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Determination of Forward and Futures Prices
Chapter 5
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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)
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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
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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
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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.
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Assumption and 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
﹡NOTATION:
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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
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(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.
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F0 = S0erT
This equation relates the forward price and the spot price for any investment asset that provides no income
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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.
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(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.
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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
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Known Income
Assume : S0 = $900 I = 40e-0.03x4/12 = $39.6 r =0.04 T = 0.75(9/12) I: ? $40 0 4 9
F0 = (900.00 – 39.6)e0.04x0.75 = $886.60
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(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
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(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
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.15
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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)
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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Futures and Forwards on Currencies
1000 units of foreign currency
at time zero
units of foreign currency at time T
Tr fe1000
dollars at time T
Tr feF01000
1000S0 dollars at time zero
dollars at time T
rTeS01000
1000 units of foreign currency
at time zero
units of foreign currency at time T
Tr fe1000
dollars at time T
Tr feF01000
1000S0 dollars at time zero
dollars at time T
rTeS01000
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.
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Futures and Forwards on Currencies
1,000 erfT F0 = 1,000 S0 erT
F0 = S0 erT / erfT
The relationship between F0 and S0
F S e r r Tf
0 0 ( )
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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
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(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.
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(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.
Futures on Consumption Assets
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(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.
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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.
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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
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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.
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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
The Cost of Carry
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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
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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
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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 )
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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.