Kyle's Lambda & Market Microstructure Lab

Model Pete Kyle (1985) insider trading equilibrium, Glosten-Milgrom (1985) sequential Bayesian quote updates, and adverse selection bid-ask spread decomposition.

MARKET REGIMES:
Kyle's Lambda (λ)
$0.000080
$0.080 per 1k shs
Market Depth (1 / λ)
12,500 shs
Required flow for $1 move
Quoted Bid-Ask Spread
$1.670
167.0 bps (95% Adverse Sel.)
Price Discovery Revealed
50.0%
$50,000 noise loss
Market Parameters
Prior Asset Price ($P_0$) $100.00
Fundamental Volatility ($\sigma_v$) $4.00
Standard deviation of unrevealed fundamental asset value
Noise Order Flow Vol ($\sigma_u$) 25,000 shs
Uninformed liquidity trader order dispersion
Number of Insiders ($N$) 1
Cournot competition among informed market participants
Microstructure & Glosten-Milgrom
Informed Trader Fraction ($\alpha$) 20%
Probability that an arriving trade is informed
Valuation Gap ($\Delta V = V_H - V_L$) $8.00
High vs low outcome spread in sequential model
Exchange & Clearing Fee ($c_{\text{proc}}$) $0.02
Inventory Risk Cost ($c_{\text{inv}}$) $0.02
Interactive Chart.js model visualization with dynamic sensitivity recalculation.
Microstructure Analytics & Parameter Breakdown
Microstructure Metric Analytical Formulation Simulated Value
Sensitivity: $\sigma_v$ vs $\sigma_u$ on Kyle's Lambda ($\lambda$ per 1k shs)
Sensitivity: Informed Proportion $\alpha$ vs $\Delta V$ on Glosten-Milgrom Spread ($)

Institutional Foundations of Market Microstructure

1. Pete Kyle's (1985) Continuous Auction Model

In Pete Kyle's landmark 1985 paper "Continuous Auctions and Insider Trading", trading occurs between three classes of market participants:

  • Informed Insider: Observes fundamental liquidation value $v \sim \mathcal{N}(p_0, \sigma_v^2)$ and chooses order quantity $x(v)$.
  • Noise Traders: Submit exogenous, price-inelastic aggregate liquidity orders $u \sim \mathcal{N}(0, \sigma_u^2)$.
  • Market Makers: Risk-neutral and competitive, observing only total aggregate order flow $\omega = x + u$ and clearing at fair expectation $P(\omega) = \mathbb{E}[v \mid \omega]$.

In linear Nash equilibrium, the market maker's pricing rule is $P(\omega) = p_0 + \lambda \omega$, and the insider's trading schedule is $x(v) = \beta(v - p_0)$. Solving the first-order conditions yields:

$$\lambda = \frac{\sigma_v}{2\sigma_u}, \quad \beta = \frac{\sigma_u}{\sigma_v}, \quad D = \frac{1}{\lambda} = \frac{2\sigma_u}{\sigma_v}$$

The parameter $\lambda$ measures illiquidity or price impact. Market depth $D$ indicates the quantity of shares necessary to move the price by \$1.00.

2. Glosten-Milgrom (1985) Sequential Trade Model

Lawrence Glosten and Paul Milgrom (1985) introduced sequential trading where traders arrive one at a time. A fraction $\alpha$ are informed traders possessing private knowledge of asset value $V \in \{V_L, V_H\}$, while fraction $1 - \alpha$ are uninformed noise traders buying or selling with equal probability.

Because informed traders buy when $V = V_H$ and sell when $V = V_L$, an incoming buy order signals positive information, driving the market maker's posterior belief upward:

$$P(V = V_H \mid \text{Buy}) = \frac{\delta_t(1 + \alpha)}{1 + \alpha(2\delta_t - 1)}$$

To avoid expected losses, the market maker sets $A_t = \mathbb{E}[V \mid \text{Buy}]$ and $B_t = \mathbb{E}[V \mid \text{Sell}]$, generating an endogenous adverse selection spread $S = (V_H - V_L)\alpha$ even without inventory costs.

3. Information Revelation & The 50% Rule

A foundational result of Kyle's model is that a monopolistic insider strategically restricts order volume to camouflage behind noise trading:

$$\text{Var}(v \mid P) = \sigma_v^2 - \frac{\beta^2 \sigma_v^4}{\beta^2 \sigma_v^2 + \sigma_u^2} = \frac{1}{2}\sigma_v^2$$

Exactly 50% of the insider's private information is incorporated into the clearing price during a single auction round. When $N$ Cournot insiders compete, the information revealed increases to $N / (N + 1)$, approaching 100% as $N \to \infty$.

4. Empirical Estimation: Roll (1984) & Amihud (2002)

When quote book data is unavailable, Richard Roll (1984) established that trade prices bounce between bid and ask, creating negative serial autocorrelation:

$$S_{\text{Roll}} = 2 \sqrt{-\min(0, \text{Cov}(\Delta P_t, \Delta P_{t-1}))}$$

Yakov Amihud (2002) formulated the illiquidity ratio $\text{ILLIQ} = \frac{1}{D} \sum \frac{|R_t|}{\text{Volume}_t}$, proxying Kyle's $\lambda$ via absolute return per dollar of volume.

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Frequently Asked Questions: Market Microstructure & Order Flow

Pete Kyle (1985) formulated lambda as the regression slope of price changes on order flow: lambda = Cov(v, w) / Var(w) = sigma_v / (2 sigma_u). It measures the dollars of price impact per share of net order flow. Market depth is defined as the reciprocal D = 1 / lambda = 2 sigma_u / sigma_v, representing the volume of orders required to move the market price by one dollar.

Glosten & Milgrom (1985) demonstrated that even in a competitive, risk-neutral market with zero transaction costs, market makers quote a positive bid-ask spread. Because informed traders buy when the asset is undervalued and sell when overvalued, the market maker incurs an expected loss against informed traders on every execution. To break even on average, the market maker sets Ask = E[V | Buy] and Bid = E[V | Sell], earning a positive spread from uninformed noise traders that exactly offsets adverse selection losses.

A monopolistic informed insider balances immediate trading profits against price impact. Trading aggressively captures a larger volume of shares but drives the market clearing price against themselves via lambda. By choosing optimal order schedule beta = sigma_u / sigma_v, the insider restricts order size so that exactly 50% of the private information variance is impounded into the market price, halving posterior variance from sigma_v^2 to 0.5 sigma_v^2.

In empirical microstructure literature (e.g., Stoll 1989, Huang & Stoll 1997), the quoted bid-ask spread is decomposed into: (1) Adverse Selection Cost compensating for trading with counterparties possessing superior private information, (2) Inventory Carrying Cost compensating the market maker for bearing overnight price volatility risk, and (3) Order Processing Cost covering exchange clearing, colocation, server technology, and regulatory fees.

Richard Roll (1984) showed that transactions bounce randomly between the bid and ask quotes. This bid-ask bounce introduces negative first-order serial covariance in consecutive price changes: Cov(Delta P_t, Delta P_{t-1}) = -S^2 / 4. The effective spread is therefore analytically estimated directly from historical trade prices without needing quote books as S_Roll = 2 * sqrt(-min(0, Cov(Delta P_t, Delta P_{t-1}))).