RBS Difference Between Hedging Speculation and Arbitrage Questions

  

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could $2,000 be withdrawn from the margin account?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be withdrawn from the margin account?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:
◼
Ae
r T
43
Continuous Compounding
With continuous compounding interest
rate r, an amount $A received at time T
has present value of:
◼
Ae
− 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
−0.051
= 1.90
Thus the futures price is:
F = (1600 + 1.90)e0.051 = $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

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