Minimal Market Spread Under Correlation Uncertainty

Two events $A$ and $B$ have known marginal probabilities $P(A) = p$ and $P(B) = q$. Define the indicator $Y = \mathbf{1}\{A \oplus B\}$, i.e., $Y = 1$ if exactly one of $A$ or $B$ occurs (exclusive or). The catch: you do not know the joint distribution. You only know that the correlation $\rho = \operatorname{Corr}(\mathbf{1}_A, \mathbf{1}_B)$ lies in a known interval $[\rho_L, \rho_U]$. 1. Derive tight (attainable) bounds on $E[Y]$ over all joint laws consistent with the given marginals and the constraint $\rho \in [\rho_L, \rho_U]$. 2. You must post a single symmetric market $(b, a)$ with mid $m$ and half-spread $s/2$, so $b = m - s/2$ and $a = m + s/2$. An informed trader knows the true $\rho \in [\rho_L, \rho_U]$ and will trade against you whenever your quote is off. Find the minimal spread $s$ (and the corresponding mid $m$) that guarantees nonnegative expected P&L against this worst-case informed trader.