Analysis of Asset Allocation

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FUTURES : Forward and Futures prices
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In this section we will try to explain how forward and futures prices are linked to the price of the underlying asset. As there is no daily settlement on forward contracts, we will first analyze the forward prices and we will try to link them to futures prices afterward.

Basic principle.

The basic difference between a spot and a forward transaction is that, in a forward transaction, you don't pay the underlying immediately and therefore you can still earn money on the amount (in case of a long position).

Notations and formulas.

The following notations will be used:

T: time in years until the delivery date.
S: price of the underlying asset of a forward contract.
K: delivery price of the forward contract.
f: value of a long position on a forward contract.
F: forward price today.
r: risk-free interest rate.
e exponent.

There is a fundamental difference between f and F. F is the delivery price would make f equal to zero taking into account the above basic principle.

As there is no premium exchanged at the initiation of the contract (unlike options), the delivery price of the contract always follows these constrains:

f=0
F=K

But this is only valid when you enter into a forward contract but no during the all life of the contract. Therefore during the life of the contract, the forward price can be computed.

Forward price on an asset that provides no income.

Examples: Stocks that don't pay dividend or zero bonds.

Base on the continuous compounding principle the forward price is:

F = S erT

If the price of the underlying asset (S) is 50, if r = 6%, and if T = 0.50 (half a year) F = 51.52.

If the contract is negotiated today, K = 51.52.

Forward price on an asset that provides known cash income.

The principle is the same but the forward price must be adjusted by the present value of the cash income (I). The above formula becomes:

F = (S - I) erT

Forward price on an asset that provides known dividend yield.

If the annual dividend yield is q, the above formula becomes:

F = S e(r-q)T

Are forward and futures prices equal?

Due to the margins principle on futures contract, forward and futures prices are not equal. If S is strongly positively correlated with interest rates, futures prices will generally be higher than forward prices. If S is negatively correlated with interest rates, futures prices will generally be lower than forward prices.

In the next section we will define some of the well known futures and forward contracts.

 


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