Determination of
Forward and
Futures Prices
Chapter 5
0
The participants
HEDGERS:
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.
1
Supply and demand for forwards and
futures will determine their market
prices.
BUT
The 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.
2
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 Speculators
And then,
the Arbitrageurs activities in the spot
and futures markets.
3
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)
a
b
Expt [St+k]
c
0
Od
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.
Quantity of long
positions
Demand for LONG futures positions by long
HEDGERS
4
Ft (k)
d
e
Expt [St + k]
f
0
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.
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.
QS
Quantity of short
positions
Supply of SHORT futures positions by
short HEDGERS.
5
Ft (k)
S
Supply
schedule
D
Ft (k)e
Premium
Expt [St + k]
Demand
schedule
S
D
0
QS
Qd
Quantity of
positions
Equilibrium in a futures market with a
preponderance of long hedgers.
6
Ft (k)
S
D
Supply schedule
Expt [St + k]
Discount
Ft (k)e
Demand schedule
S
0
D
Qd
QS
Quantity of
positions
Equilibrium in a futures market with a
preponderance of short hedgers.
7
Ft (k)
Speculators will not demand any long
positions if the settlement price
exceeds the expected future spot price.
a
Expt [St + k]
b
Speculators demand more long
positions the greater the
discount of the settlement price
below the expected future spot
price.
c
0
Quantity of long
positions
Demand for long positions in futures
contracts by speculators.
8
Ft (k)
d
Expt [St + k]
Speculators supply more short
positions
the
greater the
premium of the settlement price
over the expected future spot
price
e
f
0
Speculators will not supply any short
positions if the settlement price is
below the the expected future spot
price
Quantity of short
positions
Supply of short positions in futures contracts
by speculators.
9
Ft (k)
S
D
Expt [St + k]
Ft (k)e
Increased
supply from
speculators
Discount
Increased demand
from speculators
S
0
D
Qd QE Qs
Quantity of
positions
Equilibrium in a futures market with
speculators and a preponderance of short
hedgers.
10
Ft (k)
S
Increased
supply from
speculators
D
Ft (k)e
Increased demand
from speculators
Premium
Expt [St + k]
S
D
0
QE
Quantity of
positions
Equilibrium in a futures market with
speculators and a preponderance of long
hedgers.
11
Excess supply of the asset
when the spot market price is
St
Spot supply
}
Ft (k); St
Ft (k)e
Premium
Expt [St + k]
Spot demand
0
QE
Quantity of the asset
Equilibrium in the spot market
12
Ft (k)
Expt [St + k]
Schedule of excess demand by
hedgers and speculators
Premium
}
Ft (k)e
Excess demand for long positions by
hedgers and speculators when the
settlement price is Ft (k)e
0
Q
Net quantity of long positions held by
hedgers and speculators
Equilibrium in the futures market
13
• 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.
14
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.
15
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
16
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
17
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.
18
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
19
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:
T
2. REDEEM THE BOND
3. TAKE DELIVERY
ASSET TO CLOSE
THE LONG
FUTURES
POSITION
1. CLOSE THE SPOT SHORT POSITION
S0erT

F0,T
20
Conclusion (p.103):
When an Investment Asset
Provides NO INCOME and the
only carrying cost is the interest
F0,T = S0
rT
e
21
When an Investment Asset
Provides a Known Dollar Income
(p.105)
F0,T = (S0 – I
rT
)e
where I is the present value of
the income
22
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 life
of the contract (expressed with continuous
compounding)
23
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:
Ft,T
Fj,T
t
j
T
24
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)
25
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)
26
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
27
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
28
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
29
Stock Index Arbitrage
When
F0,T > S0e(r-q)T
an arbitrageur buys the stocks underlying
the index and sells futures.
When
F0,T < S0e(r-q)T
an arbitrageur buys futures and shorts or
sells the stocks underlying the index.
30
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
31
Futures and Forwards on
Currencies (P113)
• 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 riskfree interest rate
(r  r )T
f
F0,T  S0e
32
The same parameters used in my slides
are noted as follows:
F0,T  Ft,T (FC1/FC 2 )
S0  St (FC1/FC 2 )
r  rDOM
rf  rFOR
T Tt
33
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:
F(USD/GBP) = S(USD/GBP) e
1.94775  1.9972e
(rUS - rUK )(T - t)
(rUS rUK )(2)
rUS  rUK  1.2536%
34
.
In the following derivations of the
Theoretical Interest Rate Parity
and the practical Interest Rate Parity
in 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.
35
NO ARBITRAGE: CASH-AND-CARRY
TIME
CASH
FUTURES
t
(1) BORROW Q. rDOM
(4) SHORT FOREIGN CURRENCY
(2) BUY FOREIGN CURRENCY
FORWARD
[Q]/S(DC/FC) = [Q]S(FC/DC)]
(3) INVEST IN BONDS
DENOMINATED IN THE
Ft,T(DC/FC)
AMOUNT:
[Q]S(FC/DC )e
rFOR (T-t)
FOREIGN CURRENCY rFOR
T
(3) REDEEM THE BONDS EARN (4) DELIVER THE CURRENCY TO
rFOR (T-t)
CLOSE THE SHORT POSITION
[Q]S(FC/DC )e
(1) PAY BACK THE LOAN
[Q]e
[Q]e
rDOM (T-t)
RECEIVE:
F(DC/FC)[Q]S(FC/DC)e
IN THE ABSENCE OF ARBITRAGE:
rD (T t)
 F(DC/FC)[Q]S(FC/DC)e
Ft,T (DC/FC)  St (DC/FC)e
rFOR (T-t)
rFOR (T-t)
(rDOM - rFOR )(T-t)
36
NO ARBITRAGE: REVERSE CASH – AND - CARRY
TIME
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:
[P]S(DC/FC )e
(3) INVEST IN T-BILLS
FOR RDOM
T
REDEEM THE T-BILLS EARN
rDOM (T-t)
TAKE DELIVERY TO CLOSE
[P]S(DC/FC )e
THE LONG POSITION
PAY BACK THE LOAN
RECEIVE
[P]e
rFOR (T-t)
IN THE ABSENCE OF ARBITRAGE:
[P]e
rFOR (T-t)
R DOM (T-t)
rDOM ( T-t)
[P]S(DC/FC )e
F(DC/FC)
rDOM ( T-t)
[P]S(DC/FC )e

F(DC/FC)
Ft,T (DC/FC)  St (DC/FC)e
(rDOM rFOR )( T-t)
37
FROM THE CASH-AND-CARRY STRATEGY:
Ft,T (DC/FC) St (DC/FC)e (rDOM - rFOR )(T-t)
FROM THE REVERSE CASH-AND-CARRY STRATEGY:
(rDOM - rFOR )(T -t)
t
t,T
F (DC/FC)  S (DC/FC)e
THE ONLY WAY THE TWO INEQUALITIES HOLD
SIMULTANEOUSLY IS BY BEING AN EQUALITY:
Ft,T (DC/FC) = St (DC/FC)e
(rDOM - rFOR )(T - t)
38
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:
–[.04 - .07](.5)
F(USD/EUR) = 1.49e
F(USD/EUR) = USD1.5125185/EUR
If the Forward market rate is other than
the above, arbitrage is possible.
39
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.
40
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
41
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:
LONG FUTURES AT THE ASK PRICE:
F(ASK).
PAY TRANSACTION COST ON:
SHORT SELLING SPOT
LENDING
BUYING FUTURES
PAY MARGIN
42
rL
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
F
BL
time
43
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)
44
S0,BID(1-c)[1+f(rBID )]< F0,T< S0,ASK(1+c)(1+rASK)
S0,ASK
S0,BID
rASK
rBID
c
= $90.50 / bbl
= $90.25 / bbl
= 12 %
= 8%
= 3%
$90.25(.97)[1+f(.08)]<F0,T< $90.50(1.03)(1.12)
$87.5425 + f($7.0034) < F0,T < $104.4008
45
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: $94.5459.
46
Example 2: THE INTEREST RATES PARITY
In the real markets the forward exchange
rate fluctuates within a band of rates
without presenting arbitrage
opportunities.Only when the market
forward exchange rate diverges from this
band of rates arbitrage exists.
Given are:
Bid and Ask domestic and foreign spot
rates; forward rates and interest rates.
47
NO ARBITRAGE: CASH - AND - CARRY
TIME
CASH
FUTURES
t
(1) BORROW [Q]. rD,ASK
(4) SHORT FOREIGN CURRENCY FORWARD
(2) BUY FOREIGN CURRENCY
[Q]/SASK(DC/FC)
(3) INVEST IN BONDS
DENOMINATED IN THE
FOREIGN CURRENCY rF,BID
T
REDEEM THE BONDS
FBID (DC/FC)
[Q]/SASK (DC/FC)e
DELIVER THE CURRENCY TO CLOSE THE SHORT POSITION
EARN:
r
{[Q]/S ASK (DC/FC)}e F,BID
PAY BACK THE LOAN
[Q]e
rF,BID (T-t)
(T-t)
RECEIVE:
rD,ASK (T -t)
r
FBID (DC/FC)[Q]/SASK (DC/FC)e F,BID
(T-t)
IN THE ABSENCE OF ARBITRAGE:
[Q]e
rD,ASK (Tt)
 FBID (DC/FC)[Q]/SASK (DC/FC)e
FBID (DC/FC)  SASK (DC/FC)e
rF,BID (T-t)
(rD,ASK - rF,BID )(T-t)
48
NO ARBITRAGE:
REVERSE CASH - AND - CARRY
TIME
CASH
FUTURES
t
(1) BORROW [P] .
rF,ASK
(4) LONG FOREIGN CURRENCY FORWARD FOR
FASK(DC/FC)
(2) EXCHANGE FOR
[P]S BID (DC/FC)e
[P]SBID (DC/FC)
(3) INVEST IN T-BILLS
rD,BID (T -t)
FOR rD,BID
T
REDEEM THE T-BILLS EARN
[P]S BID (DC/FC)e
PAY BACK THE LOAN
TAKE DELIVERY TO CLOSE THE LONG POSITION
rD,BID (T -t) RECEIVE in foreign currency, the amount:
[P]e
r
rF,ASK (T -t)
[P]S BID (DC/FC)e D,BID
FASK (DC/FC)
IN THE ABSENCE OF ARBITRAGE:
r
( T - t)
( T -t)
[P]S BID (DC/FC)e D,BID
rF,ASK (T - t)

[P]e
FASK (DC/FC)
(rD,BID rF,ASK )( T-t)49
BID
FASK (DC/FC)  S
(DC/FC)e
From Cash and Carry:
(1)
FBID (DC/FC)  SASK (DC/FC)e
(rD,ASK - rF,BID )(T-t)
From reverse cash and Carry
(2)
FASK (DC/FC)  SBID (DC/FC)e
(rD,BID rF,ASK )( T-t)
(3) And FASK(DC/FC) > FBID(DC/FC)
Notice that:
RHS(1) > RHS(2)
Define: RHS(1)  BU
RHS(2)  BL
50
F($/D)
FASK(DC/FC) > FBID(DC/FC).
FASK
BU
BU
FBID (DC/FC)  BU
FASK (DC/FC)  BL
BL
BL
FBID
Arbitrage exists only if both ask and bid
futures prices are above BU,
or both are below BL.
51
A numerical example:
Given the following exchange rates:
Spot
S(USD/NZ)
Forward
F(USD/NZ)
Interest rates
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
52
Inequality (1):
FBID (USD/NZ)
 SASK (USD/NZ)e
(rUS,ASK - rNZ,BID )(T - t)
0.4450 < (0.4438)e(0.108125 – 0.05875)/12 = 0.4456 = BU
Inequality (2):
FASK (USD/NZ)  SBID (USD/NZ)e
(rUS,BID rNZ,ASK )( T- t)
0.4480 > (0.4428)e(0.106875 – 0.06000)/12 = 0.4445 = BL
No arbitrage.
Lets see the graph
53
F
FASK = 0.4480
0.4456
BU
FBID = 0.4450
BL
FBID (USD/NZ)  0.4456  BU
Clearly: FASK($/FC) > FBID($/FC).
0.4445
FASK (USD/NZ)  0.4445  BL
An example of arbitrage:
FASK = 0.4480
FBID = 0.4465
54