Quoting When the Other Side Has Seen Something

The scenario: you're about to quote a price to a specific counterparty — maybe a client who trades right before news, or a firm known for fast reactions — and you suspect there's some chance they know something about the asset's true value that you don't. How do you set your price, and by how much should you move it away from your best unconditional estimate?

The reasoning: price in the probability of facing an informed trader, not just the average case

Suppose your best unconditional estimate of fair value is $V_0$. If you quote exactly $V_0$ to everyone, an uninformed counterparty trades with you at a fair price and you break even in expectation on those trades. But an informed counterparty only trades with you when your price is wrong in their favor — they buy from you only when the true value is above $V_0$, and only when it's worth their while. Averaged across all the trades you do with informed counterparties, you lose money, because you never get the trades where the true value would have favored you. The fix is to widen your quote away from $V_0$ by enough that the profit from uninformed trades covers the expected loss from informed ones.

Worked example: solving for the break-even ask

Let $p$ be the probability any given counterparty asking for your offer is informed, and assume an informed trader who buys does so knowing the true value is, on average, $L$ higher than your current ask when they choose to trade (a simplification of the expected loss conditional on facing an informed buyer). An uninformed trader buys at whatever price you post, contributing $(a - V_0)$ profit if you quote ask $a$. Your expected profit per trade at the offer is:

$$(1-p)(a - V_0) - p \cdot L .$$

Setting this to zero (break-even) and solving for $a$:

$$a = V_0 + \frac{p \cdot L}{1 - p} .$$

Suppose V_0 = \100.00$,$p = 20%$(one in five counterparties on this name are informed), and$L = $0.50 (an informed buyer trades when the true value is \0.50 above your ask, on average). Then:

$$a = 100.00 + \frac{0.2 \times 0.50}{0.8} = 100.00 + 0.125 = \$100.125 .$$

You need to quote your offer at least $100.125, not $100.00, purely to break even given a 20% chance of facing someone who knows more than you. If $p$ rises to 50% — every other counterparty is informed — the required markup becomes \frac{0.5 \times 0.5}{0.5} = \0.50$, four times larger for only a 2.5x increase in$p$: the required markup grows faster than the probability of facing an informed trader, because you're also losing the offsetting profit from uninformed trades as they become a smaller share of your flow.

What this means in practice

This is exactly why market makers charge wider spreads to counterparties or venues associated with faster or better-informed flow, and tighter spreads to flow believed to be uninformed (retail, index rebalancers) — it isn't favoritism, it's break-even math applied with a different $p$ per counterparty type. In an interview, deriving the break-even formula from a simple expected-profit-equals-zero condition is a much stronger answer than saying "widen the spread for informed flow" without showing why the required widening isn't proportional to $p$.