EC372 Bond and Derivatives Markets
Topic #3: Futures Markets II: Speculation and Hedging
R. E. Bailey
Department of Economics
University of Essex
Outline
Contents
1
Speculation
1
2
Hedging strategies
2
2.1
2
Perfect (risk-free) hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Optimal hedging
5
4
Theories of futures prices
7
4.1
7
5
Normal backwardation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Manipulation of futures markets
8
Reading: Economics of Financial Markets, chapter 15
1
Speculation
Speculation
• Motive: to profit from expected changes in futures prices.
• Hicks Value and Capital, p. 138: ‘Futures prices are . . . nearly always made partly by speculators, who seek a profit by buying futures when the futures price is below the spot price they
expect to rule on the corresponding date . . . ’
1. Speculators partly make prices – hedgers matter too.
2. Implies that contracts will be held to maturity – ‘corresponding’ date – when f (T, T ) =
p(T )
3. Only buying futures is mentioned but speculators may sell
• Investor who expects price to rise: take a long position – buy, planning to sell later at a higher
price
• Investor who expects price to fall: take a short position – sell, planning to buy later at a lower
price
1
Comments on the Hicks’s quotation:
1. Hicks claims only that speculators are partly responsible for futures price determination.
Hedging and arbitrage also play a role, though the consequence of arbitrage is for the link
between futures and current spot prices, rather than on the general level of prices. The usual
presumption is that speculators and hedgers inhabit opposite sides of the market (one group
buying contracts from the other).
2. Hicks refers to the expected spot price at the delivery date (what Hicks calls the ‘corresponding
date’). Hicks has in mind that the futures contracts will be held to maturity when the speculator
will either take delivery (or make delivery) of the underlying asset and simultaneously sell (or
purchase) it at the spot price. The speculator’s profit or loss is then the difference in the price at
which the futures contract was acquired and the spot price upon delivery. Of course, in active
futures markets the contract may well be offset before delivery takes place. But the principle
is the same: the profit or loss depends on the change in price between the date at which the
contract is acquired and the price when it is offset.
3. Only speculation in the form of buying futures is mentioned. There is no reason, in principle,
why speculation by selling futures should be excluded. The direction — purchase or sale —
depends on the whether the investor believes that the price will rise or fall. However, Hicks
was describing a market in which most hedgers seek to reduce risk by selling futures contracts.
Speculators take the other side of the bargains, that is, they buy the contracts sold by hedgers.
2
Hedging strategies
Hedging strategies
• Principle: reduce the risk associated with one asset by holding a second asset such that, together, the payoffs cancel out across states of the world.
• Hedgers reduce the price risk of an underlying asset, or just ‘asset’
• Hedge instrument (or hedge-asset) is used to offset the risk
– ‘short hedge’: sell the hedge instrument
– ‘long hedge’: buy the hedge instrument
• Perfect hedge (risk-free hedge): eliminates price risk
• Most hedging is risky (imperfect): some risk remains
Hedging: a simple example.
Consider a world with two states (labelled 1 and 2) and two assets (labelled A and B) with payoffs as
follows:
Asset A
Asset B
State 1
−2
6
State 2
3
−4
Assets A and B are risky: their returns differ across states. But any strategy that holds A and B in
the ratio 2:1 results in a payoff that is identical across states, thereby eliminating risk. Thus asset A
could be used as a perfect hedge for asset B, and conversely.
2
2.1
Perfect (risk-free) hedging
Perfect (risk-free) hedging
• Example: a perfect hedge
– A firm plans to sell oil 11 months from today but price on the delivery date is unknown
– It sells oil futures contracts for delivery in 11 months
– In 11 months it delivers the oil and receives the price agreed 11 months previously.
– Price risk has been eliminated
• Why are perfect hedges rare?
1. A suitable hedge instrument may not exist
2. Futures contracts are standardized
3. The firm may be uncertain about its planned delivery date.
4. Tailing the hedge – complication as a result of marking to market.
Effective price†
6
No hedging
pb
Risk-free hedge
-
Market price†
pb
Figure 1: What a (perfect) hedge can achieve.
† = prices when the hedge is lifted, i.e., liquidated or concluded.
In Figure 1, pb is the price secured with a perfect (risk-free) hedge, i.e., it is independent of the
observed market price. If the hedge is risky, as most are, then the horizontal line at pb should be
replaced with a band (a ‘thick’ line) reflecting uncertainty about the effective price, when the hedge
is initiated, for whatever market price happens to be observed when the hedge is lifted.
Notice that there is always the possibility of regret on the part of the hedger: this is in the nature of
hedging. A short hedger would have gained by not hedging if the market price when the hedge is
lifted is greater than pb. Conversely, a long hedger would have gained by not hedging if the market
price when the hedge is lifted is lower than pb. But the crucial consideration is that the market prices
3
are unknown in advance. Investors have expectations about future prices but how confident can they
be about whether the expectations will be fulfilled? Confidence is inherently subjective.
Comments on the rarity of risk-free hedges:
1. A market in a suitable hedge instrument may not exist. Futures contacts (and options, for that
matter) are traded for only a limited range of commodities or financial securities. Although
in principle a forward contract could be negotiated, it may be too expensive, or impossible, to
find a counterparty to accept the other side of the bargain.
2. Futures contracts, where available, are standardized. The specification of the futures contract
may not correspond exactly with the requirements of the company seeking to acquire a hedge.
For example, the grade of crude oil that a short-hedger plans to supply may not match that
specified in the futures contract. Similarly, date or location of delivery may differ from that of
the futures contract: the company may wish to deliver in December for storage in New York,
while the (NYMEX) contract specifies delivery to Cushing, Oklahoma.
3. The company adopting the short-hedge might not be certain about the date at which the oil
will be ready for delivery. A futures contract maturing in November may be available but
the short-hedger might not know whether the oil will have arrived at a designated location by
then. Similarly, the company with a long-hedge might not know whether it will be ready to
take delivery in November.
4. Tailing the hedge. Even if the company is sure to make (or take) delivery of the commodity
according to the terms of the futures contract, a complication arises because futures contracts
are marked-to-market. If the price of the futures contract changes on any day prior to delivery,
outstanding contracts are revalued at the new market price, any gain (or loss) being credited (or
debited) to the holder’s margin account. Although interest accrues to the funds in the margin
account, the gain or loss for any investor holding an open futures position (short or long) will
depend upon the pattern of price changes while the position remains open. Hence, the amount
of interest earned or foregone as a result of marking to market is unknown at the outset and
may require adjustment (rebalancing) of the futures position. This may seem trivial but may
be significant for investors with large futures positions.
Perfect hedge: formal treatment
Take care: the formal analysis below corresponds to a short hedge, for which the underlying asset
is to be sold at a later date (typically it is owned today, for sale later). The notation for a long hedge
straightforwardly by re-interpreting the symbols: thus N becomes the amount of the underlying asset
to be purchased at a later date, M is the amount of the hedge asset purchased today, etc.
Check your understanding by rewriting the following notes for a long hedge.
• Notation:
– Today’s date = 0. Hedge liquidated at date = 1.
– Underlying asset price, p0 and p1
– Hedge instrument price, f0 and f1
– N = amount of underlying asset to be sold
– M = amount of hedge instrument sold
• Payoff at date 1: W1 = p1 N + (f0 − f1 )M
4
• Perfect hedge occurs when p1 = f1 (choose M = N )
– payoff: W1 = f0 N , known for sure at 0.
• More generally: ∆W ≡ W1 − W0 = (p1 − p0 )N + (f0 − f1 )M
• So that: ∆W = ∆p · N − ∆f · M
• Perfect hedge: choose M such that M/N = ∆p/∆f
• Hedge ratio ≡ M/N , equal to ∆p/∆f for perfect hedge only
Terminology: the ‘basis’. The basis is commonly used to refer to the difference between spot
and futures prices for the same asset. Confusingly, it is sometimes defined as p(t) − f (t, T ) and
sometimes as f (t, T ) − p(t). Hence, be careful to state which you are using.
Also note that the two prices may refer to the same asset but there is a crucial difference: p(t) is
the price for immediate delivery, while f (t, T ) is the price for delivery at T > t, in the future. Arguably, the relevant comparison is between f (t, T ) and p(T ) rather than p(t), but p(T ) is unknown
for every t < T . More on this later (section 4).
3
Optimal hedging
Risky hedging
• From the previous slide: ∆W = ∆p · N − ∆f · M
∆W
= ∆p − h∆f
N
where h = M/N is the hedge ratio
• Hence:
• Objective: choose hedge to minimize ‘risk’
• Formally: choose h to minimize var (∆W/N )
• Result: pure hedge ratio ≡ h∗ =
σpf
σf2
• h∗ is the slope coefficient of a regression of ∆p on ∆f
∆p = intercept + h∗ · ∆f + residual
Risk Minimization
When hedging is risky, the number, M , of units of the hedge instrument sold or purchased
generally differs from the number of units of the asset to be sold or purchased. To understand why
this should be so, consider an investor who plans to sell N units of the asset at date 1 and who hedges
by selling of M units of the hedge instrument.
The risk associated with the hedged position stems from uncertainty about ∆p (because p1 is
unknown at date 0) and ∆f (because f1 is unknown at date 0). Given that ∆p and ∆f are unknown
at date 0, it follows that no choice of h can guarantee to eliminate the risk of fluctuations in ∆W/N .
The variance of ∆W/N can be written as
∆W
var
= σp2 + h2 σf2 − 2hσpf
N
5
(1)
where σp2 is the variance of ∆p, σf2 is the variance of ∆f , and σpf is the covariance between ∆p and
∆f . The value of h that minimizes the variance, var(∆W/N ), is found by differentiating (1) with
respect to h and setting the derivative to zero.
The resulting value, h∗ , here called the pure hedge ratio, is given by
h∗ =
σpf
σf2
(2)
(Sometimes h∗ is called the optimal hedge ratio but, here, this term is reserved for a different concept,
introduced later.) Another way of writing h∗ is h∗ = ρσp /σf where ρ is the correlation coefficient
between ∆p and ∆f . The value of ρ2 has been proposed to measure the effectiveness of hedging.
The closer the linear relationship between ∆p and ∆f , the less risky is the hedge. In the limit as
ρ2 → +1 the hedge becomes perfect.
The pure hedge ratio, h∗ , is nothing more than the slope coefficient in an ordinary least squares
regression of ∆p against ∆f . That is, the pure hedge ratio could be constructed from
∆p = θ + h∗ ∆f + ε
(3)
where ε is an unobserved random error, such that E[ε | ∆f ] = 0, and θ = E[∆p] − h∗ E[∆f ]. The
presence of the random error is just another way of expressing the riskiness of the hedge – if the
random error is identically zero, a perfect hedge could be constructed.
Given that, for a risky hedge, the relationship between ∆f and ∆p is not exact, it is also unlikely
that h∗ can be known for sure. The regression equation, (3), however, suggests a way of estimating h∗
from data on ∆f and ∆p. A graphical illustration appears in figure 2. Each dot in figure 2 denotes a
pair of realized values of (∆f, ∆p). The slope of the ordinary least squares line is, by construction,
equal to the pure hedge ratio, where sample values of the covariance, σpf , and variance, σf2 , are
obtained from the observations.
∆p
6
q
q
q
q
q
q q
q
q
- ∆f
q
q
Figure 2: The slope of the fitted line is an estimate of the pure hedge ratio, h∗ .
Each dot represents an observed pair (∆p, ∆f ) measured for a particular time interval.
The line depicts the Ordinary Least Squares ‘line of best fit’, the slope of which is an
estimate of σpf /σf2 , the pure hedge ratio. The closer the observations to the fitted line,
the more accurate the estimates and the less risky the hedge.
Hedging as Portfolio Choice
6
• ‘Pure hedge’ places all the weight on risk: ignores expected return
• Portfolio choice: allows trade-off between expected return and risk
• Formally, choose the hedge-ratio, h, to maximize:
U = E (∆W/N ) − α · var (∆W/N )
where α expresses risk preferences (risk aversion)
• Result: h̃ that maximizes U is
h̃ = h∗ +
f0 − E(f1 )
2ασf2
(h̃ is called the optimal hedge ratio)
• ‘Pure hedge’ component: ≡ h∗ =
• ‘Speculative’ component:
4
σpf
σf2
f0 − E(f1 )
2ασf2
Theories of futures prices
Theories of futures prices
• Objective: to predict f (t, T ) − p(t)
• Two approaches:
1. Storage costs and convenience yields:
f (t, T ) − p(t) depends on c(t, T ), y(t, T ) and R(t, T )
(derived from arbitrage principle)
2. Price expectations, Et [p(T )].
• Write: f (t, T ) − p(t) = {f (t, T ) − Et [p(T )]} + {Et [p(T )] − p(t)}
– {f (t, T ) − Et [p(T )]} = ‘bias’ or ‘premium’
– {Et [p(T )] − p(t)} = expected change in the spot price
• Common assumption: f (t, T ) = Et [p(T )]
– Sometimes justified as ‘Efficient Markets Hypothesis’
– But ignores risk aversion (and hence possible risk premia)
• To be testable, the approach requires a model to predict Et [p(T )].
4.1
Normal backwardation
Normal Backwardation
• ‘Backwardation’ defined by f (t, T ) < p(t)
• Keynes-Hicks Theory: asserts that backwardation is normal
– Justification: futures markets are dominated by short-hedgers
7
– Short-hedgers sales drive down futures compared with spot prices
– Hedgers: benefit from reduced risk
– Speculators: benefit from risk premium (buy futures from hedgers)
• Ambiguity: is the prediction f (t, T ) < p(t) or f (t, T ) < Et [p(T )]?
– Usual interpretation: f (t, T ) < Et [p(T )]
• Comments:
1. Empirical studies are inconclusive – need a theory of Et [p(T )]
2. Not all futures markets are dominated by short-hedgers
3. Futures markets are not isolated from other asset markets
4. Evidence (Houthakker) that speculators tend to lose money
More detail on Normal Backwardation
Hicks developed Keynes’s argument as follows:
. . . while there is likely to be some desire to hedge planned purchases, it tends to be less
insistent than the desire to hedge planned sales. If forward markets consisted entirely of
hedgers, there would always be a tendency for a relative weakness on the demand side;
a smaller proportion of planned purchases than of planned sales would be covered by
forward contracts. [footnote omitted] . . .
[This provides an opportunity for speculators whose] action tends to raise the futures
price to a more reasonable level. But it is of the essence of speculation, as opposed to
hedging, that the speculator puts himself into a more risky position as a result of his
forward trading . . . . He will therefore only be willing to go on buying futures so long
as the futures price remains definitely below the spot price he expects; for it is the
difference between these prices which he can expect to receive as a return for his riskbearing, and it will not be worth his while to undertake the risk if the prospective return
is too small. (Hicks, Value and Capital, 1939, pages137–8.)
The steps in the Keynes-Hicks theory are as follows:
1. In ‘normal’ conditions, futures markets are dominated by short-hedgers.
2. The sales of futures contracts by short-hedgers depresses the futures price relative to the spot
price. The arbitrage condition allows for this to the extent that (a) the ‘convenience yield’ is
high, or (b) stocks of the asset are limited.
3. Speculators seek to profit from the difference between the futures price and the expected spot
price, Et [p(T )], at the maturity date of the futures contract.
4. The actions of speculators (in buying futures contracts sold by short-hedgers) constrains the
extent to which the futures price can fall below the current spot price, p(t).
5
Manipulation of futures markets
Manipulation of futures markets
8
• Manipulation = ‘unfair’ practices, typically monopoly power (traders become ‘price-makers’).
• Cornering the market
– An investor buys a large proportion of futures contracts
and stocks of the underlying asset
Given that other investors must have sold the contracts (i.e. short positions), this implies
that those investors must find quantities of the underlying asset to make delivery, unless
they can offset their positions by buying contracts.
– Holds futures to await delivery (which means that investors with short positions are in
trouble, assuming that they don’t intend to default)
– Rapid and big rise in futures (and spot) prices
Sometimes the term ‘squeeze’ is used to denote circumstances in which one investor
holds a high proportion of all outstanding futures contracts but not the underlying asset
itself. In this case, prices may still rise but not by as much as when there is a corner.
• Manipulation generally regarded as improper:
– Provides unfair advantage
– Discourages trade (i.e. weakens market mechanism)
• Puzzles and problems:
– Difficult to prove: evidence relies on motive
– Do manipulators profit from manipulation?
– prices fall as they try to realise their gains
Summary
Summary
1. Speculators seek to profit by exploiting their beliefs: they take bets on prices.
2. Hedgers seek to minimize price risk.
3. Hedgers rarely eliminate risk altogether
4. Hedging is a portfolio decision (trade-off risk and expected return)
5. Hedging, speculation and arbitrage all affect futures prices.
Normal backwardation: a theory of limited applicability
6. Futures markets widely believed to be vulnerable to manipulation
9