0 DERIVATIVES WORKBOOK By Ramon Rabinovitch. 1 DERIVATIVES ARE CONTRACTS Two parties Agreement...

1

DERIVATIVES

WORKBOOKBy

Ramon Rabinovitch

2

DERIVATIVES

ARE

CONTRACTS

Two parties

Agreement

Underlying security

3

DERIVATIVES

FORWARDS

FUTURES

OPTIONS

SWAPS

4

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.

5

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

6

AN OPTION ISA 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.

7

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.

8

WHY TRADE DERIVATIVES?

THE FUNDAMENTALREASON FOR TRADING

FORWARDS AND FUTURES IS :

PRICE RISK or

VOLATILITY

9

PRICE RISK IS THE

VOLATILITYASSOCIATED WITH THE

COMMODITY’S

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 VOLATILITYNO DERIVATIVES!!!!

100

Pr

S0

t time

St

PRICE RISK

At time zero the commodity’s price at time t is not known.

110

Pr

S0

t time

St

PRICE RISK

The larger the volatility, the more

need for derivatives

12

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

$/ B

bl

1940 1950 1960 1970 1980 1990 2000

Crude Oil PricesWellhead / First Sale

13

20.0

40.0

60.0

80.0

100.0

120.0

140.0

cent

s/ g

allon

1940 1950 1960 1970 1980 1990 2000

Gasoline PricesRetail

14

0.00

0.50

1.00

1.50

2.00

2.50

3.00

$/M

M B

tu

1940 1950 1960 1970 1980 1990 2000

Natural Gas PricesWellhead

15

1.0

2.0

3.0

4.0

5.0

6.0

7.0

cent

s/ k

Wh

1940 1950 1960 1970 1980 1990 2000

Electricity PricesComposite Retail

16

THE ECONOMIC PURPOSES OF DERIVATIVE MARKETS

HEDGINGPRICE DISCOVERYSAVING

HEDGING IS THE ACTIVITY OF MANAGING PRICE RISK EXPOSURE

PRICE DISCOVERY IS THE REVEALINGOF INFORMSTION ABOUT THE FUTURECASH MARKET PRICE FOR A PRODUCT.

SAVING IS THE COST SAVING ASSOCIATED WITH SWAPING CASH FLOWS

17

ALTHOUGH THE ECONOMIC PURPOSES OF

DERIVATIVE MARKETS ARE

HEDGINGPRICE DISCOVERYSAVING

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

18

FORWARDS AND FUTURES

The CONTRACTSThe MARKETS

PRICING FUTURES

SpeculationArbitrageHedging

19

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.

20

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.

21

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.

22

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

23

Compounded InterestAny 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.

24

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 = $1001. 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.

25

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:

.m

r1PA

mn

n

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.

26

This reasoning implies that in order to impose the concept of continuous time on the above compounding expression, we need to solve:

}m

r1{PLimitA

mn

mn

This expression may be rewritten as:

.PeA

:yearsn after P of valuecompounded

ly continuous for the expression the

yieldslimit thisofsolution The

}.

rm1

1{(P)LimitA

rnn

rn)r

m(

m n

27

EXAMPLE, continued:First, we remind you that the number eis defined as:

}x

11{Limite

x

x

For example:

x e

1 2

10 2.59374246

100 2.70481382

1,0002.71692393

10,000 2.71814592

1,000,000 2.71828046

In the limit e = 2.718281828…..

28

Recall that in our example:N = 10 years and r = 12% and P=$100.Thus, P=$100 invested at a 12% annual rate, continuously compounded for ten years will grow to:

01.332$$100e PeA (.12)(10)rnn

Continuous compounding yields the highest return to the investor:

CompoundingFactor

Simple3.105848208

Quarterly3.262037792

Monthly3.300386895

Daily3.319462164

continuously 3.320116923

29


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

30


rate.interest compounded

ly continuous theisr where

,e :byit gmultiplyin

bypresent for the discounted

lycontinuous be can ,CF flow, casht

period any time generally, More

.eA P

:is A of value

discountedly continuous the

n, andr ,A given clear that

isit ,PeA formula,

gcompoundinly continuous theFrom

rt -

t

rn -n

n

n

rnn

Continuous Discounting

31

EXAMPLE, continued:First, we remind you that the

number e is defined as:

}.x

11{Limite

x

x

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: 01.332$$100e PeA (.12)(10)rn

n

Therefore, we can write the continuously discounted value of $320.01 is:

.100$$332.01e eAA (.12)(10) --rnn0

32

This expression may be rewritten as: t.any timefor A

calculate can we t,andr

, Pgiven clear that isit

,PeA

formula, gcompoundin

lycontinuous theFrom

t

rtt

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% ==> t = 6.931yrs.

33

PURE ARBITRAGE PROFIT:

A PROFIT MADE

1. WITHOUT EQUITYand

2. WITHOUT ANY RISK.

34

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.

35


PURE ARBITRAGE PROFIT:A PROFIT MADE

1.WITHOUT EQUITY INVESTMENTand

2. 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

36


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 bond’s 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 bill’s maturity. Hence, its current market price is the NPV of the bill’s Face Value:

Pt = NPV{the T-bill Face-Value}

We will only use continuous discounting

37


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.

38


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.

39


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.

40

We are now ready to calculate the current value of a T-Bill.

Pt = NPV{the T-bill Face-Value}.

Thus:the current time, t, T-bill price, Pt , which pays FV upon its maturity on date T, is:

Pt = [FV]e-r(T-t)

Clearly, r is the risk-free rate in the economy.

41

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 = $1,000r = .05

T-t = 276/365yrs

Pt = [FV]e-r(T-t)

Pt = [$1,000]e-(.05)276/365

Pt = $962.90.

42

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 = [FV]e-r(T-t) for r:

r = [365/100]ln[$1,000/$945]

r = 10.324%.

]P

FVln[

t-T

1 r

t

43

SHORT SELLING STOCKSAn 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 stock’sshare 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 thebroker’s or a client’s account and sell itin the investor’s name. The investor’s obligation is to hand over the shares some time in the future, or upon the

broker’s request.

44

SHORT SELLING STOCKSOther conditions:The proceeds from the short sale cannot

be used by the short seller. Instead, they

are deposited in an escrow account in the investor’s 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 sale’s proceeds in the escrow

account. This additional amount guarantees that

thereis enough capital to buy back the

borrowed shares and hand them over back to the broker, in case the shares price

increases.

45

SHORT SELLING STOCKSThere 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.

46

THE FORWARDS AND FUTURES MARKETS

47

CASH OR SPOT MARKET

THE MARKET FOR IMMEDIATE

DELIVERY 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

48

A FORWARD MARKET

THE MARKET FOR DEFERRED DELIVERY

AND DEFFERED PAYMENT.

SELLER = SHORT

BUYER = LONGTHE TWO PARTIES MAKE

A CONTRACT THAT DETERMINES THE

DELIVERY AND PAYMENT PLACE AND TIME IN THE

FUTURE.

49

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.

50

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

51

NYMEX Light Crude Oil Futures

Trading Unit 1,000 U.S. barrels (42,000 gallons)

Tick Size

cent per barrel ($10 per contract)

Daily Price Limit

$7.50 per barrel ($7,500 per contract) for the first two contract months. Initial back-mont limits of $1.50 per barrel rise to $3 per barrel if the previous day’s settlement price is at the $1.50 limit. In the event of a $7.50 move in either of the first two contract months, back-month limits are expanded to $7.50 per barrel from the limit in place in the direction of the move.

Contract Months 18 consecutive months plus four long-dated futures that are initially listed 21,24,30, and 36 months prior to delivery.

Trading Hours

9:45 a.m. to 3:10 p.m. (New York Time)

Last Trading Day Third business day prior to the 25th calender day of the month preceding the delivery month.

Deliverable Grades Specific crudes with 0.5 percent sulfer by weight or less, not less than 34 degress API gravity nor more than 45 degrees API gravity. The following crude streams are deliverable: West Texas Intermediate, Mid-Continent Sweet, Low Sweet Mix, New Mexico Sweet, North Texas Sweet, Oklahoma Sweet, South Texas Sweet, Brent Blend, Bonny Light, and Oseberg Blend. Contact the exchange for details on price discounts and premiums.

52

CBOT Corn Futures

Trading Unit 5,000 bushels

Tick Size ¼ cent per bushel ($12.50 per contract)

Daily Price Limit 12 cents per bushel ($600 per contract) above or below the previous day’s settlement price (expandable to 18 cents per bushel). No limit in the spot month.

Contract Months December, March, May, July, September

Trading Hours 9:30 a.m. to 1:15 p.m. (Chicago time), Monday through Friday. Trading in expiring contracts closes at noon on the last trading day.

Last Trading Day Seventh business day preceding the last business day of the delivery month.

Deliverable Grades No. 2 Yellow at par and substitution at differentials established by the exchange.

53

CBOT U.S. Treasury Bond Futures Trading Unit $100,000 face value U.S. Treasury

bonds

Tick Size 1/32 of a point ($31.25 per contract); par is on the basis of 100 points

Daily Price Limit Three points ($3,000) per contract above or below the previous day’s settlement price (expandable to 4 ½ points). Limits are lifted the second business day preceding the first day of the delivery month.

Contract Months March, June, September, December

Trading Hours 7:20 a.m. to 2:00 p.m. (Chicago time), Monday through Friday. Evening trading hours are 5:20 p.m. to 8:05 p.m. (Chicago time), or 6:20 p.m. to 9:05 p.m. (central daylight savings time), Sunday through Thursday. Contract also trades on the GLOBEX® system

Last Trading Day Seven business days prior to the last business day of the delivery month.

Deliverable Grades U.S. Treasury bonds maturing at least 15 years from the first business day of the delivery month, if not callable; if callable, not so for at least 15 years from the first day of the delivery month. Coupon based on an 8 percent standard

Delivery Federal Reserve book-entry wire-transfer system

54

CME Standard & Poor’s 500 Stock Index Futures

Trading Unit $500 times the Standard & Poor’s500 Stock Index

Tick Size .05 index points ($25 per contract)

Daily Price Limit Coordinated with trading halts ofthe underlying stocks listed fortrading in the securities markets.Contact exchange for details of thisrule.

Contract Months March, June, September,December

Trading Hours 8:30 a.m. to 3:15 p.m. (Chicagotime). The contract also trades onthe GLOBEX ® trading system.

Last Trading Day The business day immediatelypreceding the day of determinationof the final settlement price(normally, the Thursday prior to thethird Friday of the contract month)

Delivery Cash settled

55

NIKKEI 225 Stock Index Futures

Trading Unit 1,000 times Nikkei stock average

Tick Size 10 per Nikkei stock average (minimum value 10,000)

Daily Price Limit Plus or minus 3 percent of the previous day’s closing price

Contract Months March, June, September, December cycle (five contract months traded at all times)

Trading Hours 9:00 a.m. to 11:00 a.m. and 12:30 p.m. to 3:00 p.m. (Osaka time)

Last Trading Day The business day before the second Friday of each contract month

Delivery Cash settled

56

The Delivery Sequence for T-Bond Futures

Before DeliveryTh e sh ort req u ires th e fin an c ia l

In s tu rm en t fo rd e live ry

After DeliveryTh e lon g c an :

*h o ld th e fin an c ia l in s tru m en t an d re ta in ow n ers h ip*red e live r in s tru m en ts

Day 3 Delivery DayTh e sh ort d e live rs th e fin an c ia l in s tru m en t to th e lon g

Th e lon g m akes p aym en t to th e sh ortT it le p ass es * Th e lon g assu m es a ll ow n ersh ip rig h ts an d resp on s ib lit ies

Day 2 Notice of Intention DayTh e C learin g C orp era tion m atch es th e o ld es t lon g to th e d e live rin g

sh ort th en n o tifies b o th p a rt iesTh e sh ort in vo ices th e lon g .

Day 1 Position DayTh e sh ort d ec la res h is o r h e r p os it ion b y

n o tifyin g th e C learin g C orp era tion th a t h e o r sh e in ten d s tom ak e d e live ry

First Position DayTh e lon g d ec la res h is o r h e r op en p os it ion s

Trad er n o tifies th e C learin g C orp era tiontw o b u s in ess d ays b e fo re th e firs t d ay a llow ed fo r d e live ries in th a t m on th

Source: Chicago Board of Trade

57

HOW ARE FUTURESCONTRACTSCREATED ?

FUTURES CONTRACTS ARE SUGGESTED BY THE FUTURES

EXCHANGESTHE PROPOPSALS ARE SENT

FOR APPROVALTO THE REGULATORY

AUTHORITY:

THE FUTURESCOMMODITY TRADING

COMMISSION.(FCTC)

58

WHY TRADE FUTURES AND NOT FORWARDS?

FORWARDS ARE CONTRACTS WITH:

Credit risk

Operational risk

Liquidity risk

59

1.Credit Risk

Does the other party have the mean to

pay?

60

2. Operational Risk:

Will the other party deliver the

commodity?

Will the other party take delivery?

Will the other party pay?

61

3.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.

62

The exchanges understood that there will exist no efficient

markets until the above problems are resolved. So they

created the:

CLEARINGHOUSE

63

CLEARING

MEMBERS

NONCLEARING

MEMEBRS

EXCHANGE CORPORATION

CLEARINGHOUSE

Futures Commission Merchants

CLIENTES

THE CLEARINGHOUSE PLACE IN THE MARKET

64

The clearinghouse is a non profit corporation. It gives every trading party an absolute guarantee of

the

completion of its side of the contract

65

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.

66

816 seats, 749 individual members

NYMEX Membership

COMEX Membership

772 seats, 663 individual members

Executive

committee

Board

of

Directors

Chairmen of the Board

President

Planning &

developmentCompliance Clearing

Market

Surveillance

Financial surveillance

Trade

Surveillance

Strategic Planning

Research Marketing Banking & Delveries

Position processing

NYMEX Organization

67

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

68

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.

69

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

Confirms Clearinghouse Confirms

Purchase sale

1 1

Obligation Obligation

long short

Buyer Total open interest 1 contract Seller

now now

long long

1 1

contract contract

70

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

Member Reports Reports Member

firm sale purchase firm

Confirms Clearing House Confirms

Sale 1 Obligation 1 Obligation purchase

or long or short

sold purchased

canceling canceling

Buyer has buy sell Seller has

offset obligation obligation offset

obligation Total open interest obligation

by sale- 0 contracts by purchase-

no market no market

position position

71

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

Customer 1 Customer 2 Customer 3

100 long 90 short 20 short

Non-clearing FCM

Customer’s accounts:

Long Short

150 140

Customer 4 Customer 5

150 long 140 short

72

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.

73

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.

74

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

75

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!!!!!!!

76

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

SETTLE PRICE

VALUE

MARKET-TO-MARKET

MARGIN BALANCE

$18.50 $185,000 $5,550

DAY 1 $18.42 $184,200 - $800 $4,750

DAY 2 $18.75 $187,500 + $3,300 $8,050

DAY 3 $ 18.32 $183,200 - $4,300 $3,750

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

77

* 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.

Date Settlement price

Dollar settlement price*

Mark-to-Market for the long

Margin Account **

June 2 92.23 980,575 50,000

3 92.73 981,825 $1250 51,250

4 92.83 982,075 250 51,500

5 93.06 982,650 575 52,075

6 93.07 982,675 25 52,100

9 93.48 983,700 1025 53,125

10 93.18 982,850 -750 52,375

11 93.32 983,300 350 52,725

12 93.59 983,975 675 53,400

13 93.84 984,600 625 54,025

16 93.71 984,275 -325 53,700

17 93.25 983,126 -1150 52,550

18 93.12 982,800 -325 52,225

SETTLEMENT PRICES AND MARK-TO-MARKET SETTLEMENTS ON 90-DAY TREASURY BILL FUTURES

FOR JUNE 19,1999, SETTLEMENT.

78

JUNE WTI FUTURE 1,000 bbls PER CONTRACT

DATE PARTY NUM PRICE PARTY NUM PRICE OI*

Th.5.16 A:LONG 10 $20 CH B:SHORT 10 $20 10

5.16 C:LONG 25 $21 CH D:SHORT 25 $21 35

5.16 SETTLE $21 $21

Fr.5.17 E:LONG 10 $22 CH A:SHORT 10 $22 35

5.17 SETTLE $22 $22

Mo.5.20 D:LONG 25 $22.5 CH F:SHORT 25 $22.5 35

5.20 B:LONG 10 $21.5 CH C:SHORT 10 $21.5 25

5.20 SETTLE $21.5 $21.5

Tu.5.21 F:LONG 10 $21 CH E:SHORT 10 $21 15

5.21 SETTLE $21 $21

We.5.22 F:LONG 10 $20 CH C:SHORT 10 $20 5

5.22 SETTLE $20 $20

* OI = Open Interest

79

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.

80

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.

81

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 105.17 SHORT 10 $22 +$10,000 0 TOTAL $20,000

A’s profit is = $20,000

PARTY B:


5.16 SHORT 10 $20 Initial margin SHORT 10 $21 -$10,000 SHORT 105.17 $22 -$10,000 SHORT 105.20 LONG 10 $21.5 +$5,000 0 TOTAL -$15,000

B’s loss is = $15,000

82

PARTY C:


5.16 LONG 25 $21 $21 Initial margin LONG 255.17 $22 +$25,0005.20 SHORT 10 $21.5 -$5,000 $21.5 -$7,500 LONG 155.21 $20.5 -$15,000 LONG 155.22 SHORT 10 $20 -$5,000 $20 -$2,500 LONG 55.23 TAKE DELIVERY OF 5,000 BARRELS for $20/bbl -$100,000 0

C’s 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) = $5,0005.20 (5,000)(-$.5) = -$2,5005.21 (5,000)(-$1) = -$5,0005.22 (5,000)(-$.5) = -$2,5005.23 (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.

83

PARTY D:


5.16 SHORT 25 $21 Initial margin SHORT 25 $21 0 SHORT 255.17 $22 -$25,000 SHORT 255.20 LONG 25 $22.5 -$12,500 0 TOTAL -$37,500

D’s total loss is = $37,500

PARTY E:


5.17 LONG 10 $22 Initial margin LONG 10 $22 0 LONG 105.20 $21.5 -$5,000 LONG 105.21 SHORT 10 $21 -$5,000 0 TOTAL -$10,000

E’s total loss is = $10,000

84

PARTY F:


5.20 SHORT 25 $22.5 Initial margin SHORT 25 $21.5 +$25,0005.21 LONG 10 $21 +$5,000 $20.5 +$15,000 SHORT 155.22 LONG 10 $20 +$5,000 $20 +$2,500 SHORT 55.23 DELIVER 5,000 BARRELS for $20/bbl +$100,000 0

F’s total profit up to and including 5.22 is $52,500.

Note that the 5 contracts that were delivered has accumulated thefollowing amount over the period:

5.20 (5,000)($1) = $5,0005.21 (5,000)($1) = $5,0005.22 (5,000)($.5) = $2,5005.23 (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.

85

THE MARKET PARTICIPANTS: TRADERS OF FUTURES

MAY 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.

86

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

and

simultaneously

SHORT CONTRACT 2

87

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

SPREAD = JUNE FUTURES - SEPT FUTURES

PRICE PRICE

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 EQUITYand

2. WITHOUT ANY RISK.

92

ARBITRAGE WITH FUTURES:

SPOT FUTURESMARKET 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

AT MATURITY: 2. COLLECT CAPITAL FROM INVESTMENT IN THE GOVERNEMENT SECURITIES

3. TAKE DELIVERY AGAINST LONG

FUTURES POSITION.

1. CLOSE THE SHORT SPOT 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 < 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

NOTICE: NO EQUITY IS USED and

NO RISK IS TAKEN

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 > 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

SURE PROFIT: 103.42

NOTICE: NO EQUITY IS USED and

NO RISK IS TAKEN

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

TRANSACTION COSTS

DIFFERENT BORROWING AND LENDING RATES

MARGINS REQUIREMENTS

RESTRICTED SHORT SALES AN USE OF PROCEEDS

STORAGE LIMITATIONS

* BID - ASK SPREADS

** MARKING - TO - MARKET

* BID - THE HIGHEST PRICE ANY ONE IS WILLING TO BUY AT NOW

ASK - THE LOWEST PRICE ANY ONE IS WILLING TO SELL AT NOW.

** MARKING - TO - MARKET: YOU MAY BE FORCED TO CLOSE YOUR POSITION BEFORE ITS MATURITY.

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 < F < 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

AT MATURITY: 2. COLLECT CAPITAL FROM INVESTMENT IN THE GOVERNEMENT SECURITIES

3. TAKE DELIVERY AGAINST LONG

FUTURES POSITION.

1. CLOSE THE SHORT SPOT POSITION.

103

Prove the following bounds on a futures price:St, BID (1 - k)[1 + f(RL )] < Ft,T

< St,ASK (1 + k)(1 + RB)Where:St, is the commodity’s spot price today , t. Note that you buy at the ASK price and sell at the BID price.Ft,T is today’s 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 commodity’s short sale that the arbitrageur may use. Note that the remainder, 1 - f must remain in the arbitrageur’s escrow account.

104

S0,BID (1 - T)[1 + f(CL )] < F0, t < 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)[1+f(.08)]<F0,t< $20.50(1.03)

(1.12)

$19.6425 + f($1.57) < F0,t < $23.6488

DEPENDING ON f, ANY FUTURES PRICE BETWEEN THETWO 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 f=0.5 THE LOWER BOUND

IS$20.45. IN THE REAL MARKET, f = 1, FOR SOME LARGEARBITAGE FIRMS AND THEIR LOWER BOUND IS $21.26.THUS, IT IS CLEAR THAT THERE ARE DIFFERENT ARBITRAGE BOUNDS APPLICABLE TO DIFFERENT INVESTORS. THE TIGHTER THE BOUNDS, THE GREATER ARE THE ARBITRAGE OPPORTUNITIES.

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

THE COMMODITY SPOT PRICE INCREASE IN THE

FUTURE.

106

FUTURES and CASH PRICES:

AN 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

Demand for LONG futures positions by long HEDGERS

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.c

b

a

Od0 Quantity of long positions

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)

Expt [St+k]

108

Supply of SHORT futures positions by short HEDGERS.

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.

f

e

d

QS0 Quantity of short positions

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)

Expt [St + k]

109

Equilibrium in a futures market with a preponderance of long

hedgers.

D

S

D

Qd0 Quantity of

positions

Ft (k)

Expt [St + k]

S

Ft (k)e

Supply schedule

Demand schedule

Premium

QS

110

Equilibrium in a futures market with a preponderance of short

hedgers.

S

D

Qd0 Quantity of positions

Ft (k)

Expt [St + k]

S

Ft (k)e

Supply schedule

Demand schedule

Discount

D

QS

111

Demand for long positions in futures contracts by speculators.

0 Quantity of long positions

Ft (k)

Expt [St + k]

Speculators will not demand any long positions if the settlement price exceeds the expected future spot price.

Speculators demand more long positions the greater the discount of the settlement price below the expected future spot price.

c

b

a

112

Supply of short positions in futures contracts by speculators.

0Quantity of short positions

Ft (k)

Expt [St + k]

Speculators supply more short positions the greater the premium of the settlement price over the expected future spot price

Speculators will not supply any short positions if the settlement price is below the the expected future spot price

f

e

d

113

Equilibrium in a futures market with speculators and a

preponderance of short hedgers.

S

D

Qd QE Qs0 Quantity of positions

Ft (k)

Expt [St + k]

S

Ft (k)e

Increased supply from speculators

Discount

D

Increased demand from speculators

114

Equilibrium in a futures market with speculators and a

preponderance of long hedgers.

S

D

0 Quantity of positions

Ft (k)

Expt [St + k]

S

Ft (k)e

Increased supply from speculators

Premium

D

QE

Increased demand from speculators

115

Equilibrium in the spot market

0

Quantity of the asset

Ft (k); St

Ft (k)e

Premium

QE

Spot demand

Excess supply of the asset when the spot market price is St

}

Spot supply

Expt [St + k]

116

Equilibrium in the futures market

0Net quantity of long positions held by hedgers and speculators

Ft (k)

Expt [St + k]

Ft (k)e

Premium

Q

}

Excess demand for long positions by hedgers and speculators when the settlement price is Ft (k)e

Schedule of excess demand by hedgers and speculators

117

HEDGING IS ONE COMPONENT OFCORPORATE 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 storyTaxes 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 BarrelBefore-tax profits = ($25 - $25)(1M)

= $0.0 millionIncome tax = $0.0 millionAfter-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 BarrelBefore-tax profits = ($35 - $25)(1M)

= $10.0 millionIncome tax = (.30)($10M)-$1M= $2.0MAfter-tax profits = $8.0 million

The firm pays only $2M in taxes, because it fully utilizes its tax credit of $1M.

The firm’s 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 millionIncome tax = (0.30)($5M) – 1M

= $0.5 millionAfter-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 firm’s 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.

A LONG HEDGE

LONG FUTURES IN ORDER TO HEDGE THE PRODUCT PURCHASE TO BE MADE

AT A LATER DATE.

I.E.,, LOCK IN THE PURCHASE PRICE.

RECALL:

THERE ARE TWO TYPES OF HEDGING:

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 < t k = current timet = 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 < 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:

1. 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.

2. The basis converges to zero at the futures maturity.

3. The basis is less volatile than either price

4. 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

October November December

FUTURES PRICE

BASIS = [CASH - FUTURES]

CASH PRICEEXPIRATION = DELIVERY

82

81

80

79

78

Convergence of Cash and Futures-Heating Oil

C

EN

TS

P

ER

G

AL

LO

N

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.

B0

O

Pr

S0

F0,t

k t time

Bt = 0

Bk

Sk

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

B0

O

Pr

S0

F0,t

k t time

Bt = 0

Bk

Sk

We thus, proved that hedging amounts to the reducing the firm’s risk

exposure because the basis is less risky that the spot

price risk.

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 RATIOS

NAÏVE HEDGE RATIO: ONE - FOR - ONE

QUANTITIY OF CASH POSITIONQUANTITY 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:

Vp = VP1 VP0

= (S1 + NF1,t) - (S0 + NF0,t ).

Define: S = S1 - S0 and F = F1,t - F0,t ,

then: VP = S + N(F).

136

AGAIN,

VP = S + N(F)

PROBLEM:

GIVEN THE CASH AND FUTURES VALUE CAHNGES, FIND A NUMBER

N*, SO AS TO MINIMIZE THE VOLATILITY OF THE CHANGE IN

THE HEDGER’S COMBINED

CASH – FUTURES

POSITION VALUE.

137

THE MATHEMATICS

.VAR (VP) = VAR (S) + VAR (NF)

+ 2COV (S ; NF)

= VAR (S) + N2VAR(F)

+ 2NCOV(S ; F).

TO MINIMIZE {VAR(VP)}

Take it’s derivative with respect to N and equate it to zero:

2N*VAR (F) + 2COV (S;F) = 0

N* = - COV(S;F) / VAR(F)

138

THE NUMBER OF FUTURES THAT MINIMIZES THE RISK OF

THE HEDGED POSITION IS:

.σ

σρN*

thus,,σσ

y)cov(x;ρ

: yandx , variablesany two

for But .F)VAR(

F)S;COV(N*

ΔF

ΔSΔS,ΔF

yxyx,

139

.1

2ρ

)VVar(Min

2

ΔSΔF

2

ΔS

ΔFΔS

ΔF

ΔS

2

ΔF2

ΔF

2

ΔS22

ΔS

ρσ

σσσσ

σσσρσP

To evaluate the risk of the position at its minimum level, substitute N* into the formula of

the position’s value change variance:

How to calculate N* in practice?

140

S1 F1,t S1 F1

S2 F2,t S2 F2

S3 F3,t S3 F3

. . . .

. . . .

. . . .

. . Sn Fn

Sn+1 Fn+1,t

DATA (SAY DAILY) n+1 DAYS.

*Nβ

n. ..., 1,2,i α eβΔFΔS iii

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 ΔS and ΔF over a 2 month interval yield:

SD(ΔS) = 0.025;

SD(ΔF) = 0.033; ρ = 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.

0 DERIVATIVES WORKBOOK By Ramon Rabinovitch. 1 DERIVATIVES ARE CONTRACTS Two parties Agreement...

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