You are a market maker who begins each day with zero inventory. Over 10 sequential rounds, each round you post a bid and ask. In each round a single order arrives: with probability $0.5$ it is a buy (you sell one unit, inventory decreases by 1) and with probability $0.5$ it is a sell (you buy one unit, inventory increases by 1). All orders are uninformed. Your revenue per round is the half-spread $s$ you earn (assume you always post the same half-spread $s$). At the end of 10 rounds, you pay a penalty of $c \cdot I^2$ where $I$ is your final inventory and $c = 2$. What constant half-spread $s$ maximizes your expected total profit (total spread revenue minus expected end-of-day inventory penalty)?

Sequential Quote Updating with Bayesian Learning

Optimal Spread for a Single-Asset Market Maker

Two-Sided Inventory Penalty Game

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5489. Two-Sided Inventory Penalty Game

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