Expected Value of Information Before Quoting

Worked Solution

How to Think About It: You are a market maker choosing where to center your quotes. Without any signal, your best guess for $V$ is the prior mean $\mu_0$, and you quote $[\mu_0 - s/2, \, \mu_0 + s/2]$. Every trade that hits you exposes you to adverse selection: when $V$ is far from $\mu_0$, you lose money on average because the spread may not cover the pricing error. The signal $Y$ lets you re-center your quotes closer to the true $V$, reducing that error. The question is: how much is that re-centering worth?

The key intuition is that the EVI comes from the reduction in your pricing uncertainty. The variance of your pricing error drops from $\sigma_0^2$ (blind) to the posterior variance $\hat{\sigma}^2$ (with signal). The expected improvement in P&L is tied to this variance reduction.

Quick Estimate: If $\sigma_0 = 2$ and $\tau = 1$, the posterior variance is $\hat{\sigma}^2 = 1/(1/4 + 1/1) = 1/(5/4) = 4/5 = 0.8$. So the variance drops from 4 to 0.8, a reduction of 3.2. The standard deviation of the pricing error drops from 2 to about 0.89. In a model where P&L loss scales with the absolute pricing error, the EVI is proportional to the reduction in $E[|V - m|]$, which drops from $\sigma_0 \sqrt{2/\pi} \approx 1.60$ to $\hat{\sigma}\sqrt{2/\pi} \approx 0.71$. The EVI is roughly $0.89$ in absolute pricing error units. So if $c < 0.89$, buying the signal pays for itself.

Approach: We use the conjugate normal-normal Bayesian update, then compute the expected P&L with and without the signal to derive the EVI.

Formal Solution:

*Part 1: Posterior distribution.*

The prior is $V \sim N(\mu_0, \sigma_0^2)$ and the likelihood is $Y | V \sim N(V, \tau^2)$. By the conjugate normal update:

$V | Y \sim N\!\left(\hat{\mu}(Y),\; \hat{\sigma}^2\right)$

where the posterior precision is the sum of the prior and likelihood precisions:

$\frac{1}{\hat{\sigma}^2} = \frac{1}{\sigma_0^2} + \frac{1}{\tau^2} = \frac{\sigma_0^2 + \tau^2}{\sigma_0^2 \tau^2}$

so

$\hat{\sigma}^2 = \frac{\sigma_0^2 \tau^2}{\sigma_0^2 + \tau^2}$

and the posterior mean is the precision-weighted average:

$\hat{\mu}(Y) = \hat{\sigma}^2 \left(\frac{\mu_0}{\sigma_0^2} + \frac{Y}{\tau^2}\right) = \frac{\tau^2 \mu_0 + \sigma_0^2 Y}{\sigma_0^2 + \tau^2}$

*Part 2: Expected value of information.*

In a one-shot trade with symmetric spread $s$, the market maker posts mid-quote $m$ and earns the half-spread $s/2$ on every trade but suffers adverse selection loss proportional to $|V - m|$. Specifically, the expected loss from mis-centering is $E[|V - m|]$ (the expected absolute pricing error).

Without the signal: The optimal mid-quote is $m = \mu_0$. The pricing error $V - \mu_0 \sim N(0, \sigma_0^2)$, so:

$E[|V - \mu_0|] = \sigma_0 \sqrt{\frac{2}{\pi}}$

With the signal: After observing $Y$, the optimal mid-quote is $m = \hat{\mu}(Y)$. The residual pricing error $V - \hat{\mu}(Y) | Y$ has variance $\hat{\sigma}^2$, so:

$E[|V - \hat{\mu}(Y)| \;|\; Y] = \hat{\sigma} \sqrt{\frac{2}{\pi}}$

Since $\hat{\sigma}$ does not depend on $Y$, this holds unconditionally as well.

The EVI is the reduction in expected adverse selection loss:

$\text{EVI} = E[|V - \mu_0|] - E[|V - \hat{\mu}(Y)|] = \sqrt{\frac{2}{\pi}} \left(\sigma_0 - \hat{\sigma}\right)$

Substituting $\hat{\sigma} = \sigma_0 \tau / \sqrt{\sigma_0^2 + \tau^2}$:

$\text{EVI} = \sigma_0 \sqrt{\frac{2}{\pi}} \left(1 - \frac{\tau}{\sqrt{\sigma_0^2 + \tau^2}}\right)$

*Part 3: Threshold for purchasing the signal.*

Buying the signal is optimal when $\text{EVI} > c$. The threshold is:

$c^{*} = \sigma_0 \sqrt{\frac{2}{\pi}} \left(1 - \frac{\tau}{\sqrt{\sigma_0^2 + \tau^2}}\right)$

Buy the signal if and only if $c < c^{*}$.

Limiting cases: - If $\tau \to 0$ (perfect signal), $c^{*} \to \sigma_0 \sqrt{2/\pi}$: the maximum possible value of information, which equals the full expected absolute pricing error. - If $\tau \to \infty$ (useless signal), $c^{*} \to 0$: the signal teaches you nothing. - If $\sigma_0 \to 0$ (you already know $V$), $c^{*} \to 0$: no signal is needed.

Answer: The posterior is $V|Y \sim N(\hat{\mu}(Y), \hat{\sigma}^2)$ with $\hat{\sigma}^2 = \sigma_0^2 \tau^2 / (\sigma_0^2 + \tau^2)$. The expected value of information is $\text{EVI} = \sqrt{2/\pi}\,(\sigma_0 - \hat{\sigma})$, and you should buy the signal whenever $c < c^{*} = \sigma_0 \sqrt{2/\pi}\,\bigl(1 - \tau/\sqrt{\sigma_0^2 + \tau^2}\bigr)$.