Price range & distribution

A candlestick reduces thousands of trades to four numbers: open, high, low, close. This compression discards the path — and the path contains information that the endpoints don't.

The case for distributional analysis

The idea that the distribution of prices within a bar carries information beyond OHLC goes back to the range-based volatility estimators. Parkinson (1980) showed that the high-low range is a more efficient estimator of volatility than close-to-close returns, because it uses more of the available price information. Garman & Klass (1980) extended this with an estimator that incorporates open and close alongside the range, achieving near-theoretical efficiency bounds.

But these classical estimators still reduce the distribution to a scalar. The full distributional shape — its skewness, kurtosis, and multi-modal structure — tells you about the process that generated the bar's price action.

What we compute

From all trade prices within an interval:

• Mean and median trade price — where trading actually concentrated (mean is pulled by outlier prints; median is robust)
• Standard deviation — intra-bar dispersion, directly related to Parkinson's range-based volatility
• Skewness — asymmetry of the price distribution within the bar
• Excess kurtosis — tail behavior relative to a Gaussian reference

The high-low range is computed in both absolute terms and basis points for cross-asset comparability.

Why distributional shape is a regime signal

Negative skewness in a bullish bar. Price closed up, but the distribution of all trades was tilted toward the lower half of the range. Buyers only won at the close; most of the bar saw trading below the midpoint. This is the microstructure signature of a late-session squeeze — potentially a closing auction effect or a short-covering rally — rather than sustained buying throughout the bar.

High kurtosis (leptokurtic). Most trades clustered tightly around a central price, with a few outlier prints at the extremes. This is an acceptance pattern: the market found a value area and consolidated there, probing the edges but snapping back quickly. Andersen et al. (2001) showed that intraday returns exhibit substantially higher kurtosis than daily returns, suggesting that tail events are concentrated in short intervals — exactly what intra-bar kurtosis captures.

Low kurtosis (platykurtic). Price swept continuously through the range without concentrating at any level. This is a trending regime: no price level attracted enough two-sided flow to establish a temporary equilibrium.

The combination of these distributional statistics with OHLC creates a much richer characterization of each bar — the kind of decomposition that gives machine learning models the raw material to distinguish regime types that look identical in price data alone.

References:

• Parkinson, M. (1980). "The Extreme Value Method for Estimating the Variance of the Rate of Return." Journal of Business, 53(1), 61–65.
• Garman, M.B. & Klass, M.J. (1980). "On the Estimation of Security Price Volatilities from Historical Data." Journal of Business, 53(1), 67–78.
• Andersen, T.G., Bollerslev, T., Diebold, F.X. & Ebens, H. (2001). "The Distribution of Realized Stock Return Volatility." Journal of Financial Economics, 61(1), 43–76.