# Futures Delta and Forward Delta

July 23, 2013

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 $$e^{r (T-t)}$$

Category: Futures, Quantitative Finance General