First-passage horizons in horizontal visibility graphs: a rank-invariant estimator of path roughness for rough volatility models

TL;DR

This paper introduces a novel, rank-invariant estimator based on horizontal visibility graphs to measure path roughness in time series, with applications to financial volatility modeling.

Contribution

It develops a new estimator for path roughness using horizon analysis in HVGs, validated through theoretical predictions and extensive Monte Carlo simulations.

Findings

Estimator accurately recovers the roughness parameter H for rough volatility models.

The method distinguishes rough Bergomi volatility from classical models.

Applied to VIX data, the estimator indicates significant roughness in financial volatility.

Abstract

Horizontal visibility graphs (HVGs) encode the ordinal structure of time series and provide graph-local summaries of path topology. This article introduces L+(t), the forward visibility horizon at node t, with finite-sample terminal non-crossings treated as right-censored observations. For paths without ties, each uncensored L+(t) is identical to the first-passage time {\tau}+(t) = inf{k ≥ 1 : x_{t+k} ≥ x_t}. For an i.i.d. sequence with a continuous distribution, the survival law is exactly Pr[L+ ≥ k] = 1/k, equivalent to R\'enyi's record statistic and implying infinite mean and variance. Hence roughness is estimated on a power-law survival scale through a single tail exponent {\theta}. Combining the identity L+ = {\tau}+ with discrete-grid persistence theory for fractional Brownian motion gives the prediction {\theta}(H) = 1 − H. For rough Bergomi-type volatility, the same prediction is derived under an explicit persistence hypothesis for Riemann–Liouville fBm increments and verified numerically. In Monte-Carlo experiments (N = 10,000, T = 2^16), a Hill-MLE with Clauset–Shalizi–Newman threshold selection recovers {\theta}(H) within one cross-replicate standard deviation for H ≤ 0.2 and reveals a positive finite-size bias for smoother paths. The rank-invariant, parameter-free estimator separates rough Bergomi volatility from classical Heston, GARCH, and FIGARCH benchmarks. Applied to daily FRED VIX data from 2000–2026, the rolling estimate is {\theta}̂ = 0.91 ± 0.19 across 45 four-year windows and lies far below an overlapping-window i.i.d. Monte-Carlo null (p < 0.001). The statistic offers an ordinal diagnostic of roughness for financial volatility and other complex time-series systems.