Returns & volatility

Does price tend to continue or reverse? This is the most consequential question for any directional strategy. Rather than inferring the answer from backtests of momentum strategies (which conflate the signal with execution, risk management, and survivorship), you can measure it directly from the return series.

Return autocorrelation: the first-order test

The autocorrelation of returns at lag 1 is the most direct measure of short-horizon momentum or mean reversion. Lo & MacKinlay (1988) documented significant positive weekly return autocorrelation in US equities — evidence against the random walk hypothesis and the foundation for contrarian and momentum strategies.

At the microstructure level, return autocorrelation has a different character. Roll (1984) showed that the bid-ask bounce mechanically induces negative first-order autocorrelation in trade-to-trade returns: a buy at the ask is followed by either another buy (no price change) or a sell at the bid (negative return), creating a systematic negative serial correlation equal to the squared half-spread.

We compute lag-1 autocorrelation per interval from trade-to-trade log returns. The interpretation:

• Positive autocorrelation → momentum regime. Price moves are self-reinforcing. Consistent with informed order flow creating persistent impact (Lillo & Farmer, 2004) or algorithmic trend-following amplifying directional moves.
• Negative autocorrelation → mean-reversion regime. Price moves are self-correcting. Dominated by the bid-ask bounce (Roll, 1984) and market maker inventory management (Ho & Stoll, 1981). Positions based on recent direction face adverse selection.
• Near-zero autocorrelation → the random walk benchmark. No exploitable short-horizon serial dependence.

Trendiness: measuring path efficiency

$\text{Trendiness} = \left|\sum r_i\right| / \sum |r_i|$

This ratio measures the efficiency of price movement — how much of the total distance traveled was "useful" (in the direction of the net move). It's related to the variance ratio test (Lo & MacKinlay, 1988) but operates on a single bar rather than across multiple horizons.

• Trendiness = 1.0 — price moved in a straight line. Every individual trade-to-trade return pointed in the same direction. Perfect trend.
• Trendiness near 0 — price ended roughly where it started despite substantial total movement. Pure oscillation.
• Intermediate values — partial trend with noise.

Combined regime classification

The two metrics together provide a richer regime characterization than either alone:

| Autocorrelation | Trendiness | Regime | Strategy Implication | | --------------- | ---------- | -------------------------- | -------------------------------------------------------------- | | High positive | High | Sustained directional flow | Trend-following profitable | | Low/negative | Low | Choppy oscillation | Mean-reversion or market-neutral | | Low | High | Gap/jump regime | Price jumped rather than drifted — single large displacement | | High positive | Low | Oscillating drift | Momentum in micro-returns but frequent reversals — noisy trend |

This characterization captures within-bar price dynamics that OHLC completely discards. A bar with identical OHLC values can be a clean trend (high autocorrelation, high trendiness) or a choppy fight that happened to close at the high (low autocorrelation, moderate trendiness). The distinction matters for everything from volatility estimation to strategy selection.

References:

• Lo, A.W. & MacKinlay, A.C. (1988). "Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test." Review of Financial Studies, 1(1), 41–66.
• Roll, R. (1984). "A Simple Implicit Measure of the Effective Bid–Ask Spread in an Efficient Market." Journal of Finance, 39(4), 1127–1139.
• Ho, T. & Stoll, H.R. (1981). "Optimal Dealer Pricing Under Transactions and Return Uncertainty." Journal of Financial Economics, 9(1), 47–73.
• Lillo, F. & Farmer, J.D. (2004). "The Long Memory of the Efficient Market." Studies in Nonlinear Dynamics & Econometrics, 8(3).