OHLC-Vol

A Python toolkit for computing OHLC-based volatility estimators, studying their statistical properties, and evaluating their ability to forecast future realized volatility using non-negative least squares (NNLS) regression.

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Overview

Daily OHLC (Open, High, Low, Close) prices contain more information about intraday volatility than close-to-close returns alone. This project implements a suite of classical OHLC estimators, compares them via in-sample regressions and walk-forward out-of-sample evaluation, and experiments with several functional forms for the vol-on-vol forecasting relationship.

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Volatility estimators

Column	Estimator
vol_adj_cc_ann	Adjusted close-to-close
vol_cc_ann	Unadjusted close-to-close
vol_parkinson_ann	Parkinson (1980)
vol_gk_ann	Garman–Klass (1980)
vol_rs_ann	Rogers–Satchell (1991)
vol_yz_ann	Yang–Zhang (2000)
vol_meilijson_ann	Meilijson (2011)
vol_ht_ann	Hodges–Tompkins (2002) (bias-corrected CC)
vol_tr_ann	True Range
vol_on_rs_ann	Overnight + Rogers–Satchell
neg_ret_adj_cc	Negative adjusted return (leverage proxy)
neg_ret_adj_cc_pos	max(neg_ret_adj_cc, 0) (down-day leverage)

All estimators are annualized (multiplied by √252) and expressed as percentage volatility.

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Regression models

Four functional forms are supported for predicting h-day-ahead realized volatility from linearly-weighted moving averages (LWMA) of past estimators:

Model	Specification	Notes
level	vol = c₀ + Σ cⱼ·xⱼ	Linear NNLS; fastest
sqrt-var	vol = √(c₀ + Σ cⱼ·xⱼ²)	Non-linear NLS; slowest
log-vol	log(vol) = c₀ + Σ cⱼ·log(xⱼ)	Log-log NNLS + Duan smearing
var-space	vol² = c₀ + Σ cⱼ·xⱼ²	Linear NNLS on squared quantities + ratio smearing

R², residual statistics, and BIC are reported on the vol scale for all four models, enabling direct comparison. Non-negative constraints ensure forecasts are always non-negative. Predictors are selected via a t-statistic pruning loop and an optional one-per-group filter that retains only the best LWMA window for each estimator.

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Files

Library modules

File	Purpose
ohlc_vol.py	Compute all OHLC vol estimators from raw OHLC data
ohlc_io.py	Read and write the multi-symbol OHLC CSV format
vol_analysis.py	Summary stats, ACF tables, correlation matrices, predictor builders (lag, MA, LWMA), forward vol target (RMS)
nnls_reg.py	NNLS regression engine: fitting, pruning, SE, BIC, and the four model types

Analysis scripts

Script	Purpose
xohlc_vol.py	Main analysis: in-sample NNLS regressions for each estimator and forecast horizon; regression model comparison summary
xohlc_vol_oos.py	Walk-forward out-of-sample evaluation; OOS R², RMSE, MAE, bias per model and symbol
xohlc_vol_measures.py	Lightweight script: compute and display vol measures with summary stats and ACF; optionally write to CSV
xohlc_vol_var.py	Extends xohlc_vol.py with VAR models (OLS and NNLS, full and restricted)
xreturn_stats.py	Return and vol summary statistics and autocorrelations; useful for comparing original vs. resampled data
xresample_ohlc.py	Bootstrap resampler: draws daily OHLC rows with replacement to create a null dataset that destroys temporal autocorrelation while preserving marginal distributions
vector_autoreg.py	VAR fitting library used by xohlc_vol_var.py: OLS and NNLS VAR estimation, companion matrix, RMSE and IC tables

Data

File	Description
prices_ohlc.csv	Multi-symbol daily OHLC prices (SPY, QQQ, GLD, USO, HYG; 2010–2026)

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Input data format

The CSV uses a two-row header: the first row repeats the ticker symbol for each field; the second row contains field names (Open, High, Low, Close, Adj Close). The first column is the date index. This is the format produced by yfinance when downloading multiple tickers.

         SPY                          QQQ          ...
         Open   High    Low  Close  Adj Close  Open ...
2010-01-04  ...

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Usage

Compute vol measures and print summary statistics

python xohlc_vol_measures.py
python xohlc_vol_measures.py --symbol GLD --output gld_vol.csv

In-sample regression analysis

# All symbols, default horizons (1, 5, 21 days)
python xohlc_vol.py

# Single symbol, 21-day horizon, SPY as external predictor
python xohlc_vol.py --symbol QQQ --horizons 21 --external-symbols SPY

# Predict each measure only from its own past values (baseline)
python xohlc_vol.py --same-measure --horizons 21

# Write vol measures to CSV
python xohlc_vol.py --output vol_measures.csv

Walk-forward out-of-sample evaluation

# Default: level model, all symbols, horizons 1 5 21, annual refit
python xohlc_vol_oos.py

# Compare level and var-space models, monthly refit
python xohlc_vol_oos.py --models level var-space --step 21

# Same-measure baseline OOS
python xohlc_vol_oos.py --same-measure --horizons 21

# With external SPY predictors
python xohlc_vol_oos.py --external-symbols SPY --horizons 21

Resampling null check

# Create resampled prices and compare return statistics
python xresample_ohlc.py --input prices_ohlc.csv --output prices_ohlc_resampled.csv
python xreturn_stats.py --file prices_ohlc_resampled.csv

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Key options

Most options are available as both a constant at the top of each script (for persistent defaults) and as a command-line argument (for one-off overrides).

xohlc_vol.py

Option	Default	Description
--symbol SYM	all	Analyse one symbol
--horizons H [H ...]	1 5 21	Forecast horizons in trading days
--external-symbols SYM [...]	none	Add another symbol's vol measures as predictors
--same-measure	off	Restrict predictors to the same estimator as the target
--output FILE	none	Write vol measures to CSV

xohlc_vol_oos.py

Option	Default	Description
--symbol SYM	all	Analyse one symbol
--horizons H [H ...]	1 5 21	Forecast horizons
--models MODEL [...]	level	Models: level sqrt-var log-vol var-space
--min-train N	504	Minimum training days (expanding window)
--step N	252	Refit interval in trading days
--external-symbols SYM [...]	none	External predictor symbols
--same-measure	off	Restrict predictors to the same estimator

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Dependencies

numpy
pandas
scipy

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Selected findings (SPY, QQQ, GLD, USO, HYG; 2010–2026)

• Level model wins in-sample and OOS for equity-like assets (SPY, QQQ, HYG, USO). The variance-based models (sqrt-var, var-space) do not consistently improve on the simpler linear specification.
• GLD is an exception: variance-based models show a meaningful in-sample improvement (R² +0.03), consistent with gold volatility having more symmetric dynamics than equities.
• OOS R² is 3–10 points below in-sample R², a typical degradation from predictor selection.
• 5-day horizon forecasts best on average (OOS R² ≈ 0.45); 1-day forecasts are hardest (R² ≈ 0.25).
• External SPY predictors help USO OOS but hurt GLD and HYG, suggesting the cross-asset vol relationship is non-stationary.
• Same-measure baseline (predicting vol from its own past only) is substantially weaker than pooling all OHLC estimators, confirming that the other estimators add genuine predictive value.