Adverse Selection and the Break-Even Spread

- \alpha$, the client is uninformed noise and buys or sells with equal probability. You quote a bid at $m - s$ and an ask at $m + s$, where $m$ is the current midprice and $s \geq 1$ is the half-spread in ticks. After the trade, you unwind your position one step later at the new mid $m + M$. 1. Compute $E[M \mid \text{you sell at the ask}]$ and $E[M \mid \text{you buy at the bid}]$. 2. Compute the unconditional expected P&L per trade, $E[\text{P\&L}]$. 3. Find the smallest integer half-spread $s$ that makes $E[\text{P\&L}] \geq 0$, and explain the adverse-selection intuition.