1. Forwards contracts
A Forwards contract is a contract made today for delivery of an assets at a
prespecified time in the future at a price agreed upon today.
The buyer of the Forwards contract agrees to take delivery of an
underlying assets at a future time (T) at a price agreed upon today. No money
changes hands until time expiry. The seller agrees to deliver the underlying
asset at a future time, at a price agreed upon today.
2. Meaning of Forwards
contracts
A Forwards contract is a contract between two
parties who agree to buy/sell a specified quantity of a
financial instruments/commodities at a certain price at a
certain date in future.
3. UnderlyingAssets of
Forwards contracts
• Traditional agricultural or physical commodities
• Currencies (Foreign exchange forward)
• Interest rates (Forward rate agreements FRA)
4. Why Forwards contracts
They are customized contracts unlike futures
Tailor-made and more suited for certain purpose
Useful when Futures do not exist for commodities and
financial being considered
Useful in cases futures standard may be different from the
actual
5. FEATURES OF FORWARD
CONTRACTS
They are bilateral negotiated contract between two parties and
hence exposed to counter party risk.
Each contract is custom designed and hence is unique in terms
of contract size, expiration date, and the asset type, quality etc.
Acontract has to be settled in delivery or cash on expiry date.
The contract price is generally not available in the public
domain.
If the party wishes to reverse the contract, it has to compulsory
go to the same counter-party, which often results in high prices
being charged.
6. Payoff on Forward Contracts
Forward contracts are privately executed between
two parties.
The obligation to buy
the asset at the
agreed price on the
specified future
date is referred to as
the long position
A long position profits
when prices rise.
Buyer
The obligation to sell the
asset at the agreed price
on the specified future
date is referred to
as the short position.
A short position profits
when prices go down
seller
7. Let T denote the expiration date,
K denote the forward price, and
PT denote the spot price (or market price) at the
delivery date. Then
For the long position: the payoff of a forward contract
on the delivery date is PT> K
For the short position: the payoff of a forward
contract on the delivery date is K >PT
9. What is A Futures Contract
A futures contract is a standardized agreement between the seller
(short position)of the contract and thebuyer ( long position ), traded
ona futures exchange, to buy or sell a certain underlying
instruments at a certain date in future, at a prespecified price.
The future date is called the delivery date or final settlement date.
The pre-set price is called the futures price. The price of the underlying
asset on the delivery date is called the settlement price.
(Thus, futures is a standard contract in which the seller is obligated to
deliver a specified asset (security, commodity or foreign exchange) to the
buyer on a specified date in future and the buyer is obligated to pay the
seller the then prevailing futures price upon delivery. Pricing can be
based on an ‘open outcry system’, or bids and offers can be matched
electronically.
10. Characteristics of Futures contracts
Futures are highly standardised contracts that provide for
performance of contracts through either deferred delivery of asset or
final cash settlement.
These contracts trade on organized futures exchanges with a clearing
association that acts as a middleman between the contracting parties.
Contract seller is called ‘short’ and buyer ‘long’. Both parties pay
margin to the clearing association. This is used as performance bond by
contracting parties
Margins paid are generally marked to market price everyday;
Each Futures contract has an associated month that represents the
month of contract delivery or final settlement. These contracts are
identified with their delivery months like July-T-Bill, December $/
derivative etc.
Every futures contract represents a specific quantity. It is not
negotiated by the parties to the contract.
11. • Identified with Underlying assets
• Identified with contract size
• Delivery arrangements- Place of delivery, Transfer cost
• Identified with Delivery month
• Identified with prespecified price
• Position limits
• Margin requirements
12. Abrief discussion of basic terms and institutions involved in
futures trading is presented below;
Clearing House ; Also known as clearing corporation, plays an
important role in the trading of futures contracts. It acts as an
intermediary for the parties who trade in futures contracts. It becomes
the seller of the contract for the long position and buyer of the contract
for the short position.
Open Interest ; Open interest on the contract is the number of
contract outstanding (No. of either long or short positions). When
contracts begin trading, open interest is zero.As time passes, open
interest increases as progressively more contracts are entered. Instead
of actually taking or making delivery of the commodity, virtually all
market participants enter reversing trades to cancel their original
positions, then open interest will be considered.
Mechanism of Trading in Futures Market
13. Margin requirement ; The futures exchange requires some good faith
money from both, to act as a guarantee that each will abide by the terms
of the contract, this is margin.
The margins are three types;
I. Initial Margin ; is required at the start of a new transaction. For
example in NSE they maintain % as initial margin for the initial
transactions.An exchange can change the required margin anytime. If
price volatility increases or if the price of the underlying commodity
rises substantially, the initial margin will be increased
II. Maintenance Margin ; The maintenance margin represents the
minimum margin which needs to be maintained by individual margin
accounts. It is akin to the minimum balance prescribed by banks in the
case of saving deposit accounts.
III. Variable Margin ; is calculated on a daily basis for the purpose of
marking-to-market all outstanding positions at the end of each day. This
is to be deposited most often in cash only. The day’s closing price is
generally used as the basis for the purpose of marking-to-market.
Continued……….
14. Marking-to-market (M2M) ; the process of marking profits or losses
that accrue to traders on daily basis is called M2M. Futures prices may
rise or fall everyday. Instead of waiting until the maturity date for traders
to realize all gains and losses, the clearing house requires all positions to
recognize profits as they accrue daily. If the futures price of Cotton rises
from Rs. 4,000 to Rs. 4,100 per quintal, the clearing house credits the
margin account of the long position for 500 Quintals times Rs. 100 per
quintals or Rs. 50,000 per contract.
Conversely, for the short position, the clearing house takes this
amount from the margin account for each contract held. This daily settling is
called marking-to-market. It means we do not need to wait for our losses or
gains until maturity date, it will be settle daily.
15. 15
Institutional Factors of Futures
Contracts
• Since futures contracts are traded on formal
exchanges, margin requirements, marking to market,
and margin calls are required; forward contracts do
not have these requirements.
• The purpose of these requirements is to ensure
neither party has an incentive to default on their
contract.
• Thus futures contracts can safely be traded on the
exchanges between parties that do not know each other.
16. Mark to Market
• MTM or mark-to-market in futures is a process of revaluing
open futures contracts at the end of each trading day to
determine the profit or loss that has occurred due to changes
in the price of the underlying asset.
17. Mark to market process
• The mark-to-market process involves calculating the
difference between the entry price of the contract and the
current market price of the contract and settling the profit or
loss in the trader's account.
• Traders have enough margin in their account to cover the
potential losses from their open positions.
18. MTM Calculations
• After the trading hours, the MTM calculations are performed
daily based on the day's closing price. On the same day, the
P&L is settled to the trading account and will not be reflected
in the positions on the following day.
19. Steps in Calculation
Step 1: Change in the value of a futures contract- calculated as
the difference between the futures contract price of the
current day and the closing price of the prior day.
Step 2: The P&L for the day can be calculated by multiplying the
price change in the futures contract value by the number of
lots.
Step 3: Total P&L can be obtained by summing up all the daily
P&L until the futures contract position is held.
20. Calculation of MTM
• Example Scenario
• Buy price - Rs100.
• Sell price - Rs102.
• Lot Size - 9500.
• Profit on the trade: Rs102 - Rs100 = Rs2.
• Total Profit: 9500 * Rs2 = Rs19,000.
• While the above gives the overall P&L, let’s apply MTM for the
same position as a table
21. Day
Ref price for
MTM (a)
Closing price
(b)
Profit and
loss (b-a)
Daily MTM
(P&L * Lot
size)
1 100 101 1 9500
2 101 100 -1 -9500
3 100 101.5 1.5 14250
4
101.5 and
102.3
102.3 0.5 4750
Total P&L 19000
22. MTM on the fourth day is
calculated as under:
Day
The
referenc
e price
for MTM
Sell price
Closing
price
Profit
Daily
MTM
4 101.5 102 102.3 0.5 4750
Two reference values are available - ₹101.5 as the previous day's close, i.e.
3rd day's close, and ₹102 as the price at which the position was squared
off.
23. MTM Calculation
• Each contract represents 100 kg of rice. Thus, the farmer is
hedging against a price decline on 1,000 kg of rice. The price
of each contract is Rs10. Thus, the account of the farmer
would be recorded as Rs10,000 (Rs10 x 1,000 kg of rice).
• Given that the farmer holds a short position in the rice
futures, when there is a fall in the value of the contract, an
increase to the account is witnessed. Similarly, if there is an
increase in the value of the futures, there will be a resultant
decrease in his account.
25. Once having established a futures position traders have an
obligation under the terms of the futures contract either to
take delivery ( a long position) or to make delivery ( a short
position) of the underlying commodity. However, making
or taking physical delivery is only one of several ways that
futures contracts can be
26. 26
The initial margin requirement
• Both the long and the short parties must deposit money in
their brokerage accounts.
• Typically 10% of the total value of the contract
• Not a down payment, but instead a security deposit to ensure the
contract will be honored
27. 27
Initial Margin Requirement –
Example
• Jawahar has just taken a long position in a futures
contract for 100 ounces of gold to be delivered in
January. Jaitley has just taken a short position in the
same contract. The futures price is $380 per ounce.
• The initial margin requirement is 10%
• What is Jawahar’s initial margin requirement?
• What is Jaitley’s initial margin requirement?
28. 28
Marking to market
• At the end of each trading day, all futures contracts are
rewritten to the new closing futures price.
• I.e., the price on the contract is changed.
• Included in this process, cash is added or subtracted
from the parties’ brokerage accounts so as to offset the
change in the futures price.
• The combination of the rewritten contract and the cash
addition or subtraction makes the investor indifferent to the
marking to market and allows for standardized contracts for
delivery at the same time to trade at the same price.
29. 29
Marking to market example
• Consider Jawahar (who is long) and Jaitley (who is short) in the contract for
100 ounces of gold. At the beginning of the day, the contract specified a price
of $380 per ounce At the end of the day, the futures price has risen to $385
so the contracts are rewritten accordingly.
• What is the effect of marking to market for Jawahar (long)?
• What would be the effect on Jaitley (short)?
• Who makes the marking to market payments or withdrawals from Jawahar’s
and Jaitley’s brokerage accounts?
• How does marking to market affect the net amount Jawahar will pay and
Jaitley will receive for the gold?
• What would have happened if the futures price had dropped by $10 instead of
rising by $5 as described above?
30. 30
Recap on Marking to Market
• After marking to market, the futures contract holders essentially have
new futures contracts with new futures prices
• They are compensated or penalized for the change in contract terms by the
marking to market deposits/withdrawals to their accounts.
•
31. 31
The dreaded Margin Call
• In addition to the initial margin requirement, investors are
required to have a maintenance margin requirement for their
brokerage account
• typically half of the initial margin requirement % or 5% of the value of the
futures contacts outstanding.
• Marking to market may result in the brokerage account balance
rising or falling. If it falls below the maintenance margin
requirement, then a margin call is triggered.
• The investor is required to bring the account balance back to the initial
margin requirement percentage.
32. 32
Margin Call Example
• Consider Jawahar’s initial margin requirement, the
futures price increased to $385 at the end of the first
day and now the futures price decreased to $350.
• What are the cumulative effects of marking to market?
• If the maintenance margin requirement is 5% of
$350/ounce x 100 ounces, what will be the margin call to
bring the account back to 10% of $350/ounce x 100
ounces?
• What does the margin call mean?
33. Future Contract Example
Contract Name: West Texas Intermediate Crude Oil Futures
Contract Size: 1,000 barrels
Delivery Date: June 22, 2023
Price Quotation: per barrel
Tick Size: $0.01
Margin Requirements: $5,000
Settlement Method: Physical delivery
This contract is for the delivery of 1,000 barrels of West Texas Intermediate
(WTI) crude oil on June 22, 2023. The price of the contract is quoted in dollars
per barrel. The tick size is $0.01, which means that the price of the contract
can only move in increments of $0.01. The margin requirement is $5,000,
which means that a trader must deposit $5,000 with the exchange in order to
open a position in this contract. The settlement method is physical delivery,
which means that the buyer of the contract will be required to take physical
delivery of 1,000 barrels of WTI crude oil on the delivery date.
Position limits : The maximum number of contracts that a trader can trade (
maximum number of contracts held and maximum number of contract
expiring in any particular month
34. Once having established a futures position traders have an
obligation under the terms of the futures contract either to
take delivery ( a long position) or to make delivery ( a short
position) of the underlying commodity. However, making
or taking physical delivery is only one of several ways that
futures contracts can be
35.
36.
37.
38.
39.
40.
41.
42. SETTLING A FUTURE POSITION
• There are 3 common ways of liquidating a future
position;
Physical Delivery ; Liquidating a futures position by making or
taking physical delivery is usually the most cumbersome way to
fulfil contractual obligations. It requires actually purchasing or
selling a commodity. A firm, which deals in commodities, might
very well wish to settle by physical delivery. It imposes obvious
costs on traders; warehousing expenses, insurance costs,
possible shipping costs and brokerage fees.
43. 43
Offsetting Positions
• Most investors do not hold their futures contracts until maturity
• Instead over 95% are effectively cancelled by taking an
offsetting position to get out of the contract.
• E.g., Jawahar (who was long for 100 ounces) can now enter
into another contract to go short for 100 ounces
• The two contracts cancel out
• There is no more marking to market or margin calls
• Jawahar may withdraw the remaining money in his
brokerage account.
44. OFFSETTING ; in effect, to reverse the initial transaction which is established
the futures position.
Suppose that on Jan 1, Mr.Atakes up a long position in the future market for
1 kg of Gold for April month: for Rs. 2,700/gm of Gold. On 25th Feb, he decides to
close his position, and hence, enters into another future contract.
Now for short position, at Rs. 2,800/gm of gold for the same delivery month i.e,April
Mr.A’sAccount Quantity Cash flow onApril 30th
To pay for long position +1000gm -27,00,000
To receive for short position -1000 gm +28,00,000
Gain Nil 1,00,000
45. CASH DELIVERY ; This procedure is a substitute for physical
delivery and completely eliminates having to make or take physical
delivery. Contracts on stock index futures use cash delivery to settle
contracts. Exchange have adopted cash delivery as an alternative to
physical delivery for 2 reasons;
1. The nature of underlying commodity may not permit feasible
physical delivery
2. Cash delivery avoids the problem that it may be difficult for traders
to acquire the physical commodity at the time of delivery because
of a temporary shortage of supply.
56. Spot Futures Parity
• Spot futures parity is the relationship between the current
spot price of an asset and the price of a futures contract on
that asset. The futures price is the price at which the asset can
be bought or sold for delivery at a future date.
• Spot futures parity is based on the idea that the futures price
should be equal to the spot price plus the cost of carry. The
cost of carry includes the cost of financing the asset, the cost
of storage, and any other costs associated with owning the
asset.
58. • The diagram shows that the futures price is equal to the spot
price plus the cost of carry. The cost of carry is represented by
the vertical axis, and the futures price is represented by the
horizontal axis.
59. Formula
• S= F/ (1 + r * t)
• F = S * (1 + r * t) --- Cost of Carry
• F= Spot + Cost of carry
• F is the futures price
• S is the spot price
• r is the risk-free interest rate
• t is the time to expiration
• c is the cost of carry
• The cost of carry includes the cost of financing the asset, the
cost of storage, and any other costs associated with owning
the asset.
60. 60
Consumption vs Investment Assets
• Investment assets are assets held by significant
numbers of people purely for investment
purposes (Examples: gold, silver)
• Consumption assets are assets held primarily
for consumption (Examples: copper, oil)
61. Investment Asset
• The underlying shares purchased in the cash market entitles
the holder to dividend declared by the company
• The expected dividend income from the asset during the
futures can be incorporated in the analysis
• A. Price of the underlying asset
• B. Rate of return expected from investment
• C. Risk free rate of interest
• D. Time to maturity
62. Notation for Valuing Futures and
Forward Contracts
S0: Spot price today
F0: Futures or forward price today
T: Time until delivery date
r: Risk-free interest rate for
maturity
y: % yield on investment per
futures period.
Fo= So+ So( r-y) or So(1+r-y)
63. Example
Suppose an investor borrows funds to purchase one unit of
asset X resulting in no intial cash outlay for his strategy. At the
end of 3 months period, Rs. 3 will be received from holding the
asset X and would be required to pay interest ( financing cost) of
Rs. 2
F= 100+100(0.02-0.03)
= Rs. 99
64. 64
Short Selling
• Short selling involves selling securities
you do not own
• Your broker borrows the securities from
another client and sells them in the
market in the usual way
65. Short Selling
(continued)
• At some stage you must buy the
securities back so they can be
replaced in the account of the client
• You must pay dividends and other
benefits the owner of the securities
receives
66. Arbitrage Opportunity?
• Suppose that:
• The spot price of dividend paying stock is Rs. 100
• The 3-month forward price is Rs. 92
• The expected return form the asset is 3 % per quarter
• Risk free rate of interest is 2 % per quarter
• Is there an arbitrage opportunity?
67. If Fo= Rs. 92
• An Investor will buy one future contract – Rs. 92
• Short sell one unit of Asset X for Rs. 100
• Rs. 100 so received will be invested @ 8% for 3 months
• After 3 months
• Will receive proceeds ( Rs. 100+2) = 102
X He will not receive yield of Rs. 3 from asset
Total Cost = Rs. 92 ( purchase of asset) + Rs. 3 ( no yield since
sold the asset) = Rs. 95
So total gain is = Rs. 102- 95 = Rs. 7
68. Arbitrage Opportunity?
• Suppose that:
• The spot price of a non-dividend paying stock is Rs. 100
• The 3-month forward price is Rs. 107
• The expected return form the asset is 3 % per quarter
• Risk free rate of interest is 2 % per quarter
• Is there an arbitrage opportunity?
69. If Fo= Rs. 107
• An Investor will sell one future contract – Rs.107
• Borrow Rs. 100 now to buy one unit of asset X @ 2 % per
quarter
• After 3 months
• Will pay ( Rs. 100+2) = 102
Total Sale value = 107 ( sale value) + Rs. 3 (yield from the asset)
= Rs. 110
So total gain is = Rs. 110- 102 = Rs. 8
70. The Forward Price
If the spot price of an investment asset is S0
and the futures price for a contract deliverable
in T years is F0, then
F0 = S0erT
where r is the 1-year risk-free rate of interest.
In our examples, S0 =40, T=0.25, and r=0.05 so
that
F0 = 40e0.05×0.25 = 40.50
71. If Short Sales Are Not Possible..
Formula still works for an investment asset
because investors who hold the asset will sell it
and buy forward contracts when the forward
price is too low
72. Continuous Compounding
• Continuous compounding is a method used to calculate the
future value of an investment or loan when the interest is
compounded continuously. In continuous compounding, the
interest is calculated and added to the principal an infinite
number of times throughout the compounding period.
• FV = P * e^(rt)
• Where: FV is the future value
• P is the principal amount (initial investment or loan)
• e is the mathematical constant approximately equal to
2.71828 (Euler's number)
• r is the annual interest rate (expressed as a decimal)
• t is the time period in years
74. 74
When an Investment Asset
Provides a Known Dollar
Income
F0 = (S0 – I )erT
where I is the present value of the income
during life of forward contract
75. When an Investment Asset
Provides a Known Yield
F0 = S0 e(r–q )T
where q is the average yield during the life of the contract
(expressed with continuous compounding)
76. 76
Valuing a Forward Contract
• Suppose that
K is delivery price in a forward contract and
F0 is forward price that would apply to the
contract today
• The value of a long forward contract, ƒ, is
ƒ = (F0 – K )e–rT
• Similarly, the value of a short forward contract
is
(K – F0 )e–rT
77. Index Futures
• Index futures are a type of derivative where the holder has
the right to buy or sell an index at a certain time in the future.
Futures on 3 indices: Nify50, Nifty Bank (BankNifty) and Nifty
Financial Services (FinNifty).
The purpose of these futures is to give traders more degree of
control over the fluctuations in their investment portfolio.
Futures contracts can also be used to hedge portfolios and
protect them from losses when the markets fall.
78. Hedging Through Futures
A hedge is an action taken out specifically to reduce or cancel
out risk in an investment
Futures contract is a mechanism to reduce risk associated with
exposure in the underlying assets.
79. Hedging Process
• Step 1: Build a portfolio
• Step 2: Find an Index Future that is closely correlated to the
portfolio
• Step 3: Sell the number of lots of Index futures which is
equivalent to (or near to) the value of the portfolio you have.
80. Hedging process
Stock portfolio of Rs 10, 00, 000
to hedge it with Index
futures of Nifty50
Nifty50 is 17500 and the lot size is 50.
sell 1 lot of Nifty50 Index
futures worth Rs 8, 75,000
81. Hedging process ( Fall in price)
market falls by 10% and as a result,
the Index Futures also falls by 10%.
Now let us assume that the value of
the portfolio also falls by 10%.
Loss from portfolio = (10% of Rs 10, 00,
000) = Rs 1, 00, 000.
82. Short Hedge
• Owner of an asset expects the price to fall and inorder to
counter this risk, he sells the futures now.
• Protect a future selling price on an existing asset.
• Long position in asset is hedged by creating a short position in
futures.
• Asset is not owned but will be owned in future
83. Long Hedge
• Prospective future buyer buys the futures today and thereby
lock the prices at which he would be acquiring the asset in
future, irrespective of what price would be prevailing in the
market on that day.
85. Basis
Basis Future price- Spot price
bo Fo – So (Intial Basis)
b1 Ft – St ( Basis at expiration)
● Basis risk arises because of the uncertainty about the basis
when the hedge is closed out
● Hedging involves the substitution of basis risk for spot
price risk.
86. Perception and Hedging
Strategy
Present Position Perceptions about
Future
Hedging Action
Holding Asset Price may fall Short Futures
Holding Asset Price may increase Do nothing
About to buy asset Price may increase Long futures
Short the asset Price may increase Long futures
Short the asset Price may fal Do nothing s
87. 3.12
Long Hedge
● Suppose that
F1: Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
● Hedge via a long futures contract the
future purchase of an asset, risk of S2
● Cost of Asset=S2+(F1
–F2
) = F1+ Basis2
88. 3.13
Short Hedge
● Suppose that
F1: Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
● Hedge via a short futures the future sale of
an asset, risk of S2
● Price Realized=S2
+ (F1– F2
) = F1+ Basis2
90. 3.14
Choice of
Contract
● Choose a delivery month that is as close as possible to, but
later than, the end of the life of the hedge
● When there is no futures contract on the asset being hedged,
choose the contract whose futures price is most highly
correlated with the asset price, aka Cross- hedging. There are
then 2 components to the basis.
91. Hedge Ratio
• Hedge ratio is a measure of the relationship between the price
of an asset and the price of a hedging instrument. It is used to
determine how much of the hedging instrument to buy in
order to offset the risk of the asset.
• For example, if the hedge ratio for a stock is 0.5, then for
every $1 decrease in the stock price, the hedging instrument
should increase in value by $0.5. This means that the hedging
instrument will offset half of the risk of the stock.
92. Hedge Ratio
No of contracts = Value of Portfolio / Value of 1 Future Contract
Suppose a person has 10000 shares of SAIL with market price of
Rs. 85 per share. Value of his underlying portfolio is Rs.
8,50,000.
Futures are traded at Rs. 85.35 and one contract of futures has
1900 shares.
Value of one contract = 1900 X 85.35
No of contracts =
93. Hedge Ratio
Suppose the hedger has a portfolio consisting of 20 shares and
and this porfolio is being hedged by NIFTY futures, both value
will differ
Where we calculate the beta value
No of contracts = Value of Portfolio X Beta of portfolio
Value of 1 Future contract
94. 94
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
95. 95
Stock Index
• Can be viewed as an investment asset
paying a dividend yield
• The futures price and spot price relationship
is therefore
F0 = S0 e(r–q )T
where q is the average dividend yield on the
portfolio represented by the index during
life of contract
96. 96
Stock Index
• 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
97. 97
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
98. 98
Index Arbitrage
(continued)
• Index arbitrage involves simultaneous trades in
futures and many different stocks
• Very often a computer is used to generate the
trades
99. Forward Pricing:
Cash and Carry Arbitrage
• Ignore, for now, the carry return (CR), as well as carrying costs such as
storage and insurance costs
• What if F > S + CC = S(1+h(0,T)) = S + h(0,T)S?
• Today
borrow +S
buy the good -S
sell the good forward ___
CF0 = 0
• At delivery
repay loan w/ interest -S(1+h(0,T))
sell at forward price +F
CFT = F-S(1+h(0,T)) > 0 ARBITRAGE PROFIT!
100. Forward Pricing:
Cash and Carry Arbitrage
• h(0,T) = the unannualized interest rate = rT/365
• T = days until delivery
• r = the annual interest rate
• If the set of cash and carry trades entails no cash flow at time 0, there must be
no cash flow at time T (delivery).
• Arbitrage: A set of trades requiring no initial investment, no risk, and a
positive return.
• If F-S(1+h(0,T)) >0, then by borrowing to buy the spot good, and selling it
forward, you can arbitrage.
• Conclusion: F cannot be greater than S(1+h(0,T)); F<S(1+h(0,t))
101. ThePerfectMarketAssumptionsforthe
CostofCarryRelationship
• There are no commissions.
• There are no bid-ask spreads.
• There are no taxes.
• Market participants have no influence over price (price takers).
• Market participants want to maximize wealth.
• There are no impediments to short-selling.
• Short-sellers have full use of the short-sale proceeds.
• There is an unlimited ability to borrow or lend money.
• All borrowing and lending is done at the same interest rate.
• There is no default risk associated with buying or selling in either the
forward or spot market.
• Commodities can be stored indefinitely without any change in the
characteristics of the commodity (such as its quality).
102. Forward Pricing:
Reverse Cash and Carry
Arbitrage
• What if F<S(1+h(0,T))?
• Today
sell the good +S
lend the proceeds -S
buy the good forward ___
CF0 = 0
• At delivery
the loan repays you w/ interest +S(1+h(0,T))
buy at forward price -F
CFT = -F+S(1+h(0,T)) > 0 ARBITRAGE!
Conclusion: F cannot be less than S(1+h(0,T))
104. Conclusions about Forward
Pricing
• Assume no transaction costs, no carry return and no
costs of storing, (like insurance).
• For non-carry commodities: F<S(1+h(0,T)). Prices cannot
permit cash and carry arbitrage.
• For gold and financials (carry commodities):
F=S(1+h(0,T))=S + Sh(0,T) = S + interest. Prices cannot
permit either cash and carry arbitrage or reverse cash
and carry arbitrage.
105. An Example
• Spot gold sells for $403/oz. The six month interest rate is 4.5%; the
one year interest rate is 5% (both are annual rates).
• Assume no transaction costs and no storage, etc. costs.
• For there to be no arbitrage, the forward price of gold for delivery
six months hence must be:
403(1.0225) = 412.0675.
• The forward price of gold for delivery one year hence must be:
403(1.05) = 423.15.
106. Cash and Carry Arbitrage: An
Example
• What if the actual forward price of gold for delivery 6 months hence is 413?
[that is, FP0 > S(1+h(0,T))]
• Today (at zero cash flow):
borrow: $403
buy 1 oz. of gold for $403
sell gold forward at FP0 = $413
• Six months hence:
repay loan: 403(1.0225) = -$412.0675
sell gold: +$413
arbitrage profit = $0.9325/oz.
107. Reverse Cash and Carry
Arbitrage:
An Example
• What if the forward price of gold for delivery one year hence is 422?
[that is, FP0 < S(1+h(0,T))]
• Today (at zero cash flow):
sell gold for: $403
lend: $403
buy gold forward at FP0 = $422
• One year hence
get repaid: 403(1.05) = +423.15
buy gold: -422.00
arbitrage profit = $1.15/oz.
108. Cash and Carry Arbitrage
With Storage and Insurance
Costs (CC0)
• Today
borrow +S+CC0
buy the good -S
pay storage and insurance costs -CC0
sell the good forward _________
CF0 = 0
• At delivery
repay loan w/ interest -(S+CC0)(1+h(0,T))
sell at forward price _____F____
CFT = F-(S+CC0)(1+h(0,T))
Hence, F must equal (S+CC0)(1+h(0,T)), or there will be an arbitrage
opportunity.
