# Is commodity futures's delta equal to 1?

According to John Hull's book, equity futures delta does not equal to one. For commodities futures, since there is no centralized exchange for physical commodities, do commodities futures' delta equal to one?

• Which Delta do you care about $\frac{\partial}{\partial S}$ or $\frac{\partial}{\partial F}$? For the Delta with respect to the spot price $S$ the exact same reasoning applies to equity futures as to commodity futures. It is less than one due to interest rates and storage costs. – noob2 Jun 19 '17 at 14:50
• Could you give a bit more of context: why does Hull say that future's delta does not equal 1? Why do you think having no centralized exchange changes something? – Daneel Olivaw Jun 19 '17 at 14:50
• @DickJ please feel free to accept my answer if you feel I have correctly addressed your question. – Daneel Olivaw Jul 2 '17 at 12:39

In a risk-neutral framework, the price at $t$, $\text{Fut}(t,T,X)$, of a future of maturity $T$ written on a asset $X$ whose price process is $(X_t)_{t \geq 0}$ is given by its conditional risk-neutral expectation:

$$\text{Fut}(t,T,X) = \mathbb{E}^{\, \mathbb{Q}\, }[X_T|\mathcal{F}_t]$$

For further details on this result, you can consult Stochastic Calculus for Finance II: Continuous-Time Models by Shreve. The delta $\Delta_{\text{Fut}}$ of the future is then defined to be:

$$\Delta_{\text{Fut}} = \frac{\partial \, \text{Fut}}{\partial X_t}$$

Now, to obtain a specific expression for the delta, it is normally necessary to have a model for the evolution of the price $X_t$. Given that you are interested in a commodity, a plausible model for the price process $(X_t)_{t \geq 0}$ is a Geometric Brownian Motion (GBM), so that under $\mathbb{Q}$:

\begin{align} dX_t= rX_tdt+\sigma X_tdW_t \end{align}

The price $X_T$ at $T$ can be explicitly written as :

$$X_T = X_te^{\left(r-\frac{\sigma^2}{2}\right)(T-t)+\sigma W_{T-t}}$$

Where $W_{T-t}\sim\mathcal{N}(0,T-t)$ is a Gaussian variable. $X_T$ is lognormal, its expectation under $\mathbb{Q}$ is given by $X_te^{r(T-t)}$ thus:

$$\Delta_{\text{Fut}} = \frac{\partial \, \text{Fut}}{\partial X_t} = \frac{\partial}{\partial X_t}\mathbb{E}^{\, \mathbb{Q}\, }[X_T|\mathcal{F}_t] = e^{r(T-t)}$$

For short maturities and/or low interest rate $r$, we have $e^{r(T-t)} \approx 1$.

On a more qualitative note, as it has been said in the commentaries the delta is not strictly one, it is slightly lower or higher depending on costs and benefits generated by the underlying asset: indeed, a future or forward contract involves the future delivery of an underlying asset $X$ at a price agreed in advance, the future price $\text{Fut}(t,T,X)$ or the forward price. Basically, to hedge this position you simply need to buy the asset today and hold it until maturity, when you will deliver it to your client. Holding this position has:

• Costs: you might have to borrow money at a rate $r$ to buy the asset; for a commodity, you might have storage costs to pay for until the maturity; etc.
• Benefits: the underlying might generate cash-flows that accrue to you, for example dividends for a stock.

So actually, the delta of a future is conceptually equal to $1$, adjusted for the costs and benefits of hedging the position:

$$\Delta_{\text{Fut}} \approx 1+\text{Costs from holding the underlying}-\text{Benefits from holding the underlying}$$

As far as I know, whether the contract is a future or a forward $-$ i.e. traded through a centralized exchange or bilaterally $-$ does not fundamentally change this $-$ although it can change the details, i.e. the exact costs and benefits.

Speaking in qualitative terms:

I would treat a commodity futures as delta one w.r.t spot if the maturity is not too long. In the case of rather short futures this can be seen as basis risk.

For longer terms you have term structure risk too, which I would not model/see as linear w.r.t. the spot market.