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A standardized maturity-and-quantity agreement whose quote changes generate margin payments as gains and losses are marked to market. A newly settled contract has zero value, while its quoted delivery price is generally nonzero. In the continuous-settlement idealization, futures pricing uses a pricing-measure martingale quote.

A futures contract fixes a maturity and underlying quantity, but its quoted delivery price is marked to market: gains and losses from quote changes are settled through the margin account, typically daily. A newly settled position has zero contract value; the quoted futures price is not an upfront purchase price for that position. At maturity the quote equals the spot price. In the continuous-settlement idealization, discounted futures gains must be local martingales under the money-market risk-neutral measure. Thus the quote has zero pricing-measure drift. Assuming the relevant integrability makes it a true martingale, $\boxed{F_{tT}={\mathbb{E}}_Q[S_T\mid {\mathcal{F}}_t]}$.

This is a futures pricing formula. In contrast, an unsettled forward contract has delivery quote ${\mathbb{E}}_Q[D_{tT}{S}_T\mid {\mathcal{F}}_t]/{\mathbb{E}}_Q[D_{tT}\mid {\mathcal{F}}_t]$, where $D_{tT}=e^{-{\int }_t^T{r}_{u}du}$. Stochastic interest rates can make the two quotes differ. Contango means the futures quote is above spot for the maturity considered, while backwardation means it is below spot; futures curves are correspondingly described as rising or falling when comparing maturities.

For the Multivariate Ornstein-Uhlenbeck process, multiplication by the matrix exponential $e^{At}$ gives $d(e^{At}{X}_t)=e^{At}d{B}_t$, henceFor deterministic $X_0=x_0$, the Gaussian distribution has mean $e^{-At}{x}_0$ and covarianceEquivalently $\dot{C}=I-AC-C{A}^{\mathsf{T}}$, $C_0=0$. No symmetry or invertibility of $A$ is required. This covariance can also be evaluated from a single block matrix exponential: if the upper-right block of $\exp (t\left(\begin{array}{cc}-A& I\\ 0& A^{\mathsf{T}}\end{array}\right))$ is $H_t$, then $C_t=H_t{e}^{-A^{\mathsf{T}}t}$. If $X_0$ is random, these assertions hold conditionally on $X_0$; the unconditional law need not be Gaussian. A stationary Gaussian law exists when the eigenvalues of $A$ have positive real parts, but stability is unnecessary for the finite-time formulas.

For $\tau =T-t$, the conditional mean of $X_T$ is $e^{-A\tau }{X}_t$ and its conditional covariance is $C_{\tau }$. The Gaussian exponential-moment formula therefore gives the explicit quoteIt tends to $e^{b\cdot X_t}=S_t$ as $T\downarrow t$, as required.