Posterior from a Private Gaussian Signal for Optimal Quotes

Options Pricing · Hard · Free problem

Before quoting a market, you observe a private signal $Y = V + \varepsilon$, where $V \sim N(\mu, \sigma^2)$ is the unknown asset value and $\varepsilon \sim N(0, \tau^2)$ is independent observation noise.

  1. Compute the posterior distribution of $V$ given $Y$. State the posterior mean $m$ and posterior variance $\rho^2$ in terms of $\mu$, $\sigma^2$, $\tau^2$, and $Y$.
  1. Suppose you set a bid $b(Y)$ and ask $a(Y)$ subject to a zero-expected-profit-conditional-on-fill rule. Incoming order flow is partially informed: with probability $\alpha$, the counterparty trades on the true value $V$; with probability
    - \alpha$, the counterparty trades randomly (uninformed noise). Derive the optimal bid and ask in terms of the posterior parameters and $\alpha$.
  1. Discuss how the quote skew $a(Y) - b(Y)$ depends on the signal accuracy (the ratio $\sigma^2 / \tau^2$) and the informed-flow fraction $\alpha$.

Hints

  1. Gaussian-Gaussian conjugacy: the posterior of $V$ given $Y$ is normal with precision equal to the sum of prior and signal precisions.
  2. For the zero-profit condition, think about adverse selection: when someone lifts your ask, the expected value of $V$ conditional on a fill is higher than $m$ because informed traders only buy when $V$ is high.
  3. The bid-ask spread is
\rho\delta(\alpha)$ where $\delta$ is an implicit function of $\alpha$. For small $\alpha$, linearize to get $\delta \approx \alpha / \sqrt{2\pi}$. Note $\rho^2 = \sigma^2\tau^2/(\sigma^2+\tau^2)$ rises in both variances, so frame accuracy as smaller $\tau^2$, not a higher $\sigma^2/\tau^2$ ratio.

Worked Solution

How to Think About It: You are a market maker who gets a noisy peek at the true value before setting quotes. The posterior mean $m$ is your best estimate of $V$ - it is a precision-weighted average of your prior mean $\mu$ and your signal $Y$. The bid-ask spread compensates you for adverse selection: when someone hits your bid, there is a chance they know $V$ is low. The spread should be wider when adverse selection is worse (high $\alpha$) and when your signal is noisier (high $\tau^2$ relative to $\sigma^2$, meaning your posterior is uncertain).

Quick Estimate: If $\sigma^2 = \tau^2$ (signal and prior equally informative), the posterior mean is $(\mu + Y)/2$ and posterior variance is $\sigma^2/2$. If $\alpha = 1$ (all flow is informed), you would need to set bid = ask = $V$ to avoid losses, which is impossible since you do not know $V$ - you would just not quote. If $\alpha = 0$ (all flow is noise), there is no adverse selection and $a = b = m$ (zero spread). Reality is in between.

Approach: Derive the Gaussian posterior, then set up the zero-profit condition for each side.

Formal Solution:

Part 1: Posterior distribution.

With $V \sim N(\mu, \sigma^2)$ and $Y \mid V \sim N(V, \tau^2)$, the posterior is Gaussian:

$V \mid Y \sim N(m, \rho^2)$

where the posterior mean and variance are given by precision-weighting:

$m = \frac{\sigma^{-2} \cdot \mu + \tau^{-2} \cdot Y}{\sigma^{-2} + \tau^{-2}} = \frac{\tau^2 \mu + \sigma^2 Y}{\sigma^2 + \tau^2}$

$\rho^2 = \frac{1}{\sigma^{-2} + \tau^{-2}} = \frac{\sigma^2 \tau^2}{\sigma^2 + \tau^2}$

Equivalently, $m = \mu + \frac{\sigma^2}{\sigma^2 + \tau^2}(Y - \mu)$. The posterior mean is a shrinkage estimator: it pulls $Y$ toward the prior mean $\mu$, with the shrinkage factor depending on relative precision.

Part 2: Optimal quotes under zero-profit-conditional-on-fill.

Consider the ask side. When someone lifts your ask at price $a$, they either: - With probability $\alpha$: an informed trader who buys only if $V > a$ (or more generally, if $V$ exceeds their threshold). - With probability

- \alpha$: a noise trader who buys regardless.

The zero-profit condition on the ask says: expected value of $V$ conditional on someone buying at $a$ equals $a$.

$a = E[V \mid \text{fill at ask}]$

The fill event is: informed trader buys (prob $\alpha$, happens when $V > a$) or noise trader buys (prob

-\alpha$, always). Conditional on a fill:

$a = \frac{\alpha \cdot E[V \cdot \mathbf{1}_{V > a} \mid Y] + (1-\alpha) \cdot E[V \mid Y]}{\alpha \cdot P(V > a \mid Y) + (1-\alpha)}$

Since $V \mid Y \sim N(m, \rho^2)$, let $\Phi$ and $\phi$ denote the standard normal CDF and PDF. Write $Z = (V - m)/\rho$. Then:

$E[V \cdot \mathbf{1}_{V > a} \mid Y] = m \cdot P(V > a \mid Y) + \rho \cdot \phi\left(\frac{a - m}{\rho}\right)$

using the truncated normal mean formula. So the zero-profit condition becomes:

$a = \frac{\alpha \left[ m \cdot \bar{\Phi}\left(\frac{a-m}{\rho}\right) + \rho \cdot \phi\left(\frac{a-m}{\rho}\right) \right] + (1-\alpha) \cdot m}{\alpha \cdot \bar{\Phi}\left(\frac{a-m}{\rho}\right) + (1-\alpha)}$

where $\bar{\Phi}(x) = 1 - \Phi(x)$. Rearranging:

$a \left[ \alpha \bar{\Phi}\left(\frac{a-m}{\rho}\right) + (1-\alpha) \right] = \alpha m \bar{\Phi}\left(\frac{a-m}{\rho}\right) + \alpha \rho \phi\left(\frac{a-m}{\rho}\right) + (1-\alpha)m$

$(a - m) \left[ \alpha \bar{\Phi}\left(\frac{a-m}{\rho}\right) + (1-\alpha) \right] = \alpha \rho \phi\left(\frac{a-m}{\rho}\right)$

Let $\delta = (a - m)/\rho$ (the ask offset in standard deviations). Then:

$\delta \left[ \alpha \bar{\Phi}(\delta) + (1-\alpha) \right] = \alpha \phi(\delta)$

This is an implicit equation for $\delta$ that must be solved numerically for given $\alpha$. By symmetry, the bid satisfies $b = m - \rho \delta$, so the spread is:

$a - b = 2 \rho \delta(\alpha)$

For small $\alpha$ (little adverse selection), $\delta \approx \alpha \phi(0) / (1-\alpha) = \alpha / (\sqrt{2\pi}(1-\alpha))$, so the half-spread is approximately $\alpha \rho / \sqrt{2\pi}$.

Part 3: Dependence of spread on signal accuracy and $\alpha$.

The spread $a - b = 2\rho \delta(\alpha)$ depends on two factors:

  • Posterior standard deviation $\rho = \sigma\tau / \sqrt{\sigma^2 + \tau^2}$: This measures your residual uncertainty about $V$ after seeing the signal. When $\tau^2 \ll \sigma^2$ (very accurate signal), $\rho \approx \tau$ is small and the spread narrows - you know $V$ almost as well as the informed traders. When $\tau^2 \gg \sigma^2$ (noisy signal), $\rho \approx \sigma$ and the spread approaches what it would be with no private signal at all.
  • Informed fraction $\alpha$: As $\alpha \to 0$, $\delta \to 0$ and the spread collapses (no adverse selection). As $\alpha \to 1$, $\delta \to \infty$ and the spread blows up (nearly all flow is informed, so you need an infinite spread to avoid being picked off). In the limit $\alpha = 1$, no finite spread yields zero profit.

Be careful interpreting "signal accuracy" through the ratio $\sigma^2/\tau^2$: the posterior variance $\rho^2 = \sigma^2\tau^2/(\sigma^2+\tau^2)$ is increasing in BOTH $\sigma^2$ and $\tau^2$ (its partials are $\tau^4/(\sigma^2+\tau^2)^2 > 0$ and $\sigma^4/(\sigma^2+\tau^2)^2 > 0$), so raising the ratio is direction-ambiguous. Raising it by inflating $\sigma^2$ at fixed $\tau^2$ actually raises $\rho$ and WIDENS the spread. The clean statement is in terms of $\tau^2$: a more accurate signal means a sharper observation, i.e. smaller $\tau^2$ holding $\sigma^2$ fixed, which lowers $\rho$ and narrows the spread. In practice, a market maker with a better signal (lower $\tau^2$) can quote tighter and capture more flow.

Answer: The posterior is $V \mid Y \sim N\left(\frac{\tau^2 \mu + \sigma^2 Y}{\sigma^2 + \tau^2}, \; \frac{\sigma^2 \tau^2}{\sigma^2 + \tau^2}\right)$. The optimal quotes are $a = m + \rho \delta$ and $b = m - \rho \delta$, where $\delta$ solves $\delta[\alpha \bar{\Phi}(\delta) + (1-\alpha)] = \alpha \phi(\delta)$. The spread

\rho\delta$ widens with the informed-flow fraction $\alpha$ and with posterior uncertainty $\rho$, and narrows as the signal becomes sharper (smaller $\tau^2$ at fixed $\sigma^2$, which lowers $\rho$); the bare ratio $\sigma^2/\tau^2$ is not a clean accuracy measure since $\rho$ rises in both $\sigma^2$ and $\tau^2$.

Intuition

This problem captures the fundamental tension in market making: your private signal helps you center your quotes, but you still face adverse selection from better-informed traders. The spread compensates for adverse selection risk, and its size depends on two things -- how uncertain you remain after seeing your signal (posterior variance $\rho^2$) and what fraction of incoming flow is informed ($\alpha$).

The precision-weighting formula for the posterior mean is one of the most important results in quantitative finance. It says: your best estimate of an unknown quantity is the precision-weighted average of your prior and your signal. This exact formula underlies Kalman filtering, Bayesian portfolio optimization, and the Kyle (1985) model of informed trading. The quote-setting problem here is a simplified version of the Glosten-Milgrom model, where the market maker's spread arises endogenously from the adverse selection problem rather than being set exogenously.

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