DERIVATIVES

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DERIVATIVES WORKBOOK By Ramon Rabinovitch
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DERIVATIVES ARE CONTRACTS Two
parties Agreement Underlying security
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DERIVATIVES FORWARDS FUTURES OPTIONS SWAPS
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• A FORWARD IS
• A BILATERAL AGREEMENT IN WHICH ONE PARTY COMMITS
  TO BUY AND THE OTHER PARTY COMMITS TO SELL A
  SPECIFIED AMOUNT OF AN AGREED UPON COMMODITY FOR
  A PREDETERMINED PRICE ON A SPECIFIC DATE IN THE
  FUTURE.

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A FUTURES IS NOTHING MORE THAN A
STANDARDIZED FORWARD TRADED ON AN ORGANIZED
EXCHANGE. STANDARDIZATION THE COMMODITY TYPE
AND QUALITY THE QUANTITY PRICE QUOTES DELIVERY
DATES DELIVERY PROCEDURES
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• AN OPTION IS
• A BILATERAL AGREEMENT IN WHICH ONE PARTY HAS THE
  RIGHT, BUT NOT THE OBLIGATION, TO BUY OR SELL A
  SPECIFIED AMOUNT OF AN AGREED UPON COMMODITY FOR
  A PREDETERMINED PRICE BEFORE OR ON A SPECIFIC
  DATE IN THE FUTURE. THE OTHER PARTY HAS THE
  OBLIGATION TO DO WHAT THE FIRST PARTY WISHES TO
  DO. THE FIRST PARTY, HOWEVER, MAY CHOOSE NOT TO
  EXERCISE ITS RIGHT AND LET THE OPTION EXPIRE
  WORTHLESS.

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• A SWAP IS
• A BILATERAL AGREEMENT IN WHICH THE TWO PARTIES
  COMMIT TO EXCHANGE A SERIES OF CASH FLOWS. THE
  CASH FLOWS ARE BASED ON AN AGREED UPON PRINCIPAL
  AMOUNT. NORMALLY, ONLY THE NET FLOW EXCHANGES
  HANDS.

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WHY TRADE DERIVATIVES? THE FUNDAMENTAL REASON
FOR TRADING FORWARDS AND FUTURES IS PRICE
RISK or VOLATILITY
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PRICE RISK IS THE VOLATILITY ASSOCIATED WITH THE
COMMODITYS PRICE IN THE CASH MARKET REMEMBER
THAT THE CASH MARKET IS WHERE FIRMS DO THEIR
BUSINESS. I.E., BUY AND SELL THE COMMODITY. ZERO
PRICE VOLATILITY NO DERIVATIVES!!!!
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PRICE RISK At time zero the commoditys price at
time t is not known.
Pr
St
S0
0
t
time
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PRICE RISK The larger the volatility, the more
need for derivatives
Pr
St
S0
0
t
time
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THE ECONOMIC PURPOSES OF DERIVATIVE
MARKETS HEDGING PRICE DISCOVERY SAVING
HEDGING IS THE ACTIVITY OF MANAGING PRICE RISK
EXPOSURE PRICE DISCOVERY IS THE REVEALING OF
INFORMSTION ABOUT THE FUTURE CASH MARKET PRICE
FOR A PRODUCT. SAVING IS THE COST SAVING
ASSOCIATED WITH SWAPING CASH FLOWS
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ALTHOUGH THE ECONOMIC PURPOSES OF DERIVATIVE
MARKETS ARE HEDGING PRICE
DISCOVERY SAVING WE WILL SEE THAT SPECULATIVE
AND ARBITRAGE ACTIVITIES ARE NOT ONLY BENEFICIAL
IN THESE MARKETS, THEY ARE NECESSARY FOR
MAINTAINING MARKET EFFICIENCY AND EFFICIENT
MARKET PRICES
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FORWARDS AND FUTURES The CONTRACTS The
MARKETS PRICING FUTURES Speculation Arbitrage Hedg
ing
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Some Financial Economics Principles

• 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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• Valuation The current market value (price) of
  any project or investment is the net present
  value of all the future expected cash flows from
  the project.
• One-Price Law Any two projects whose cash flows
  are equal in every possible state of the world
  have the same market value.
• Domination Let two projects have equal cash
  flows in all possible states of the world but
  one. The project with the higher cash flow in
  that particular state of the world has a higher
  current market value and thus, is said to
  dominate the other project.

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• A proof by contradiction is a method of proving
  that an assumption, or a set of assumptions, is
  incorrect by showing that the implication of the
  assumptions contradicts these very same
  assumptions.
• Risk-Free Asset is a security of investment
  whose return carries no risk. Thus, the return
  on this security is known and guaranteed in
  advance.
• Risk-Free Borrowing And Landing By purchasing
  the risk-free asset, investors lend their capital
  and by selling the risk-free asset, investors
  borrow capita at the risk-free rate.

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• The One-Price Law
• There exists only one risk-free rate in an
  efficient economy.

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Compounded Interest

• Any principal amount, P, invested at an
• annual interest rate, r, compounded
• annually, for n years would grow to
• An P(1 r)n.
• If compounded Quarterly
• An P(1 r/4)4n.
• In general, with m compounding periods
• every year, the periodic rate becomes
• r/m and nm is the total compounding
• periods. Thus, P grows to
• An P(1 r/m)nm.

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• Monthly compounding becomes
• An P(1 r/12)n12
• and daily compounding yields
• An P(1 r/12)n12.
• EXAMPLES
• n 10 years r 12 P 100
• 1. Simple compounding yields
• A10 100(1 .12)10  310.58
• 2. Monthly compounding yields
• A10 100(1 .12/12)120   330.03
• 3. Daily compounding yields
• A10 100(1 .12/365)3650 331.94.

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• In the early 1970s, banks came up with
• the following economic reasoning Since
• the bank has depositors money all the
• time, this money should be working for
• the depositor all the time!
• This idea, of course, leads to the concept of
• continuous compounding.
• We want to apply this idea in the formula

Observe that continuous time means that the
number of compounding periods, m, increases
without limit, while the periodic interest rate,
r/m, becomes smaller and smaller.
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• This reasoning implies that in order to impose
  the concept of continuous time on the above
  compounding expression, we need to solve

This expression may be rewritten as
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• EXAMPLE, continued
• First, we remind you that the number e
• is defined as

For example x e 1 2 10 2.59374246 100
2.70481382 1,000 2.71692393 10,000 2.718145
92 1,000,000 2.71828046 In the limit e
2.718281828..
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• Recall that in our example
• N 10 years and r 12 and P100.
• Thus, P100 invested at a 12 annual
• rate, continuously compounded for ten
• years will grow to

Continuous compounding yields the highest return
to the investor Compounding Factor Simple 3.
105848208 Quarterly 3.262037792 Monthly 3.30
0386895 Daily 3.319462164 continuously
3.320116923
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Discrete Discounting Clearly, any stream of cash
flows may be discounted to the present by
discounting every future cash flow for today. P
An(1 r/m)-nm.
i
This expression may be rewritten as
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Continuous Discounting
This expression may be rewritten as
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• EXAMPLE, continued
• First, we remind you that the number e
• is defined as

Recall that in our example P 100 n 10
years and r 12 Thus, 100 invested at an
annual rate of 12 , continuously compounded
for ten years will grow to
Therefore, we can write the continuously
discounted value of 320.01 is
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This expression may be rewritten as
But first, QUESTION Given P and r, how long it
takes to double our money? - the 72
rule Ans. 2P Pert t ln2/r t
69.31/r. r 10 gt t 6.931yrs.
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PURE ARBITRAGE PROFIT A PROFIT MADE 1.
WITHOUT EQUITY and 2. WITHOUT ANY RISK.
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Risk-free lending and borrowing

• Arbitrage A market situation in
• which 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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• Risk-free lending and borrowing
• PURE ARBITRAGE PROFIT
• A PROFIT MADE
• 1.WITHOUT EQUITY INVESTMENT
• and
• WITHOUT ANY RISK
• We will assume that
• the options market is efficient.
• This assumption implies that one cannot make
  arbitrage profits in the options markets

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• Risk-free lending and borrowing
• Treasury bills are zero-coupon bonds, or pure
  discount bonds, issued by the Treasury.
• A T-bill is a promissory paper which promises its
  holder the payment of the bonds Face Value (Par-
  Value) on a specific future maturity date.
• The purchase of a T-bill is, therefore, an
  investment that pays no cash flow between the
  purchase date and the bills maturity. Hence, its
  current market price is the NPV of the bills
  Face Value
• Pt NPVthe T-bill Face-Value
• We will only use
• continuous discounting

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• Risk-free lending and borrowing
• Risk-Free Asset is a security whose return is a
  known constant and it carries no risk.
• T-bills are risk-free LENDING assets. Investors
  lend money to the Government by purchasing
  T-bills (and other Treasury notes and bonds)
• We will assume that investors also can borrow
  money at the risk-free rate. I.e., investors may
  write IOU notes, promising the risk-free rate to
  their buyers, thereby, raising capital at the
  risk-free rate.

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• Risk-free lending and borrowing
• The One-Price Law
• There exists only one
• risk-free rate in an efficient economy.
• Proof If two risk-free rates exist in
• the market concurrently, all investors
• will try to borrow at the lower rate
• and simultaneously try to invest at
• the higher rate for an immediate
• arbitrage profit. These activities will
• increase the lower rate and decrease
• the higher rate until they coincide to
• one unique risk-free rate.

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• Risk-free lending and borrowing
• By purchasing the risk-free asset,
• investors lend capital.
• By selling the risk-free asset, investors borrow
  capital.
• Both activities are at the
• risk-free rate.

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• We are now ready to calculate the current value
  of a T-Bill.
• Pt NPVthe T-bill Face-Value.
• Thus
• the current time, t, T-bill price, Pt , which
  pays FV upon its maturity on date T, is
• Pt FVe-r(T-t)
• Clearly, r is the risk-free rate in the economy.

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• EXAMPLE Consider a T-bill that promises its
  holder FV 1,000 when it matures in 276 days,
  with a yield-to-maturity of 5
• Inputs for the formula
• FV 1,000
• r .05
• T-t 276/365yrs
• Pt FVe-r(T-t)
• Pt 1,000e-(.05)276/365
• Pt 962.90.

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• EXAMPLE The yield-to -maturity of a bond which
  sells for 945 and matures in 100 days, promising
  the FV 1,000 is
• r ?
• Pt 945 FV 1,000 T-t 100 days.
• Inputs for the formula
• FV 1,000 Pt 945 T-t 100/365.
• Solving Pt FVe-r(T-t) for r
• r 365/100ln1,000/945
• r 10.324.

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• SHORT SELLING STOCKS
• An Investor may call a broker and ask to sell a
  particular stock short.
• This means that the investor does not own shares
  of the stock, but wishes to sell it anyway.
• The investor speculates that the stocks
• share price will fall and money will be
• made upon buying the shares back at a
• lower price. Alas, the investor does not
• own shares of the stock. The broker
• will lend the investor shares from the
• brokers or a clients account and sell it
• in the investors name. The investors
• obligation is to hand over the shares
• some time in the future, or upon the
• brokers request.

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• SHORT SELLING STOCKS
• Other conditions
• The proceeds from the short sale cannot be
• used by the short seller. Instead, they are
• deposited in an escrow account in the
• investors name until the investor makes
• good on the promise to bring the shares
• back.
• Moreover, the investor must deposit an
• additional amount of at least 50 of the
• short sales proceeds in the escrow account.
• This additional amount guarantees that there
• is enough capital to buy back the borrowed
• shares and hand them over back to the
• broker, in case the shares price increases.

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• SHORT SELLING STOCKS
• There are more details associated with short
  selling stocks. For example, if the stock pays
  dividend, the short seller must pay the dividend
  to the broker. Moreover, the short seller does
  not gain interest on the amount deposited in the
  escrow account, etc.
• We will use stock short sales in many of
• strategies associated with options
• trading.
• In all of these strategies, we will assume that
  no cash flow occurs from the time the strategy is
  opened with the stock short sale until the time
  the strategy terminates and the stock is
  repurchased.
• In terms of cash flows
• St is the cash flow from selling the stock
• short on date t, and
• -ST is the cash flow from purchasing the
• back on date T.

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THE FORWARDS AND FUTURES MARKETS

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CASH OR SPOT MARKETTHE MARKET FOR
IMMEDIATEDELIVERY AND PAYMENT

• GAS STATION, GROCERY STORE, DEPARTMENT
  STORE..
• SELLER BUYER DELIVERS ACCEPTS
• COMMODITY COMMODITY
• NOW NOW
• ACCEPTS PAYS
• PAYMENT NOW
• NOW
• The SELLER is said to be LONG
• The BUYER is said to be SHORT

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• A FORWARD MARKET
• THE MARKET FOR DEFERRED DELIVERY AND DEFFERED
  PAYMENT.
• SELLER SHORT
• BUYER LONG
• THE TWO PARTIES MAKE
• A CONTRACT THAT DETERMINES THE
• DELIVERY AND PAYMENT PLACE AND TIME IN THE FUTURE.

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• A FORWARD IS
• A BILATERAL AGREEMENT IN WHICH ONE PARTY COMMITS
  TO BUY AND THE OTHER PARTY COMMITS TO SELL A
  SPECIFIED AMOUNT OF AN AGREED UPON COMMODITY FOR
  A PREDETERMINED PRICE ON A SPECIFIC DATE IN THE
  FUTURE.

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A FUTURES IS NOTHING MORE THAN A
STANDARDIZED FORWARD TRADED ON AN ORGANIZED
EXCHANGE. STANDARDIZATION THE COMMODITY TYPE
AND QUALITY THE QUANTITY PRICE QUOTES DELIVERY
DATES DELIVERY PROCEDURES
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NYMEX Light Crude Oil Futures
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CBOT Corn Futures
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CBOT U.S. Treasury Bond Futures
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CME Standard Poors 500 Stock Index Futures
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NIKKEI 225 Stock Index Futures
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The Delivery Sequence for T-Bond Futures
Source Chicago Board of Trade
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HOW ARE FUTURES CONTRACTS CREATED ? FUTURES
CONTRACTS ARE SUGGESTED BY THE FUTURES
EXCHANGES THE PROPOPSALS ARE SENT FOR APPROVAL TO
THE REGULATORY AUTHORITY THE FUTURES COMMODITY
TRADING COMMISSION. (FCTC)
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WHY TRADE FUTURES AND NOT FORWARDS? FORWARDS
ARE CONTRACTS WITH Credit risk Operational
risk Liquidity risk
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• Credit Risk
• Does the other party have the mean to pay?

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2. Operational Risk Will the other party deliver
the commodity? Will the other party take
delivery? Will the other party pay?
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• Liquidity Risk.
• In case either party wishes to get out of its
  side of the contract, what are the obstacles?
• Find another counterparty. It may not be easy to
  do that. Even if you find someone who is willing
  to take your side of the contract, the other
  party may not agree.

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The exchanges understood that there will exist no
efficient markets until the above problems are
resolved. So they created the CLEARINGHOUSE
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THE CLEARINGHOUSE PLACE IN THE MARKET
EXCHANGE CORPORATION
CLEARINGHOUSE
CLEARING MEMBERS
NONCLEARING MEMEBRS
Futures Commission Merchants
CLIENTES
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The clearinghouse is a non profit corporation. It
gives every trading party an absolute guarantee
of the completion of its side of the contract
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The Clearinghouse guarantee LONG will be able
to take delivery and pay the agreed upon
price. SHORT will be able to deliver and
receive the agreed upon price.
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• Outside Customers
• A B C D E
• Customer
• FCM a FCM b FCM c
  Margins
• Clearing Clearing
• member 1 member 2
• Clearinghouse Clearinghouse
  Clearing margins
• B A

FCM FUTURES COMMODITY MERCHANT
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• A. BUYER LONG B. SELLER SHORT
• 10 OIL FUTURES 10 OIL FUTURES
• FOR 20/ bbl
• A BUY CH SELL B
• CLEARINGHOUSE GUARANTEE
• LONG SHORT
• BUY 10 JUNE CRUDE 20 SELL 10 JUNE CRUDE
• THE CH GIVES BOTH A AND B AN ABSOLUTE
  GUARANTEE OF THEIR SIDE OF
• THE AGREEMENT.
• THUS,
• 1. THERE IS NO CREDIT PROBLEM !
• 2. LIQUIDITY PROBLEMS ARE MINIMIZED.

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• Buyer Seller
• Member Buying Selling Member
• firm floor floor firm
• broker broker
• Trading Ring
• Buying Orders executed by open Selling
• floor outcry by buying and selling floor
• broker floor brokers, recorded and broker
• confirms placed on ticker confirms
• purchase sale
• Member Reports Reports
  Member
• firm purchase sale firm

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• Seller-long Buyer-short
• with obligation to pay with
  obligation
• for and take delivery to deliver
• Member Selling Buying Member
• firm floor floor firm
• broker broker
• Trading Ring
• Selling Buying
• floor Orders executed by floor
• broker open outcry by buying
  and broker
• confirms selling floor brokers,
  recorded confirms
• sale and placed on
  ticker purchase

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• Clearing association
• Member accounts
• Long Short
• FCM (A) 250 230
• FCM (B) 20
• Member FCM (A) Member FCM (B)
• Customers accounts Customers accounts
• Long Short Long Short
• 100 90 0 20
• Omnibus accounts
• Long Short
• 150 140

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MARGINS A MARGIN is an amount of money that must
be deposited in a margin account in order to open
any futures position. It is a good will
deposit. The clearinghouse maintains a system of
margin requirements from all traders, brokers and
futures commercial merchants.
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Most of the time, Initial margins are between 3
to 10 of the position value. Maintenance (or
variable) margin is usually around 70 of the
initial margin. If, for example, you open a
position in 10 CBT treasury bonds futures
(100,000 face value each) at a price of 75,000
each, your initial margin deposit of 5 of
750,000 will stand at 37,500. You will receive
a MARGIN CALL when the margin in your margin
account will drop to below 75 of this amount or,
26,250.
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How does your margin changes in the margin
account? MARKING TO MARKET Every day, upon the
market close, all profits and losses for that day
must be SETTLED in cash. The capital in the
margin accounts is used in order to settle the
accounts, using the SETTLEMENT PRICES
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A SETTLEMENT PRICE IS the average price of
trades during the last several minutes of the
trading day. Every day, when the markets close,
SETTLEMENT PRICES for the futures of all products
and for all months of delivery are set. They are
then compared with the previous day settlement
prices and the difference must be settled
overnight!!!!!!!
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OPEN A LONG POSITION IN 10 JUNE CRUDE OIL
FUTURES AT 18.50/bbl.VALUE (10)(1,000)(18.50)
185,000INITIAL MARGIN (.03)(185,000)
5,550
3,750/5,550 .676
MARGIN CALL ADD 1,800 TO MARGIN
ACCOUNT TO BRING IT UP TO 5,550 5,550 DAY
4 18.97 189,700 6,500 12,050
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A contract 1M face value of 90-day T-bills.
The implied settlement price is 100 - (100 -
P)(90/360), where P is the quoted settlement
price. Without interest earned Margin is
assumed to be 5 of contract fee.
SETTLEMENT PRICES AND MARK-TO-MARKET SETTLEMENTS
ON 90-DAY TREASURY BILL FUTURES FOR JUNE 19,1999,
SETTLEMENT.
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JUNE WTI FUTURE 1,000 bbls PER CONTRACT
DATE PARTY NUM PRICE
PARTY NUM PRICE OI Th.5.16
ALONG 10 20 CH BSHORT
10 20 10 5.16
CLONG 25 21 CH DSHORT
25 21 35 5.16
SETTLE 21

21 Fr.5.17 ELONG 10 22
CH ASHORT 10 22
35 5.17 SETTLE 22

22 Mo.5.20 DLONG 25 22.5
CH FSHORT 25 22.5
35 5.20 BLONG 10
21.5 CH CSHORT 10 21.5
25 5.20 SETTLE
21.5
21.5 Tu.5.21 FLONG 10
21 CH ESHORT 10 21
15 5.21 SETTLE
21
21 We.5.22 FLONG 10
20 CH CSHORT 10 20
5 5.22 SETTLE
20
20 OI Open Interest
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CLEARINGHOUSE ACCOUNTING A LONG 10 SHORT 10
OUT B SHORT 10 LONG 10 OUT C LONG
25 SHORT 10 SHORT 10 C remains LONG
5. D SHORT 25 LONG 25 OUT E LONG 10
SHORT 10 OUT F SHORT 25 LONG 10 LONG
10 F remains SHORT 5.
5.23 F DECIDES TO DELIVER 5 FUTURES
C ACCEPTS DELIVERY OF 5 CONTRACTS. The
actual delivery is now scheduled for June 23.

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CLEARINGHOUSE PROFIT/LOSS ZERO
LONG PRICE SHORT PRICE
TOTAL PROFIT A 10 20
10 22 20,000 B
10 21.5 10
20 -15,000 C 10
21 10 21.5
5,000
10 20 -10,000 D
25 22.5 25
21 -37,500 E 10
22 10 21
-10,000 F 10 21
25 22.5 15,000
10 20
25,000
TOTAL
-7,500 C TAKES DELIVERY 5 PAYS
21 -105,000 F DELIVERS 5
RECEIVES 22.5 112,500

7,500 TOTAL 0
This calculation accounts for buying and selling
only. It does not account for cash movements
resulting from the daily marking-to-market
process.
THE ACTUAL PROFITS AND LOSSES OF MARKET
PARTICIPANTS ARE ACCUMULATED IN THE MARGIN
ACCOUNTS.
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The following exhibits illustrate the activity
in the margin account of each of the traders
focusing only on cash flow resulting from the
daily marking-to-market process. Thus, possible
margin calls are ignored. PARTY
A DATE ACTION PRICE SETTLE CASH FLOW
POSITION 5.16 LONG 10 20
Initial margin LONG 10
21
10,000 LONG 10 5.17
SHORT 10 22
10,000 0
TOTAL 20,000
As profit is 20,000 PARTY
B DATE ACTION PRICE SETTLE CASH FLOW
POSITION 5.16 SHORT 10 20
Initial margin SHORT 10
21
-10,000 SHORT 10 5.17
22
-10,000 SHORT 10 5.20 LONG 10
21.5 5,000
0 TOTAL
-15,000 Bs loss is
15,000
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PARTY C DATE ACTION PRICE
SETTLE CASH FLOW POSITION 5.16 LONG
25 21 21 Initial
margin LONG 25 5.17
22 25,000 5.20
SHORT 10 21.5
-5,000
21.5 -7,500
LONG 15 5.21
20.5 -15,000 LONG
15 5.22 SHORT 10 20
-5,000
20 -2,500
LONG 5 5.23 TAKE
DELIVERY OF 5,000 BARRELS
for 20/bbl -100,000
0 Cs total loss up to and and
including 5.22 is 10,000. Note that the 5
contracts that were delivered has accumulated the
following amount over the period 5.17
(5,000)(1) 5,000 5.20
(5,000)(-.5) -2,500 5.21
(5,000)(-1) -5,000 5.22
(5,000)(-.5) -2,500 5.23
(5,000)(-20) -100,000 Payment upon delivery
TOTAL.-105,000 The five contracts have
accumulated total payment of 105,000. Observe
105,000/5,000 21/bbl
AS PER THE INITIAL COMMITMENT.
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PARTY D DATE ACTION PRICE
SETTLE CASH FLOW POSITION 5.16 SHORT 25
21 Initial margin
SHORT 25 21
0
SHORT 25 5.17
22 -25,000
SHORT 25 5.20 LONG 25 22.5
-12,500 0
TOTAL
-37,500
Ds total loss is 37,500 PARTY
E DATE ACTION PRICE SETTLE CASH FLOW
POSITION 5.17 LONG 10 22
Initial margin LONG 10
22
0 LONG 10 5.20
21.5
-5,000 LONG 10 5.21 SHORT 10
21 -5,000
0
TOTAL -10,000 Es total loss is
10,000
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PARTY F DATE ACTION PRICE
SETTLE CASH FLOW POSITION 5.20 SHORT 25
22.5 Initial margin
SHORT 25
21.5 25,000 5.21 LONG
10 21 5,000

20.5 15,000 SHORT 15 5.22
LONG 10 20
5,000
20 2,500
SHORT 5 5.23 DELIVER 5,000
BARRELS for 20/bbl
100,000 0 Fs
total profit up to and including 5.22 is
52,500. Note that the 5 contracts that were
delivered has accumulated the following amount
over the period 5.20 (5,000)(1)
5,000 5.21 (5,000)(1)
5,000 5.22 (5,000)(.5)
2,500 5.23 (5,000)(20) 100,000
Payment upon delivery
TOTAL..112,500 The five contracts that
party F delivers accumulated a total of
112,500. Observe 112,500/5,000
22.5/bbl AS PER INITIAL COMMITMENT.
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THE MARKET PARTICIPANTS TRADERS OF
FUTURESMAY BE CLASSIFIED BYTHEIR GOALS

• SPECULATORS
• WILL OPEN A RISKY FUTURES POSITION FOR
  EXPECTED PROFITS.
• ARBITRAGERS
• WILL OPEN SIMULTANEOUS FUTURES AND CASH
  POSITIONS IN ORDER TO MAKE AN ARBITRAGE
  PROFIT.
• HEDGERS WILL OPEN A
• FUTURES POSITION IN ORDER MINIMIZE
• OR ELIMINATE ALL PRICE RISK.

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• SPECULATORS
• TAKE RISK FOR EXPECTED PROFIT.
• ON THE MARKET FLOOR, WE FIND EXCHANGE MEMBERS
  WHO TRADE FOR THEIR ON ACCOUNTS. THESE ARE
  SPECULATORS.
• SCALPERS LARGE POSITIONS
• SMALL PRICE MOVEMENTS
• NEVER STAY OPEN OVERNIGHT
• DAY TRADERS OPEN A POSITION IN THE
• MORNING CLOSE AT THE
• CLOSE OF THE SAME DAY.
• POSITION TRADERS HOLD OPEN POSITIONS
• FOR LONGER PERIODS THEY USUALLY OPEN
  SPREAD POSITIONS.
• OUTRIGHT SPECULATION GO LONG or GO SHORT
• A SPREAD LONG CONTRACT 1

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• PROFIT IN SPREADS MISALIGNMENT OF TWO
  DIFFERENT FUTURES PRICES
• CROSS COMMODITY SPREAD
• SHORT JUNE CRUDE OIL CONTRACT
• LONG JUNE HEATING OIL CONTRACT
• CROSS EXCHANGE SPREAD
• LONG WHEAT CBT
• SHORT WHEAT KCB
• TIME OR, CALENDAR SPREAD
• LONG CONTRACT MONTH 1, SAY JUNE
• SHORT CONTRACT MONTH 2, SAY SEPT.
• CALENDAR SPREAD
• SPREAD F 0,t1 - F 0,t2

88

• How to open a calendar spread?
• Rule 1 If the spread between two contracts
  narrows, a profit will occur if the lower-priced
  contract has been purchased and the higher-priced
  contract sold. A loss occurs if the lower-priced
  contract is sold and the higher-priced contract
  is purchased.
• Rule 2 If the spread between two contracts
  widens, a profit will occur if the lower-priced
  contract has been sold and the higher priced
  contract purchased. A loss occurs if the
  lower-priced contract is purchased and the higher
  priced contract is sold.

89
THEREFORE in deciding which contracts to buy and
sell Rule 1 If the spread is expected to
narrow SELL THE SPREAD! i.e., buy the low
priced contract and sell the high priced
contract Rule 2 If spread is expected to
widen BUY THE SPREAD! i.e., buy the high
priced contract and sell the low priced contract.
90
CALENDAR SPREADTHE SPECULATOR EXPECTS THE
SPREAD TO NARROW
ACTION SELL THE SPREAD July
December Heating Oil
Heating Oil Spread Initial Position
buy .80 sell .92
.12 Terminal Position sell .84, (.65)
buy .89, (.89) - .05, (.24)
gain .04 gain
.03 net gain .07 ( -.12
loss) sell 1.00
buy 1.05 - .05 IN
GALLONS July
December Spread Initial Position buy 42,000
gal. sell 42,000 gal. .12
.80/gal .92/gal
value, 33,600
value, 38,640 Terminal Position sell
42,000 gal. buy 42,000 gal.
.05 .84/gal.
.89/gal. value, 35,280
value, 37,380 gain .04
x gain .03 x 42,000
1,680 42,000 1,260 net
gain .07 x 42,000 2,940 TO TERMINATE THE
POSITION BUY THE SPREAD.
91
PURE ARBITRAGE PROFIT A PROFIT MADE 1.
WITHOUT EQUITY and 2. WITHOUT ANY RISK.
92

• ARBITRAGE WITH FUTURES
• SPOT FUTURES
• MARKET MARKET
• Buy the Sell futures
• product
• Or
• Sell the Buy futures
• product
• short

93

• ARBITRAGE BUY AND SELL THE SAME COMMODITIY
  SIMULTANEOUSLY IN TWO DIFFERENT MARKETS FOR A
  (RISK-FREE) SURE PROFIT, WITHOUT ANY INVESTMENT.
• THE CLASSICAL EXAMPLE
• SO , NY .9 /GALLON OF HEATING OIL
• SO , LONDON .8/GALLON OF HEATING OIL
• COST .05/GALLON.
• ARBITRAGE
• BUY IN LONDON -80 CENTS/GALLON
• SELL IN NY 90 CENTS/GALLON
• SHIP TO NY - 5 CENTS/GALLON
• ARBITRAGE PROFIT 5 CENTS/GALLON
• NO INVESTMENT IS REQUIRED!
• NO RISK IS TAKEN !
• MARKETS MUST ADJUST

94

• ARBITRAGE IN PERFECT MARKETS
• CASH -AND-CARRY
• NOW 1. BORROW CAPITAL
• 2. BUY IN THE SPOT MARKET AND CARRY
  IT TO DELIVERY
• 3. SELL FUTURES AGAINST THE
• STORED COMMODITY
• AT MATURITY 3. DELIVER THE STORED
• COMMODITY AGAINST THE
• SHORT FUTURES.
• 1. REPAY THE LOAN
• REVERSE CASH-AND-CARRY
• 1. SELL COMMODITY SHORT IN
• THE SPOT MARKET
• NOW 2. INVEST THE PROCEEDS IN
• GOVERNMENT SECURITIES
• 3. OPEN A LONG FUTURES
• POSITION

95

• EXAMPLE CASH - AND - CARRY ON AUG
  15, 2001
• SPOT CRUDE OIL 20/ bbl SO
• AUGUST 2002 FUTURES 23/ bbl FO , AUG 02
• ANNUAL RATE 10 CC
• 20e .1 22.10342 lt 23 FO , AUG 02
• TRANSACTION
• t 0 CASH FLOW
• BORROW 20,000 FOR 1 YR AT 10 20,000
• BUY 1,000 BARRELS OF CRUDE -20,000
• SELL ONE AUGUST 02 WTI FUTURES 0
• 0
• t 1 (AUGUST 2002)
• DELIVER THE 1,000 BARRELS TO CLOSE
• THE SHORT FUTURES POSITION 23,000
• REPAY THE LOAN -22,103.42
• SURE PROFIT 897.58

96

• EXAMPLE REVERSE CASH - AND - CARRY
  ON AUG 15, 2001
• SPOT CRUDE OIL 20 / bbl SO
• AUGUST 2002 FUTURES 22/ bbl FO , AUG 02
• ANNUAL RATE 10 CC
• 20e .1 22.10342 gt 22 FO , AUG 02
• TRANSACTION
• t 0 CASH FLOW
• SELL 1,000 BARRELS SHORT 20,000
• LEND 20,000 FOR 1YR AT 10 -20,000
• BUY ON AUGUST 1997 FUTURES 0
• 0
• t 1 (AUGUST 2002)
• COLLECT 20,000e .1 22,103.42
• TAKE DELIVERY OF 1,000 BARRELS -22,000.00
• DELIVER 1,000 TO CLOSE THE
• SHORT SPOT POSITION 0
• 103.42

97

• IN THE ABSENCE OF ARBITRAGE OPPORTUNITIES
• F0 , T S0 (1 COST-OF-CARRY)
• IN OUR EXAMPLE THE SPOT PRICE IS 20/bbl.
• THEREFORE, THE THEORETICAL FUTURES PRICE
  SATISFIES
• FO, AUG 02 20e.1 22.10342 /bbl
• ANY OTHER FUTURES PRICE WILL LEAD
• TO ARBITRAGE OPPORTUNITIES

98
ARBITRAGE IN THE REAL WORLD IMPEDIMENTS TRA
NSACTION COSTS DIFFERENT BORROWING AND
LENDING RATES MARGINS REQUIREMENTS RESTRICTE
D 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.
99

• FOR THE CASH - AND - CARRY
• BORROW AT THE BORROWING RATE
• CB
• 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

100

• FOR 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 AT THE LENDING RATE
• CL
• LONG FUTURES AT THE ASK PRICE
• F(ASK).
• PAY TRANSACTION COST ON
• SHORT SELLING SPOT
• LENDING
• BUYING FUTURES
• PAY MARGIN

101
With these market realities, a new no-arbitrage
condition emerges BL lt F lt BU.
F BU
F BL
time
As long as F fluctuates between the upper and
lower bounds there are no arbitrage profits.
102

• ARBITRAGE IN IMPERFECT MARKETS
• CASH -AND-CARRY
• NOW 1. BORROW CAPITAL
• 2. BUY IN THE SPOT MARKET AND CARRY
  IT TO DELIVERY
• 3. SELL FUTURES AGAINST THE
• STORED COMMODITY
• AT MATURITY 3. DELIVER THE STORED
• COMMODITY AGAINST THE
• SHORT FUTURES.
• 1. REPAY THE LOAN
• REVERSE CASH-AND-CARRY
• 1. SELL COMMODITY SHORT IN
• THE SPOT MARKET
• NOW 2. INVEST THE PROCEEDS IN
• GOVERNMENT SECURITIES
• 3. OPEN A LONG FUTURES
• POSITION

103
Prove the following bounds on a futures
price St, BID (1 - k)1 f(RL ) lt Ft,T lt
St,ASK (1 k)(1 RB) Where St, is the
commoditys spot price today , t. Note that you
buy at the ASK price and sell at the BID
price. Ft,T is todays futures price for
delivery at T. For trading futures purposes,
assume that F is used for buying and selling.
That is, no BID or ASK price. k is the
transaction cost associated with trading the spot
commodity. k is a percentage of the price per
unit. RL and RB are the annual lending and
borrowing rates, respectively. f is the fraction
of the proceeds from the commoditys short sale
that the arbitrageur may use. Note that the
remainder, 1 - f must remain in the
arbitrageurs escrow account.
104

• S0,BID (1 - T)1 f(CL ) lt F0, t lt S0,ASK (1
  T)(1 CB)
• EXAMPLE
• S0 , ASK 20.50 / bbl
• S0, BID 20.25 / bbl
• CB 12
• CL 8
• T 3
• 20.25(.97)1f(.08)ltF0,tlt 20.50(1.03)(1.12)
• 19.6425 f(1.57) lt F0,t lt 23.6488
• DEPENDING ON f, ANY FUTURES PRICE BETWEEN THE
• TWO LIMITS WILL LEAVE NO ARBITRAGE
• OPPORTUNITIES. THE CASH-AND-CARRY WILL COST
• 23.6488/bbl. THE REVERSE CASH-AND-CARRY WILL
• COST 19.6425 f(1.62). IF f0.5 THE LOWER BOUND
  IS
• 20.45.
• IN THE REAL MARKET, f 1, FOR SOME LARGE

105

• HEDGERS
• HEDGERS TAKE FUTURES POSITIONS IN ORDER TO
  ELIMINATE PRICE RISK.
• THERE ARE TWO TYPES OF HEDEGES
• A LONG HEDGE
• TAKE A LONG FUTURES POSITION IN ORDER
• TO LOCK IN THE PRICE OF AN ANTICIPATED
• PURCHASE AT A FUTURE TIME
• A SHORT HEDGE
• TAKE A SHORT FUTURES POSITION IN ORDER
• TO LOCK IN THE SELLING PRICE OF
• AN ANTICIPATED SALE AT A FUTURE TIME.
• ANTICIPATORY HEDGES ARE HEDGES, LONG OR
• SHORT, THAT HEDGERS OPEN IN ANTICIPATION OF

106
FUTURES and CASH PRICESAN ECONOMICS MODEL

• SPECULATORS WILL OPEN RISKY FUTURES
  POSITIONS FOR EXPECTED PROFITS.
• HEDGERS WILL OPEN FUTURES POSITIONS IN
  ORDER TO ELIMINATE ALL PRICE RISK.
• ARBITRAGERS WILL OPEN SIMULTANEOUS FUTURES
  AND CASH POSITIONS IN ORDER TO MAKE
  ARBITRAGE PROFITS.

107
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
108
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.
109
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.
110
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.
111
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.
112
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.
113
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.
114
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.
115
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
116
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
117
HEDGING IS ONE COMPONENT OF CORPORATE FINANCIAL
POLICY BY HEDGING THE FIRM MAY LOWER
EXPECTED TRANSACTION COST REDUCE THE
PROBABILITY OF BANKRUPCY SIGNAL TO CREDITORS
THAT FIRM IS SAFER REDUCE EXPECTED TAX
LIMITATIONS LOWER COST OF AGENCY CONTROL
PROBLEMS BENEFIT MANAGERS DIRECTLY
118
Example The Tax story Taxes and the Gain from
Hedging Consider an oil company whose assets
consist solely of 1 million barrels of oil
reserves that the firm intends to extract in one
year at a cost of 25 per barrel. The current
futures price for oil is 30 per barrel, and the
oil price in one year has an equal chance of
being 25 or 35 per barrel. For simplicity,
assume that the current futures price equals the
expected future spot price. The firm faces a 30
percent income tax rate and has a 1 million tax
credit that it can apply up to the amount of
income taxes paid.
119

• If the firm does not hedge, its after-tax
  profits under each oil price scenario will be
  I. 25 per Barrel
• Before-tax profits (25 - 25)(1M)
• 0.0 million
• Income tax 0.0 million
• After-tax profits 0.0 million
• The firm pays no taxes, because its taxable
  income is zero. It loses the
• 1 million tax credit.
• II. 35 per Barrel
• Before-tax profits (35 - 25)(1M)
• 10.0 million
• Income tax (.30)(10M)-1M 2.0M
• After-tax profits 8.0 million
• The firm pays only 2M in taxes, because it
  fully utilizes its tax credit of 1M.
• The firms expected after-tax profits in one
  year are
• (0.5) (0.0M) (0.5) (8.0M) 4.0M

120

• If the firm hedges with a short position in oil
  futures, its after-tax profits under the two oil
  price scenarios will be
• Before-tax profits (30 - 25)(1M)
• 5.0 million
• Income tax (0.30)(5M) 1M
• 0.5 million
• After-tax profits 4.5 million
• The expected after-tax profit is greater for the
  hedged firm than for the non hedged firm. The
  0.5 million difference is exactly equal to the
  non hedged firms expected loss of the 1 million
  tax credit. The hedged firm always utilized its
  tax credit fully, so its value is higher than
  that of the non hedged firm.

121

• In general, the effect of hedging when tax
  credits and deductions are available is
• Unhedged Hedged Expected loss
• firm firm - of credit
  and
• value value deductions
• The benefit of hedging when tax benefits could
  be lost will be mitigated if firms can carry tax
  credits and deductions forward and backward in
  time. Further, firms that will surely have ample
  income to use all of their credits and deductions
  will gain little value form hedging due to this
  tax effect.

122
A LONG HEDGE
RECALL THERE ARE TWO TYPES OF HEDGING

• LONG FUTURES IN ORDER TO HEDGE THE PRODUCT
  PURCHASE TO BE MADE
• AT A LATER DATE.
• I.E.,, LOCK IN THE PURCHASE PRICE.

A SHORT HEDGE
SHORT FUTURES IN ORDER TO HEDGE THE SALE OF
THE PRODUCT TO BE MADE AT A LATER DATE. I.E.,
LOCK IN THE SALE PRICE
123

• NOTATIONS
• F k,t THE FUTURES PRICE AT TIME k
  FOR DELIVERY AT TIME t.
• k lt t k current time t delivery time
• Sk THE SPOT PRICE AT TIME k.
• THE TERMS SPOT AND CASH
• ARE USED INTERCHANGEABLY.

124
BASIS AT ANY POINT IN TIME, k BASISk SPOT
PRICEk - FUTURES PRICEk NOTATIONALLY Bk Sk -
Fk,t k lt t Bt St - Ft, t 0 k t. t is
the nearest month of delivery which is at or
following k. The latter equation indicates that
the basis converges to zero on the delivery date.
Ft, t is the price of the commodity on date t for
delivery and payment on date t. Hence, Ft, t is
the spot price on date t Ft, t St .

125

• The relationship between the cash and the futures
  price over time
• The basis is the difference between two random
  variables. Thus, it varies in an unpredictable
  way. Over time, it narrows, widens and may change
  its sign.
• The basis converges to zero at the futures
  maturity.
• The basis is less volatile than either price
• Futures and spot prices of any underlying asset,
  co vary over time. Although not in tandem and not
  by the same amount, these prices move up and down
  together most of the time, during the life of the
  futures.
• RESULT 4. IS THE KEY TO THE SUCCESS OF HEDGING
  WITH FUTURES!

126
Convergence of Cash and Futures-Heating Oil
82 81 80 79 78
FUTURES PRICE
CENTS PER
GALLON
BASIS CASH - FUTURES
EXPIRATION DELIVERY
CASH PRICE
October November
December
127

• We now prove that hedging is the transfer of
  outright PRICE RISK
• to BASIS RISK.

Generally, the basis fluctuates less than both,
the cash and the futures prices. Hence, hedging
with futures reduces risk.
Sk
Pr
Bk
F0,t
S0
Bt 0
B0
time
k
t
O
128

• A LONG HEDGE
• TIME CASH FUTURES
• 0 DO NOTHING LONG F 0,t
• k BUY Sk SHORT F k,t
• t delivery
• ACTUAL PAYMENT Sk F0,t - Fk,t
• F0,t Sk - Fk,t
• F0,t BASIS k

129
A SHORT HEDGE TIME CASH FUTURES 0 DO
NOTHING SHORT F0,t k SELL Sk LONG Fk,t t
delivery ACTUAL SELLING PRICE Sk F0,t -
Fk,t F0,t Sk - Fk,t F0,t
BASISk
130
The last two slides prove that for both types of
hedge A SHORT HEDGE or A LONG HEDGE, The final
cash flow to the hedger is F0,t
BASISk Notice that this cash flow consists of two
parts the first - F0,t is KNOWN when the hedge
is opened. The second part - BASISk is a random
element. Conclusion At time 0, the firm faces
the cash-price risk. Upon opening a hedging
position, the firm locks in the futures price,
but it still remains exposed to the basis risk,
because the basis at time k is random.
131

• We thus, proved that hedging amounts to the
  reducing the firms risk exposure because the
  basis is less risky that the spot price risk.

Sk
Pr
Bk
F0,t
S0
Bt 0
B0
time
k
t
O
132
HEDGE RATIOS Open a hedge. Questions Long or
Short? Delivery month? Commodity to use? How many
futures to use? The number of futures in the
position is determined by the HEDGE RATIO
133
HEDGE RATIOSNAÏVE HEDGE RATIO ONE - FOR - ONE

• QUANTITIY OF CASH POSITION
• QUANTITY IN ONE FUTURE
• Examples
• Intend to sell 50,000 bbls of crude oil. Short
  50 NYMEX futures.
• Intend to borrow 10M for ten years. Short 100
  CBT T-bond futures.
• Intend to buy 17,000 pounds of gold. Long 170
  NYMEX futures

134
OPTIMAL HEDGE RATIOS THE MINIMUM VARIANCE HEDGE
RATIO GOAL TO MINIMIZE THE RISK ASSOCIATED WITH
VALUE CHANGE OF THE CASH - FUTURES POSITION. RISK
IS MEASURED BY VOLATILITY. THE VOLATILITY
MEASURE IS THE VARIANCE OF THE VALUE
CHANGE OBJECTIVE FIND THE NUMBER OF FUTURES
THAT MINIMIZES THE VARIANCE OF THE CHANGE OF THE
HEDGED POSITION VALUE.
135

• THE MATHEMATICS
• S CASH VALUE
• F FUTURES PRICE
• N NUMBER OF FUTURES EMPLOYED IN THE HEDGE.
• The initial and terminal hedged position values
• VP0 S0 NF0,t
• VP1 S1 NF1,t
• The position value change
• DVp VP1 - VP0
• (S1 NF1,t) - (S0 NF0,t ).
• Define DS S1 - S0 and DF F1,t - F0,t ,
  then DVP DS N(DF).

136

• AGAIN,
• DVP DS N(DF)
• PROBLEM
• GIVEN THE CASH AND FUTURES VALUE CAHNGES, FIND A
  NUMBER N, SO AS TO MINIMIZE THE VOLATILITY OF
  THE CHANGE IN THE HEDGERS COMBINED
• CASH FUTURES
• POSITION VALUE.

137

• THE MATHEMATICS
• .VAR (DVP) VAR (DS) VAR (NDF)
• 2COV (DS NDF)
• VAR (DS) N2VAR(DF)
• 2NCOV(DS DF).
• TO MINIMIZE VAR(DVP)
• Take its derivative with respect to N and
  equate it to zero
• 2NVAR (DF) 2COV (DSDF) 0
• N - COV(DSDF) / VAR(DF)

138
THE NUMBER OF FUTURES THAT MINIMIZES THE RISK OF
THE HEDGED POSITION IS
139
To evaluate the risk of the position at its
minimum level, substitute N into the formula of
the positions value change variance
How to calculate N in practice?
140
DATA (SAY DAILY) n1 DAYS.
141

• EXAMPLE
• A company needs to buy 800,000 gallons of diesel
  oil in 2 months. It opens a long hedge using
  heating oil futures. An analysis of price changes
  ?S and ?F over a 2 month interval yield
• SD(?S) 0.025
• SD(?F) 0.033 ? 0.693.
• The risk minimizing hedge ratio
• h (.693)(0.025)/0.033 0.525. One heating oil
  contract is for 42,000 gallons, so purchase
• N (0.525)(800,000)/42,000
• 10 futures.

142

• EXAMPLE, continued
• Notice that in this case, a NAÏVE HEDGE
• ratio would have resulted in taking a long
• position in
• 800,000/42,000 19 futures.
• Taking into account the correlation
• between the spot price changes and the
• futures price changes, allows the use of
• only 10 futures.
• Of course, if the correlation and the
• standard deviations take on other
• values the risk-minimizing hedge ratio
• may require more futures than the naïve
• ratio.

143
EXAMPLE A company knows that it will buy 1
million gallons of jet fuel in 3 months. The
company chooses to long hedge with heating oil
futures. The standard deviation of the change in
the price per gallon of jet fuel over a 3-month
period is calculated as 0.04. The standard
deviation of the change in the futures price over
a 3-month period is 0.02 and the coefficient of
correlation between the 3-month change in the
price of jet fuel and the 3-month change in the
futures price is 0.42. The optimal hedge
ratio H (0.42)(0.04)/(0.02) 0.84, And the
risk-minimizing number of futures N
(0.84)(1,000,000)/42,000 20.
144
HEDGE RATIOS As we move from one type of
underlying asset to another, we will use these
hedge ratios as well as new ones to be developed
later.
145
Delivery month? Normally, the hedge is opened
with futures for the delivery month closest to
the firm operation date in the cash market or the
nearest month beyond that date. The key factor
here is the correlation between the cash and
futures prices or price changes. Statistically,
it is known that in most cases, the highest
correlation is with the futures prices of the
delivery month nearest to the cash activity.