KylesLambda

The rolling OLS slope of mid-price changes on signed trade volume — Kyle's λ, the canonical measure of price impact and market depth. Big λ means a thin book that moves per unit traded; small λ a deep one.

Quick reference

Item	Value
Family	Microstructure
Input type	TradeQuote — a trade plus the mid prevailing at execution
Output type	f64 (price move per unit signed volume)
Output range	unbounded; usually ≥ 0
Default parameters	window required (≥ 2)
Warmup period	window + 1
Interpretation	Price impact / market depth

Formula

q      = size · D                       (signed volume, D = aggressor sign)
Δmid   = midₜ − midₜ₋₁
cov    = (1/n) · Σ q·Δmid − q̄·Δ̄mid
var    = (1/n) · Σ q²      − q̄²
λ      = cov / var

The rolling OLS slope of Δmid regressed on q over the trailing window paired observations. Four running sums keep each update O(1). See crates/wickra-core/src/indicators/kyles_lambda.rs.

Parameters

Name	Type	Default	Constraint	Description
window	usize	none	≥ 2	Rolling number of (Δmid, signed-volume) pairs.

Inputs / Outputs

Indicator<Input = TradeQuote, Output = f64>. Bindings: update(price, size, is_buy, mid). Python / Node batch take four equal-length arrays and return a 1-D array (NaN during warmup). WASM streaming-only.

Warmup

warmup_period() == window + 1: one trade-quote seeds the previous mid, then window paired observations fill the regression.

Edge cases

• Constant signed volume. Zero variance in the regressor; λ is undefined and the indicator returns 0 rather than NaN.
• Negative λ. Possible but unusual; signals mid moves opposite to flow over the window (mean-reverting microstructure or mislabelled aggressor).
• Window below 2. Rejected at construction.

Examples

Rust

use wickra::{Indicator, KylesLambda, Side, Trade, TradeQuote};

// A book that moves 0.5 per unit of signed volume gives λ = 0.5.
let mut lambda = KylesLambda::new(8).unwrap();
let mut mid = 100.0;
let mut last = None;
for i in 0..20 {
    let side = if i % 2 == 0 { Side::Buy } else { Side::Sell };
    let size = 1.0 + f64::from(i % 3);
    mid += 0.5 * size * side.sign();
    let trade = Trade::new(mid, size, side, 0).unwrap();
    last = lambda.update(TradeQuote::new(trade, mid).unwrap());
}
assert!((last.unwrap() - 0.5).abs() < 1e-9);

Python

import wickra as ta

kl = ta.KylesLambda(50)
# feed (price, size, is_buy, mid) per trade; reads price impact per unit flow

Node

const { KylesLambda } = require('wickra');
const kl = new KylesLambda(50);
// kl.update(price, size, isBuy, mid)

Interpretation

Kyle's λ is the slope of the price-impact line — how far the mid moves per unit of net order flow. It is the canonical inverse of market depth.

• Large λ. A thin, fragile book: a small signed order shifts the mid a lot. Trading into it is expensive and impact-dominated.
• Small λ. A deep, resilient book: it absorbs size with little mid movement. The cheap regime for execution.
• Rising intraday. Liquidity is evaporating — λ climbs ahead of scheduled events and in stress, then recovers afterwards. Watching its trajectory is a practical liquidity-risk gauge.

λ is the empirical analogue of the depth parameter in Kyle's insider-trading model; Amihud's illiquidity ratio is the daily, low-frequency cousin.

Common pitfalls

• Not scale-free. λ has units of price per unit volume, so its magnitude depends on tick size and contract size. Compare it only across comparable instruments and size conventions — never as a bare cross-asset number.
• Window tuning. Too short and the regression is noisy; too long and it blurs regime changes. Match the window to your trade rate, not a fixed clock.
• Signed-flow quality. λ regresses Δmid on signed volume; a mislabelled aggressor flag corrupts the regressor and can even flip λ negative. Sign trades reliably (tick/quote rule) before feeding them in.

References

• Albert S. Kyle, Continuous Auctions and Insider Trading, Econometrica, 1985 — the original λ as the price-impact / depth parameter.
• Yakov Amihud, Illiquidity and Stock Returns, Journal of Financial Markets, 2002 — the daily illiquidity ratio, a low-frequency λ.
• Joel Hasbrouck, Empirical Market Microstructure, 2007 — estimating price impact from trade-and-quote data.

See also

• RealizedSpread — effective spread minus the impact λ measures.
• DepthSlope — depth read from the resting book instead of from executed flow.
• Microprice — fair value tilted by book imbalance.
• Indicators-Overview — the full taxonomy.