Direct evidence for inversion formula in multifractal financial volatility measure

TL;DR

This paper empirically verifies the inversion formula for conservative multifractal measures using high-frequency financial data, demonstrating its validity in turbulent financial markets.

Contribution

It provides the first empirical test of the inversion formula in real financial systems, confirming its applicability to market volatility measures.

Findings

Both measures exhibit multifractal properties.

The inversion formula holds in financial market data.

Scaling ranges are consistent for direct and inverse measures.

Abstract

The inversion formula for conservative multifractal measures was unveiled mathematically a decade ago, which is however not well tested in real complex systems. In this Letter, we propose to verify the inversion formula using high-frequency turbulent financial data. We construct conservative volatility measure based on minutely S&P 500 index from 1982 to 1999 and its inverse measure of exit time. Both the direct and inverse measures exhibit nice multifractal nature, whose scaling ranges are not irrelevant. Empirical investigation shows that the inversion formula holds in financial markets.