You are a market maker on a single asset whose true value is uniformly distributed on $[0, 100]$. Each period, a single trader arrives and is either informed (knows the true value) with probability $p = 0.3$, or uninformed (trades randomly, buying or selling with equal probability) with probability $0.7$. You must post a symmetric bid-ask spread centered at your mid-price of $50$, i.e., bid $= 50 - s$ and ask $= 50 + s$ for some half-spread $s \geq 0$. An informed trader buys if the true value exceeds your ask, and sells if the true value is below your bid. An uninformed trader buys or sells with equal probability regardless of price. Your profit per trade is $+s$ when you trade with an uninformed trader, and your loss equals the absolute difference between the true value and your quote when you trade with an informed trader. What is the optimal half-spread $s^*$ that maximizes your expected profit per trade?

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5488. Optimal Spread for a Single-Asset Market Maker

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