Impact

Market impact — the price displacement caused by trading — is the central cost of active investing. Two foundational academic measures capture it from different angles, and both reveal something the other misses.

Kyle's Lambda (1985): The price of information

Kyle's seminal model of insider trading introduced lambda (λ) as the slope of the price-impact function: how much prices move per unit of signed order flow. In the model, a single informed trader optimally camouflages their trades among noise traders, and the market maker sets lambda to reflect the expected information content per dollar of flow.

$\lambda = \Delta P / \Delta(\text{signed volume})$

Lambda is directional — it distinguishes buys pushing price up from sells pushing it down. A higher lambda means each dollar of order flow moves prices more, implying either thin liquidity or high information content in the flow. Hasbrouck (2009) showed that Kyle's lambda remains one of the most effective low-frequency proxies for the true cost of trading, outperforming many more complex alternatives.

Amihud Illiquidity (2002): The price of size

Amihud's measure takes an elegantly simpler approach:

$\text{ILLIQ} = |r_t| / V_t$

Where $r_t$ is the return and $V_t$ is dollar volume. It answers: how much return per dollar of trading? High ILLIQ means a small trade moves price disproportionately — the hallmark of an illiquid market.

Despite its simplicity, Amihud (2002) demonstrated that ILLIQ predicted expected stock returns cross-sectionally: illiquid stocks commanded a return premium, consistent with the liquidity risk framework of Acharya & Pedersen (2005).

Where they diverge — and why it matters

• Rising Amihud without rising Kyle lambda → price moves are increasing but aren't driven by signed order flow → exogenous news events, not informed trading
• Rising Kyle lambda without much Amihud change → directional flow is becoming more impactful per unit → informed traders increasing participation (the Kyle model's core prediction)

We also compute directional impact asymmetry — whether buy-side lambda differs from sell-side. Markets frequently absorb sells more efficiently than buys (Chordia, Roll & Subrahmanyam, 2002), particularly in trending environments where sell pressure is expected but buy pressure can destabilize.

Both metrics are computed per interval using covariance-based estimation for Kyle lambda and absolute-return normalization for Amihud, allowing real-time tracking of intraday liquidity cycles.

References:

• Kyle, A.S. (1985). "Continuous Auctions and Insider Trading." Econometrica, 53(6), 1315–1335.
• Amihud, Y. (2002). "Illiquidity and Stock Returns: Cross-Section and Time-Series Effects." Journal of Financial Markets, 5(1), 31–56.
• Hasbrouck, J. (2009). "Trading Costs and Returns for U.S. Equities: Estimating Effective Costs from Daily Data." Journal of Finance, 64(3), 1445–1477.
• Acharya, V.V. & Pedersen, L.H. (2005). "Asset Pricing with Liquidity Risk." Journal of Financial Economics, 77(2), 375–410.
• Chordia, T., Roll, R. & Subrahmanyam, A. (2002). "Order Imbalance, Liquidity, and Market Returns." Journal of Financial Economics, 65(1), 111–130.