# 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

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

where

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

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', 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 ${r}$.