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Determination of Forward and Futures Prices · Arbitrage arguments are used to determine the...
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Determination of Forward
and Futures Prices
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
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A popular trading (arbitrage) strategy is the short-
selling or shorting.
This trade involves selling an asset that is not owned
by the seller, and that the seller has borrowed.
Short selling is motivated by the belief that an asset’s
price will decline, enabling it to be bought back at a
lower price to make a profit.
Short selling is largely prompted by speculation.
Short sellers take advantage of any likely arbitrage
opportunities.
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Short selling
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Investor A instructs a broker to short 500 shares who
does not own.
The broker will borrow the shares from another client,
say Investor B, and sell them in the market.
Investor A can maintain the short position for as long
as desired, provided there are always shares to
borrow.
At some stage, Investor A will close out the position by
purchasing the 500 shares.
The 500 shares are replaced in the account of Investor
B.
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Short selling
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Investor A takes a profit if the stock price has declined
and a loss if it has risen.
If at any time while the contract is open the broker is
not able to borrow shares, investor A is forced to close
out the position, even if not ready to do so.
Investor A must pay dividends and other benefits to the
owner of the shares, i.e. to Investor B via the broker.
A broker fee is also charged to Investor A for borrowing
the shares.
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Short selling
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Investor A is required to maintain a margin account
with the broker.
The margin account consists of cash or marketable
securities deposited by the Investor with the broker to
guarantee that the investor will not walk away from the
short position if the share price increases.
An initial margin is required and if there are adverse
movements (i.e., increases) in the price of the asset
that is being shorted, additional margin may be
required.
If the additional margin is not provided, the short
position is closed out.
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Short selling
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Consider the position of Investor A who shorts 500
shares in April when the price per share is $120.
Investor A closes out the position by buying them back
in July when the price per share is $100.
Suppose that a dividend of $1 per share is paid in May.
Investor A receives 500 x $120 = $60,000 in April when
the short position is initiated.
The dividend leads to a payment by Investor A of 500
x $1 = $500 in May.
Investor A also pays 500 x $100 = $50,000 for shares
when the position is closed out in July.
The net gain is: $60,000 - $500 - $50,000 = $9,500
assuming there is no fee for borrowing the shares. 6
Short selling: Example
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Massive short selling is a practice that is often
observed after the burst of a price bubble.
Examples are the Dutch Tulip-mania in the
seventeenth century, the U.S. stock price crash in
1929, the NASDAQ price bubble of 1998-2000, etc.
Since the practice of short selling is alleged to magnify
the decline of asset prices, it has been banned and
restricted many times during history.
As such, short selling bans have been commonly used
as a regulatory measure to stabilise prices during
downturns in the economy.
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Short selling
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The most recent example was in September of 2008 with
the prohibition of short selling by the U.S. Securities and
Exchange Commission (SEC) for 799 financial companies in
an effort to stabilise those companies.
At the same time the U.K. Financial Services Authority
(FSA) prohibited short selling for 32 financial companies.
On September 22, Australia enacted even more extensive
measures with a total ban of short selling.
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Short selling
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S0: Spot price today
F0: Futures or forward price today
T: Time until delivery date
r: Risk-free interest rate for maturity T
r is the rate at which money is borrowed or lent when
there is no credit risk, so that the money is certain to be
repaid.
Participants in derivatives markets have traditionally
assumed that LIBOR rates are the relevant risk-free
rates.
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Pricing Futures and Forward Contracts
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LIBOR is a reference interest rate; it is designed to reflect
the rate of interest at which banks are prepared to make
large wholesale deposits with other banks.
LIBOR is quoted in all major currencies for maturities up to
12 months:
E.g., 1-month LIBOR is the rate at which 1-month deposits
are offered.
A deposit with a bank can be regarded as a loan to that
bank.
A bank must satisfy certain creditworthiness criteria in order
to be able to receive deposits from another bank at LIBOR.
Typically, it must have a AA credit rating.
LIBID is the rate which a AA bank is prepared to pay on
deposits from another bank
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Pricing Futures and Forward Contracts
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Arbitrage arguments are used to determine the forward
and futures prices of an asset from its spot price and
other observable market variables.
In other words, it is the trading activities of market
participants to take advantage of arbitrage
opportunities as they occur that determine the
relationship between forward and spot prices.
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Pricing Futures and Forward Contracts
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Suppose an asset that provides the holder with no
income: non-dividend-paying stocks, or zero-coupon
bonds.
Consider a long forward contract to purchase a non-
dividend-paying stock in 3 months.
Assume the current stock price is $40 and the 3-month
risk-free interest rate is 5% per annum.
Suppose that the forward price is relatively high at $43.
An arbitrageur can borrow $40 at the risk-free interest
rate of 5% per annum, buy one share, and short a
forward contract to sell one share in 3 months.
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Forward price of an asset
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Forward price of an asset
If the spot price of an investment asset is S0 and the
futures price for a contract deliverable in T years is F0,
then:
F0 = S0erT
where r is the T-year risk-free rate of interest.
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Interest Rates – Compounding
When we compound m times per year at rate R an
amount A grows to A(1+R/m)m in one year
Compounding frequency Value of $100 in one year at 10%
Annual (m=1) 110.00
Semiannual (m=2) 110.25
Quarterly (m=4) 110.38
Monthly (m=12) 110.47
Weekly (m=52) 110.51
Daily (m=365) 110.52
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Interest Rates – 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.
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At the end of the 3 months, the arbitrageur delivers the
share and receives $43.00
The sum of money required to pay off the loan is:
Hence, the arbitrageur locks in a profit of $43.00 -
$40.50 = $2.50 at the end of the 3-month period.
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Forward price of an asset
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Suppose that the forward price is at $39.00
An arbitrageur can short one share, invest the money
brought in from a transaction of the short sale at 5%
per annum for 3 months, and take a long position in a
3-month forward contract.
The proceeds of the short sale in 3 months grow to:
At the end of the 3 months, the arbitrageur pays $39,
takes delivery of the share under the terms of the
forward contract, and uses it to close out the short
position.
A net gain of $40.50 - $39.00 = $1.50 is made at the
end of the 3 months. 17
Forward price of an asset
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A net gain of $40.50 - $39.00 = $1.50 is made at the
end of the 3 months.
Overall:
a) The first arbitrage works when the forward price is
greater than $40.50.
b) The second arbitrage works when the forward price is
less than $40.50.
We deduce that for there to be no arbitrage the forward
price must be exactly $40.50.
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Forward price of an asset
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Short sales are not possible for all assets and
sometimes is banned from authorities.
This does not matter: we can still derive and use the
same formula (F0 = S0erT ).
All that we require is that there be a significant number
of people who hold the asset purely for investment
(and by definition this is always true of an investment
asset).
If the forward price is too low, they will find it attractive
to sell the asset and take a long position in a forward
contract.
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Forward price of an asset
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Consider a forward contract on an asset that will
provide a perfectly predictable cash income to the
holder.
Examples are stocks paying known dividends and
coupon-bearing bonds.
We adopt the same approach as before and obtain:
F0 = (S0 – I )erT
where I is the present value of the income during life of
forward contract.
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A Known Income
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Consider the situation where the asset underlying a
forward contract provides a known yield.
This means that the income is known when expressed
as a percentage of the asset’s price at the time the
income is paid.
Suppose that an asset is expected to provide a yield of
5% per annum.
This could mean that income is paid once a year and is
equal to 5% of the asset price at the time it is paid.
The formula is: F0 = S0 e(r–q )T where q is the average
yield during the life of the contract (expressed with
continuous compounding).
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A Known Yield
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When the short-term risk-free interest rate is constant,
the forward price for a contract with a certain delivery
date is in theory the same as the futures price for a
contract with that delivery date.
When interest rates vary, forward and futures prices
are in theory no longer the same.
We can get a sense of the nature of the relationship by
considering the situation where the price of the
underlying asset, S, is strongly positively correlated
with interest rates.
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Forward Prices and Futures Prices
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When S increases, an investor who holds a long
futures position makes an immediate gain because of
the daily settlement procedure.
The positive correlation indicates that it is likely that
interest rates have also increased.
Similarly, when S decreases, the investor will incur an
immediate loss.
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Forward Prices and Futures Prices
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An investor holding a forward contract rather than a
futures contract is not affected in this way by interest
rate movements.
A long futures contract will be slightly more attractive
than a similar long forward contract.
Hence, when S is strongly positively correlated with
interest rates, futures prices will tend to be slightly
higher than forward prices.
When S is strongly negatively correlated with interest
rates, a similar argument shows that forward prices will
tend to be slightly higher than futures prices.
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Forward Prices and Futures Prices
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We have introduced futures on stock indices and
showed how a stock index futures contract is a useful
tool in managing equity portfolios.
We now consider how index futures prices are
determined.
A stock index can usually be regarded as the price of
an investment asset that pays dividends.
The investment asset is the portfolio of stocks
underlying the index, and the dividends paid by the
investment asset are the dividends that would be
received by the holder of this portfolio.
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Futures Prices of Stock Indices
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It is usually assumed that the dividends provide a
known yield rather than a known cash income.
The following formula gives the futures price of stock indices:
F0 = S0 e(r–q )T
where q is the average dividend yield on the portfolio represented by the index during life of contract.
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Futures Prices of Stock Indices
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When F0 > S0e(r-q)T an arbitrageur buys the stocks
underlying the index and sells futures.
When F0 < S0e(r-q)T an arbitrageur buys futures and
shorts or sells the stocks underlying the index.
To do index arbitrage, a trader must be able to trade
both the index futures contract and the portfolio of
stocks underlying the index very quickly at the prices
quoted in the market.
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Index Arbitrage
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Forward and Futures Prices: Summary