Cross-Asset Market Making with Correlated Fundamentals

You are a market maker quoting two correlated assets simultaneously. The latent (true) values $(V_1, V_2)$ are jointly normal: $(V_1, V_2) \sim N\!\left(\begin{pmatrix}\mu_1 \\ \mu_2\end{pmatrix},\ \begin{pmatrix}\sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2\end{pmatrix}\right)$ with correlation $\rho \in (-1, 1)$. You quote one share on each side for both assets. Your bid/ask for asset 1 are shifted from the mid by a skew term $k_1 I_1 + k_2 I_2$, where $I_1, I_2$ are your current inventory positions in each asset. Order flow is adversarial: each incoming trader observes a Gaussian signal $S_j = V_j + \epsilon_j$ with $\epsilon_j \sim N(0, \sigma^2)$ (independent of $V_j$) and trades against the most favorable quote. Derive the inventory-optimal hedging coefficients $(k_1, k_2)$ -- the skew parameters that equalize the conditional expected loss on asset 1 fills given the joint inventory $(I_1, I_2)$. In particular, explain why $k_2 \neq 0$: why does inventory in asset 2 affect how you skew asset 1?