DERIVATIVE INSTRUMENTS
RIJKSUNIVERSITEIT GRONINGEN – MASTER FINANCE 2022-2023
CHAPTER 1: INTRODUCTION
A derivative involves two parties agreeing to a future transaction. Its value (payoff) depends on the
(uncertain) values of other underlying variables (frequently, the prices of traded assets). Example: the
value of a stock option depends on the price of a stock.
Derivative exchange: market where individuals and companies trade standardized contracts that have
been defined by the exchange.
• Once two traders have agreed to trade a product offered by an exchange, it is handled by the
exchange clearing house → stands between the two traders and manages the risks.
• Advantage: traders do not have to worry about the creditworthiness of the people they are trading
with. Clearing house takes care of credit risk by requiring both parties to deposit funds (margin)
with the clearing house to ensure that they will live up to their obligations.
Many trades take place in the over-the-counter (OTC) market, instead of on exchanges. Banks, large
financial institutions, fund managers, and corporations are the main participants in OTC markets. Once
an OTC trade has been agreed, the two parties can either present it to a central counterparty (CCP)
or clear the trade bilaterally.
• CPP stands between the two parties to the derivatives transaction so that one party does not have
to bear the risk that the other party will default.
• When trades are cleared bilaterally, the two parties have usually signed an agreement covering
all their transactions with each other. The issues covered in the agreement include the
circumstances under which outstanding transactions can be terminated, how settlement amounts
are calculated in the event of a termination, and how the collateral (if any) that must be posted
by each side is calculated.
Large banks often act as market makers for the more commonly traded instruments → they are
always prepared to quote a bid price (at which they are prepared to take one side of a derivatives
transaction) and ask price (at which they are prepared to take the other side).
Systemic risk: the risk that a default by one financial institution will create a “ripple effect” that leads
to default by other financial institutions and threatens the stability of the financial system.
Forward contract: an agreement to buy or sell an asset at a certain future time for a certain price.
• In contrast, a spot contract is an agreement to buy or sell an asset almost immediately.
• Forward contract is traded in OTC market – usually between two financial institutions or between
a financial institution and one of its clients.
o One of the parties to a forward
contract assumes a long position:
agrees to buy the underlying asset
on a certain specified date for a
certain specified price.
▪ Payoff: 𝑆𝑇 − 𝐾
o The other party assumes a short
position: agrees to sell the asset on
the same date for the same price.
▪ Payoff: 𝐾 − 𝑆𝑇
,Futures contract: an agreement between two parties to buy or sell an asset at a certain time in the
future for a certain price.
• Difference with forward contract: futures contracts are traded on an exchange. The exchange
specifies certain standardized features of the contract. As the two parties to the contract do not
necessarily know each other, the exchange clearing house stands between them.
• Futures price is determined by the laws of supply and demand: if more traders want to go long
than go short, the price goes up; if the reverse is true, then the price goes down.
Options are traded both on exchanges and in the OTC market.
• There are two types of options:
o Call option gives the holder the right to buy the underlying asset by a certain date for a
certain price.
o Put option gives the holder the right to sell the underlying asset by a certain date for a
certain price.
• American options can be exercised at any time up to the expiration date. European options can
be exercised only on the expiration date itself.
• An option gives the holder the right to do something → the holder does not have to exercise this
right. This is what distinguishes options from forwards and futures, where the holder is obligated
to buy or sell the underlying asset. Whereas it costs nothing to enter into a forward or futures
contract (except for margin requirements), there is a cost to acquiring an option.
• Properties of options:
o The price of a call option decreases as the strike price increases (less worth to buy at
higher price), while the price of a put option increases as the strike price increases (more
worth to sell at higher price).
o Both types of options tend to become more valuable as their time to maturity increases.
• Four types of participants in the option market:
1. Buyers of calls
2. Sellers of calls
3. Buyers of puts
4. Sellers of puts
▪ Buyers are referred to as having long positions; sellers are referred to as having
short positions. Selling an option is also known as writing the option.
Three broad categories of traders can be identified:
1. Hedgers use derivatives to reduce the risk that they face from potential future movements in a
market variable.
o If an investor is concerned about a possible share price decline (increase) in the next
months and wants protection, the investor could buy a put (call) option contract with a
strike price.
o Fundamental difference between the use of forward contracts and options for hedging:
▪ Forward contracts are designed to neutralize risk by fixing the price that the
hedger will pay or receive for the underlying asset.
▪ Option contracts provide insurance. They offer a way for investors to protect
themselves against adverse price movements in the future while still allowing
them to profit from favourable price movements.
2. Speculators use derivatives to bet on the future direction (either up or down) of a market variable.
o Futures and options are similar instruments for speculators in that they provide a way in
which a type of leverage can be obtained. However, an important difference is:
, ▪ When a speculator uses futures, the potential loss as well as the potential gain
is very large.
▪ When options are purchased, no matter how bad things get, the speculator’s
loss is limited to the amount paid for the options.
3. Arbitrageurs take offsetting positions in two or more instruments to lock in a (riskless) profit
o Transaction costs would probably eliminate the profit for a small trader.
o Arbitrage opportunities cannot last for long → the forces of supply and demand will cause
the price to go to an equilibrium (equivalent to the current exchange rate)
▪ Buying the stock forces the price to rise
▪ Selling the stock will drive the price down
CHAPTER 2: FUTURES MARKETS AND CENTRAL COUNTERPARTIES
Both futures and forward contracts are agreements to buy or sell an asset at a future time for a certain
price.
• A futures contract is traded on an exchange, and the contract terms are standardized by that
exchange. (settled daily)
• A forward contract is traded in the over-the-counter market and can be customized to meet the
needs of users. (settled at the end of its life)
A trader who agrees to buy has a long futures position in one contract; a trader who agrees to sell has
a short position in one contract. The price agreed to is the current futures price for the product which
is bought at a certain time in the future (for a price agreed upon today).
The vast majority of futures contracts do not lead to delivery. The reason is that most traders choose
to close out their positions (i.e., enter into the opposite trade to the original one) prior to the delivery
period specified in the contract.
• Trader who bought a futures contract can close out the position by selling a futures contract.
• Trader who sold (i.e., shorted) a futures contract can close out the position by buying one contract.
• In each case, the trader’s total gain or loss is determined by the change in the futures price
between the dates of buying (selling) and the day when the contract is closed out.
When developing a new contract, the exchange must specify in some detail the exact nature of the
agreement between the two parties. In particular, it must specify the asset, the contract size (exactly
how much of the asset will be delivered under one contract), where delivery can be made, and when
delivery can be made.
• As a general rule, the party with the short position chooses what will happen when alternatives
are specified by the exchange.
• When the party with the short position is ready to deliver, it files a notice of intention to deliver
with the exchange. This notice indicates any selection it has made with respect to the grade of
asset that will be delivered and the delivery location.
Price limits and position limits: daily price movement limits are specified by the exchange.
• If in a day the price moves down from the previous day’s close by an amount equal to the daily
price limit, a contract is said to be limit down. If it moves up by the limit, it is said to be limit up. A
limit move is a move in either direction equal to the daily price limit. The purpose of daily price
limits is to prevent large price movements from occurring because of speculative excesses.
• Position limits are the maximum number of contracts that a speculator may hold. The purpose of
these limits is to prevent speculators from exercising undue influence on the market.
,As the delivery period for a futures contract is approached, the futures price converges to the spot
price of the underlying asset. When the delivery period is reached, the futures price equals – or is very
close to – the spot price.
One of the key roles of the exchange is to organise trading so that contract defaults (one of the traders
regretting the deal and trying to back out) are avoided. This is where margin accounts come in.
• A trader has to keep funds in a margin account. The amount that must be deposited at the time
the contract is entered into is known as the initial margin. At the end of each trading day, the
margin account is adjusted to reflect the trader’s gain or loss. This practice is referred to as daily
settlement or marking to market. A trade is first settled at the close of the day on which it takes
place. It is then settled at the close of trading on each subsequent day.
• Variation margin is the daily flow of funds between traders to reflect gains and losses. Daily
settlement leads to funds flowing each day between traders with long and short positions.
o If the futures price increases from one day to the next, funds flow from traders with short
positions to traders with long positions (i.e., long positions make profit).
▪ Payoff long position is 𝑆𝑇 − 𝐾, so if 𝑆𝑇 increases, long position makes profit.
▪ Payoff short position is 𝐾 − 𝑆𝑇 , so if 𝑆𝑇 increases, short position makes loss.
o If the futures price decreases from one day to the next, funds flow from traders with long
positions to traders with short positions (i.e., long positions make loss).
• Most individuals have to contact their brokers to trade and are subject to a maintenance margin.
This is somewhat lower than the initial margin. If the balance in the margin account falls below
the maintenance margin, the trader receives a margin call and is expected to top up the margin
account to the initial margin level within a short period of time. If the trader does not provide this
variation margin, the broker closes out the position.
• Minimum levels for the initial and maintenance margin are set by the exchange clearing house
and are determined by the variability of the price of the underlying asset → the higher the
variability, the higher the margin levels. The maintenance margin is usually about 75% of the initial
margin.
Whereas a forward contract is settled at the end of its life, a futures contract is settled daily. At the
end of each day, the trader’s gain (loss) is added to (subtracted from) the margin account, bringing
the value of the contract back to zero.
In determining margin requirements, the number of contracts outstanding is usually calculated on a
net basis rather than a gross basis: short positions the clearing house member is handling for clients
are netted against long positions.
• Clearing house members are required to contribute to a guaranty fund: may be used by the
clearing house in the event that a member defaults and the member’s margin proves insufficient
to cover losses.
The whole purpose of the margining system is to ensure that funds are available to pay traders when
they make a profit (and limit credit risk → not being able to pay debt).
Credit risk has traditionally been a feature of OTC derivatives markets. To reduce credit risk, the OTC
market has borrowed some ideas from exchange-traded markets:
• Central counterparties (CCPs) are clearing houses for standard OTC transactions that perform
much the same role as exchange clearing houses.
o CCP members have to provide both initial margin and daily variation margin. They are also
required to contribute to a guaranty fund.
, o Once an OTC derivative transaction has been agreed upon between two parties, it can be
presented to a CCP. Assuming the CCP accepts the transaction, it becomes the
counterparty to both A and B (it takes on the credit risk for both parties):
▪ Buy the asset from B in one year for the agreed upon price
▪ Sell the asset to A in one year for the agreed upon price
o If an OTC market participant is not a member of a CCP, it can arrange to clear its trades
through a CCP member. It will then have to provide margin to the CCP member (similar to
the relationship between a broker and a futures exchange clearing house member).
• OTC transactions that are not cleared through CCPs are cleared bilaterally: the two parties enter
into a master agreement covering all their trades. This agreement includes an annex (or credit
support annex; CSA), requiring A and/or B to provide collateral (similar to required margin).
o A simple two-way agreement between companies A and B might work as follows: if (from
one day to the next) the transaction between A and B increases in value to A by X (and
therefore decrease in value to B by X), B is required to provide collateral worth X to A.
• Transactions in the OTC market, whether cleared through CCPs or cleared bilaterally, are usually
not settled daily. For this reason, the daily variation margin that is provided by the member of a
CCP or, as a result of a CSA, earns interest when it is in the form of cash.
Future quotes are available from exchanges and several online sources, they include the asset
underlying the futures contract, the contract size, the price, and the maturity month of the contract.
• Settlement price: the price used for calculating daily gains and losses and margin requirements. It
is usually calculated as the price at which the contract traded immediately before the end of a
day’s trading session. Then, last trade represents the most recent trading price, and change shows
the price change from the previous day’s settlement price.
• Trading volume: the number of contracts traded in a day. In contrast, open interest is the number
of contracts outstanding (i.e., the number of long positions or short positions).
o if there is a large amount of trading by day traders (traders who enter into a position and
close it out on the same day), the volume of trading in a day can be greater than either
the beginning-of-day or end-of-day open interest.
Futures prices can show a number of different patterns:
• Normal market: futures prices increase with the maturity of the futures contract
• Inverted market: futures prices decrease with the maturity of the futures contract
• Mixed pattern: futures prices sometimes increase and sometimes decrease with maturity.
Few of the futures contracts that are entered into lead to delivery of the underlying asset; most are
closed out early. Nevertheless, the futures price is determined by the possibility of eventual delivery.
• The period during which delivery can be made is defined by the exchange and varies from contract
to contract. The decision on when to deliver is made by the party with the short position.
• Notice of intention is issued to the exchange clearing house by the trader with the short position
when he decides to deliver. The notice states how many contracts will be delivered and, in the
case of commodities, also specifies where delivery will be made and what grade will be delivered.
• The exchange chooses a party with a long position to accept delivery, usually the party with the
oldest outstanding long position. Parties with long positions must accept delivery notes (they have
only a short period of time for finding another party with a long position that is prepared to take
delivery in place of them).
There are three critical days for a contract:
1. First notice day is the first day on which a notice of intention to make delivery can be submitted
to the exchange.
,2. Last notice day is the last such day
3. Last trading day is generally a few days before the last notice day
To avoid the risk of having to take delivery, a trader with a long position should close out his or her
contracts prior to the first notice day.
When a contract is settled in cash, all outstanding contracts are declared closed on a predetermined
day. The final settlement price is set equal to the spot price of the underlying asset at either the open
or close of trading on that day.
Two main types of traders executing trades:
• Futures commission merchants (FCMs) are following the instructions of their clients and charge
a commission for doing so.
• Locals are trading on their own account
Individuals taking positions (locals or clients of FCMs) can be categorized as hedgers, speculators, or
arbitrageurs. Speculators can be classified as:
• Scalpers are watching for very short-term trends and attempt to profit from small changes in the
contract price. They usually hold their positions for only a few minutes.
• Day traders hold their positions for less than one trading day. They are unwilling to take the risk
that adverse news will occur overnight.
• Position traders hold their positions for much longer periods of time. They hope to make
significant profits from major movements in the market.
Types of orders:
• Market order: a request that a trade be carried out immediately at the best price available in the
market (the simplest type of order placed with a broker)
• Limit order: specifies a particular price. The order can be executed only at this price or at one
more favourable to the trader. There is no guarantee that the order will be executed, because the
limit price may never be reached.
• Stop order (or stop-loss order): order that is executed at the best available price once a bid or ask
is made at that particular price or a less favourable price. In effect, a stop order becomes a market
order as soon as the specified price has been hit. The purpose of a stop order is usually to close
out a position if unfavourable price movements take place (limits loss that can be incurred).
• Stop-limit order: a combination of a stop order and a limit order. The order becomes a limit order
as soon as a bid or ask is made at a price equal to or less favourable than the stop price. Two prices
must be specified in a stop-limit order: the stop price and the limit price.
o if the stop and limit price are the same, the order is called stop-and-limit order.
• Market-if-touched (MIT) order: executed at the best available price after a trade occurs at a
specified price or at a price more favourable than the specified price. In effect, an MIT becomes a
market order once the specified price has been hit.
o An MIT is also known as a board order. Consider a trader who has a long position in a
futures contract and is issuing instructions that would lead to closing out the contract.
▪ A stop order is designed to place a limit on the loss that can occur in the event of
unfavourable price movements
▪ A MIT order is designed to ensure that profits are taken if sufficiently favourable
price movements occur.
• Discretionary order or market-not-held order: traded as a market order except that execution
may be delayed at the broker’s discretion in an attempt to get a better price
• Time-of-day order: specifies a particular period of time during the day when the order can be
executed.
,• Open order or good-till-cancelled order: is in effect until executed or until the end of trading in
the particular contract.
• Fill-or-kill order: must be executed immediately on receipt or not at all.
Futures markets in the US are currently regulated federally by the Commodity Futures Trading
Commission (CFTC) which was established in 1974. However, with the formation of the National
Futures Association (NFA) in 1982, some of the responsibilities of the CFTC were shifted to the futures
industry itself.
Corner the market: a trading irregularity that occurs when a trader group takes a huge long futures
position and tries to exercise some control over the supply of the underlying commodity.
• As the maturity of the futures contract is approached, the trader group does not close out its
position, so that the number of outstanding futures contracts may exceed the amount of the
commodity available for delivery.
• The holders of short positions realize that they will find it difficult to deliver and become desperate
to close out their positions. The result is a large rise in both futures and spot prices.
• Regulators usually deal with this type of abuse of the market by increasing margin requirements
or imposing stricter positions limits or prohibiting trades that increase a speculator’s open position
or requiring market participants to close out their positions.
Accounting standards require changes in the market value of a futures contract to be recognized when
they occur unless the contract qualifies as a hedge. Hedge accounting: if the contract qualifies as a
hedge, gains or losses are generally recognized for accounting purposes in the same period in which
the gains or losses from the items being hedged are recognized.
A hedge transaction is defined by tax regulations as a transaction entered into in the normal course
of business primarily for one of the following reasons:
1. To reduce the risk of price changes or currency fluctuations with respect to property that is held
or to be held by the taxpayer for the purpose of producing ordinary income
2. To reduce the risk of price or interest rate changes or currency fluctuations with respect to
borrowings made by the taxpayer.
Comparison of forward and futures contracts
Forward Futures
Private contract between two parties Traded on an exchange
Not standardized Standardized contract
Usually one specified delivery date Range of delivery dates
Settled at end of contract Settled daily
Delivery/final cash settlement usually happens Contract is usually closed out prior to maturity
Some credit risk Virtually no credit risk
Both contracts are agreements to buy or sell an asset for a certain price at a certain future time.
• A forward contract is traded in the OTC market and there is no standard contract size or standard
delivery arrangements. A single delivery date is usually specified and the contract is held to the
end of its life and then settled. Whole gain or loss realized at the end of the life of contract.
• A futures contract is a standardized contract traded on an exchange. A range of delivery dates is
usually specified. It is settled daily and usually closed out prior to maturity. The gain or loss is
realized day by day because of daily settlement procedures.
CHAPTER 3: HEDGING STRATEGIES USING FUTURES
Perfect hedge: a hedge that completely eliminates risk.
,A study of hedging using futures contracts is a study of the ways in which hedges can be constructed
so that they perform as close to perfectly as possible.
When an individual or company chooses to use futures markets to hedge a risk, the objective is often
to take a position that neutralizes the risk as far as possible.
• If the price of the commodity goes up, the loss on the futures position is offset by the gain on the
rest of the company’s business.
• If the price of the commodity goes down, the loss of the company’s business is offset by the gain
on the futures position.
Short hedge: hedge that involves a short position in a futures contract.
• Appropriate when the hedger already owns an asset and expects to sell it at some time in the
future.
• Can also be used when an asset is not owned right now but will be owned and ready for sale at
some time in the future.
Long hedge: hedge that involves taking a long position in a futures contract.
• Appropriate when a company knows it will have to purchase a certain asset in the future and
wants to lock in a price now.
• Hedgers with long positions usually avoid any possibility of having to take delivery by closing out
their positions before the delivery period.
Arguments for hedging:
• Hedge risks associated with variables such as interest rates, exchange rates, and commodity
prices, that are difficult for companies to predict (so avoid unpleasant surprises). The company
can then focus on their main activities.
• Hedging is likely to be less expensive when carried out by the company than by individual
shareholders (commission and transaction costs).
o Shareholders can more easily than a corporation diversify risk. A shareholder with a well-
diversified portfolio may be immune to many of the risks faced by a corporation.
Therefore, it is also argued that hedging is unnecessary in many situations.
Arguments against hedging:
• Competitive pressures within the industry may be such that the prices of the goods and services
produced by the industry fluctuate to reflect raw material costs, interest rates, exchange rate, and
so on. If hedging is not the norm, then it might not make sense to be different from competitors.
o A company that does not hedge can expect its profit margins to be roughly constant: the
price it pays for the materials will be reflected in the price that it sells the products for.
o A company that does hedge can expect its profit margins to fluctuate:
▪ Price goes up but company has hedged its buy price, so profit margin increases
due to the increase in the market price it can sell (as competitors do not hedge)
▪ Price goes down but company has hedged its buy price, so profit margin decreases
due to decrease in market price it sells the product for
• A hedge using futures contracts can result in a decrease or increase in a company’s profits relative
to the position it would be in with no hedging.
• Hedging may increase risks for treasurers if others do not fully understand what is being done.
Ideally, hedging strategies are set by a company’s board of directors and are clearly communicated
to both the company’s management and the shareholders.
In practice, hedging is not as easy for the following reasons:
• The asset whose price is to be hedged may not be exactly the same as the asset underlying the
futures contract.
,• There may be uncertainty as to the exact date when the asset will be bought or sold
• The hedge may require the futures contract to be closed out before its delivery month
These problems give rise to what is termed basis risk.
• Basis = Spot price of asset to be heged − Futures price of contract used
• If the asset to be hedged and the asset underlying the futures
contract are the same, the basis should be zero at the
expiration of the futures contract. Prior to expiration, the basis
may be positive or negative.
• As time passes, the spot price and the futures price for a
particular month do not necessarily change by the same
amount. As a result the basis increases (strengthens) or
decreases (weakens).
Assume a hedge is put in place at time 𝑡1 and closed out at time 𝑡2 .
From the definition of the basis we have: 𝑏1 = 𝑆1 − 𝐹1 and 𝑏2 = 𝑆2 − 𝐹2 . We assume that 𝑆2 < 𝑆1
and 𝐹2 < 𝐹1
• Consider the situation of a hedger who knows that the asset will be sold at time 𝑡2 and takes a
short futures position at time 𝑡1 . The price realised for the asset is 𝑆2 and the profit on the futures
option is 𝐹1 − 𝐹2 . The effective price that is obtained for the asset with hedging is therefore:
o 𝑆2 + 𝐹1 − 𝐹2 = 𝐹1 + 𝑏2
▪ The value of 𝐹1 is known at time 𝑡1 .
▪ If 𝑏2 were known at time 𝑡1 , a perfect hedge would be the result. The hedging risk
is the uncertainty associated with 𝑏2 and is known as the basis risk.
• Consider the situation where a company knows it will buy the asset at time 𝑡2 and initiates a long
hedge at time 𝑡1 . The price paid for the asset is 𝑆2 and the loss on the hedge is 𝐹1 − 𝐹2 . The
effective price that is paid with hedging is therefore:
o 𝑆2 + 𝐹1 − 𝐹2 = 𝐹1 + 𝑏2
Basis changes can lead to an improvement or worsening of a hedger’s position:
• Consider a company that uses a short hedge because it plans to sell the underlying asset:
o If the basis strengthens (i.e., increases) unexpectedly, the company’s position improves
because it will get a higher price for the asset after futures gains or losses are considered.
o If the basis weakens (i.e., decreases) unexpectedly, the company’s position worsens
• Consider a company that uses a long hedge because it plans to buy the asset:
o If the basis strengthens unexpectedly, the company’s position worsens, because it will pay
a higher price for the asset after futures gains or losses are considered
o If the basis weakens unexpectedly, the company’s position improves.
Cross hedging: the asset that gives rise to the hedger’s exposure is sometimes different from the asset
underlying the futures contract that is used for hedging. This leads to an increase in basis risk. Define
𝑆2∗ as the price of the asset underlying the futures contract at time 𝑡2 . As before, 𝑆2 is the price of the
asset being hedged at time 𝑡2 . By hedging a company ensures that the price that will be paid (or
received) for the asset is:
• 𝑆2 + 𝐹1 − 𝐹2 → 𝐹1 + (𝑆2∗ − 𝐹2 )
⏟ + ⏟ (𝑆2 − 𝑆2∗ )
basis if asset being basis arising from
hedged is the same the difference
as the asset underlying between the two
assets
the futures contract
Basis risk is affected by the choice of the futures contract in two components:
, 1. The choice of the asset underlying the futures contract → it requires careful analysis to determine
which of the available futures contracts has futures prices that are most closely correlated with
the price of the asset being hedged.
2. The choice of the delivery month → basis risk increases as the time difference between the hedge
expiration and the delivery month increases. Rule of thumb: choose a delivery month that is as
close as possible to, but later than, the expiration of the hedge.
Cross hedging occurs when the two assets (asset underlying future contract and the asset whose price
is being hedged) are different.
Hedge ratio: the ratio of the size of the position taken in futures contracts to the size of the exposure.
• Hedge ratio of 1.0: the asset underlying the futures contract is the same as the asset being hedged
• With cross hedging, the hedger should choose a value for the hedge ratio that minimizes the
variance of the value of the hedge position.
Calculating the minimum variance hedged ratio
The minimum variance hedge ratio depends on the relationship between changes in the spot price
(∆𝑆) and changes in the futures price (∆𝐹) during a period of time equal to the life of the hedge.
If we assume that the relationship between ∆𝑆 and ∆𝐹 is approximately linear, we can write:
• ∆𝑆 = 𝑎 + 𝑏∆𝐹 + 𝜖
Suppose that the hedge ratio is ℎ (i.e., a percentage ℎ of the exposure to 𝑆 is hedged with futures).
Then the change in the value of the position per unit of exposure to 𝑆 is:
• ∆𝑆 − ℎ∆𝐹 = 𝑎 + (𝑏 − ℎ)∆𝐹 + 𝜖
o The standard deviation of this is minimized by setting ℎ = 𝑏 (so that the second term on
the right-hand side disappears)
Denote the minimum variance hedge ratio by ℎ∗. We have shown that ℎ∗ = 𝑏. It follows from the
𝜎
formula for the slope in the linear regression that: ℎ∗ = 𝜌 𝜎𝑆 .
𝐹
• If 𝜌 = 1 and 𝜎𝐹 = 𝜎𝑆 , the hedge ratio ℎ∗ = 1.0. This result is
expected because the futures price mirrors the spot price perfectly.
• If 𝜌 = 1 and 𝜎𝐹 = 2𝜎𝑆 , the hedge ratio ℎ∗ = 0.5. This result is also
expected because the futures price always changes by twice as much
as the spot price.
The hedge effectiveness is defined as the proportion of the variance that is eliminated by hedging.
This is the 𝑅 2 from the regression of ∆𝑆 against ∆𝐹 and equals 𝜌2 .
𝑜𝑝𝑡𝑖𝑚𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓
𝑎𝑠𝑠𝑒𝑡𝑠
ℎ ∗ 𝑄𝐴 ⏞∗ 𝑄𝐴
ℎ
The optimal number of futures contracts required in hedging: 𝑁 ∗ = 𝑄𝐹
= 𝑄⏟𝐹
ℎ𝑜𝑤 𝑚𝑎𝑛𝑦 𝑎𝑠𝑠𝑒𝑡𝑠
𝑝𝑒𝑟 𝑓𝑢𝑡𝑢𝑟𝑒𝑠 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡
• 𝑄𝐴 : Size of position being hedged (units)
• 𝑄𝐹 : Size of one futures contract (units)
• 𝑁 ∗ : Optimal number of futures contracts for hedging
Daily settlement of futures contracts mean that when futures contracts are used, there are a series
of one-day hedges, not a single hedge. Define:
• 𝜎̂𝑆 : Standard deviation of percentage one-day changes in the spot price
• 𝜎̂𝐹 : Standard deviation of percentage one-day changes in the futures price
• 𝜌̂: Correlation between percentage one-day changes in the spot and futures
The standard deviation of one-day changes in spot and futures are 𝜎̂𝑆 𝑆 and 𝜎̂𝐹 𝐹. It follows that the
̂ 𝑆
𝜎 ̂ 𝑆𝑄
𝜎
optimal one-day hedge is: ℎ∗ = 𝜌̂ 𝜎̂𝑆𝐹, so 𝑁 ∗ = 𝜌̂ 𝜎̂𝑆 𝐹𝑄𝐴
𝐹 𝐹 𝐹
