0 determination of forward and futures prices chapter 3
DESCRIPTION
2 SHORT SELLING STOCKS An 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.TRANSCRIPT
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