A forward contract gives linear exposure to an underlying. At expiry the payoff for the Long side is:
$$ N(S-K) $$
where:
For the Short side the payoff will be the exact opposite: $N(K-S)$.
Payoff profile of the forward for $N=1$ and $K=100$
Payoff profile of the forward for $N=1$ and $K=100$
As you may notice, the payoff is very similar to holding the actual underlying, for example you could buy $N$ units of the underlying paying $K$ for each, and then your payoff will be similar to the Long side above. The main difference is that in the Forward you don’t have to actually hold the underlying as the contract can be settled directly in USD.
The forward is also very similar to the futures contract, which many CEXs such as FTX or Binance list for different expiries. The main difference is that the futures are standardized with margin requirements, while the forward are customizable and don’t have margining.
Note that the payoff above is that of an idealized forward, i.e. one with infinite collateral. In practice, Long and Short sides deposit an amount of collateral at the beginning, which means that the maximum profit and loss for each side are capped.
In particular, given $C_L$ and $C_S$ the collateral amounts deposited by the Long and Short side, the final payoff for the Long side will be:
$$ \max(-C_L, \min(C_S, N(S-K)) $$
so that the Long side can lose at most its capital, $C_L$, but the max profit is also capped at $C_S$. For the Short side it’s opposite, which can be written as $\max(-C_S, \min(C_L, N(K-S))$.
Payoff profile of the forward for $N=1$, $K=100$, $C_L=C_S=50$
Payoff profile of the forward for $N=1$, $K=100$, $C_L=C_S=50$