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

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.