ($/D)F (2)

t)-)(Tr - (rASK

BIDF,ASKD,($/D)eS From Cash and Carry:

($/D)F (1) BID

From reverse cash and Carry

Notice that

The RHS(1) > RHS(2)

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

(3) And FASK($/D) > FBID($/D) Always!

63

BU

BL

FASKFASK($/D) > FBID($/D).

CONCLUSION:

Arbitrage exists only if both ask and bid futures prices are above BU, or both are below BL.

FBID

t)-T)(r(rBIDASK

ASKF,BIDD,($/D)eS ($/D)F

t)-)(Tr - (rASK

BIDF,ASKD,($/D)eS ($/D)FBID

F($/D)

BU

BL

64

A numerical example:

Given the following exchange rates:

Spot Forward Interest ratesS(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

65

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):

66

BU

BL

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

An example of arbitrage: FBID = 0.4465

FASK = 0.4480

FBID = 0.4450

4445.0 (USD/NZ)FASK

0.4456(USD/NZ)FBID

F

BU

FASK = 0.4480

BL

0.4445

0.4456