Optimal Bid-Ask Quotes Against an Informed Trader
You are making a market in a stock whose true value $V$ is uniformly distributed on $[0, 100]$. You must quote a bid $b$ and an ask $a$, with $b < a$. The trader you face is perfectly informed: they buy from you at your ask $a$ if $V > a$, and they sell to you at your bid $b$ if $V < b$. No trade occurs when $b \leq V \leq a$.
What bid and ask should you quote to maximize your expected profit? What does the optimal solution tell you about market making against a fully informed counterparty?
Hints
- Ask yourself first: when a trade happens, who benefits? If the trader only transacts when it is favorable to them, every trade hurts you -- so widening the spread always reduces your expected losses.
- Write expected profit as $\Pi(b, a) = \Pi_b(b) + \Pi_a(a)$, where the two sides are separable. Each piece is a simple integral over the $\text{Uniform}[0, 100]$ density.
- Take $\partial \Pi / \partial a = 0$ and $\partial \Pi / \partial b = 0$ independently. You will find $a^{*} = 100$ and $b^{*} = 0$ -- the widest possible spread, consistent with the no-profit conclusion.
Worked Solution
How to Think About It: Before writing any integrals, ask yourself: when does a trade happen, and who benefits? The trader only buys from you when the asset is worth MORE than your ask -- so every trade on the ask side loses you money. Similarly, they only sell to you when the asset is worth LESS than your bid -- so every trade on the bid side also loses you money. Every single trade goes against you. Your expected profit from any trade is negative by construction. The only way to avoid losing money is to avoid trading entirely, which means quoting a spread so wide that the trader never finds a favorable price. That gut check already tells you where this is heading: the only "optimal" strategy in this adversarial setting is $b = 0$, $a = 100$ -- i.e., post no meaningful quotes at all.
Quick Estimate: Take $b = 25$ and $a = 75$. The ask side: if $V > 75$, the trader buys at 75. The asset is worth on average $87.5$ in that region, so you lose $87.5 - 75 = 12.5$ per trade, and this happens with probability $25/100 = 0.25$. Ask-side loss: $0.25 \times (-12.5) = -3.125$. The bid side: if $V < 25$, the trader sells at 25. The asset is worth on average $12.5$, so you lose $25 - 12.5 = 12.5$ per trade, probability $0.25$. Bid-side loss: $0.25 \times (-12.5) = -3.125$. Total expected profit: $-6.25$. Now try $b = 10$, $a = 90$: the adverse selection losses shrink because the trader finds fewer favorable prices. The pattern is clear -- widen the spread, reduce losses. The widest possible spread, $b = 0$, $a = 100$, gives expected profit of exactly $0$.
Approach: Set up expected profit as a function of $b$ and $a$, take first-order conditions, and verify the answer is consistent with the economic intuition above.
Formal Solution:
The market maker's expected profit has two pieces.
*Ask side:* The trader buys at $a$ when $V > a$. You sell an asset worth $V$ for $a$, so your profit is $a - V < 0$.
$$\Pi_a(a) = \int_a^{100} (a - V) \cdot \frac{1}{100}\, dV = \frac{1}{100}\left[aV - \frac{V^2}{2}\right]_a^{100} = \frac{1}{100}\left(100a - 5000 - a^2 + \frac{a^2}{2}\right) = \frac{100a - 5000 - a^2/2}{100}$$
*Bid side:* The trader sells at $b$ when $V < b$. You buy an asset worth $V$ for $b$, so your profit is $V - b < 0$.
$$\Pi_b(b) = \int_0^{b} (V - b) \cdot \frac{1}{100}\, dV = \frac{1}{100}\left[\frac{V^2}{2} - bV\right]_0^b = \frac{-b^2/2}{100}$$
*Total:*
$$\Pi(b, a) = \frac{1}{100}\left(-\frac{b^2}{2} + 100a - 5000 - \frac{a^2}{2}\right)$$
Optimize over $a$:
$$\frac{\partial \Pi}{\partial a} = \frac{1}{100}(100 - a) = 0 \implies a^{*} = 100$$
Optimize over $b$:
$$\frac{\partial \Pi}{\partial b} = \frac{1}{100}(-b) = 0 \implies b^{*} = 0$$
Substituting back: $\Pi(0, 100) = \frac{1}{100}(0 + 10000 - 5000 - 5000) = 0$.
Answer: The optimal quotes are $b^{*} = 0$, $a^{*} = 100$, yielding expected profit of exactly $\$0$. In a world with only perfectly informed traders, there is no profitable market to make.
Intuition
This problem illustrates the fundamental theorem of market microstructure: a market maker facing only perfectly informed traders cannot survive. Every trade is adversely selected -- the informed trader has a free option to transact only when it benefits them, and that option is entirely funded by the market maker. The math formalizes what intuition says immediately: widening the spread always reduces losses, so the "optimal" spread is infinitely wide (capped here at [0, 100] by the support of $V$).
In practice, real market makers profit because their order flow is a mix of informed and uninformed (noise) traders. The uninformed traders -- hedgers, index rebalancers, retail flow -- pay the spread without having an informational edge, and their losses subsidize the market maker's losses on the informed trades. The optimal spread then balances the revenue from noise traders against the adverse selection cost from informed ones. Models like Glosten-Milgrom and Kyle make this explicit. When you see a wide bid-ask spread in the real market, it is often a signal that the market maker believes a large fraction of incoming flow is informed -- they are widening defensively, exactly as this model predicts.