Hasbrouck Information Share Bounds for Two-Venue Price Discovery

Understand why correlated venue innovations require upper and lower Cholesky bounds

When the same equity or futures contract trades on two cointegrated venues, Hasbrouck (1995) measures each venue’s share of efficient random-walk variance var_rw = (psi1*sig1)^2 + 2*rho*psi1*psi2*sig1*sig2 + (psi2*sig2)^2. Whenever reduced-form innovations correlate contemporaneously (rho <> 0), ordering Venue 1 first in a Cholesky factorization attributes shared covariance to Venue 1 (Upper Bound), while ordering Venue 1 second strips it out via (1 - rho^2) (Lower Bound).

Verify by hand with permanent impact weights psi1 = 0.70 (B2), psi2 = 0.30 (B3), innovation SDs sig1 = 0.10 (B4), sig2 = 0.10 (B5), and correlation rho = 0.60 (B6): psi1*sig1 = 0.07, psi2*sig2 = 0.03, and var_rw = 0.0049 + 0.00252 + 0.0009 = 0.00832. Venue 1’s Upper Bound is (0.07 + 0.03*0.60)^2 / 0.00832 = 0.007744 / 0.00832 = 0.9308, while its Lower Bound is 0.0049*(1 - 0.36) / 0.00832 = 0.3769, yielding midpoint IS = 0.6538.

Hasbrouck Midpoint Information Share Formula

Guard |rho| >= 1 or non-positive random-walk variance before computing the Cholesky bounds.

=LET(p_1, B2, p_2, B3, s_1, B4, s_2, B5, rho_v, B6, a_1, p_1 * s_1, a_2, p_2 * s_2, var_rw, a_1^2 + 2 * rho_v * a_1 * a_2 + a_2^2, IF(OR(ABS(rho_v) >= 1, s_1 <= 0, s_2 <= 0, var_rw <= 0), "Degenerate Covariance", LET(is_up, (a_1 + rho_v * a_2)^2 / var_rw, is_lo, (a_1^2 * (1 - rho_v^2)) / var_rw, ROUND(0.5 * (is_up + is_lo), 4))))

Bound-Width Diagnostic & Zero-Correlation Convergence Check

Track the Cholesky ambiguity spread IS1_upper - IS1_lower. When timestamp resolution eliminates contemporaneous correlation (set B6 = 0), upper and lower bounds collapse identically to 0.0049 / 0.0058 = 0.8448 with bound width = 0.

=LET(a_1, B2 * B4, a_2, B3 * B5, rho_v, B6, var_rw, a_1^2 + 2 * rho_v * a_1 * a_2 + a_2^2, ROUND(((a_1 + rho_v * a_2)^2 - a_1^2 * (1 - rho_v^2)) / var_rw, 4))

Further reading

• Arte Sheets Lesson — Kyle’s Lambda Order-Flow Price Impact and Market Depth
• Arte Sheets Lesson — Corwin-Schultz High-Low Two-Day Bid-Ask Spread Estimator