The delta of a futures contract is not the same as the delta of a forward contract. This is often a point of confusion for students. The difference between a forward contract and a futures contract is that a forward contract is an OTC contract settled at maturity and a futures contract is settled daily using a margin account with a clearing house.
## Forward Delta

The payoff from a long position in a forward contract on an asset expiring at time \( T \) is \( S_T – K\) where \(S_T\) is the final price of the asset and \( K \) is the agreed purchase or strike price. Assuming constant interest rates, using elementary pricing theory the present value of this payoff at time t \( V_t \) is

\begin{align*}

V_t = e^{-r (T – t)}\; (\mathbb{E}[S_T|\mathcal F_t] – K) = S_t – e^{-r(T-t)} K

\end{align*}

The Delta \( \frac{\partial V_t}{\partial S_t} \)is 1.

## Futures Delta

Let \( F(t,T) \) denote the futures price at time \( t \) for delivery of the asset at time \( T \). The futures contract is settled daily by an amount given by the change in futures price so the delta is \( \frac{\partial F(t,T)}{\partial S_t} \). In this simple situation \( F(t,T) = S_t e^{r (T-t)} \) so the delta is \( e^{r (T-t)} \)

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