Derivatives
„
Forwards and Futures
„
Forward
„
„
„
The most common derivative securities are
forward, futures and options.
A forward contract is an agreement to buy or
sell an asset at a fixed future time for a predetermined price.
For example, you agree to buy 100 ounces of
gold for $30,000 on Dec. 30, 2003.
Options
„
„
„
There are two basic types of options.
A call option gives the holder the right (but no
obligation) to buy the underlying asset by (or
at) a certain date for a certain price.
A put option gives the holder the right (not
obligation) to sell the underlying asset by (or
at) a certain date for a certain price.
A derivative security is a security whose value
depends on the values of other more basic
underlying variables.
E.g., a security that pays off a dollar if
Microsoft is closing above $30 tomorrow and
a payoff of zero if otherwise.
Futures
„
A futures contract, like a forward contract, is
an agreement between two parties to buy or
sell an asset at a certain time in the future for
a pre-specified price. Unlike forward
contracts, futures contracts are traded on an
exchange and they are standardized.
Initial Cost
„
„
Forward and futures cost nothing to enter.
For options, you have to pay something up
front, which is called premium.
1
Why Futures and Forwards?
„
„
The uncertainty faced by the oil producers is
the future oil price.
The uncertainty faced by the oil consumers is
also the future oil price.
Key Terms for Futures
Contracts
„
„
„
„
Futures price Agreed-upon price at maturity.
Long position Agree to purchase.
Short position Agree to sell.
Profits on positions at maturity
Long: spot minus original futures price (ST – F0)
Short: original futures price minus spot (F0 – ST)
Margin and Trading
Arrangements
„
„
„
Initial Margin Funds deposited to provide
capital to absorb losses.
Marking to Market Each day the profits or
losses from the new futures price are
reflected in the margin account.
Maintenance or variance margin
Established value below which a trader’s
margin may not fall.
Key Differences Between
Futures and Forwards
„
Futures
„ Secondary trading - liquidity
„ Marked to market or “pay-as-you-go”
„ Standardized contract units
„ Clearinghouse warrants performance
Types of Contracts
„
„
„
„
Agricultural commodities
Metals and minerals (including energy
contracts)
Foreign currencies
Financial futures
„ Interest rate futures
„ Stock index futures
Trading Strategies
„
„
Speculation
„ Short – You believe price will fall
„ Long - You believe price will rise
Hedging
„ Long hedge - Protecting against a rise in
price
„ Short hedge - Protecting against a fall in
price
2
Payoffs
„
„
Payoffs
Suppose that you have a long position in gold
futures with a delivery date in three months.
The delivery price is $300.
What is your payoff if the gold price in three
months is
„
„
„
If the ending gold price is $250
If the ending gold price is $300
If the ending gold price is $350
a) $250
b) $300
c) $350
Payoffs from Forwards &
Futures
Payoffs
„
So when you long a futures contract, your
payoff is higher the higher the future price is.
Payoff
Payoff= ST – F0
0
F0
ST
Long Position
Payoffs
„
„
„
„
Suppose that you agree to sell an ounce of
gold in three months for $300.
If the ending gold price is $250
If the ending gold price is $300
If the ending gold price is $350
Payoffs
„
So when you short a futures contract, your
payoff is higher the lower the future price is.
3
Payoffs from Forwards &
Futures
A Zero-Sum Game
„
Payoff
Payoff= F0 - ST
„
„
0
F0
You may have noticed that what the long
position wins is exactly what the short
position loses and vice versa.
(ST - F0) + (F0 - ST ) = 0
This is a zero-sum game.
ST
Short Position
A Zero-Sum Game
Payoff
Futures Price
„
Long Position
„
Payoff= ST - F0
„
„
Sum=0
0
ST
F0
How do we determine the futures price?
Is the futures price the expected future price?
Should the futures price be close to the spot
price?
It turns out we can determine the futures price
of many contracts by imposing the no
arbitrage condition.
Short Position
Payoff= F0 - ST
Futures Prices
„
„
There are two ways to acquire an asset for
some date in the future:
„ Purchase it now and store it.
„ Take a long position in futures.
The Spot-Futures Parity Theorem says that
these two strategies should have the same
market determined costs.
Parity Example I
„
„
„
„
Consider gold.
„ It has basically no storage costs
Strategy A: Buy the gold and hold it for a year.
Strategy B: Put funds aside today and buy a gold
futures contract with one year to maturity.
$400 is the current gold price, ST is the gold price
at T, which is of course unknown at t. $440 is the
futures price. 10% is the risk free rate.
4
Example I
Parity Example II
Strategy A
Action
Buy gold
Cash Flow at t
($400)
Cash Flow at T
ST
„
Strategy B
Action
Long gold futures
Invest $400 in TBill
Total For B
Cash Flow at t
0
($400)
($400)
Cash Flow at T
ST –$440
$400(1.1)=$440
ST
„
„
Consider gold.
„ It has basically no storage costs
Strategy A: Buy the gold and hold it until time
T.
Strategy B: Put funds aside today and buy a
gold futures contract.
St is the current gold price, ST is the gold
price at T, which is of course unknown at t. Ft
is the futures price. rf is the risk free rate.
Strategy A and B have the same cash flow at t and T.
The key here is that the futures price $440 is exactly
$400 (1+10%).
„
Example II
Parity Theorem
Strategy A
Action
Buy gold
Cash Flow at t
-St
Cash Flow at T
ST
Strategy B
Action
Long gold futures
Cash Flow at t
0
Cash Flow at T
ST –Ft
Ft
T-t
Invest Ft /(1+rf )
T-t
in TBill
-Ft /(1+rf )
Total For B
T-t
-Ft /(1+rf )
„
„
Strategy A and Strategy B give identical
payoffs at T, regardless what the future gold
price is.
Hence they should cost the same at t.
− St =
ST
− Ft
(1 + rf )T −t
⇒ Ft = St (1 + rf )
T −t
$440 = $400 (1 + 10%)
Price of Futures with Parity
Ft = S t (1 + r f )
T −t
„
„
This formula is also called the cost of carry
formula as the futures price is the spot price
plus the cost of carry the gold.
Note that for gold, the futures price is always
greater than the spot price regardless of your
expected future gold price.
Arbitrage
„
What if the futures price ($420) is less than
what the cost of carry formula gives ($440)?
Short gold
Long gold futures
T-t
Invest Ft /(1+rf )
Total
in TBill
$400
0
$420/(1+10%)
-ST
ST –$420
$420
$400-$420 /
(1+10%) = $18.18
0
5
Arbitrage
„
What if the futures price is less than what the
cost of carry formula gives?
Action
Short gold
Long gold futures
T-t
Invest Ft /(1+rf )
Total
Futures Pricing
in TBill
Cash Flow at t
St
0
T-t
-Ft /(1+rf )
T-t
St -Ft /(1+rf )
Cash Flow at T
-ST
ST –Ft
Ft
„
The cost of carry formula does not work for
agriculture products and many commodities.
„
„
„
Perishable
High storage costs
seasonality
0
6