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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.
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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)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
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• 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.
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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.
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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
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ARBITRAGE IN PERFECT MARKETS
REVERSE CASH -AND-CARRY
DATE SPOT MARKETFUTURES 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
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NotationSt = 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 MARKETNOW 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 STOREDT COMMODITY TO CLOSE THE SHORT FUTURES POSITION
S0erT F0,T
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ARBITRAGE IN PERFECT MARKETS REVERSE CASH -AND-CARRY
DATE SPOT MARKETFUTURES MARKETNOW 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)
)Tfr(reSF 00,T
34tTT
rrrr
)/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%rr1.9972e1.94775
.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/DCt)-(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 - (r
tFORDOM(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:BORROWINGBUYING SPOTSELLING FUTURESPAY CARRYING COSTPAY MARGINS
43
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
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 < $104.4008THE 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: $94.5459.
48
Example 2: THE INTEREST RATES PARITY
In the real markets the forward exchange rate fluctuates within a band of rates without presenting
arbitrage opportunities.Only when the market forward exchange rate diverges from this band of rates
arbitrage exists.Given are:
Bid and Ask domestic and foreign spot rates; forward rates and interest rates.
49
NO ARBITRAGE: CASH - AND - CARRYTIME CASH FUTURES
t (1) BORROW [Q]. rD,ASK (4) SHORT FOREIGN CURRENCY FORWARD
(2) BUY FOREIGN CURRENCY
[Q]/SASK(DC/FC) FBID (DC/FC)
(3) INVEST IN BONDS
DENOMINATED IN THE
FOREIGN CURRENCY rF,BID
T REDEEM THE BONDS DELIVER THE CURRENCY TO CLOSE THE SHORT POSITION
EARN:
PAY BACK THE LOAN RECEIVE:
IN THE ABSENCE OF ARBITRAGE:
t)-(TrASK
BIDF,(DC/FC)}e{[Q]/S
t)-(TrASKBID
BIDF,(DC/FC)e/S(DC/FC)[Q]Ft)-(Tr ASKD,[Q]e
t)-(TrASKBID
t)(Tr BIDF,ASKD, (DC/FC)e/S(DC/FC)[Q]F [Q]e
t)-)(Tr - (rASKBID
BIDF,ASKD,(DC/FC)eS (DC/FC)F
t)-(TrASK
BIDF,(DC/FC)e[Q]/S
50
NO ARBITRAGE:
REVERSE CASH - AND - CARRYTIME CASH FUTURESt (1) BORROW [P] . rF,ASK (4) LONG FOREIGN CURRENCY FORWARD FOR FASK(DC/FC)
(2) EXCHANGE FOR [P]SBID (DC/FC)
(3) INVEST IN T-BILLS FOR rD,BID
T REDEEM THE T-BILLS EARN TAKE DELIVERY TO CLOSE THE LONG POSITION
RECEIVE in foreign currency, the amount:
PAY BACK THE LOAN
IN THE ABSENCE OF ARBITRAGE:
t)-(TrBID
BIDD,(DC/FC)e[P]S
(DC/FC)F(DC/FC)e[P]S
ASK
t)-T(rBID
BIDD,
t)-(Tr ASKF,[P]e
t)-T)(r(rBIDASK
ASKF,BIDD,(DC/FC)eS (DC/FC)F
t)-(TrBID
BIDD,(DC/FC)e[P]S
t)-(Tr ASKF,[P]e (DC/FC)F(DC/FC)e[P]S
ASK
t)-T(rBID
BIDD,
51
t)-T)(r(rBIDASK
ASKF,BIDD,(DC/FC)eS (DC/FC)F (2)
t)-)(Tr - (rASK
BIDF,ASKD,(DC/FC)eS
From Cash and Carry:
(DC/FC)F (1) BID
From reverse cash and Carry
Notice that: RHS(1) > RHS(2) Define: RHS(1) BU RHS(2) BL
(3) And FASK(DC/FC) > FBID(DC/FC)
52
BU
BL
FASKFASK(DC/FC) > FBID(DC/FC).
Arbitrage exists only if both ask and bid futures prices are above BU,
or both are below BL.
FBID
LASK B(DC/FC)F
UBID B(DC/FC)F
F($/D)
BU
BL
53
A numerical example:
Given the following exchange rates:
Spot Forward Interest 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
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
