Futures Price

Posted on April 26, 2011


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

\displaystyle  F(t)e^{-r\tau}N(d_{+})-Ke^{-r\tau}N(d_{-})


\displaystyle  d_{\pm} = \frac{\ln F(t)/K \pm \frac{1}{2}\sigma^{2}\tau}{\sigma \sqrt{\tau}}.

The above can be obtained if we substitute {S(t)} by {F(t)} and {q} by {r} into the Black-Scholes call price formula; that is if we regard {F(t)} as the price of a stock which pays dividend yield {q=r}.

Alternatively, let us think of {F(t)} as a price of a hypothetical asset, say A. Let {T} denote the maturity date of the futures contract. Since {F(t)=S(t)e^{(r-q)(T-t)}}, owning one unit of the asset A with price {F(t)} per unit is like owning {e^{(r-q)(T-t)}} units of the index underlying the futures contract.

Now, at time {t'<T}, the value of our position in the index is

\displaystyle  S(t')e^{(r-q)(T-t)} \times e^{q(t'-t)} \ \ \ \ \ (1)

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

\displaystyle  S(t')e^{(r-q)(T-t')}e^{(r-q)(t'-t)}e^{q(t'-t)} = F(t')e^{r(t'-t)} \ \ \ \ \ (2)

since {F(t') = S(t')e^{(r-q)(T-t')}}.

The right-hand side of (2) shows that if we long one unit of asset A at time {t}, our position would grow to {e^{r(t'-t)}} units at time {t'}, i.e. asset A pays a dividend yield of {r}.