  We study a single-period optimal transport problem on $\mathbb{R}^2$ with a
covariance-type cost function $c(x,y) = (x_1-y_1)(x_2-y_2)$ and a backward
martingale constraint. We show that a transport plan $\gamma$ is optimal if and
only if there is a maximal monotone set $G$ that supports the $x$-marginal of
$\gamma$ and such that $c(x,y) = \min_{z\in G}c(z,y)$ for every $(x,y)$ in the
support of $\gamma$. We obtain sharp regularity conditions for the uniqueness
of an optimal plan and for its representation in terms of a map. Our study is
motivated by a variant of the classical Kyle model of insider trading from
Rochet and Vila (1994).
