Lee and Ready’s trade classification

Trade records do not say who initiated a trade. Charles Lee and Mark Ready (1991) signed trades above the midquote as buys and below it as sells, used the tick test for trades at the midquote, and compared each trade with the quote of five seconds earlier: on the NYSE the quote revision a trade caused was often time-stamped before the trade itself, so the contemporaneous quote had already moved toward the trade price.

import matplotlib.pyplot as plt
import numpy as np

from financialkit.market_microstructure import align_quotes, lee_ready, quote_rule, tick_test

rng = np.random.default_rng(2)

A market with early-stamped quote updates

Trades arrive every 15 seconds on average, 15% of them at the midquote. Each one moves the midquote by 4 cents in its direction, plus noise, and the new quote is recorded two seconds before the trade that caused it. Without the delay, the quote rule compares trades with a quote that has already moved past them.

n, half = 3000, 0.05
t = np.cumsum(3.0 + rng.exponential(12.0, n))
side = rng.choice([-1.0, 1.0], n)
step = 0.04 * side + 0.02 * rng.standard_normal(n)
mid_before = 20 + np.concatenate([[0.0], np.cumsum(step)[:-1]])
at_mid = rng.random(n) < 0.15
price = np.where(at_mid, mid_before, mid_before + side * rng.uniform(0.3, 1.0, n) * half)
mid_after = mid_before + step
quote_t = np.concatenate([[0.0], t - 2.0])
quote_mid = np.concatenate([[mid_before[0]], mid_after])

results = {}
for delay in (0.0, 5.0):
    b, a = align_quotes(t, quote_t, quote_mid - half, quote_mid + half, delay=delay)
    results[f"quote rule, {delay:g}s"] = quote_rule(price, b, a)
    results[f"Lee-Ready, {delay:g}s"] = lee_ready(price, b, a)
results["tick test"] = tick_test(price)

for name, est in results.items():
    print(f"{name:17s} correct {np.mean(est == side):.1%}")

quote rule, 0s    correct 29.9%
Lee-Ready, 0s     correct 29.9%
quote rule, 5s    correct 85.6%
Lee-Ready, 5s     correct 91.5%
tick test         correct 81.0%

Accuracy by rule

fig, ax = plt.subplots(figsize=(7, 3.5))
ax.barh(list(results), [np.mean(e == side) for e in results.values()], color=["C0", "C2", "C0", "C2", "C1"])
ax.set_xlim(0.4, 1.0)
ax.set_xlabel("fraction of trades correctly signed")
fig.tight_layout()

plt.show()