The Mathematics of Financial Modelingand Investment Management

2-Financial Markets Page 62 Wednesday, February 4, 2004 1:15 PM

62 The Mathematics of Financial Modeling and Investment Management

The theoretical futures price may be at a premium to the cash market

price (higher than the cash market price) or at a discount from the cash

market price (lower than the cash market price) depending on P(r – y).

The term r – y, which reflects the difference between the cost of financing

and the asset’s cash yield, is called the net financing cost. The net financ-

ing cost is more commonly called the cost of carry or, simply, carry. Posi-

tive carry means that the yield earned is greater than the financing cost;

negative carry means that the financing cost exceeds the yield earned.

At the delivery date, the futures price must be equal to the cash market

price. Thus, as the delivery date approaches, the futures price will con-

verge to the cash market price. This can be seen by looking at the equation

for the theoretical futures price. As the delivery date approaches, the

financing cost approaches zero, and the yield that can be earned by hold-

ing the investment approaches zero. Hence the cost of carry approaches

zero, and the futures price will approach the cash market price.

To derive the theoretical futures price using the arbitrage argument,

several assumptions are made. When the assumptions are violated, there

will be a divergence between the actual futures price and the theoretical

futures price as derived above; that is, the difference between the two

prices will differ from carry. The reasons for the deviation of the actual

futures price from the theoretical futures price are as follows.

First, no interim cash flows due to variation margin are assumed. In

addition, any cash flows payments from the underlying asset are assumed

to be paid at the delivery date rather than at an interim date. However, we

know that interim cash flows can occur for both of these reasons. Because

we assume no variation margin, the theoretical price for the contract is

technically the theoretical price for a forward contract that is not marked

to market, not the theoretical price for a futures contract. This is because,

unlike a futures contract, a forward contract that is not marked to market

at the end of each trading day does not require additional margin.

Second, in deriving the theoretical futures price it is assumed that

the borrowing rate and lending rate are equal. Typically, however, the

borrowing rate is greater than the lending rate. Letting rB denote the

borrowing rate and rL denote the lending rate, then the following

boundaries would exist for the theoretical futures price:

Upper boundary: F = P + P(rB – y)

Lower boundary: F = P + P(rL – y)

Third, in determining the theoretical futures price, transaction costs

involved in establishing the positions are ignored. In actuality, there are