Determination of Forward and Futures Prices Chapter 5

1
Determination of Forward and Futures
PricesChapter 5
2

• 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.

3

• 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.

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 Speculators
• And then,
• the Arbitrageurs activities in the spot
• and futures markets.

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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
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.
b
Expt Stk
c
Od
0
Quantity of long positions
Demand for LONG futures positions by long
HEDGERS
6
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)
d
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.
e
Expt St k
f
QS
0
Quantity of short positions
Supply of SHORT futures positions by short
HEDGERS.
7
Ft (k)
S
Supply schedule
D
Ft (k)e
Premium
Expt St k
Demand schedule
S
D
Qd
0
Quantity of positions
QS
Equilibrium in a futures market with a
preponderance of long hedgers.
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Ft (k)
S
D
Supply schedule
Expt St k
Discount
Ft (k)e
Demand schedule
S
D
Qd
0
QS
Quantity of positions
Equilibrium in a futures market with a
preponderance of short hedgers.
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Ft (k)
Speculators will not demand any long positions if
the settlement price exceeds the expected future
spot price.
a
Speculators demand more long positions the
greater the discount of the settlement price
below the expected future spot price.
Expt St k
b
c
0
Quantity of long positions
Demand for long positions in futures contracts
by speculators.
10
Ft (k)
Speculators supply more short positions the
greater the premium of the settlement price over
the expected future spot price
d
Expt St k
e
Speculators will not supply any short positions
if the settlement price is below the the expected
future spot price
f
0
Quantity of short positions
Supply of short positions in futures contracts
by speculators.
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Ft (k)
S
D
Increased supply from speculators
Expt St k
Discount
Ft (k)e
Increased demand from speculators
S
D
Qd QE Qs
0
Quantity of positions
Equilibrium in a futures market with speculators
and a preponderance of short hedgers.
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Ft (k)
S
Increased supply from speculators
D
Ft (k)e
Increased demand from speculators
Premium
Expt St k
S
D
0
Quantity of positions
QE
Equilibrium in a futures market with speculators
and a preponderance of long hedgers.
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Ft (k) St
Excess supply of the asset when the spot market
price is St
Spot supply

Ft (k)e
Premium
Expt St k
Spot demand
0
QE
Quantity of the asset
Equilibrium in the spot market
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Ft (k)
Schedule of excess demand by hedgers and
speculators
Expt St k

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
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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 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

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• 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

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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.

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• ARBITRAGE IN PERFECT MARKETS(P103)
• CASH -AND-CARRY
• DATE SPOT MARKET FUTURES MARKET
• NOW 1. BORROW CAPITAL S0 3. SHORT FUTURES
• t0 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

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• ARBITRAGE IN PERFECT MARKETS
• REVERSE CASH -AND-CARRY
• DATE SPOT MARKET FUTURES MARKET
• NOW 1. SHORT SELL ASSET S0 3. LONG FUTURES
• t0 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

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Conclusion (p.103)When an Investment Asset
Provides NO INCOME and the only carrying cost is
the interest F0,T S0erT
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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

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When an Investment Asset Provides a Known annual
Yield, q. (P.107)

• F0,T S0e(rq )T
• where q is the average yield during the life
• of the contract (expressed with continuous
• compounding)

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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.

Ft,T Fj,T
Date t j T
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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, ƒjL, is
• fjL (Fj,T Ft,T )er(T-j)

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Valuing a Forward Contract(p.108)

• At j, t j T, the value of a short forward
  contract fjSH is
• fjSH (Ft,T Fj,T )er(T-j)

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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

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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(rq )T
• where q is the dividend yield on the portfolio
  represented by the index

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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

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Stock Index Arbitrage

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

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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

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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 risk-free
  interest rate

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The same parameters used in my slides are noted
as follows
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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
36

• 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.

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NO ARBITRAGE CASH-AND-CARRY TIME CASH FUTURE
S t (1) BORROW Q. rDOM (4) SHORT FOREIGN
CURRENCY (2) BUY FOREIGN CURRENCY
FORWARD Ft,T(DC/FC) Q/S(DC/FC)
QS(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
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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)
PS(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
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FROM THE CASH-AND-CARRY STRATEGY
FROM THE REVERSE CASH-AND-CARRY STRATEGY
THE ONLY WAY THE TWO INEQUALITIES HOLD
SIMULTANEOUSLY IS BY BEING AN EQUALITY
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• 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.

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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.
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• 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

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• 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

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With these market realities, a new no-arbitrage
condition emerges BL lt FBID lt FASK lt BU As
long as the futures price fluctuates between the
bounds there is no possibility to make arbitrage
profits
BU
BU
F
BL
BL
time
45

• Example 1
• S0,BID (1 - c)1 f(rBID ) lt F0, T lt 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)(1rBID)
• F0,T ? S0,BID(1-c)(1 frBID)

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• S0,BID(1-c)1f(rBID )lt F0,Tlt S0,ASK(1c)(1rASK)
  
• S0,ASK 90.50 / bbl
• S0,BID 90.25 / bbl
• rASK 12
• rBID 8
• c 3
• 90.25(.97)1f(.08)ltF0,Tlt 90.50(1.03)(1.12)
• 87.5425 f(7.0034) lt F0,T lt 104.4008

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• EXAMPLE 1.
• 87.5425 f(7.0034) lt F0,T lt 104.4008
• THE CASH-AND-CARRY costs 104.4008/bbl.
• THE REVERSE CASH-AND-CARRY costs
• 87.5425 f(7.0034).
• IF f0.5 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.

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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.
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NO ARBITRAGE CASH - AND - CARRY TIME CASH FU
TURES 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
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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 PSBID
(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
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From Cash and Carry
From reverse cash and Carry
(3) And FASK(DC/FC) gt FBID(DC/FC)
Notice that RHS(1) gt RHS(2)
Define RHS(1) ? BU RHS(2) ? BL
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F(/D)
FASK(DC/FC) gt FBID(DC/FC).
FASK
BU BL
BU
BL
FBID
Arbitrage exists only if both ask and bid futures
prices are above BU, or both are below BL.
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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) gt F(bid). (USD0.4480NZ
gt 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
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Inequality (1)
0.4450 lt (0.4438)e(0.108125 0.05875)/12
0.4456 BU
Inequality (2)
0.4480 gt (0.4428)e(0.106875 0.06000)/12
0.4445 BL

• No arbitrage.
• Lets see the graph

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F
FASK 0.4480
0.4456
BU BL
Clearly FASK(/FC) gt FBID(/FC).
FBID 0.4450
0.4445
An example of arbitrage FASK 0.4480 FBID
0.4465