How Rates and Dividends Shape Futures Maturity Sensitivity

# Greeks for Futures


# Greeks for Futures












Is there some general result on the sensitivity of futures price to its maturity? For example, I have two futures on the same underlying, but maturing at different dates. Can I say which one is more expensive? Or it depends on the model/parameters?

## Answer by StackG (score 5, accepted)

https://quant.stackexchange.com/a/59168

This is a slightly deeper question that it appears at first. Depending on your treatment of rates (deterministic vs. stochastic), it can indeed be model-dependent.

Let's first think about a forward contract $F_T(t)$ locks you in to a transaction at price $F_T(t)$ on an underlying $S(t)$ at time $T$. The well-known price of this contract at time $t$ is

\begin{align} F_T(t) = S(t) \cdot e^{\int^T_t ( r(s) - q(s) ) ds} \end{align}

where I've assumed that the underlying $S$ is freely tradable (note this sometimes doesn't strictly hold for indices, VIX, etc.), and that the underlying pays continuous dividends at rate $q$, and that both $r(t)$ and $q(t)$ are deterministic (otherwise we need some expectation terms to appear).

We can differentiate this contract by $T$ to calculate the effect on its price of a longer time-to-expiry:

\begin{align} {\frac {\partial} {\partial T}} F_T(t) &= {\frac {\partial} {\partial T}} S(t) \cdot e^{\int^T_t ( r(s) - q(s) ) ds} \\ &= S(t) \cdot e^{\int^T_t ( r(s) - q(s) ) ds} \cdot {\frac {\partial} {\partial T}} \int^T_t ( r(s) - q(s) ) ds \\ &= S(t) \cdot e^{\int^T_t ( r(s) - q(s) ) ds} \cdot \bigl( r(T) - q(T) \bigr) \\ &= F_T(t) \cdot \bigl( r(T) - q(T) \bigr) \end{align}

So this, as expected, says that $F_T(t)$ will increase with $T$ if rates are positive (as by trading the future we are 'avoiding' funding costs, and need to pay for that), but will decrease with $T$ if dividends are positive (as we're missing out on dividends by not holding the underlying, and need compensation for that).

How do forwards and futures relate? Well, it turns out that they are the same when both:

- rates and dividends are deterministic

- there is no counterparty credit risk

So as a first approximation, you can use this expression for your futures $T$-greek. If you want to extend this to stochastic rates things become a little more tricky due to correlation terms, see for example here and here.