FORWARD
AND
FUTURES
ARBITRAGE
RELATIONSHIPS
Introduction
We start with the discussion of determinants
of forward and futures prices.
We use arbitrage arguments to understand
(1) the relationship between forward and spot
prices of the underlying;
(2) the relationship between futures and spot
prices of the underlying.
Introduction
Forwards and futures are similar contracts.
Except for some institutional features.
Futures are easier to use.
Forwards are easier to price.
Only one single payment at maturity date.
A forward price can be easily determined
from the spot price.
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Arbitrage
Arbitrage is the adhesive that links the two
prices together.
Arbitrage is
Any trading strategy requiring no cash input.
That has some probability of making profits.
Without any risk of a loss.
Arbitrage
To clarify the meaning of arbitrage,
consider starting with a zero cash portfolio.
You formulate an investment strategy
involving a portfolio of securities.
Because you have zero cash, the purchase
of any securities must be financed by
borrowing or short selling other securities.
Arbitrage
You design an investment strategy such that
the worst possible outcome will leave where
you started.
With zero cash.
In other possible outcomes, the strategy
generates positive profits.
Although it is uncertain how much your
wealth will increase, there is no risk of a
loss.
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Arbitrage
This is the key of investment strategy.
Such trading strategy is called arbitrage
opportunity.
If arbitrage opportunity exists, arbitrageurs
correct such economic disequilibria through
profit taking.
From economic perspective, existence of
arbitrage opportunity implies economy is in
an economic disequilibrium.
Example I: Arbitrage
GM is selling for $60 in US.
But is selling for $61 in Japan.
Brokerage cost is $0.25/share.
“Buy cheap, sell dear”.
Get arbitrage profit of $0.50.
Assumptions
Four assumptions are needed for this:
A1. No market frictions
No transaction costs,
No bid/ask spreads,
No margin requirements,
No restrictions on short sales,
No taxes.
Reasonable approximation for large traders.
Benchmark model- frictions can be added
later.
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Assumptions (cont’d)
A2. No counterparty risks.
Reasonable approximation for exchangetraded assets due to clearing houses.
A3. Competitive markets.
Standard postulate- traders are price takers.
A4. Prices have adjusted so that there are no
arbitrage opportunities.
FORWARD AND SPOT PRICES
No Cash Flows on the
Underlying
Consider forward contracts written on
financial assets.
There is no cash flow on the underlying
asset over forward’s life.
Focus only on the start and end dates.
Examples would be non-dividend paying
stocks and zero-coupon Treasury Bonds.
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CASH-AND-CARRY
Cash-and-Carry
Forward price can be solved from a “cashand-carry” strategy for generating a
forward contract synthetically.
Buy asset through borrowing (cash)
Carry asset to the future
Creating a contract synthetically means
constructing a portfolio of traded assets that
duplicate both the cash flow and value of
the contract under consideration.
Cash-and-Carry: Notations
t = 0 is start date and t = T is maturity date.
S(t) = stock price at time t.
F(t,T) = forward price at time t for a
contract with delivery date T.
B(t,T) = value at time t of a T-Bill that pays
$1 at time T = 1/[1 + iS  T] where
iS = simple interest rate.
Stock has no dividends over contract life.
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Example II: Cash-and-Carry
Price of a stock with no dividends is $25.
Interest rate is 7.12%.
Investor writes 6-mo forward (price is F).
Portfolio
Net Outflow (t=0)
a) Buy stock
25
b) Borrow to finance
-25
c) Write a forward
0
Net payoff
0
Example II: Cash-and-Carry
(cont’d)
Consider the outcomes from this strategy
in six months.
The payouts of strategy at maturity are,
Portfolio
a) Stock’s value
b) Repay borrowing
c) Forward’s payoff
Net payoff
Net Inflow (in 6 months)
S(6-mo)
–25[1 + 0.0712  (1/2)]
–[S(6-mo) – F]
F – 25[1 + 0.0712  (1/2)]
Example II: Cash-and-Carry
(cont’d)
Net payoff is known today (t=0)
It is independent of S(6-mo)
Initial outlay of investor was zero.
To prevent arbitrage, net payoff must be 0.
Arbitrage-free forward price F should be,
F = 25[1 + 0.0712  (1/2)] = $25.89
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Cash-and-Carry Strategy: Steps
1) Buy the stock through borrowing (cash).
2) Hold it until the delivery date of the
forward contract (carry).
At delivery date, initial borrowing is paid
off.
Thus, strategy replicates the forward’s
contract cash flows at delivery date.
Strategy generates total ownership of the stock
at delivery date when borrowing is paid off.
Cash-and-Carry: Observations
Why sell forward when buying the asset?
Long stock will be there at maturity.
Short forward will get rid off the stock but will
bring in some cash at maturity.
That cash is brought “back to the present” to
finance the stock purchase.
Any break in this chain opens up arbitrage
opportunities.
Cash-and-Carry: Observations
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Cash-and-Carry: Observations
Cash-and-Carry: Derivation
Time 0 (start) Time T (end)
Portfolio
Net Outflow Net Inflow
a)Buy share
-S(0)
S(T)
b)Borrow
S(0)
-S(0)[1 + iS  T]
c)Write forward 0
-[S(T) - F(0,T)]
Total
0 F(0,T) - S(0)[1+iS  T]
Cash-and-Carry: Derivation
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Cash-and-Carry: Derivation (cont’d)
Cash-and-Carry: Derivation
(cont’d)
Net outflow at start date is 0 for sure.
Result # 1: For an asset with no dividends,
cash-and-carry gives
S(t) = B(t,T)F(t,T)
Or,
S = BF
Example III: Arbitraging a
Cash-and-Carry Mispricing
S(0) = $45.
No dividends in 90 days.
F(0,90) = $46.54 is 90-day forward price.
iS = 4.85% per annum.
F(0,90) = 45[1 + 0.0485(90/365)] = 45.54
Theoretical forward price is less than the
quoted price => arbitrage opportunity.
Sell overvalued forward.
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Example III: Arbitraging a Cashand-Carry Mispricing (cont’d)
But that alone will not do.
Can lose money if spot falls.
Carefully create portfolio as before.
Time 0 (start)
90 days later
Portfolio
Net Outflow Net Inflow
a) Buy share
-45
S(T)
b) Borrow
45
-45[1+0.0485(90/365)]
c) Write forward
0
-[S(T) - 46.54]
Net profit
0
1.00
Example III: Arbitraging a Cashand-Carry Mispricing (cont’d)
Need profit be always taken at maturity?
No.
Borrow present value (PV) of $46.54.
Get 0 cash flow for sure at maturity.
Get$0.99 in arbitrage profits today.
Example III: Arbitraging a Cashand-Carry Mispricing (cont’d)
Portfolio PV of F=46.54.
Time 0 (start)
Portfolio
Net Outflow
a) Buy share
-45
b) Borrow
45.99*
c) Write forward
0
Net profit
0.99
90 days later
Net Inflow
S(T)
-46.54
-[S(T) - 46.54]
0
*45.99 = 46.54 / [1+0.0485(90/365)]
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Example III: Arbitraging a Cashand-Carry Mispricing (cont’d)
What if forward price was $45?
Buy the relatively undervalued forward.
Sell the relatively overpriced stock.
Reverse all trades to capture arbitrage profits.
Example III: Arbitraging a Cashand-Carry Mispricing (cont’d)
Portfolio F=45.
Time 0 (start)
90 days later
Portfolio
Net Outflow Net Inflow
a) Sell share
45
-S(T)
b) Lend
-45
45[1+0.0485(90/365)]
c) Long forward
0
[S(T) - 45]
Net profit
0
0.54
Example III: Arbitraging a Cashand-Carry Mispricing (cont’d)
Portfolio PV of F=45.
Time 0 (start)
90 days later
Portfolio
Net Outflow
a) Sell share
45
b) Lend
-44.47*
c) Long forward
0
Net profit
0.53
Net Inflow
-S(T)
45
[S(T) - 45]
0
*44.47 = 45/ [1+0.0485(90/365)]
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VALUE OF A
FORWARD CONTRACT
THAT BEGAN EARLIER
Value of a Forward Contract
That Began Earlier
At start, value of a forward,
V(0) = 0
At maturity, value of a forward to Long is
V(T) = [S(T) – F(0,T)]
Opposite to Short.
Long gains, Short loses if spot soars.
Value of a Forward Contract
That Began Earlier (cont’d)
At intermediate date t, value of a forward to
Long is
V(t) = PV of [S(T) – F(0,T)]
= S(t) – B(t,T)F(0,T)
Result # 2: Value of a forward that began
earlier is
V(t) = S(t) – B(t,T)F(0,T)
Or,
V = S – BF(0)
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Example IV: Value of a Forward
That Began Earlier
F(0,6) is $25.89.
Delivery in 6 months.
After 3 months, stock is $23
After 3 months, iS = 8.08%.
f(3,6) = 23*[1 + 0.0808  (1/4)] = $23.46
Current value of the original forward is
V(3) = 23 – 25.89/[1 + 0.0808  (1/4)] = –$2.38
V(3) = [23.46 – 25.89]*1/[1 + 0.0808  (1/4)] = –$2.38
Long must pay Short $2.38 to close this
position.
CASH-AND-CARRY
WITH KNOWN CASH FLOWS
TO UNDERLYING ASSET
Example V: Cash-and-Carry
with a Dividend
Spot is S(t) = $25 today.
B(t,T) = $0.96 pays $1 in 6 months.
Dividend d(t1) = $0.50 is due in 3 months.
B(t,t1) = $0.98 pays $1 in 3 months.
PV of dividend = B(t,t1)d(t1) = 0.98  0.50 =
$0.49.
Now, 0.96F = 25 – 0.49
=> F = $25.53.
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Cash-and-Carry with Known
Cash Flows to Underlying Asset
Result # 3a: At time t, cash-and-carry gives
B(t,T)F(t,T) = S(t)  PVt [of all cash flows over
the remaining life of the forward]
Or,
BF = S – [PV of future cash flows]
Cash-and-Carry with Known
Cash Flows to Underlying Asset
Result # 3b: Value of a forward that began earlier
is
V(t) = S(t)  PVt [of all cash flows over the
remaining life of the forward] - B(t,T)F(0,T)
Or,
V = S – [PV of future cash flows] – BF(0)
Example VI: Treasury Bond
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Example VI: Treasury Bond
(cont’d)
Want to determine the forward price of a 9-mo
forward contract on 12-mo Treasury bond.
Investment Strategy I.
Time 0 (start)
9 months later
Portfolio
Net Outflow
Net Inflow
a) Buy Treasury -1,021.39
Bc(9,12) + 50
b) Borrow PV(C) 50/[1+0.0718(1/2)]
-50
Total Cost
-1,021.39+ PV (C)
Total Value
Bc(9,12)
Example VI: Treasury Bond
(cont’d)
Investment Strategy II.
Time 0 (start)
9 months later
Portfolio
Net Outflow
Net Inflow
a) Long Forward F(0,9)
0
Bc(9,12) - F(0,9)
b) Invest PV(F) F(0,9)/[1+0.0766(9/12)] F(0,9)
Total Cost
F(0,9)/[1+0.0766(9/12)]
Total Value
Bc(9,12)
Example VI: Treasury Bond
(cont’d)
Payoffs are identical in nine months.
To avoid arbitrage, the date 0 costs must be the same.
Implies,
Bc(0,12) – PV(C) = PV [F(0,9)]
1,021.39 - 50/[1+0.0718(1/2)] = F(0,9)/[1+0.0766(9/12)]
Then,
F(0,9) = 1,029.03
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Cash-and-Carry for
Forward on a Stock Index
For indexes like S&P 500, a continuous
dividend rate is a good approximation.
Result # 4: At time 0,
For an index with value I(0), and
Constant dividend yield ,
Cash-and-carry gives
F(0,T)B(0,T) = I(0)e-T
Or,
BF = Ie-T
Example VI: Forward Price
of a Stock Index
Index level I(0) = 436.00.
 = 2.80%. Then e-T= 0.9908.
iS = 3.5%. Then B(0,120) = 0.9886.
BF = Ie-T gives
F(0,120) = 436.00  0.9908/0.9886 = 436.97.
Foreign Currency Forward Contract Notation
At time 0, S = S(0) is spot exchange rate.
F = F(0,T) is forward exchange rate.
BF(0,T) = BF is a foreign zero that pays 1
unit of foreign currency at time T.
B(0,T) = B is an American zero coupon.
BFS is the $ value of the foreign bond now.
F.1 is dollar value of 1 unit of foreign
currency in the future.
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Foreign Currency Forward Contract
Time 0 (start)
Maturity
Portfolio
Net Outflow Net Inflow
a)Buy foreign bond -BFS
S(T)
b) Borrow PV of F
BF
–F
c)Write forward
0
–[S(T) – F]
Net profit
-BFS + BF
0
To prevent arbitrage, -BFS + BF must be 0.
Cash-and-Carry for Foreign
Currency Forwards
This gives cash-and-carry for foreign
currency forwards which is also known as
interest rate parity:
Result # 5: At time 0, for foreign currency
forwards, cash-and-carry gives
B(0,T)F(0,T) = BF(0,T)S(0)
Or,
BF = BFS
Example VII: Foreign Currency
Forward Price
US-based company buys goods from a
Swiss firm.
Cost of goods = 62,500 Swiss francs.
The US firm must pay for goods in 120
days.
The US firm is concerned about the Swiss
franc appreciation against the dollar.
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Example VII: Foreign Currency
Forward Price
S(0) = 0.7032 dollars per Swiss franc.
If the exchange increased to 0.7532 $/SF.
The dollar cost of buying increases by
$3,125 => 62,500*(0.7532 – 0.7032)
To hedge risk, company enters into forward
contract.
Buy 62,500 SF in 120 days at F(0,120) ($/SF)
Forward rate set to make value of contract to
zero.
Example VII: Foreign Currency
Forward Price
S(0) = 0.7032 dollars per Swiss franc.
BF(0,120) = 0.9854 SF.
Pays 1 SF in 120 days.
=> Foreign rate is 4.5%.
B(0,120) = 0.9894 $.
Pays $1 in 120 days.
=> Domestic rate is 3.25%.
Example VII: Foreign Currency
Forward Price
– Strategy One – Date 0
Long forward
Initial cost is zero
Contract to buy 62,500 Swiss francs
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Example VII: Foreign Currency
Forward Price
– Strategy Two – Date 0
a) Buy PV 62,500 Swiss francs
Buy 62,500 / [1+0.045*(120/365)] Swiss francs.
Invest SF in Swiss riskless asset for 120 days.
Dollar cost to invest = 0.7032 * PV 62,500 SF.
b) Borrow PV of forward price in dollars.
62,500*F(0,120)/[1+0.0325*(120/365)]
Domestic interest rate used (3.25%) for 120
days.
Example VII: Foreign Currency
Forward Price
– Strategy One – Date 120 Days
Value Long forward
62,500 * [S(120) – F(0,120)] dollars
Example VII: Foreign Currency
Forward Price
– Strategy Two – Date 120 Days
a) Pays 62,500 Swiss francs
62,500*S(120) dollars
b) Repay amount borrowed
-62,500*F(0,120) dollars
Net payoff is,
62,500*[S(120)-F(0,120)] dollars
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Example VII: Foreign Currency
Forward Price – Strategies
Strategy One Payoff (date 120 days)
62,500 * [S(120) – F(0,120)] dollars
Strategy Two Payoff (date 120 days)
62,500 * [S(120) – F(0,120)] dollars
Therefore, the cost of two strategies at date
0 must be the same to avoid arbitrage.
Example VII: Foreign Currency
Forward Price – Strategies
F(0,120) = 0.7004 ($/SF)
Example VII: Foreign Currency
Forward Price
S(0) = 0.7032 dollars per Swiss franc.
BF(0,120) = 0.9854 SF. Pays 1 SF in 120
days.
B(0,120) = 0.9894 $. Pays $1 in 120 days.
BFS = $0.6929 is the $ value of the Swiss
bond. This pays 1 SF in the future.
BF = 0.9894F
Also pays 1 SF in 120 days.
So, F = 0.7004($/SF).
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Conclusion
Forwards and futures are similar contracts
except for some institutional features.
A forward price can be easily determined
from the spot by cash-and-carry and costof-carry relationships.
“No-arbitrage” principle also helped to
value a contract that began earlier and
exploit an arbitrage opportunity.
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