The expected P&L of quoting both sides

Question

You quote a two-sided market with half-spread $h = \$ 0.05 $. A fraction$ \alpha = 0.15 $of the traders who hit you are informed (they take about$ d = $0.20 $out of you); the rest are noise, from whom you simply earn the half-spread. You get filled on about$ 500 $round-lots ($ 100$ shares each) per day.

What's your expected P&L per share, and per day?

Accepted Answer

Decompose the expected profit on a single share you trade. Two kinds of counterparty: - Uninformed (probability 1-\alpha): you buy on the bid / sell on the ask and the price doesn't systematically move, you keep the half-spread +h. - Informed (probability \alpha): they trade toward the true value, so instead of earning h you lose roughly d - h (you got the spread but the price ran d against you). Net on those is -(d - h). Expected P&L per share: \mathbb{E}[	ext{PnL}] = (1-\alpha)\,h - \alpha\,(d - h) = h - \alpha d. The messy terms collapse to a clean result: you keep the half-spread minus t…

The expected P&L of quoting both sides

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