First-passage horizons in horizontal visibility graphs: a rank-invariant estimator of path roughness for rough volatility models
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918483709657088 |
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| author | Sikorski, Michał |
| author_facet | Sikorski, Michał |
| contents | 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 τ+(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 Rényi's record statistic and
implying infinite mean and variance. Hence roughness is estimated on a
power-law survival scale through a single tail exponent θ. Combining the
identity L+ = τ+ with discrete-grid persistence theory for fractional
Brownian motion gives the prediction θ(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 θ(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 θÌ‚ = 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02352 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | First-passage horizons in horizontal visibility graphs: a rank-invariant estimator of path roughness for rough volatility models Sikorski, Michał Statistical Finance Computational Finance General Finance Trading and Market Microstructure 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 τ+(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 Rényi's record statistic and implying infinite mean and variance. Hence roughness is estimated on a power-law survival scale through a single tail exponent θ. Combining the identity L+ = τ+ with discrete-grid persistence theory for fractional Brownian motion gives the prediction θ(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 θ(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 θÌ‚ = 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. |
| title | First-passage horizons in horizontal visibility graphs: a rank-invariant estimator of path roughness for rough volatility models |
| topic | Statistical Finance Computational Finance General Finance Trading and Market Microstructure |
| url | https://arxiv.org/abs/2512.02352 |