Optimal and sub-optimal market makers
Companion demo to On a Simple Relationship Between Order Imbalance, Skew and Width in Over-The-Counter Trading.
Three dealers respond to the same stream of sealed-bid enquiries against the same competing quotes, so every difference in outcome is policy. Customers are sellers with probability q; the best competing quote is exponential with mean w = 1; inventory costs c(x) = c₂x² per period. The optimal dealer quotes from the solved indifference cost and knows the flow. The CWLS dealer keeps a constant width and skews linearly in inventory, calibrated to the balanced problem, with no flow term: the benchmark policy of the literature. The no-skew dealer quotes a constant width around fair value.
Look for: the optimal dealer's edge over CWLS (the flow term, a few percent), and the no-skew dealer's inventory pinned at the position limit while carry bleeds her — imbalance acting as a carrying cost, exactly as the corollary says.