Cross-Asset Market Making with Correlated Fundamentals

Worked Solution

How to Think About It: This is a cross-asset adverse selection problem. In single-asset market making, you skew your quotes to manage inventory risk -- the bigger your long position, the more you shade your bid down to attract sellers. With two correlated assets, it is more subtle: a large long position in asset 2, combined with positive correlation $\rho$, means you are also effectively long on asset 1 (in a value sense). You should shade asset 1 quotes too, even if your asset 1 inventory is zero. The hedging coefficients $(k_1, k_2)$ capture this cross-asset effect.

The key is to work with the conditional distribution of $V_1$ given the signal and inventory context, then equalize the expected P\&L on each side of the market.

Key Insight: Under a Gaussian model, the optimal skew is linear in inventories. The coefficient $k_2$ on asset 2 inventory is proportional to $\rho \cdot \frac{\sigma_1}{\sigma_2}$ -- the correlation scaled by the relative volatilities. This is exactly the hedge ratio you would use to hedge asset 1 risk with asset 2.

The Method:

Step 1: Posterior on $V_1$ given signal $S_1$. The trader observes $S_1 = V_1 + \epsilon_1$ with $\epsilon_1 \sim N(0, \sigma^2)$. By standard Gaussian updating, the posterior mean of $V_1$ given $S_1$ is:

$$E[V_1 | S_1] = \mu_1 + \frac{\sigma_1^2}{\sigma_1^2 + \sigma^2}(S_1 - \mu_1)$$

Let $\alpha = \frac{\sigma_1^2}{\sigma_1^2 + \sigma^2} \in (0,1)$ be the signal weight (informativeness).

Step 2: Inventory-adjusted mid. Your effective mid for asset 1, accounting for inventory risk, should be:

$$m_1^{\text{adj}} = \mu_1 - (k_1 I_1 + k_2 I_2)$$

The skew shifts the mid: a long position ($I_1 > 0$) lowers your effective mid so you sell cheaper and buy more expensively, reducing inventory.

Step 3: Equalize expected loss. A buy order on asset 1 arrives at ask $a_1 = m_1^{\text{adj}} + s$ (half-spread $s$). The informed trader buys only if $E[V_1|S_1] > a_1$. Your expected loss per informed buy is:

$$E[V_1 - a_1 | \text{buy}] = E\left[E[V_1|S_1] - a_1 \,\Big|\, E[V_1|S_1] > a_1\right]$$

For the symmetric case (zero inventory, symmetric spread), this equals $\alpha \cdot \phi(s/\tilde{\sigma}) / (1 - \Phi(s/\tilde{\sigma}))$ times $\tilde{\sigma}$ -- the standard adverse selection cost, where $\tilde{\sigma}^2 = \sigma_1^2 \sigma^2/(\sigma_1^2 + \sigma^2)$ is the posterior standard deviation.

Step 4: Cross-asset term. With joint inventory $(I_1, I_2)$, the risk to the market maker is not just inventory in asset 1 but also correlated exposure from asset 2. The P\&L variance from holding $(I_1, I_2)$ is:

$$\text{Var}(I_1 V_1 + I_2 V_2) = I_1^2 \sigma_1^2 + 2\rho\sigma_1\sigma_2 I_1 I_2 + I_2^2 \sigma_2^2$$

To flatten this exposure, the optimal skew on asset 1 must include a term proportional to $I_2$ that reflects the marginal contribution of asset 1 trades to the portfolio variance. Differentiating the portfolio variance with respect to $I_1$:

$$\frac{\partial}{\partial I_1}\text{Var} = 2(\sigma_1^2 I_1 + \rho\sigma_1\sigma_2 I_2)$$

Setting this proportional to the skew gives the optimal hedging coefficients:

$$k_1 = \gamma \sigma_1^2, \qquad k_2 = \gamma \rho\sigma_1\sigma_2$$

where $\gamma > 0$ is a risk aversion parameter (chosen to balance inventory cost against spread revenue).

Practical Considerations: The ratio $k_2 / k_1 = \rho \sigma_2/\sigma_1$ is exactly the regression coefficient of $V_2$ on $V_1$ -- i.e., the minimum-variance hedge ratio. This makes intuitive sense: you are hedging your asset 2 inventory risk by skewing asset 1 quotes in the direction that reduces your net correlated exposure. When $\rho = 0$, $k_2 = 0$ and the assets decouple. When $|\rho| \to 1$, the cross-skew dominates.

Answer: The inventory-optimal hedging coefficients are $k_1 = \gamma\sigma_1^2$ and $k_2 = \gamma\rho\sigma_1\sigma_2$, where $\gamma$ is the market maker's risk aversion. The cross-asset skew $k_2 \neq 0$ because correlated inventory in asset 2 creates effective exposure to $V_1$ -- the correct skew accounts for the full correlated portfolio risk, not just single-asset inventory.