Zero-Profit Bid-Ask Quotes for a Uniform Value
An asset has an efficient value $V \sim \text{Unif}[0,1]$. When a trader arrives, with probability $\alpha$ they are informed -- they buy if and only if $V > a$ (the ask) and sell if and only if $V < b$ (the bid). With probability
A market maker posts a bid $b$ and ask $a$, and wants zero expected P&L on each side. This means the ask must satisfy $a = E[V \mid \text{buy at } a]$ and the bid must satisfy $b = E[V \mid \text{sell at } b]$.
- Derive explicit formulas for $a$ and $b$ in terms of $\alpha$.
- Prove that the spread $a - b > 0$ whenever $\alpha > 0$.
Hints
- Use Bayes' rule to express $E[V \mid \text{buy}]$ and $E[V \mid \text{sell}]$ in terms of $a$, $b$, and $\alpha$. Remember to account for both informed and noise traders in the total probability of each trade direction.
- For $V \sim \text{Unif}[0,1]$, you will need $E[V \mid V > a] = (1+a)/2$ and $E[V \mid V < b] = b/2$. Cross-multiplying the conditional expectation equation gives a quadratic in $a$ (and separately in $b$).
- After solving, note that $a + b = 1$ by symmetry of the uniform distribution around /2$. This means proving $a - b > 0$ reduces to showing $a > 1/2$, which follows from $\sqrt{1 - \alpha^2} < 1$ for $\alpha > 0$.
Worked Solution
How to Think About It: This is the Glosten-Milgrom model in its simplest form. You are a market maker quoting a bid and ask for an asset whose true value you do not know. Some fraction $\alpha$ of traders know $V$ and will pick you off -- buying when the asset is worth more than your ask, selling when it is worth less than your bid. The rest are noise traders who trade randomly. Your quotes must be set so that, on average, the gains from trading with noise traders exactly offset the losses from trading with informed traders. The zero-profit conditions are Bayesian -- the ask equals the expected value conditional on someone wanting to buy at that price, and similarly for the bid.
Quick Estimate: Suppose $\alpha = 0.5$. Half the time you face an informed trader, half the time noise. The ask should be above
/2$ because buys are more likely when $V$ is high. Rough guess: maybe $a \approx 0.63$, so the bid is around $b \approx 0.37$ by symmetry, giving a spread of about $0.26$. When $\alpha = 0$ (all noise), there is no adverse selection, so $a = b = 1/2$ and the spread is zero. When $\alpha = 1$ (all informed), the market maker is always picked off, so $a \to 1$ and $b \to 0$ -- the spread blows out to 1 and no trading happens.Approach: Write the conditional expectations using Bayes' rule, then solve the resulting quadratic equations.
Formal Solution:
*Part 1: Deriving the ask $a$.*
The probability that the arriving trader buys at ask $a$:
$P(\text{buy}) = \alpha \cdot P(V > a) + (1 - \alpha) \cdot \tfrac{1}{2} = \alpha(1 - a) + \tfrac{1 - \alpha}{2}$
The expected value of $V$ weighted by the buy event:
$E[V \cdot \mathbf{1}_{\text{buy}}] = \alpha \cdot E[V \mathbf{1}_{V > a}] + (1 - \alpha) \cdot \tfrac{1}{2} \cdot E[V] = \frac{\alpha(1 - a^2)}{2} + \frac{1 - \alpha}{4}$
Setting $a = E[V \mid \text{buy}]$ and cross-multiplying:
$a \left[\alpha(1 - a) + \frac{1 - \alpha}{2}\right] = \frac{\alpha(1 - a^2)}{2} + \frac{1 - \alpha}{4}$
Expand and simplify. After collecting terms, one obtains:
$2\alpha(1 - a)^2 = (1 - \alpha)(2a - 1)$
Substitute $u = 1 - a$:
$2\alpha u^2 + 2(1 - \alpha)u - (1 - \alpha) = 0$
Apply the quadratic formula, taking the root with $0 < u < 1$:
$u = \frac{-(1 - \alpha) + \sqrt{(1 - \alpha)(1 + \alpha)}}{2\alpha} = \frac{\sqrt{1 - \alpha^2} - (1 - \alpha)}{2\alpha}$
So the ask is:
$\boxed{a = \frac{1 + \alpha - \sqrt{1 - \alpha^2}}{2\alpha}}$
*Part 1 (continued): Deriving the bid $b$.*
By an analogous calculation for the sell side:
$P(\text{sell}) = \alpha b + \frac{1 - \alpha}{2}$
$E[V \mid \text{sell at } b] = \frac{\alpha b^2 / 2 + (1 - \alpha)/4}{\alpha b + (1 - \alpha)/2}$
Setting $b = E[V \mid \text{sell}]$ and simplifying yields the same quadratic:
$2\alpha b^2 + 2(1 - \alpha)b - (1 - \alpha) = 0$
The positive root gives:
$\boxed{b = \frac{\sqrt{1 - \alpha^2} - (1 - \alpha)}{2\alpha}}$
Note that $a + b = 1$, which is expected by the symmetry of $V \sim \text{Unif}[0,1]$ around
/2$.*Part 2: Proving $a - b > 0$ when $\alpha > 0$.*
The spread is:
$a - b = \frac{1 + \alpha - \sqrt{1 - \alpha^2}}{2\alpha} - \frac{\sqrt{1 - \alpha^2} - (1 - \alpha)}{2\alpha} = \frac{1 - \sqrt{1 - \alpha^2}}{\alpha}$
For $\alpha \in (0, 1]$, we have $\alpha^2 > 0$, so
- \alpha^2 < 1$, which gives $\sqrt{1 - \alpha^2} < 1$. Therefore:$1 - \sqrt{1 - \alpha^2} > 0 \quad \text{and} \quad \alpha > 0$
so $a - b > 0$. $\blacksquare$
*Sanity checks:* - $\alpha = 0$: By L'Hopital or direct substitution, $a, b \to 1/2$, spread $\to 0$. No adverse selection, no spread. - $\alpha = 1$: $a = 1$, $b = 0$, spread $= 1$. Fully informed traders, market breaks down. - $\alpha = 0.5$: $a \approx 0.634$, $b \approx 0.366$, spread $\approx 0.268$.
Answer: The zero-profit ask and bid are $a = \frac{1 + \alpha - \sqrt{1 - \alpha^2}}{2\alpha}$ and $b = \frac{\sqrt{1 - \alpha^2} - (1 - \alpha)}{2\alpha}$. The spread is $a - b = \frac{1 - \sqrt{1 - \alpha^2}}{\alpha} > 0$ for all $\alpha > 0$, since $\sqrt{1 - \alpha^2} < 1$.
Intuition
This problem captures the fundamental tension in market making: you must quote prices that protect you from informed traders while still attracting enough noise flow to stay in business. The zero-profit condition is really a Bayesian break-even -- your ask equals the expected asset value conditional on someone wanting to buy at that price, which is an adverse selection-weighted average. The informed traders only buy when the asset is worth more than your ask, dragging the conditional expectation up. To compensate, you raise the ask above the unconditional mean. The same logic pushes the bid below
/2$.The elegant result $a + b = 1$ comes from the symmetry of the uniform distribution, but the key economic insight is that the spread $a - b$ is monotonically increasing in $\alpha$. More adverse selection means a wider spread. In practice, this is exactly what you see: stocks with high insider trading activity or around earnings announcements have wider bid-ask spreads. This Glosten-Milgrom framework is the foundation for understanding why spreads exist and what drives their size -- it is not about inventory risk or transaction costs, but about information asymmetry.