Why a Forward Contract Has Delta One

Summary

The document explains a common distinction in forward pricing: the forward price is set when the contract is initiated, while the contract’s value changes as the underlying price moves. Confusing these quantities can lead to the mistaken conclusion that delta equals one plus the interest rate.

Under continuous compounding, the forward price reflects spot grown at the risk-free rate to maturity. For an existing contract, its value is the current spot price minus the discounted delivery price. Differentiating that value with respect to spot gives a delta of one. The exchange provides a concise formula-based clarification, but does not discuss dividends, storage costs, changing rates, or other market conventions that can affect forward valuation.

Key ideas

• A forward price is a contract parameter, while a forward contract value changes over time.
• The forward price under continuous compounding grows spot by the risk-free rate to maturity.
• The contract value is current spot less the discounted delivery price.
• The forward contract’s delta with respect to spot is one under the stated valuation setup.

Tags

# Delta of a forwards contract


# Delta of a forwards contract












in university's lecture notes, from what I understand using the replication of portfolio principle to price derivates, the forward price of a contract K should be: $K = P_0(1+r)$ where $P_0$ is the spot price of the underlying and $r$ is the risk-free rate. However, with this definition, I do not see how the delta of a forwards contract = 1 (which is what many sources are claiming). Indeed, $\frac{dK}{dP_0}=1+r$ and this isn't 1 unless $r=0$. Can someone help clarify my misunderstanding of this issue? Many thanks

## Answer by Kermittfrog (score 1)

https://quant.stackexchange.com/a/60688

I think your error is in confusing forward contract and forward price.

The forward price, with continuous interest rates, is $K=P_0e^{rT}$. It is a fixed parameter of your forward contract.

The forward contract value, on the other hand, is $V_t=P_t-Ke^{-r(T-t)}$. Its derivative w.r.t. the underlying is then indeed $dV_t/dP_t=1$.

HTH?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.