Determination of Forward
and Futures Prices
Chapter 5
Consumption vs Investment Assets
Investment Assets:
That is held for investment purposes by significant
numbers of investors.
(Examples: stocks, bonds, gold, silver)
 Consumption Assets:
That is held by primarily for consumption.
(Examples: copper, oil, pork)

Short Selling




Short selling involves selling securities
you do not own
Your broker borrows the securities
from another client and sells them in
the market in the usual way
Required to maintain a margin account
with the broker
You must pay dividends and other
benefits the owner of the securities
receives
Cash flows form short sale and
purchase of shares
Purchase of shares
April : Purchase 500 shares for $120
-$60,000
May : Receive dividend
+$500
July : Sell 500 shares for $100 per share
+$50,000
Net profit= -$9,500
----------------------------------------------------------------------------Short sale of shares
April : Borrow 500 shares and sell them for $120
+$60,000
May : Pay dividend
- $500
July : Buy 500 shares for $100 per share
-$50,000
Replace borrowed shared to short position
Net profit= +$9,500
Assumption and Notation
 Assumption:
1.No transaction costs when they trade.
2.The same tax rate on all net trading profits.
3.Borrow money at the same risk-free rate
of as they can lend money.
4.Take advantage of arbitrage opportunities
as they occur.
Assumption and Notation
﹡NOTATION:
S0: Price of the asset underlying the
forward or futures contract today
F0: Futures or forward price today
T: Time until delivery date
r: Risk-free interest rate for maturity T
Forward Price For an Investment Asset
Assume : S0 = $40,r = 5%,t = 3 months
(a)If F0 =$43 > S0ert
1.Borrow $40 at risk-free interest rate of 5% per annum.
2.Short a forward contract to sell one share in 3-months.
$40e0.05x3/12 = $40.5
$43 - $40.5 = $2.5
(b)If F0 =$39 < S0ert
1.Short one share, invest the proceeds of the short sale
at 5% per annum for 3 months.
2.Take a long position in a 3-months forward contract.
$40e0.05x3/12 = $40.5
$40.5 - $39 = $1.5
∴We deduce that for there to be no arbitrage the forward
price must be exactly $40.5.
F0 = S0erT
This equation relates the forward price and the
spot price for any investment asset that provides
no income
What If Short Sale Are Not Possible?
(a)If F0 > S0ert
1.Borrow S0 dollars at an interest rate r for T years.
2.Buy 1 ounce of gold.
3.Short a forward contract on 1 ounce of gold.
The investor make a profit of F0 - S0ert.
(b)If F0 < S0ert
1.Sell the gold for S0.
2.Invest the proceeds at interest rate r for time T.
3.Take a long position in a forward contract on
1 ounce of gold.
The investor make a profit of S0ert - F0.
When an Investment Asset Provides a
Known Dollar Income
F0 = (S0 – I )erT
where I is the present value of the income
during life of forward contract
Known Income
Assume : S0 = $900 I = 40e-0.03x4/12 = $39.6
r =0.04 T = 0.75(9/12)
I:
?
0
$40
4
9
F0 = (900.00 – 39.6)e0.04x0.75 = $886.60
(a)If F0 = $910 > (S0 - I)ert = $886.60
1.Borrow $900 to buy the bond.
2.Short a forward contract.
→
→
→
900.00 - 39.6 = $860.40
860.40e0.04x0.75 = $886.60
910.00 -886.60 = $23.40
(b)If F0 = $870 < (S0 - I)ert = $886.60
1.Short the bond.
2.Enter into a long forward contract.
→
900 - 39.6 = $ 860.4
→ 860.40e0.04x0.75 = $886.60
→
886.60 - 870 = $16.60
∴The forward price must be $886.60
5.15
Options, Futures, and Other Derivatives 6th Edition,
Copyright © John C. Hull 2005
When an Investment Asset
Provides a Known Yield
F0 = S0 e(r–q )T
where q is the average yield during the life of the
contract (expressed with continuous
compounding)
Known Yield

Assume : S0 = 25,r = 0.1,and T = 0.5,
the yield is 4% per annum with semiannual
compounding.
1+0.04 = (1+q/2)2
q = 3.96%
F0 = 25e(0.10 – 0.0396)x0.5 = $25.77
Valuing a Forward Contract

K is delivery price in a forward contract
F0 is forward price today
ƒ :Value of forward contract today

The value of a long forward contract, ƒ, is
ƒ = (F0 – K )e–rT
Similarly, the value of a short forward contract is
(K – F0 )e–rT


The value of a forward contract on an investment asset that
provides no income:
ƒ = (F0–K)e-rt
Equation shows that F0 = S0ert
ƒ = (S0ert–K)e-rt
ƒ = S0 – Ke-rt

The value of a long forward contract on an investment asset that
provides a known income with present value I:
ƒ = S0 – I – Ke-rt

The value of a long forward contract on an investment asset that
provides a known yield at rate q:
ƒ = S0e-qt – Ke-rt
Forward vs Futures Prices




A strong positive correlation between interest
rates and the asset price implies the futures price
is slightly higher than the forward price
A strong negative correlation implies the reverse
Last only a few months are in most circumstances
sufficiently small to be ignored
Forward and futures prices are usually assumed
to be the same. When interest rates are uncertain
they are, in theory, slightly different
Futures Prices Of Stock Index


Can be viewed as an investment asset paying a
dividend yield
The futures price and spot price relationship is
therefore
F0 = S0 e(r–q )T
where q is the average dividend yield on the
portfolio represented by the index during life
of contract
Futures Prices Of Stock Index

F0 = S0e(r-q)T
q:The dividend yield
Example:
r = 0.05 S0 = 1,300 T = 3/12 (0.25) q = 0.01
F0 = 1,300e(0.05-0.01)x0.25 = $1,313.07
Index Arbitrage

If F0 > S0e(r-q)T
1.Buying the stocks underlying the index at
the spot price
2.Shorting futures contracts
By a corporation holding short-term money
market investment.
Index Arbitrage

If F0 < S0e(r-q)T
1.Shorting or selling the stocks underlying
the index
2.Taking a long position in futures contracts
By a pension fund that owns an indexed
portfolio of stocks
Index Arbitrage


Program trading
Occasionally (e.g., on Black Monday)
simultaneous trades are not possible and
the theoretical no-arbitrage relationship
between F0 and S0 does not hold
Futures and Forwards on Currencies
Two ways of converting 1,000 units of a foreign
currency to dollars at time T. Here, S0 is spot exchange
rate, F0 is forward exchange rate, and r and rf are the
dollar and foreign risk-free rates.
1000 units of
foreign currency
at time zero
1000 e
rf T
units of foreign
currency at time T
1000 F0 e
rf T
dollars at time T
1000S0 dollars
at time zero
1000 S 0 e rT
dollars at time T
Futures and Forwards on Currencies
1,000 erfT F0 = 1,000 S0 erT
F0 = S0 erT / erfT

The relationship between F0 and S0
F0  S0e
( r rf ) T
Futures on Commodities

Income and Storage Costs
(a)In the absence of storage costs and income,
the forward price of a commodity that is an
investment asset is give by:
F0 = S0erT
(b)If U is the present value of all the storage costs,
net of income, during the life of a forward contract:
F0 = (S0 + U)erT
(c)If the storage costs net of income incurred at any
time are proportional to the price of the commodity,
they can be treated as negative:
F0=S0e(r+u)T
Where u denotes the storage costs per annum as
proportion of the spot price net of any yield earned on
the asset.
Futures on Consumption Assets
(a) F0 > (S0 + U)erT
1. Borrow an amount S0 + U at the risk-free rate and
use it to purchase one unit of the commodity and
to pay storage costs.
2. Short a forward contract on one unit of the
commodity.
(b) F0 < (S0 + U)erT
1. Sell the commodity, save the storage costs,
and invest the proceeds at the risk-free
interest rate.
2 . Take a long position in a forward contract.
Futures on Consumption Assets
F0  S0 e(r+u )T
where u is the storage cost per unit time as a
percent of the asset value.
Alternatively,
F0  (S0+U )erT
where U is the present value of the storage
costs.
Convenience Yield
* The benefits from holding the physical asset are
sometimes referred to as the convenience yield.
If the dollar amount of storage costs is known and has a
present value U, that the convenience yield y is defined
such that:
F0eyT = ( S0 + U )erT
If the storage costs per unit are a constant proportion, u, of the
spot price, then y is defined so that:
F0eyT = S0e(r+u)T
or
F0 = S0e(r+u-y)T
The Cost of Carry

The cost of carry, c, is the storage cost plus
the interest costs less the income earned

For a non-dividend-paying stock, it is r.
For a stock index, it is r - q.
For a currency, it is r - rf.
For a commodity that provide income at rate q and
require storage costs at rate u, it is r - q + u.



The Cost of Carry


Define the cost of carry as c.
For an investment asset , the futures price is
F0 = S0ecT

For a consumption asset, The convenience yield
on the consumption asset, y, is defined
so that
F0 = S0 e(c–y )T
Delivery Options

Form equation (F0 = S0 e(c–y )T ) that c > y, the benefits
from holding the asset (including convenience yield and
net of storage costs) are less than the risk-free rate .

If futures prices are decreasing as time to maturity
increase (c < y).It is then usually optimal for the party
with the short position to deliver as late as possible, and
futures prices should, as a rule, be calculated on this
assumption
The Risk in a Futures Position
The cash flow to the speculator are as follow :
Today : - F0e-rT
End of futures contract : +ST
The futures prices today : F0
The prices of the asset at time T : ST
The risk-free return on funds invested for time : T
The investor's required return : k
The expected value : E
The PV of this investment : - F0e-rT + E(ST)e-kT
Assume net present value = 0
- F0e-rT + E(ST)e-kT = 0
F0 = E(ST)e(r-k)T
The Risk in a Futures Position

If the asset has
 no systematic risk, then k = r
F0 = E(ST)
and F0 is an unbiased estimate of ST
 positive systematic risk, then k > r and
F0 < E (ST )
 negative systematic risk, then k < r and
F0 > E (ST )
Normal Backwardation and Contango
Normal backwardation:
When the futures price is below the expected
future spot price.
Contango:
When the futures price is above the expected
future spot price.