Most Efficient Homogeneous Volatility Estimators

TL;DR

This paper develops a comprehensive theory for creating the most efficient homogeneous volatility estimators using OHLC prices, applicable to various stochastic processes, and introduces quasi-unbiased estimators with explicit analytical forms for Wiener processes.

Contribution

It introduces a unified framework for optimal homogeneous volatility estimators based on OHLC data, including the derivation of explicit formulas for Wiener processes with drift.

Findings

New estimators outperform Garman-Klass and Roger-Satchell in efficiency.

Explicit analytical expressions derived for Wiener processes with drift.

The framework is adaptable to different stochastic processes.

Abstract

We present a comprehensive theory of homogeneous volatility (and variance) estimators of arbitrary stochastic processes that fully exploit the OHLC (open, high, low, close) prices. For this, we develop the theory of most efficient point-wise homogeneous OHLC volatility estimators, valid for any price processes. We introduce the "quasi-unbiased estimators", that can address any type of desirable constraints. The main tool of our theory is the parsimonious encoding of all the information contained in the OHLC prices for a given time interval in the form of the joint distributions of the high-minus-open, low-minus-open and close-minus-open values, whose analytical expression is derived exactly for Wiener processes with drift. The distributions can be calculated to yield the most efficient estimators associated with any statistical properties of the underlying log-price stochastic process. Applied to Wiener processes for log-prices with drift, we provide explicit analytical expressions for the most efficient point-wise volatility and variance estimators, based on the analytical expression of the joint distribution of the high-minus-open, low-minus-open and close-minus-open values. The efficiency of the new proposed estimators is favorably compared with that of the Garman-Klass, Roger-Satchell and maximum likelihood estimators.