1. Forward Contracts
A forward contract is a particularly simple derivative. It is an agreement to buy or sell an
asset at a certain future time for a certain price. It can be contrasted with a spot contract,
which is an agreement to buy or sell an asset today. A forward contract is usually traded
in the over-the-counter market—usually between two financial institutions or between a
financial institution and one of its clients.
One of the parties to a forward contract assumes a long position and agrees to buy the
underlying asset on a certain specified future date for a certain specified price. The other
party assumes a short position and agrees to sell the asset on the same date for the same
price. The price in a forward contract is known as the delivery price. At the time the
contract is entered into, the delivery price is chosen so that the value of the forward
contract to both sides is zero. This means that it costs nothing to take either a long or a
short position.
Assumptions and Notation
We assume that there are some market participants for which the following are true:
1. The market participants are subject to no transaction costs when they trade
2. The market participants are subject to the same tax rate on all net trading profits
3. The market participants can borrow and lend at the same risk-free rate of interest
4. The market participants take advantage of arbitrage opportunities as they occur
The following notation is used:
T: Time when the forward contract matures (years)
S0: Price of asset underlying the forward contract today
F0: Forward price today
r: Risk-free rate of interest per annum.
In general, the payoff from a long position in a forward contract on one unit of an asset is
ST-K
Where K is the delivery price and ST is the spot price of the asset at maturity of the
contract. This is because the holder of the contract is obligated to buy an asset worth ST
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for K. Similarly, the payoff from a short position in a forward contract on one unit of an
asset is
K- ST
These payoffs can be positive or negative. They are illustrated in the figure below.
Because it costs nothing to enter into a forward contract, the payoff from the contract is
also the trader’s total gain or loss from the contract.
Payoff
Payoff
0
0
K
ST
K
a) Long Position
ST
b) Short Position
Generalization
For an investment asset providing no income:
F0= S0er*T
In our example S0=30, r=0.05 and T=2, so that F0= S0er*T = 30e0.05*2 =33.16.
If F0 > S0er*T, arbitrageurs can buy the asset and short forward contacts on the asset
If F0 < S0er*T, they can short the asset and buy forward contracts on it.
Example
Consider a four-month forward contract to buy a zero-coupon bond that will mature one
year from today. The current price of the bond is 930. (Because the bond will have eight
months to go when the forward contract matures, we can regard the contract as on an
eight-month zero-coupon bond.) We assume that the four-month risk-free rate of interest
is 6% per annum. Then the forward price is obtained by
F0= P0er*T = 930e0.06*4/12 =948.79
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2. Future Contracts
Like a forward, a future contract is an agreement between two parties to buy or sell an
asset at a certain time in the future for a certain price. Unlike forward contracts, future
contracts are traded on an exchange. To make trading possible, the exchange specifies
certain standardized features of the contract. As the two parties to the contract do not
necessarily know each other, the exchange also provides a mechanism that gives the two
parties a guarantee that the contract will be honored.
The vast majority of future contracts do not lead to delivery. The reason is that most
traders choose to close out their positions prior to the delivery period specified in the
contract. Closing out a position means entering into the opposite type of trade from the
original one. For example, the New York investor who bought a July corn future contract
on March 5 can close out the position by selling (shorting) one July corn futures contract
on April 20. The Kansas investor who sold (shorted) a July contract on March 5 can close
out the position by buying one July contract on April 20. In each case, the investor’s total
gain or loss is determined by the change in the futures price between March 5 and April
20.
Specification of the Futures Contract
When developing a new contract, an exchange must specify in some detail the exact
nature of the agreement between the two parties. It is the party with the short position that
chooses between these alternatives.
Asset
When the asset is a commodity, there may be quite a variation in the quality of what is
available in the marketplace. The financial assets in futures contacts are generally well
defined.
Contract Size
The contract size specifies the amount of the asset that has to be delivered under one
contract. This is an important decision for the exchange. If the contract size is too large,
many traders who wish to hedge relatively small exposures or who wish to take relatively
small speculative positions will be unable to use the exchange. On the other hand, if the
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contract size is too small, trading may be expensive because there is a cost associated
with each contract traded.
Delivery Arrangements
Although the vast majority of the futures contracts that are initiated do not lead to
delivery of the underlying asset, the delivery arrangements are nevertheless important in
understanding the relationship between the futures price and the spot price of the asset.
Delivery Months
A futures contract is referred to by its delivery month. The exchange must specify the
precise period during the month when delivery can be made. For many futures contracts,
the delivery period is the entire month.
Price Quotes
The futures price is quoted in a way that is convenient and easy to understand (ex. crude
oil futures prices are quoted in dollars per barrel with two decimal places).
Daily Price Movement Limits
For most contracts, daily price movement limits are specified by the exchange. The
purpose of daily price movements is to prevent large price movements from occurring
because of speculative excesses.
Marking to Market
If two people get in touch with each other directly and agree to trade an asset in the future
for a certain price, there are obvious risks. One of them may regret the deal and try to
back out. It is also possible that one of them may not have the financial resources to
honor the agreement. One of the key roles of the exchange is to organize trading so that
contract defaults are minimized.
To illustrate how margins work, consider a trader who contacts a broker on Monday June
3, to buy two December gold futures contracts on the New York Commodity Exchange
(COMEX). We suppose that the current futures price is 400 per ounce. Because the
contract size is 100 ounces, the trader has contracted to buy a total of 200 ounces at this
price. The broker will require the trader to deposit funds in what is termed a margin
account. The amount that must be deposit at the time the contract is entered into is known
as the initial margin. We will suppose this is 2000 per contract, or 4000 in total. At the
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end of each trading day, the margin account is adjusted to reflect the trader’s gain or loss.
This is known as marking to market the account.
Suppose, for example, that by the end of June 3, the futures price has dropped from 400
to 397. The trader has a loss of 200*3, or 600. This is because the 200 ounces of
December gold, which the trader contracted to buy at 400, can now be sold for only 397.
The balance in the margin account would therefore be reduced by 600 to 3400. Similarly,
if the price of December gold rose to 403 by the end of the first day, the balance in the
margin account would be increased by 600 to 4600. A trade is first market to market at
the close of the day on which it takes place. It is then marked to market at the close of
trading on each subsequent day.
4. Hedging with Futures/Forwards
Future prices on stock indices
An index can be thought of as an investment asset that pays dividends. The asset is the
portfolio of stocks underlying the index, and the dividends are the dividends that would
be received by the holder of this portfolio. Often there are many stocks underlying the
index providing dividends at different times. To a reasonable approximation, the index
can then be considered as an asset providing a continuous dividend yield. If q is the
dividend yield rate, then the futures price is
F0= S0e(r-q)*T
If F0 > S0e(r-q)*T, profits can be made by buying the stocks underlying the index and
shorting futures contracts.
If F0 < S0e(r-q)*T, profits can be made by shorting or selling the stocks underlying the index
and taking a long positions in futures contracts. These strategies are known as index
arbitrage.
Stock index futures can be used to hedge the risk in a well-diversified portfolio of stocks.
As you know, the relationship between the expected return on a portfolio of stocks and
the return on the market is described by the beta (β). When β=1, the return on the
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portfolio tends to mirror the return on the market. When β=2, the excess return on the
portfolio tends to be twice as great as the excess return on the market.
When the β of the portfolio equals 1, the position in futures contracts should be chosen so
that the value of the stocks underlying the futures contracts equals the total value of the
portfolio being hedged. When β=2, the portfolio is twice as volatile as the stocks
underlying the futures contract and the position in futures contracts should be twice as
great. In general, the correct number of contracts to short in order to hedge the risk in the
portfolio is
N=β*P/A
where P is the value of the portfolio and A is the value of the underlying one future
contract (it is 250 times the current index price). This formula assumes that the maturity
of the futures contract is close to the maturity of the hedge and ignores the daily
settlement of the futures contract.
Example
A company wishes to hedge a portfolio worth 5.000.000 over the next three months using
an S&P500 index futures contract with four months to maturity. The current level of the
S&P500 is 1000, the futures price is 1010, and the β of the portfolio is 1,5. The value of
the assets underlying one futures contract is 1000*250=250.000. The correct number of
futures contracts to short is, therefore,
1,5*5.000.000/250.000=30
To show that the hedge works, we suppose the risk-free rate is 4% per year and the value
of the S&P500 index is 900, while the futures price id 902. The risk-free rate is 1% per
three months. Assume that the dividend yield on the index is 1% per annum, or 0,25% per
three months. This means that the index declines by 9,75% during the three months. From
CAPM we find that the return of the portfolio
RP = Rf + b*( RM + qM – Rf )= 1+1.5*(-10+0.25-1) = -15.125%
The gain from the short futures position is
(1010-902)*250*30=810.000
The expected value of the portfolio at the end of the 3 months is:
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5.000.000*(1-0.15125)=4.243.750
It follows that the expected value of the hedger’s position, including the gain on the
hedge is
4.243.750+810.000=5.053.750.
The net gain is about 1%. This is expected. The return on the hedged position during the
three months is the risk-free rate. It is easy to verify that roughly the same return is
realized regardless of the performance of the market.
Performance of Stock Index Hedge
Value of index in 3 months
Futures price of Index today
Futures price of Index in 3 months
Gain on futures position
Return on Market
Expected return on portfolio
Expected portfolio value in 3 months
Total expected value in 3 months
900
1010
902
810000
-9.75%
-15.125%
4.243.750
5.053.750
950
1000
1050
1010
1010
1010
952
1003
1053
435000
52500
-322500
-4.75%
0.25%
5.25%
-7.625% -1.125% 7.375%
4.618.750 4.993.750 5.368.750
5.053.750 5.046.250 5.046.250
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1100
1010
1103
-697500
10.25%
14.875%
5.743.750
5.046.250
5. Hedging with put options
Portfolio managers Portfolio managers can use index options to limit their downside risk.
Suppose that the value of an index today is S0.
When the portfolio’s beta is 1
Consider a manager in charge of a well diversified portfolio whose beta is 1,0. A beta of
1,0 implies that the returns from the portfolio mirror those from the index. If the dividend
yield from the portfolio is the same as the dividend yield from the index, the percentage
changes in the value of the portfolio can be expected to be approximately the same as the
percentage changes in the value of the index. Each contract on the S&P 500 is on 100
times the index. It follows that the value of the portfolio is protected against the
possibility of the index falling below K if, for each 100S0 dollars in the portfolio, the
manager buys one put option contract with strike price K.
Suppose that the manager's portfolio is worth $500.000 and the value of the index is
1.000. The portfolio is worth 500 times the index. The manager can obtain insurance
against the value of the portfolio dropping below $450.000 in the next three months by
buying five put option contracts with a strike price of 900. Suppose that the risk-free rate
is 12%, and the volatility of the index is 22%. The parameters of the option are:
S0=1000,
K=900,
r=0.12,
σ=0.22,
T=0.25,
Using the Black-Scholes formula the price of this put option is $6,48. The cost of the
insurance is therefore 5*100*6,48=3.240.
To illustrate how this works, consider the situation where the index drops to 880 in three
months. The portfolio will be worth about $440.000. The payoff from the options will be
5 x (900 - 880) x 100 = $10.000, bringing the total value of the portfolio up to the insured
value of $450.000.
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When the portfolio’s beta is Not 1
If the portfolio's returns are not expected to equal those of an index, the capital asset
pricing model can be used. This model asserts that the expected excess return of a
portfolio over the risk-free interest rate equals beta times the excess return of a market
index over the risk-free interest rate. Suppose that the $500.000 portfolio just considered
has a beta of 2,0 instead of 1,0. Suppose further that the current risk-free interest rate is
12% per annum, and the dividend yield on both the portfolio and the index is expected to
be 4% per annum. As before, we assume that the S&P 500 index is currently 1.000.
Table 1. shows the expected relationship between the level of the index and the value of
the portfolio in three months.
Table 1. Relationship between value of index and value of portfolio for beta = 2,0
Value of index
Value of portfolio
in three months
in three months ($)
1.080
570.000
1.040
530.000
1.000
490.000
960
450.000
920
410.000
880
370.000
Suppose that S0 is the value of the index. It can be shown that, for each 100S0 dollars in
the portfolio, a total of beta put contracts should be purchased. The strike price should be
the value that the index is expected to have when the value of the portfolio reaches the
insured value. Suppose that the insured value is $450.000, as in the case of beta = 1,0.
Table 1 shows that the appropriate strike price for the put options purchased is 960. In
this case, 100S0 = $100,000 and beta = 2,0, so that two put contracts are required for each
$100.000 in the portfolio. Because the portfolio is worth $500.000, a total of 10 contracts
should be purchased.
Alternatively, to find the strike price we use the CAPM:
Rp + q = Rf + [RM + q – Rf]b where we find that for Rp =-10% the Rm is -4%.
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Then,
K= S0(1+ RM)=960
To illustrate that the required result is obtained, consider what happens if the value of the
index falls to 880. As shown in Table 1, the value of the portfolio is then about $370.000.
The put options pay off (960 — 880) x 10 x 100 = $80.000, and this is exactly what is
necessary to move the total value of the portfolio manager's position up from $370.000 to
the required level of $450.000.
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