The futures price of an index can be treated like the price of an asset which pays a dividend yield equal to the risk-free rate.

Recall that the Black’s price of a European futures option is

where

The above can be obtained if we substitute by and by into the Black-Scholes call price formula; that is if we regard as the price of a stock which pays dividend yield .

Alternatively, let us think of as a price of a hypothetical asset, say A. Let denote the maturity date of the futures contract. Since , owning one unit of the asset A with price per unit is like owning units of the index underlying the futures contract.

Now, at time , the value of our position in the index is

since the index pays a dividend yield of . We can write (1) as

The right-hand side of (2) shows that if we long one unit of asset A at time , our position would grow to units at time , i.e. asset A pays a dividend yield of .

*Quantitative Finance*

Posted on April 26, 20110