DescriptionRutgers Business School

Rutgers, The State University of New Jersey

Derivatives

Spring 2023

HW1

Due: February 13, 2023 (Monday)

Required (12 Questions):

Chapter 1. Introduction

Questions 1.2

Chapter 2. Mechanics of futures markets

Questions: 2.10, 2.11, 2.19, 2.22, 2.30

Chapter 5. The determination of forward and futures prices

Questions: 5.1, 5.3, 5.12, 5.14, 5.15, 5.33

1

Q1.2

Explain carefully the difference between hedging, speculation, and arbitrage.

Q 2.10

Explain how margin accounts protect futures traders against the possibility of default.

Q 2.11

A trader buys two July futures contracts on frozen orange juice concentrate. Each

contract is for the delivery of 15,000 pounds. The current futures price is 160 cents per

pound, the initial margin is $6,000 per contract, and the maintenance margin is $4,500

per contract. What price change would lead to a margin call? Under what circumstances

could $2,000 be withdrawn from the margin account?

Q 2.19

“Speculation in futures markets is pure gambling. It is not in the public interest to allow

speculators to trade on a futures exchange.” Discuss this viewpoint.

Q 2.22

“When a futures contract is traded on the floor of the exchange, it may be the case that

the open interest increases by one, stays the same, or decreases by one.” Explain this

statement.

Q2.30

A company enters into a short futures contract to sell 5,000 bushels of wheat for 750

cents per bushel. The initial margin is $3,000 and the maintenance margin is $2,000.

What price change would lead to a margin call? Under what circumstances could $1,500

be withdrawn from the margin account?

Q5.1

Explain what happens when an investor shorts a certain share.

Q5.3

Suppose that you enter into a 6-month forward contract on a non-dividend-paying stock

when the stock price is $30 and the risk-free interest rate (with continuous compounding)

is 5% per annum. What is the forward price?

Q5.12

Suppose that the risk-free interest rate is 6% per annum with continuous compounding

and that the dividend yield on a stock index is 4% per annum. The index is standing at

400, and the futures price for a contract deliverable in four months is 405. What

arbitrage opportunities does this create?

Q5.14

The 2-month interest rates in Switzerland and the United States are, respectively, 1% and

2% per annum with continuous compounding. The spot price of the Swiss franc is

2

$1.0500. The futures price for a contract deliverable in 2 months is also $1.0500. What

arbitrage opportunities does this create?

Q5.15

The spot price of silver is $25 per ounce. The storage costs are $0.24 per ounce per year

payable quarterly in advance. Assuming that interest rates are 5% per annum for all

maturities, calculate the futures price of silver for delivery in 9 months.

Q5.33

A trader owns a commodity that provides no income and has no storage costs as part of

a long-term investment portfolio. The trader can buy the commodity for $1250 per ounce

and sell it for $1249 per ounce. The trader can borrow funds at 6% per year and invest

funds at 5.5% per year (both interest rates are expressed with annual compounding.) For

what range of 1-year forward prices does the trader have no arbitrage opportunities?

Assume the bid and offer for a forward price are the same.

3

Derivatives

Professor Ken Zhong

Rutgers Business School

Rutgers University

1

1. Introduction

◼

Summary

Introduction to Derivatives

◼ Use of Derivatives

◼ Forward and Futures Contracts

◼ Options Contracts

◼ Types of Traders

◼ Size of the Markets

◼

2

What is a Derivative?

◼

A derivative is a financial instrument

whose value depends on the values

of other more basic underlying

assets/variables

◼

Forward, Futures, Options, and Swaps

3

Example

Your family grows corn

◼ Your friend’s family buys corn to mill

into cornmeal

◼

A bushel of corn is not a derivative

◼ It is a commodity

◼

4

Example

◼

You enter into an agreement with your

friend that says:

If the price of a bushel of corn in one year is

greater than $3, you will pay your friend $1

◼ If the price of corn is less than $3, then your

friend will pay you $1

◼

◼

This is a derivative

5

Example

◼

◼

You earn $1 if your family’s corn sells

for a low price; this supplement your

income

Your friend earns $1 if the corn his

family bought is expensive; this offset

the high cost of corn

6

What is a Derivative?

◼

Derivative was first created as a tool

to reduce risk

Derivatives can be thought of as bets

◼ But the bet hedges both of you against

unfavorable outcomes, thus reduce the

risk for both of you

◼

◼

Derivative is also a tool to speculate

◼

◼

Investors can also use derivatives to

speculate on the price of corn

Key point: how it is used

7

Use of Derivatives

◼

Risk management

Derivatives can be used to reduce risks

◼ Hedging

◼ Insurance (auto insurance)

◼

◼

Speculation

Derivatives can serve as investment

vehicles

◼ Making bets that are highly leveraged

◼

◼

Arbitrage

8

Forward and Futures

Contracts

A forward/futures contract is an

agreement to buy or sell an asset at a

certain time in the future for a certain

price

◼ By contrast, in a spot contract there is

an agreement to buy or sell the asset

immediately

◼

9

Forward Contracts

Forward contracts are similar to futures

except that they trade in the over-thecounter market

◼ Forward contracts are popular on

currencies and interest rates

◼

10

Futures Contracts

Futures contracts are standardized

agreements

◼ Futures contracts are traded on an

exchange, e.g.,

◼

the Chicago Board of Trade (CBOT)

◼ the Chicago Mercantile Exchange (CME)

◼

11

Electronic Trading

Traditionally futures contracts have

been traded using the open outcry

system where traders physically meet

on the floor of the exchange

◼ Electronic trading and high frequency

algorithmic trading is becoming an

increasingly important part of the

market

◼

12

Underlying Assets

◼

CBOT

◼

◼

Corn, oats, soybeans, wheat, Treasury bonds,

S&P500 Index, Nasdaq100 Index

CME

Commodities (pork bellies, live cattle, live hogs,

feeder cattle)

◼ Foreign currencies (Euro, BP, JY, SF)

◼

◼

NYM

◼

Crude oil, heating oil, natural gas

13

History of

Futures

Trading

Composition

Source: CFTC Annual Report

14

Source: Futures Industry Association (FIA)

15

Source: Futures Industry Association (FIA)

16

For Jan-Dec 2021, Data Source: Futures Industry Association (FIA)

17

Options

◼

◼

A call option is an option to buy a

certain asset by a certain date for a

certain price (the strike price)

A put option is an option to sell a

certain asset by a certain date for a

certain price (the strike price)

18

American vs. European Options

◼

◼

An American option can be exercised at

any time on or before maturity

A European option can be exercised

only at maturity

19

Types of Traders

◼

Hedgers:

Reduce a risk they face

◼ Make the outcome more certain

◼ It does not necessarily improve the

outcome

◼

20

Types of Traders

◼

Speculators:

Take a position in the market

◼ Bet that the price will go up or go down

◼ Speculator using underlying asset: require

initial cash payment

◼ Speculator using futures contracts: require

no initial cash payment (leverage)

◼

21

Types of Traders

◼

Arbitrageurs:

◼

Arbitrage is possible when the derivative’s

price of an asset gets out of line with its

cash price, e.g.,

◼

Index arbitrage

22

Dangers

The largest trading losses in history are often

linked to the misuse of derivatives

◼ See examples of Barings Bank (1995), Societe

Generale (2008; Business Snapshot 1.4 in the

textbook), and the “London Whale” (2012)

◼

(Source: Bloomberg)

Archegos Capital Management (2021), etc.

◼ Updated list

◼

23

Source: https://www.fia.org

Global Futures and Options Volume

Type

Jan-Dec 2021

Vol

Jan-Dec 2020

Vol

Vol %

Change

2021 December 2020 December OI %

OI

OI

Change

Options

33,309,394,225 21,265,962,084 56.6%

810,522,968

798,791,004

1.5%

Futures

29,275,289,895 25,549,388,053 14.6%

265,234,141

249,235,907

6.4%

Total

62,584,684,120 46,815,350,137 33.7%

1,075,757,109

1,048,026,911

2.6%

Global Futures and Options Volume

Based on the number of contracts traded and/or cleared at 86 exchanges

worldwide.

Source: Futures Industry Association (FIA)

26

For Jan-Dec 2021, Data Source: Futures Industry Association (FIA)

27

Source: Futures Industry Association (FIA)

28

Exchange Rank in 2021 (Ranked by number

of contracts traded and/or cleared)

RankExchange

Jan-Dec 2021 Vol Jan-Dec 2020 Vol

Vol %

Change

1

National Stock Exchange of India

17,255,329,463

8,850,473,823

94.97%

2

B3

8,755,773,393

6,342,883,080

38.04%

3

CME Group

4,942,738,176

4,820,589,858

2.53%

4

Intercontinental Exchange

3,317,893,282

2,788,944,012

18.97%

5

Nasdaq

3,292,840,477

2,660,595,514

23.76%

6

CBOE Holdings

3,095,692,862

2,614,108,017

18.42%

7

Zhengzhou Commodity Exchange

2,582,227,206

1,701,847,321

51.73%

8

Shanghai Futures Exchange

2,445,774,713

2,128,613,700

14.90%

9

Dalian Commodity Exchange

2,364,418,367

2,207,327,866

7.12%

10

Korea Exchange

2,281,738,234

2,184,930,969

4.43%

Source: Futures Industry Association (FIA)

29

Over-the Counter Markets

The over-the counter market is an

important alternative to exchanges

◼ Trades are usually between financial

institutions, corporate treasurers, and

fund managers

◼ Transactions are much larger than in

the exchange-traded market

◼

30

Size of Markets – OTC vs Exchange

Source: Bank for International Settlements. (Chart shows total principal amounts

for the OTC market and value of underlying assets for the exchange market)

31

Size of Markets in 2021 – OTC

Derivatives

32

2. Forward and Futures Markets

1

Review: What is a Derivative?

Definition: a contract whose value is derived

from the value of some underlying variable

Asset Value

Derivative Value

S&P 500 Index

S&P 500 Index Futures

Google Stock

Call Options on Google Stock

3-Month Treasury Yield

Interest rate Swap

The average temperature in N.Y.

Weather Derivative

(Heating/Cooling Degree Days)

2

Forward Contract

◼

◼

A forward contract is an OTC agreement

to buy or sell an asset at a certain time

in the future for a certain price

Forward contracts are privately

negotiated and are not standardized.

3

Forward Contract

Buyer

(Long)

Seller

(Short)

Broker

4

Forward Contract

Now

Determined Today:

1. Underlying Assets

2. Settlement Date

3. Price

4. Delivery Procedure

Settlement

Date

Settlement Options:

1. Physical Delivery

2. Cash Settlement

5

Forward Contract – Example

Oil Contract

Commodity:

Crude Oil

Quantity:

1000 bbl

Settlement Date:

April 15, 2023

Price:

$85 / bbl

6

Forward Contract – Example

Price of Oil

105

95

Gain

85

Loss

75

Jan 1

Feb 1

Contract

Date

Mar 1

Apr 1

May 1

Settlement

Date

Valuing a Forward Contract

Symmetry of Gains and Losses

At Settlement

Long

Position

(buyer)

Short

Position

(seller)

Contract

Price

Forward

Price of

Apr Oil

0

$75

$85

$95

Forward Contracts – Pros and

Cons

◼

Pros

◼

◼

Flexible

Cons

◼

Costs of search for agreement

◼

◼

Subject to default risk

◼

◼

Hard to find a counter-party with the exact same needs

Requires information to screen good from bad counterparty

Lack of liquidity:

◼

Thin or non-existent secondary market

9

Futures Contract

◼

◼

◼

A futures contract is a standardized

agreement

Futures are traded on organized exchanges

The exchanges clear, settle, and guarantee

all transactions that occur on the exchange

10

Forwards vs Futures

Forwards

Buyer

(long)

Problems

1. Cost of Search

2. Default Risk

3. Liquidity

Seller

(short)

Futures

Buyer

(long)

Exchange

Clearing

Corp

Advantages

1. Standardized Contracts

2. Manage Default Risk (marking to market)

3. Liquidity (easy to close out positions)

Seller

(short)

Futures Contract is a

Standardized Agreement

The asset

◼ The contract size

◼ Delivery arrangements

◼ Delivery months

◼ Price quotes

◼ Price Limits and Position limits

◼

12

Futures Contract – Example

(Corn)

Trading Unit

5,000 bu

Deliverable Grades No. 2 Yellow at par and substitutions at

differentials established by the exchange

Price Quote

Cents and quarter-cents/bu

Tick Size

1/4 cent/bu ($12.50/contract)

Daily Price Limit

25 cent/bu above or below the previous

day’s settlement price (expandable to 40

cent/bu)

Contract Months Dec, Mar, May, Jul, Sep

Ticker Symbol

C

13

Margins

A margin is cash or marketable

securities deposited by an investor with

his or her broker

◼ The balance in the margin account is

adjusted to reflect daily settlement

◼ Margins minimize the possibility of a

loss through a default on a contract

◼

14

Example of a Futures Trade

◼

An investor takes a long position in 2

December gold futures contracts on

June 5

◼

◼

◼

◼

contract size is 100 oz.

futures price is US$1,250

margin requirement is US$6,000/contract

(US$12,000 in total)

maintenance margin is US$4,500/contract

(US$9,000 in total)

15

Day

Trade

Price ($)

1

1,250.00

Settle

Price ($)

Daily

Gain ($)

Cumul.

Gain ($)

Margin

Balance ($)

12,000

1

1,241.00

−1,800

− 1,800

10,200

2

1,238.30

−540

−2,340

9,660

…..

…..

…..

……

6

1,236.20

−780

−2,760

9,240

7

1,229.90

−1,260

−4,020

7,980

8

1,230.80

180

−3,840

12,180

…..

……

…..

…..

…..

10

1,228.10

540

−4,380

11,640

11

1,211.00

−3,420

−7,800

8,220

12

1,211.00

0

−7,800

12,000

…..

…..

…..

…..

……

15

1,223.00

1,380

−5,400

14,400

780

−4,620

15,180

16

1,226.90

Margin

Call ($)

…..

4,020

3,780

16

Margin Cash Flows When Futures

Price Increases

Clearing House

Clearing House

Member

Clearing House

Member

Broker

Broker

Long Trader

Short Trader

17

Margin Cash Flows When Futures

Price Decreases

Clearing House

Clearing House

Member

Clearing House

Member

Broker

Broker

Long Trader

Short Trader

18

Other Key Points About

Futures

◼

◼

Most contracts are closed out before

maturity

Closing out a futures position involves

entering into an offsetting trade

19

Closing Out positions

◼

An example:

March 6, long one July corn futures contract

◼ April 12, short one July corn futures contract

◼ Your gain or loss is determined by the

change in the futures prices between March

6 and April 12

◼

20

The Mechanics of Offsetting Trades

Date 1

Date 2

Party A

Party A

sells one

contract

buys one

contract

Clearing

Corp

Party B

Party C

sells one

contract

buys one

contract

Delivery

If a futures contract is not closed out before

maturity, it is usually settled by delivering the

assets underlying the contract.

◼ When there are alternatives about what is

delivered, where it is delivered, and when it is

delivered, the party with the short position

chooses.

◼ A few contracts (for example, those on stock

indices and Eurodollars) are settled in cash

◼

22

Price Quotes

23

Price Quotes

Opening price

◼ The highest price

◼ The lowest price

◼ Settlement price: the price just before

the final bell each day

◼

◼

used for the daily settlement process

24

Price Quotes

◼

Open interest

the total number of contracts outstanding

◼ equal to number of long positions or

number of short positions

◼

◼

Volume of trading:

◼

the number of trades in 1 day

25

Questions

◼

◼

When a new trade is completed

what are the possible effects on

the open interest?

Can the volume of trading in a day

be greater than the open interest?

26

Reading Index Futures Quotes

Closing price of the day

Low of the day

Lifetime high

High of the day

The open price

Expiration month

“Delivery Month”

Daily change

Lifetime low

Open interest

“# of contracts

outstanding”

Contract

These quotes skip two

decimal places to save

space

Actual Index

Multiplier

OTC Market and the Recent

Financial Crisis

Despite the many advantages of exchange

trading, the over-the-counter derivatives market

is still a popular choice especially among major

financial institutions and fund managers

◼ Traditionally OTC derivatives transactions have

been cleared bilaterally

◼ This has led to an increasing concern about

systemic risk

◼ Such concern is exemplified by the recent

financial crisis

◼

28

The Lehman Bankruptcy

Lehman’s filed for bankruptcy on September

15, 2008. This was the biggest bankruptcy in

US history

◼ Lehman was an active participant in the OTC

derivatives markets

◼ It had hundreds of thousands of transactions

outstanding with about 8,000 counterparties

◼ Unwinding these transactions has been

challenging for both the Lehman liquidators and

their counterparties

◼

29

Bilateral Clearing vs.

Central Clearing

CC

CCP

CC

CP

30

New Regulations for the OTC

Derivatives Market

The OTC market is becoming more like the

exchange-traded market.

◼ New regulations introduced since the crisis

include

◼

OTC products must be traded on swap

execution facilities

◼ A central counterparty (CCP) must be used as

an intermediary for clearing

◼ Trades must be reported to a central registry

◼

31

Futures-Cash Basis

32

Futures-Cash Basis

The cash-futures basis is the difference between the

futures price of a underlying asset and the cash (spot)

price of the underlying asset.

Basis = S – F

33

Convergence of Futures to Spot

◼

◼

Futures price converges to the spot

price of the underlying asset as the

delivery month is approached

Arbitrage activity between the futures

and spot markets will make the two

prices equal on the expiration day

34

The Basis and Convergence

Case (a) Futures prices above spot price

S,F

F

Settlement

Date

S

Time

Basis

The Basis and Convergence

Case (b) Futures prices below spot price

S,F

S

Settlement

Date

F

Basis

Time

Simple vs. Continuous

Compounding

37

Simple vs. Continuous

Compounding

◼

The effect of increasing the compounding

frequency on the value of the $100 at the end of

1 year when the interest rate is r=10% per year

r m

100 (1 + )

m

m is the compounding frequency

38

Simple vs. Continuous

Compounding

◼

m =1

r m

10 % 1

100 (1 + ) = 100 (1 +

) = 110

m

1

◼

m=2

r m

10 % 2

100 (1 + ) = 100 (1 +

)

2

m

=?

39

Simple vs. Continuous

Compounding

Compounding

Frequency

Annually (m = 1)

Semiannually (m = 2)

Quarterly (m = 4)

Monthly (m = 12)

Weekly (m = 52)

Daily (m = 365)

Value of the $100 at

End of 1 Year

110.00

110.25

110.38

110.47

110.51

110.52

40

Continuous Compounding

◼

◼

◼

In the limit as we compound more and more

frequently we obtain continuously

compounded interest rates

$100 grows to $100erT when invested at a

continuously compounded rate r for time T

$100 received at time T discounts to $100e-rT

at time zero when the continuously

compounded discount rate is r

41

Continuous Compounding

10% 1 year

=?

−10% 1 year

=?

$100 e

$100 e

42

Continuous Compounding

With continuous compounding interest

rate r, an amount $A invested at time 0

for T years grows to:

◼

Ae

r T

43

Continuous Compounding

With continuous compounding interest

rate r, an amount $A received at time T

has present value of:

◼

Ae

− r T

44

Conversion Formulas

Define

Rc : continuously compounded rate

Rm: same rate with compounding m times

per year

Rm

Rc = m ln 1 +

m

(

Rm = m e

Rc / m

)

−1

45

3. Forward and Futures Prices

◼

Summary

Short Selling

◼ The Repo Rate

◼ Forward Contracts

◼ Forward Price vs. Futures Price

◼ Stock Index Futures

◼ Forward and Futures Contracts on Currencies

◼ Forward and Futures Contracts on Commodities

◼ Cost of Carry Model and Convenience Yield

◼

1

Short Selling

◼

Selling securities that you do not own and

buying them back later

◼

◼

Yield a profit if the price of a security goes

down

Yield a loss if the price of a security goes up

2

Short Selling: Example

You call your broker and instruct him/her

to short 100 shares of ATT

◼ Your broker then

◼

borrow 100 shares of ATT from another client

◼ Sell them in the open market at $30 per share

◼ Deposit the cash in your account

◼

3

Short Selling: Requirements

◼

◼

◼

Shares can only be sold on an uptick

(price increases)

You must pay dividends or other income

the owner of the securities receives

Margin requirement

4

Short Selling: Example

You instruct your broker to close out the

position 3 months later.

◼ Your broker will buy 100 shares of ATT,

and return them to the client from

whom they were borrowed

◼

If the price is $28, then your profit is $200

◼ If the price is $31, then your loss is $100

◼

5

The Repo Rate

◼

The risk-free rate available to investors is

the repo rate, repurchase agreement

You sell securities to another investor

◼ Agree to buy them back at a higher price

later

◼ The difference between the two prices is the

interest earned by the counterparty

◼

6

The Repo Rate

Repo is slightly higher than the T-bill

rate

◼ Overnight repo, term repo

◼

7

Forward Contracts

◼

Notations:

T : expiration day

t : current time

T-t : the life of the contract (in years)

T

8

Forward Contracts

◼

Notations: (cont.)

St : price of asset underlying forward

contract at t

ST : price of asset underlying forward

contract at T

F : forward price at t

r : risk-free rate per year

9

Forward Contracts

◼

Forward contracts on a security that

provides no income

◼

Determine today’s forward price using noarbitrage argument

10

Forward Contracts

Action

Buy 1 Unit

of Asset

Short 1 Forward

Contract

Borrow $ St

at rate r

Total

Cash Flow at t

Cash Flow at T

− St

ST

0

F − ST

St

− St e r (T − t )

0

ST + F − ST − S t e

r (T − t )

=0

Arbitrage activities will guarantee this

11

Forward Contracts

Therefore, the price of the forward contract

at t should be:

F = St e

r (T − t )

12

Forward Contracts

◼

If F > St e r(T-t) , you can earn an

arbitrage profit by taking the following

positions at t:

◼

Borrow St dollars at rate r

Buy 1 unit of asset

◼ Take a short position in the forward contract

(Initial cash flow is zero)

◼

13

Forward Contracts

◼

◼

At time T:

◼

Deliver the asset ST and receive F

◼

Pay back the loan, -St e r(T-t)

Realize a profit, F – St e r(T-t) > 0

14

Forward Contracts

◼

If F < St e r(T-t) , you can earn an
arbitrage profit by taking the following
positions at t :
◼
◼
Short 1 unit of asset
Invest St dollars at rate r
Take a long position in the forward contract
(Initial cash flow is zero)
◼
15
Forward Contracts
◼
◼
At time T:
◼
Take delivery of asset ST and pay F
◼
Close out the short position
◼
Receive St e r(T-t) from the investment
Realize a profit, St e r(T-t) – F > 0

16

Forward Contracts: Example 1

A forward contract is written on a nondividend paying stock. The maturity of

the contract is 3 months. The stock

price is $40 today and the risk-free rate

is 5% per year

r (T − t )

◼ Recall

F=Se

◼

t

17

Forward Contracts: Example 1

T-t = 3/12 year

r = 0.05

St = $40

Thus, the forward price is:

F =?

18

Forward Contracts: Example 2

A forward contract is written on a

discount bond. The maturity of the

contract is 4 months. The bond price is

$950 today and the risk-free rate is 8%

per year

◼ Thus, the forward price is:

◼

F = St e

r (T −t )

=?

19

Forward Contracts

◼

Forward contracts on a security that

provides a known cash income

Dividends-paying stock with known

(discrete) dividends

◼ Coupon bonds

◼ Determine today’s forward price using noarbitrage argument

◼

◼

I: present value of all known cash income

20

Forward Contracts

Action

Cash Flow at t

Cash Flow at T

Buy Underlying

Asset

− St + I

ST

Short 1 Forward

Contract

0

F − ST

Borrow

at rate r

St − I

− ( St − I )e r (T − t )

0

ST + F − ST − ( St − I )e r (T −t ) = 0

Total

21

Arbitrage activities will guarantee this

Forward Contracts

Therefore, the price of the forward

contract at t is

F = ( St − I )e r (T − t )

22

Forward Contracts: Example

A 10-month contract is written on a

dividend-paying stock. The stock price

is $50 today. The risk free rate is 8%

per year. The dividend is $2.00 per

share and will be paid after 6 months

and 9 months

◼ Recall

◼

F = ( St − I )e r (T − t )

23

Forward Contracts : Example

T-t = 0.83 year (10/12)

r = 0.08

St = $50

$2.00 dividend will be paid in 6 and 9 months

I is the present value of known cash income

I =?

F =?

24

Forward Contracts

◼

Forward contracts on a security that

provides a known (continuous) dividend

yield

Dividend-paying stocks. Dividend yield is

paid continuously at an annual rate of q

◼ Determine today’s forward price using

no-arbitrage argument

◼

25

Forward Contracts

Action

Cash Flow at t

Cash Flow at T

Buy e − q (T − t )

Unit of Asset

− St e −q(T − t )

ST

Short 1 Forward

Contract

0

F − ST

Borrow

at rate r

Total

St e −q(T − t )

0

− St e −q(T − t )e r (T − t )

ST + F − ST − St e ( r −q )(T − t ) = 0

Arbitrage activities will guarantee this26

Forward Contracts

Therefore, the price of the forward

contract at t is:

F = St e( r −q )(T − t )

27

Forward Contracts: Example

A 6-month forward contract is written

on a dividend-paying stock. The

dividend yield is 4% per year. The

stock price is $25 today. The risk free

rate is 10% per year

◼ Recall

◼

F = St e( r −q )(T − t )

28

Forward Contracts: Example

T-t = 0.5 year (6/12)

r = 0.10

St = $25

q = 0.04

Thus, the forward price is:

F =?

29

Forward Price vs. Futures Price

The forward price and futures price are

the same when the interest rate is

constant

◼ When the interest rates change over

time (unpredictable), the forward and

futures prices are different

◼

30

Forward Price vs. Futures Price

◼

◼

If asset price is positively correlated with

interest rates, futures prices tend to be

higher then forward prices

If asset price is negatively correlated with

interest rates, futures price tend to be

lower than forward prices

31

Forward Price vs. Futures Price

◼

◼

If the maturity is only a few months, the

difference between the two prices is small

and can be ignored

In general, it is reasonable to assume that

forward and futures prices are equal

32

Stock Index Futures

◼

Stock Indices

◼

◼

Track the changes in the underlying

portfolio

Stock Index Futures

◼

Can be viewed as a dividend-paying

security

33

Stock Index: Example

◼

S&P 500 Index includes 500 leading companies in leading

industries of the U.S. economy.

34

Stock Index: Example

◼

S&P 500 Index

Value-weighted portfolio

◼ Accounts for about 75% of the market

capitalization of all stocks listed on the

NYSE

◼ Each futures contract is based on 250

times the index

◼

35

Stock Index Futures

◼

Stock Index Futures

◼

Assume the dividend is paid continuously

with a dividend yield of rate q

The price of the futures contract at t is:

F = St e

( r −q )(T − t )

36

Stock Index Futures

◼

Index Arbitrage

Implementation: Program Trading

◼ If

F > Ste(r-q)(T-t) , you can make a profit by

taking the following positions:

◼

Buy stocks underlying the index

◼ Short futures contracts

◼ Done by investors who have a lot of cash or

money market investments

◼

37

Stock Index Futures

◼

If F < Ste(r-q)(T-t) , you can make a profit
by taking the following positions:
Short stocks underlying the index
◼ Long futures contracts
◼ Done by fund managers that own index
portfolios
◼
38
Forward and Futures Contracts
on Currencies
A foreign currency is analogous to a stock
paying a known dividend yield
◼ The “dividend yield” is the risk-free rate in the
foreign currency
◼ Interest earned on a foreign currency holding is
denominated in the foreign currency
◼ The “present value of the dividend” is the
present value of the interest earned on a foreign
currency
◼
39
Forward and Futures Contracts
on Currencies
St = price in dollars of one unit of the
foreign currency
r = domestic (U.S.) interest rate with
continuous compounding
rf = foreign interest rate with continuous
compounding
40
Forward and Futures Contracts
on Currencies
◼
Recall:
F = St e ( r −q )(T − t )
◼
Replace q by rf , we obtain the futures price
on a foreign currency
F = St e
( r − r f )(T − t )
41
Forward and Futures Contracts
on Currencies
When r - rf < 0 ,
futures price is less than the spot price St
When r - rf > 0 ,

futures price is greater than the spot

price St

42

Futures on Commodities

◼

◼

Commodities can be held for investment

◼ Examples: gold, silver

Commodities can be held for consumption

◼ Examples: crude oil, gas

43

Gold Futures

◼

If no storage costs, gold can be considered as

a security paying no income. Thus the

futures price is:

F = St e r (T − t )

44

Gold Futures

If there are storage costs, storage costs can

be regarded as negative income.

◼ Let U be the present value of the storage

costs that will be incurred during the life of

the contract, replace I by -U in the formula

◼

F = ( St − I )e r (T − t )

we obtain the futures price

F = ( St + U )e r (T − t )

45

Gold Futures

If the storage costs are incurred continuously, regard

the storage costs as negative dividend yield.

◼ Let storage costs be u, and replace q by -u in the

formula

◼

F = St e ( r −q )(T − t )

we obtain the futures price

F = St e ( r + u )(T − t )

where u is the storage costs per annum as a

proportion of the spot price.

46

Gold Futures: Example

A futures contract is written on gold.

The maturity of the contract is 1 year.

The storage cost is $2 per ounce per

year. The payment will be made at the

end of the year. The spot price is

$1,600, and the risk-free rate is 5% per

year

r (T − t )

F

=

(

S

+

U

)

e

◼ Recall

t

◼

47

Gold Futures: Example

T-t = 1 year

r = 0.05

St = $1,600

U = 2e

− r ( T −t )

= 2e

−0.051

= 1.90

Thus the futures price is:

F = (1600 + 1.90)e0.051 = $1684.03

48

The Cost of Carry

For all investment assets, the futures

price can be summarized by the cost of

Carry model:

F = St e

c (T − t )

49

The Cost of Carry

◼

Cost of carry c depends on

(+) the interest paid to finance the asset

◼ (-) the income earned on the asset

◼ (+) The storage cost

◼

50

The Cost of Carry

For a non-dividend paying stock, c = r

◼ For a dividend-paying stock, c = r – q

◼ For a currency, c = r – rf

◼ For a commodity, c = r + u

◼

51

Convenience Yield

◼

We have shown that for an investment asset

F = St e c (T − t )

◼

However, for a consumption asset

F S t e c (T −t )

◼

The convenience yield on the consumption

asset, y, is defined so that

F = St e

( c − y )(T −t )

52

Convenience Yield

◼

The explanation:

There is extra advantage that firms derive

from holding the underlying asset rather

than the forward/futures.

◼ Use “convenience yield” to measure the

benefit or premium associated with holding

an underlying asset, rather than the

forward/futures contracts

◼

53

Implied Convenience Yield

◼

Solve

F = St e

c’

c ‘(T −t )

Where F is the observed price of a futures contract; c’

is the implied cost of carry

◼

The implied convenience yield is calculated by

y = c − c’

Where c’ is the implied cost of carry from above, and c

is the actual cost of carry

54

4. Hedging Using Futures

◼

Summary

Why Hedge

◼ How to Hedge

◼ Basis Risk

◼ Optimal Hedge Ratio

◼ Hedging Using Stock Index Futures

◼ Rolling The Hedge Forward

◼

1

Why Hedge?

Objective: Neutralize the risk

◼ An example:

◼

A firm is due to sell an asset at a particular

time in the future.

◼ The profit/loss depends on the price in the

future

◼ The firm can hedge by shorting a futures

contract

◼

2

Why Hedge?

If the price of the asset goes down, the firm

will lose money when it sells the asset, but

gain on the futures position

◼ If the price of the asset goes up, the firm will

gain from the sale of the asset, but will take a

loss on the futures position

◼ In both scenarios, the gain in one contract

offset the loss in another

◼

3

Short Hedges

Involves a short position in the futures

contract

◼ It is appropriate when the hedger owns

the asset and expects to sell it in the

future

◼ It is also appropriate when the hedger

does not own the asset now, but will

own it in the future

◼

4

Short Hedges: Example

Today, Feb 15

◼ Exxon signed a contract to sell 1 million

barrels of oil. The selling price is the

spot price on May 15

◼ Spot Price: $75/barrel

◼ May oil futures price: $74.75/barrel

◼

5

Short Hedges: Example

◼

Hedging:

Feb 15: short 1000 May oil futures (each

contract is written on 1000 barrels)

◼ May 15: close out the position

◼

◼

Outcome:

◼

Exxon effectively locks in a selling price of

$74.75/barrel

6

Short Hedges: Example

◼

If the oil price proves to be $70.50 on

May 15 (spot price)

Exxon receives $70.50 under the sale contract

◼ Exxon receives $4.25 (74.75-70.50) from the

futures contract

◼

◼

If the oil price proves to be $79.50 on

May 15 (spot price)

Exxon receives $79.50 under the sale contract

◼ Exxon loses $4.75 (79.50-74.75) from the

futures contract

◼

◼

The effective selling price is $74.75/barrel

in either case

7

Long Hedges

Involves a long position in the futures

contract

◼ It is appropriate when the hedger has

to purchase assets in the future and

wants to lock in a price now

◼

8

Basis Risk

9

Basis Risk

◼

Hedging is not perfect:

1.

2.

3.

You do not know the date when the asset

will be purchased or sold

Mismatch between the futures expiration

date and the date when the asset will be

bought or sold

There may be a difference between the

hedged asset and the asset underlying the

futures contracts

10

Basis Risk

◼

All these issues relate to the basis risk

◼

◼

Basis risk = spot price of asset to be

hedged – futures price of contract used

If the hedged asset is the same as the

asset underlying the futures contract

The basis risk is 0 on the expiration day

◼ But before the expiration day, the basis risk

could be positive or negative

◼

11

Basis Risk

◼

Notations:

S1: spot price at t1

S2: spot price at t2

F1: futures price at t1

F2: futures price at t2

b1: basis risk at t1

b2: basis risk at t2

t1: the first day of the hedge (open the position)

t2: the last day of the hedge (close out the position)

12

Basis Risk – Short Hedge

◼

A hedger knows that he will sell assets at t2

and decide to take a short position in t1

◼ receive S2 at t2

◼ gain (or lose) F1 – F2 on the futures

position

Thus, the price realized is

S2 + (F1 – F2)

= F1 + (S2 – F2)

= F 1 + b2

13

Basis Risk – Long Hedge

◼

A hedger knows that he will buy assets at t2

and take a long position in t1

◼ pay S2 at t2

◼ gain (or lose) F2 – F1 on the futures

position

thus, the cost of asset is

S2 – (F2 – F1)

= F1 +( S2 – F2)

=F1 + b2

14

Basis Risk

◼

◼

◼

F1 is known at t1, but

b2 is unknown since F2 and S2 are

unknown at t1

The hedging risk is the uncertainty

associated with the basis risk, b2

15

Basis Risk – Continued

◼

◼

Sometimes the asset being hedged is

different from the asset underlying the

futures contract

Example: jet fuel vs. heating oil

16

Basis Risk – Continued

◼

Notation:

S1: Spot price of the asset being hedged at t1

S2: Spot price of the asset being hedged at t2

S2*: Spot price of the asset underlying the

futures contract at t2

17

Basis Risk – Continued

◼

The price that will be paid (or received) for

the asset is

S2 + F1 − F2

= S2 + F1 – F2 + S2 – S2

= F1 + ( S2 – F2 ) + ( S2 – S2 )

S2* – F2 is the basis that would exist if the asset

being hedged is the same as the asset underlying

the futures contract

*

◼ S2 – S2 is the difference between asset being

hedged and the asset underlying the futures

contract

◼

18

Choice of Contracts

◼

The choice of the asset underlying the

futures contract

The correlation between prices of the asset

being hedged and the asset underlying the

futures contract

◼ Choose the one with the highest correlation

◼

◼

The choice of the delivery month

◼

Choose the contract with the maturity as

close as possible to (but later than) your

hedging horizon

19

Basis Risk – Example

◼

It is March 1. A U.S. firm expects to

receive 50 million Japanese Yen (JY) on

July 31.

◼

JY futures:

12.5 million JY for each contract

◼ March, June, Sept., and Dec

◼

Sept. contract is chosen

◼ On March 1, the September futures are

traded at 0.68 cents per yen

◼

20

Basis Risk – Example

◼

Positions:

Short 4 Sept JY futures on March 1

◼ Close out the position on July 31

◼

21

Basis Risk – Example

Suppose:

F1 = 0.68

F2 = 0.625 (on July 31)

S2 = 0.62 (on July 31)

b2 = S2 – F2 = 0.62 – 0.625 = -0.005

The effective price received by the hedger is:

S2+(F1-F2)= 0.62+(0.68-0.625)=0.62+0.055 = 0.675

or

F1 + b2 = 0.68 + (-0.005) = 0.675

22

Optimal Hedge Ratio

23

Cross Hedging

Cross hedging occurs when the asset

being hedged and the asset underlying

the futures are different

◼ Hedge ratio: the ratio of the size of the

position taken in futures contracts to

the size of exposure

◼

◼

The optimal hedge ratio is not necessary

equal to 1

24

Optimal Hedge Ratio

◼

How to choose the optimal hedge ratio?

◼

Minimize the variance of the hedger’s

position

25

Optimal Hedge Ratio

◼

Notations:

S : changes in spot price S during the

life of the hedge

F : changes in futures price F during the

life of the hedge

S : standard deviation of S

F : standard deviation of F

: correlation between S and F

26

Optimal Hedge Ratio

◼

Notations: (cont.)

h: hedge ratio

h*: hedge ratio that minimizes the

variance of the hedger’s position

S

h =

F

27

Minimum Variance Hedge Ratio

◼

If the hedger shorts the futures and longs the

asset, the change in the value of the position

is

S − h F

◼

The variance of the changes in value of the

hedged position is

2

var = S2 + h2 F

− 2h S F

28

Minimum Variance Hedge Ratio

Variance of position

Hedge Ratio, h

h*

29

Minimum Variance Hedge Ratio

◼

To minimize the variance of hedger’s

position, choose h such that

var

2

= 2h F − 2 S F = 0

h

Thus,

S

h =

F

30

Minimum Variance Hedge Ratio

ΔS

•

•

•

•

•

• ••

ΔF

•

•

•

The optimal hedge ratio, h* is also the slope of

the best-fit line when ΔS is regressed against ΔF

31

Minimum Variance Hedge Ratio:

Example

A company will buy 2 million gallons of

jet fuel in 1 month

◼ Decide to use the heating oil contract to

hedge

◼

◼

Each heating oil contract traded on NYMEX

is on 42,000 gallons of heating oil

32

Minimum Variance Hedge Ratio:

Example

◼

Base on statistics of historical data

S = 0.0263

F = 0.0313

= 0.928

◼

The optimal hedge ratio is

h =?

33

Optimal Number of Contracts

◼

The number of contract should be used:

h QA

N =

QF

QA: Size of the position being hedged (units)

QF : Size of one futures contracts (units)

N* : Optimal number of futures contracts for hedging

34

Optimal Number of Contracts

◼

The number of contract should be used in

our previous example:

h Q A 0.78 2,000,000

N =

=

= 37.14

QF

42,000

35

Hedging Using Stock Index Futures

36

Hedging Using Stock Index Futures

◼

To hedge the risk in a portfolio, the number

of contracts that should be shorted is

P

N =b

F

*

where P is the value of the portfolio, b is its beta,

and, F is the current value of one futures contract

37

Example

Stock index is 1,000

Stock index futures price is 1,010

One contract is on $250 times the index

Size of portfolio is $5,050,000

Beta of portfolio is 1.5

What position in futures contracts on the stock

index is necessary to hedge the portfolio?

N*=?

38

Reasons for Hedging an Equity

Portfolio

◼

Desire to be out of the market for a short

period of time.

◼

◼

Hedging may be cheaper than selling the

portfolio and buying it back.

Desire to hedge systematic risk

◼

Appropriate when you feel that you have picked

stocks that will outperform the market.

39

Changing Beta

◼

◼

What position is necessary to reduce

the beta of the portfolio to 0.75?

What position is necessary to increase

the beta of the portfolio to 2.0?

40

Rolling The Hedge Forward

41

Rolling The Hedge Forward

The expiration date of the hedge is later

than the delivery dates of all the futures

contracts available

◼ The hedger should roll the hedge forward

◼ Close out one contract

◼ Take the same position in the same

contract with a later delivery date

◼ You can roll the hedge over many times

◼

42

Rolling The Hedge Forward

Example

Suppose in April 2022, a company realizes

that it will need to sell 100,000 barrels in

June 2023. The current price of oil is $99 per

barrel.

◼ Although contracts are available for every

delivery month up to one year, only the first

six delivery months provide sufficient liquidity.

The contract size is 1,000 barrels

◼

43

Rolling The Hedge Forward

Example

◼

The Outcome:

◼ October 2022 futures contract: shorted in

April 2022 at $98.20 and closed out in

September 2022 at $97.40

◼ March 2023 futures contract: shorted in

September 2022 at $97.00 and closed out

in February 2023 at $96.50

◼ July 2023 futures contract: shorted in

February 2023 at $96.30 and closed out in

June 2023 at $95.90

44

Rolling The Hedge Forward

Example

Spot oil price in June 2023: $96 per barrel

◼ The decline in oil price from April 2022 to

June 2023 is $3.

◼ The gain from the futures contracts is:

(98.20-97.40)+(97.00-96.50)

+(96.30-95.90)=1.70

◼ Thus, the gain from the futures contracts

partly offsets the $3 decline in oil prices

◼

45

Arguments in Favor of Hedging

Companies should focus on the main

business they are in and take steps to

minimize risks arising from interest

rates, exchange rates, and other market

variables

46

Arguments against Hedging

Explaining a situation where there is a

loss on the hedge and a gain on the

underlying can be difficult

◼ Shareholders are usually well diversified

and can make their own hedging

decisions

◼ It may increase risk to hedge when

competitors do not (Text p52)

◼

47

Purchase answer to see full

attachment