Volatility Modeling#
Volatility modeling sits at the intersection of financial econometrics and practical risk management. From Engle’s ARCH model through the HAR cascade to modern deep learning approaches, the field has progressively improved its ability to forecast the volatility that directly determines position sizes, risk capital allocations, and option prices. A central stylized fact — that volatility clusters, mean-reverts, and exhibits long memory — constrains the class of useful models and explains why simple GARCH(1,1) remains competitive with far more complex alternatives.
This collection spans the core modeling traditions: the ARCH/GARCH family and its extensions for conditional heteroskedasticity, the Heterogeneous Autoregressive (HAR) model as the standard realized-volatility benchmark, neural-network approaches from LSTM to convolutional architectures, regime-switching models including Markov-switching GARCH for Bitcoin, and stochastic volatility frameworks. Practical entries cover log-return normalization for options pricing and the volatility paradox in systematic crypto trading. The connecting theme is that volatility forecasting is high-stakes: getting it wrong by a factor of two has a direct and proportional impact on strategy returns.
Related topics include Risk Management for volatility-targeting position sizing frameworks, Market Regimes for regime-switching volatility models, and Portfolio Construction for covariance estimation and portfolio volatility control.
For complete worked examples of trading strategies and individual factors, see Strategy Examples and Factor Examples.
Volatility-Managed Portfolios#
Alan Moreira and Tyler Muir (Journal of Finance, 2017) show that managed portfolios which scale down exposure when volatility is high and scale up when volatility is low produce large alphas, higher Sharpe ratios, and large utility gains for mean-variance investors. They document the effect across the market factor, value, momentum, profitability, return on equity, investment, and betting-against-beta factors, as well as the currency carry trade. The mechanism is simple: changes in volatility are not offset by proportional changes in expected returns, so volatility timing adds value even without forecasting returns.
Our summary: this is the canonical reference for the idea that cutting risk in high-volatility regimes improves risk-adjusted performance almost universally across equity and macro factors. What makes the result powerful is its breadth — the same volatility-scaling overlay helps nearly every standard factor portfolio. For practitioners, the paper justifies the common CTA and risk-parity practice of volatility targeting as an ex-post optimal behaviour, and it is the natural companion to the Kim-Tse-Wald paper on TSMOM and volatility scaling, which asks how much of trend-following alpha is really the vol-scaling overlay rather than the direction signal.
Key metrics: volatility-managed versions of the market, value, momentum, and other factor portfolios produce statistically significant alphas relative to their unmanaged counterparts, with material improvements in Sharpe ratios. The effect is robust across the full sample (1926–2015 for equity factors) and across international markets and asset classes.
A Simple Approximate Long-Memory Model of Realized Volatility#
Fulvio Corsi proposes the Heterogeneous Autoregressive model of Realized Volatility (HAR-RV), an additive cascade of volatility components defined over daily, weekly, and monthly horizons. The model captures the empirically observed long-memory behavior of realized volatility through a parsimonious structure — a constrained AR model with three regressors: yesterday’s daily RV, last week’s average daily RV, and last month’s average daily RV. Despite having no true long-memory mechanism, the HAR-RV successfully reproduces slow hyperbolic decay in the autocorrelation function, fat tails, and multi-scaling properties of financial returns. Published in the Journal of Financial Econometrics, Vol. 7, No. 2, 2009.
Our summary: this is the single most important baseline paper for anyone forecasting realized volatility. The HAR-RV is the benchmark that every subsequent model — HARNet, regime-switching HAR, machine-learning volatility forecasters — must beat. Its power comes from the Heterogeneous Market Hypothesis insight that traders operating at different frequencies (daily, weekly, monthly) create a cascade of volatility components. The model’s practical appeal is that it requires only simple OLS estimation, is trivially reproducible, and still outperforms many far more complex alternatives.
Key metrics: the HAR-RV model outperforms standard AR, ARFIMA, and GARCH models in out-of-sample realized-volatility forecasting for the S&P 500 across daily, weekly, and monthly horizons.
A Multiple Regime Smooth Transition Heterogeneous Autoregressive Model for Long Memory and Asymmetries#
Michael McAleer and Marcelo C. Medeiros extend the HAR-RV model to a nonlinear multiple-regime framework using logistic smooth-transition functions. The resulting model simultaneously captures long-memory behavior through the HAR cascade structure and asymmetric, nonlinear volatility dynamics through regime-dependent coefficients. The paper develops a sequence of specification tests to determine the number of regimes and presents an estimation procedure for the smooth-transition HAR model. Published in the Journal of Econometrics, Vol. 147, No. 1, 2008.
Our summary: this is the first systematic extension of HAR to nonlinear regimes, and it answers whether adding regime-dependent behavior to HAR improves on the linear version. The smooth-transition mechanism allows model coefficients to change gradually as a function of an observable transition variable, which is more realistic than abrupt switching for capturing how markets transition between calm and turbulent states.
Key metrics: the multiple-regime smooth-transition HAR model improves in-sample fit relative to the linear HAR baseline, particularly for capturing sign and size asymmetries in volatility dynamics.
Forecasting Realised Volatility Using Regime-Switching Models#
Yi Ding, Dimos S. Kambouroudis, and David G. McMillan extend standard AR and HAR models for realised-volatility forecasting to include nonlinearity through smooth transition and Markov-switching approaches. Using daily data for eight international stock markets over 2007-2021, the paper provides a comprehensive comparison using both statistical forecast evaluation metrics and economic (risk management) metrics including Value-at-Risk and Expected Shortfall. Published in the International Review of Economics & Finance, Vol. 101, 2025.
Our summary: this is the most up-to-date comparison of regime-switching approaches applied to HAR-style realized-volatility forecasting. The headline finding is nuanced: regime-switching models do improve both in-sample fit and out-of-sample forecasts, but the out-of-sample gains are less clear-cut at the daily horizon and become stronger at longer horizons. The paper also finds that Markov switching is preferred to smooth transition.
Key metrics: regime-switching HAR models outperform linear HAR in-sample and generally out-of-sample, with Markov-switching preferred over smooth transition. Daily-horizon gains are less clear-cut than weekly and monthly.
Realized Volatility Forecasting with Neural Networks#
Andrea Bucci compares feed-forward and recurrent neural-network architectures for realized-volatility forecasting, with particular emphasis on LSTM and NARX models against more traditional econometric baselines. The paper argues that volatility is a natural fit for neural methods because the target exhibits both long memory and nonlinear dependence, and evaluates whether those properties can be captured more effectively by recurrent architectures than by classical linear or semiparametric alternatives.
Our summary: this is one of the cleaner early references showing that neural nets can be genuinely useful for volatility forecasting rather than just fashionable. The important point is not simply that an LSTM beats a benchmark once, but that architectures designed to retain long-range temporal information consistently help when the volatility process becomes more unstable.
Key metrics: recurrent neural networks outperform the traditional econometric competitors in realized-volatility forecasting, with LSTM and NARX delivering the strongest results. The advantage is especially visible in highly volatile periods.
A Machine Learning Approach to Volatility Forecasting#
Kim Christensen, Mathias Siggaard, and Bezirgen Veliyev study realized-variance forecasting for Dow Jones Industrial Average constituents using a broad set of machine-learning methods, including regularized regressions, tree models, and neural networks, benchmarked against the HAR family. A key design choice is that the ML models are not given an unfair advantage through exhaustive tuning: the authors deliberately keep implementation simple and then ask whether generic ML still improves materially on the standard realized-volatility toolkit.
Our summary: the paper is useful because it shows that the gains from ML do not have to come from exotic feature sets or aggressive optimisation. Even with only daily, weekly, and monthly realized-variance lags, ML competes strongly and often wins, which suggests the main edge comes from learning a richer persistence structure than linear HAR models can express.
Key metrics: ML beats the HAR lineage even when the predictor set is restricted to daily, weekly, and monthly realized-variance lags, and the gains become larger at longer forecast horizons.
Volatility Forecasting with Machine Learning and Intraday Commonality#
Chao Zhang, Yihuang Zhang, Mihai Cucuringu, and Zhongmin Qian forecast intraday realized volatility by pooling information across stocks and explicitly exploiting commonality in intraday volatility together with a market-volatility proxy. The paper moves beyond single-series forecasting and tests whether cross-sectional structure can be used to train models that generalize not just within a stock, but across previously unseen names as well.
Our summary: this is one of the more practically interesting volatility papers because it treats volatility as a partially shared phenomenon rather than a completely separate process for each asset. That framing matters in production settings where you want a single model to transfer across a universe instead of maintaining one bespoke forecaster per stock.
Key metrics: neural networks outperform linear regressions and tree-based models for intraday realized-volatility forecasting and the gains remain robust when applied to stocks absent from the training set.
HARNet: A Convolutional Neural Network for Realized Volatility Forecasting#
Rafael Reisenhofer, Xandro Bayer, and Nikolaus Hautsch introduce HARNet, a convolutional architecture designed as a structured deep-learning extension of the classic HAR model. The model uses a hierarchy of dilated convolutions so the receptive field grows quickly without a comparable blow-up in parameters, and it is initialized so that before training it reproduces the corresponding HAR forecast exactly, making the transition from econometric baseline to neural refinement unusually transparent.
Our summary: HARNet is compelling because it does not ask practitioners to abandon the HAR intuition that has worked well for realized-volatility forecasting. Instead it starts from HAR, then adds learnable nonlinear filters on top of the same multi-horizon logic. That makes it one of the better examples of a finance-aware deep architecture: the gains come from extending a sensible domain prior rather than replacing it with an unconstrained network.
Key metrics: across three stock-market indexes, HARNet improves meaningfully on its HAR baselines in realized-volatility forecasting. Code and data available at GitHub.
DeepVol: Volatility Forecasting from High-Frequency Data with Dilated Causal Convolutions#
Fernando Moreno-Pino and Stefan Zohren propose DeepVol, a dilated-causal-convolution model that forecasts next-day volatility directly from raw high-frequency intraday data rather than relying on precomputed realized measures alone. The paper is motivated by the idea that once the data are compressed into handcrafted realized-volatility features, some predictive information may already have been discarded, especially in the shape and timing of intraday volatility bursts.
Our summary: this is the closest match to the “raw input” version of volatility forecasting. The main contribution is not just a different architecture, but the end-to-end stance that the model should learn directly from intraday sequences instead of from manually aggregated volatility proxies.
Key metrics: using two years of NASDAQ-100 intraday data, DeepVol delivers more accurate day-ahead volatility forecasts and more accurate risk measures than traditional benchmarks. A one-day receptive field with 5-minute sampling gives the best overall forecasting accuracy.
Comparing Deep Learning Models for the Task of Volatility Prediction Using Multivariate Data#
Wenbo Ge, Pooia Lalbakhsh, Leigh Isai, Artem Lensky, and Hanna Suominen compare several deep-learning forecasters for multivariate volatility prediction across five assets: the S&P 500, NASDAQ 100, gold, silver, and oil. The paper benchmarks multilayer perceptrons, recurrent networks, temporal convolutional networks, and the Temporal Fusion Transformer against GARCH-style baselines, with the goal of understanding whether newer sequence architectures actually justify their added complexity.
Our summary: the value of this paper is comparative rather than methodological. It is useful when you already accept that deep models can forecast volatility and instead want to know which class of architecture tends to work best once you move from univariate to multivariate inputs.
Key metrics: the Temporal Fusion Transformer delivers the strongest overall performance among the tested deep-learning models, with temporal convolutional architectures typically next best.
Introducing NBEATSx to Realized Volatility Forecasting#
Hugo Gobato Souto and Amir Moradi apply NBEATSx, a neural basis expansion model with exogenous variables, to daily stock realized-volatility forecasting over multiple horizons. They compare the architecture with LSTM, TCN, HAR, GARCH, and GJR-GARCH models across six stock indexes, three error measures, four statistical tests, and several robustness checks, making the paper unusually systematic for a single-model introduction.
Our summary: this is one of the stronger “modern forecasting architecture” papers in the volatility literature because the evaluation is broad and the results are reported in both accuracy and robustness terms. NBEATSx is attractive here not simply because it is newer than LSTM or TCN, but because its basis-expansion structure appears to work well when volatility persistence and exogenous effects both matter.
Key metrics: across six stock indexes, NBEATSx produces statistically more accurate and more robust forecasts than the competing models. On average, 13% and 8% better accuracy for medium- and long-term forecasts, plus robustness improvements of 43%, 60%, and 59% for short-, medium-, and long-term horizons respectively.
The Hybrid Forecast of S&P 500 Volatility Ensembled from VIX, GARCH and LSTM Models#
Natalia Roszyk and Robert Ślepaczuk compare four approaches to forecasting S&P 500 volatility: a standalone GARCH model, a standalone LSTM, a hybrid LSTM-GARCH specification, and a hybrid model that also incorporates the VIX. The paper uses daily S&P 500 and VIX data from January 3, 2000 through December 21, 2023 and asks whether combining a classical volatility model with a neural sequence model and an implied-volatility signal produces a more useful risk forecast.
Our summary: the paper is a good example of the hybrid direction in current volatility research. Rather than framing ML and econometrics as mutually exclusive choices, it treats them as complementary components: GARCH supplies a disciplined volatility structure, LSTM adds nonlinear sequence modelling, and VIX contributes a market-implied state variable.
Key metrics: over the 2000-2023 sample, the hybrid LSTM models outperform the standalone GARCH benchmark, and adding VIX further improves forecasting performance beyond the plain LSTM-GARCH combination.
Realized Volatility and HAR Models: A New Paradigm for Volatility Forecasting#
A comprehensive guide to realized volatility and the Heterogeneous Autoregressive (HAR) model for volatility forecasting. The article explains how high-frequency intraday data can be used to construct realized volatility measures that are more precise than traditional close-to-close estimators, and how the HAR framework captures the multi-scale, long-memory properties of volatility through a hierarchical cascade of daily, weekly, and monthly components.
The post covers key extensions of the base HAR model, including handling jumps (HAR-J), leverage effects, and realized semi-variance decomposition. It also discusses practical considerations such as microstructure noise, kernel-based estimators, and model evaluation using QLIKE and MSE loss functions. A Python implementation is provided for fitting HAR models and generating forecasts.
By Simone Menaldo.
Mentioned by Ralph Sueppel in this discussion.
Using Log Returns and Volatility to Normalize Strike Distances#
A foundational tutorial on why log returns matter in financial and derivatives modeling. The article starts from first principles, explaining how the constant e represents continuous compounding and how the natural logarithm measures the time needed to reach a certain level of growth. This mathematical foundation is then applied to understanding why upside and downside price moves are not symmetric in compounded return space.
The practical application focuses on normalizing option strike distances. For a $100 stock, $150 and $50 are not equidistant in the world of compounding — $150 is closer. The article shows how to compute equivalent distances using log returns and extends this to normalizing for volatility, allowing traders to compare strikes across different assets with different volatility profiles.
By Kris Abdelmessih.
The Volatility Paradox: How Lower Risk Produces Superior Returns in Crypto#
Research Article #58 from Trading Research Hub. Examines the counterintuitive relationship between risk and return in cryptocurrency markets. The study explores how systematic approaches that target lower volatility can actually achieve superior risk-adjusted returns, challenging the traditional finance assumption that higher risk always correlates with higher returns.
The article presents a systematic trading framework that demonstrates how strategic risk management generates substantial risk-adjusted returns in crypto. By deliberately targeting lower portfolio volatility through position sizing and signal filtering, the strategy achieves better overall outcomes than unconstrained approaches, illustrating the volatility paradox in practice.
By Pedma.
Stochastic Volatility Models#
An in-depth exploration of stochastic volatility models, which offer a more sophisticated and realistic framework for capturing market dynamics than simpler constant-volatility assumptions. The article covers the foundational concepts of the basic stochastic volatility model, the widely used Heston model, and methods for their estimation and calibration.
The post explains how stochastic volatility models treat volatility as a latent random process driven by its own source of uncertainty, allowing them to capture key features of financial markets such as volatility clustering, mean reversion, and the leverage effect. Applications to derivatives pricing, risk management, and portfolio optimization are discussed.
By Simone Menaldo.
Understanding Multivariate GARCH Models#
An overview of Multivariate GARCH models for modeling and forecasting the volatility of multiple assets simultaneously. The article covers the progression from the foundational VEC and BEKK models to the more advanced Constant Conditional Correlation (CCC) and Dynamic Conditional Correlation (DCC) frameworks, with practical applications in portfolio risk management.
The post explains how these models capture time-varying correlations and volatility spillovers between assets, which are critical for portfolio construction, hedging, and risk measurement. The trade-offs between model flexibility and computational tractability are discussed, with the DCC model highlighted as the most practical choice for large-dimensional systems.
By Simone Menaldo.
Volatility Modeling: A Deep Dive into the ARCH Family#
A comprehensive exploration of the ARCH family of volatility models, from the original ARCH model by Robert Engle to its many extensions. The article begins with the stylized facts of financial returns that motivate these models, including volatility clustering, fat tails, and the leverage effect, then covers ARCH, GARCH, EGARCH, GJR-GARCH, and IGARCH specifications.
The post provides detailed mathematical formulations, explains estimation via maximum likelihood, and discusses model selection and diagnostic testing. Practical considerations for applying these models to financial risk management, Value-at-Risk estimation, and options pricing are covered, with emphasis on choosing the right specification for different financial applications.
By Simone Menaldo.
Why Your Volatility Forecast Matters More Than You Think#
Concretum Research, working with academics Walter Distaso and Antonio Mele, introduces NextVoL, a volatility forecasting framework designed for volatility targeting strategies. The core thesis builds on Charlie Munger’s observation about compounding: investor behavior during volatility spikes, not lack of returns, interrupts long-term wealth accumulation. Volatility targeting mitigates this by scaling exposure inversely with predicted volatility. The article walks through why the forecasting methodology matters as much as the targeting mechanism itself, and introduces three innovations: short-horizon smoothing to reduce noise from transient spikes, dynamic overnight adjustment accounting for regime-dependent variance composition, and stabilization mechanisms for persistent deviation correction.
Testing seven forecasting approaches on SPY data from 2008 to 2025, NextVoL achieves 10.11% realized volatility against a 10% target, a volatility-of-volatility of just 0.26 (nearly half that of competitors), and a Sharpe ratio of 0.79 (highest among tested methods). It ranks first or second in 16 of 18 calendar years. Notably, the VIX-based approach systematically underdelivers by approximately 22% due to the embedded volatility risk premium, making it a poor foundation for targeting despite its popularity.
By Concretum Research.
Forecasting Realized Volatility in Turbulent Times using Temporal Fusion Transformers#
Johannes Frank (Friedrich-Alexander University Erlangen-Nuremberg, 2023) analyses the performance of temporal fusion transformers in forecasting annualised weekly and monthly realized volatility (RV) of 355 individual stocks listed in the S&P 500 from January 2000 to June 2022. The TFT is trained on three different feature sets: company-specific data only (F1), extended with S&P 500 index returns and VIX (F2), and further augmented with macroeconomic variables including the effective federal funds rate, 30-year Treasury yield, Economic Policy Uncertainty index, credit spreads, and sectoral employment data (F3). Models are trained using three approaches — individual (per-stock), sectoral pooling, and overall pooling — and benchmarked against LSTM, Random Forest, and GARCH(1,1)/GARCH-X models using RMSE and Diebold-Mariano tests.
Our summary: this is one of the first rigorous applications of TFT to realized volatility forecasting for individual equities at scale. The headline finding is that TFT significantly outperforms LSTM, Random Forest, and GARCH benchmarks, with the advantage becoming strongest under pooling. The most important practical insight is that sectoral and overall pooling substantially improve all ML models — the data-hungry TFT architecture particularly benefits from the larger training samples that pooling provides. The best-performing configuration uses the F2 feature set (company data plus S&P 500 returns and VIX) with overall pooling on weekly data, achieving mean RMSE of 0.0699. The inclusion of macroeconomic variables (F3) consistently reduces forecast errors relative to company-only features (F1) but does not reliably beat F2.
Data: Refinitiv daily stock data for 355 S&P 500 constituents (January 2000-June 2022), 5,660 trading days. 70/30 train-test split with 10% validation holdout. Macroeconomic data from FRED and PolicyUncertainty.com. No public code repository. FAU Discussion Papers in Economics 03/2023.
Key metrics: TFT with overall pooling and F2 features achieves mean RMSE 0.0699 (weekly) and 0.1003 (monthly). Relative RMSE versus TFT: Random Forest 1.58-1.63x and LSTM 1.19-1.27x on weekly data with pooling, confirming TFT superiority. TFT consistently beats GARCH(1,1) and GARCH-X regardless of training approach and frequency.
Volatility Calibration: Structural Rigidity vs. Local Adaptivity#
Alexander Nevolin discusses the central tension in volatility surface calibration: balancing structural rigidity against local adaptivity. Too much rigidity suppresses genuine market signals, while excessive flexibility absorbs microstructure noise. The post contrasts two approaches: rigid prior geometry from classical parametric models (SSVI/SABR), which create globally smooth, low-dimensional surfaces where the admissible manifold acts as a filter preventing unwanted deformations; and flexible prior geometry from arbitrage-constrained deep learning, which relaxes geometric restrictions while maintaining no-arbitrage constraints, allowing higher-dimensional deformations.
Nevolin categorizes observed market data into three types — genuine outliers rejected by both approaches, inference-sensitive anomalies handled differently by each method, and prior-sensitive local propagation where flexibility introduces non-local consequences. He advocates for hybrid modeling that preserves analytical geometry as a backbone while learning residual deformations, referencing two recent papers: “Meta-Learning Neural Process for Implied Volatility Surfaces with SABR-induced Priors” (Zhuang & Wu, 2025) and “A Geometry-Aware Residual Correction of Hagan’s SABR Implied Volatility Formula” (Reghai, Tarsissi, Biau & Lipton, 2026).
By Alexander Nevolin.
Meta-Learning Neural Process for Implied Volatility Surfaces with SABR-induced Priors#
Jirong Zhuang and Xuan Wu (University of Macau) propose two principles for implied volatility surface (IVS) reconstruction: a meta-learning framework that trains across multiple trading days to learn a procedure mapping sparse option quotes to complete surfaces via conditional prediction, eliminating daily recalibration; and structural priors via transfer learning by pre-training on SABR-generated synthetic data then fine-tuning on historical market data. Their model, Volatility Neural Process (VolNP), uses an attention-based Neural Process architecture to produce a complete IVS from a sparse context set in a single forward pass.
Our summary: VolNP is a compelling example of the hybrid direction in volatility surface modeling — combining the structural discipline of a classical parametric model (SABR) with the flexibility of neural networks. The SABR pre-training acts as a physics-informed prior that regularises the network in data-sparse regions (long maturities, deep out-of-the-money strikes), while the meta-learning formulation across trading days means the model learns a general calibration procedure rather than memorising a single day’s surface. The result is a method that generalises better to new market conditions without the fragility of daily recalibration.
Data: SPX options data. Published on arXiv (September 2025, arXiv:2509.11928).
Key metrics: on SPX options, VolNP outperforms SABR, SSVI, and Gaussian process baselines. The SABR-induced prior reduces RMSE by approximately 40% relative to an ablation trained only on market data and suppresses large errors, with pronounced gains at long maturities where quotes are sparse.
Mentioned by Alexander Nevolin in this discussion.
A Geometry-Aware Residual Correction of Hagan’s SABR Implied Volatility Formula#
Adil Reghai, Lama Tarsissi, Gérard Biau, and Alex Lipton present a hybrid approach combining Hagan’s classical SABR analytical approximation with a neural network that learns the residual error. Rather than replacing the analytical formula entirely, the network is trained to correct higher-order effects absent from asymptotic expansions. The key innovation is augmenting the neural network input with geometric features derived from the SABR stochastic differential equations, maintaining interpretability while capturing effects that analytical approximations miss.
Our summary: this paper exemplifies the hybrid paradigm that Alexander Nevolin advocates — preserving the analytical backbone (Hagan’s SABR formula) while learning residual deformations with a lightweight neural network. The geometry-aware input representation is the critical design choice: by feeding the network features derived from the SDE structure rather than raw parameters, the model converges faster and generalises better across parameter regimes, including stressed scenarios where pure analytical approximations break down. The lightweight correction is designed to be suitable for real-time trading applications where latency matters.
Data: numerical experiments across realistic and stressed SABR parameter domains. Published on arXiv (May 2026, arXiv:2605.06604). 33 pages, 17 figures.
Key metrics: the geometry-aware residual correction demonstrates improved accuracy and robustness compared to both pure analytical methods and standard neural network approaches across realistic and stressed parameter domains.
Mentioned by Alexander Nevolin in this discussion.
On the Persistence of Market Sentiment: A Multifractal Fluctuation Analysis#
Schadner applies multifractal detrended fluctuation analysis (MF-DFA) to U.S. and European stock market sentiment derived from financial option data and surveys. The paper measures sentiment the way Mandelbrot measured prices (see A Multifractal Model of Asset Returns in Time Series Forecasting) and finds the same fingerprint: long memory, fat tails, and pronounced multifractality.
The key asymmetry: investor fear is associated with overreaction (anti-persistent sentiment), while optimism drifts closer to a random walk. The expected Hurst exponent fluctuates strongly, implying nonlinear dynamics in return expectations. The paper argues that while heavy-tailed distributions are known to destabilise markets, this nonlinear behaviour is a potentially greater amplifier of market meltdowns — crashes are more structured than rallies. The multifractal spectrum width is substantial in both markets, confirming that no single persistence parameter characterises sentiment dynamics. Published in Physica A, Volume 581, 2021.
The Implied Volatility Edge: Why a Structural Edge Is Rarely a Constant Edge#
Gilberto Pellerano builds a real-time stochastic differential equation (SDE) engine in Python to visualise a hard truth about quant finance: having a mathematically verifiable edge does not guarantee a smooth, predictable P&L. The simulation contrasts a baseline pure Geometric Brownian Motion path (the Black-Scholes world of continuous prices and constant volatility) against a “real market” path fed the same Gaussian shocks but injected with two harsh realities — stochastic volatility that clusters into calm and stressed regimes, and Poisson jumps implemented as Merton jump-diffusion with a martingale compensator so jumps add genuine variance without dragging price to zero.
Because of those jumps and fat tails, the market demands a Volatility Risk Premium (VRP): implied volatility prices higher than realized volatility on roughly 70-80% of days, so systematically selling volatility to harvest the spread is a true structural edge. The post’s central lesson comes from the short-vol P&L panel: weeks of slowly grinding upward collecting premium are punctuated by violent drawdowns when a Poisson jump triggers, giving back large chunks of profit in seconds. The edge is real over a large sample but aggressively discontinuous in the short term — surviving the jumps and sizing the book so a non-constant edge does not wipe you out is what actually pays. The environment is built entirely in Python (pygame for 60fps interactive rendering, math/random for the SDE core, imageio/threading for async MP4 encoding) rather than static charting libraries.
By Gilberto Pellerano.
OTM Crypto Options Performance Screener: The “Lottery Ticket” Experiment#
Pawel Lachowicz, PhD (CEO of Halcyon Waters) runs a continuous, live experiment screening cheap out-of-the-money (OTM) crypto options on Deribit and publishes the realised results on his Cayo Largo dashboard. In derivatives trading, cheap OTM options with high convexity per dollar are sometimes called “lottery tickets” for their asymmetric risk profile: a small fixed loss (the premium) against a large potential gain from gamma repricing or volatility expansion. Every option shown is an expired contract scored across six dimensions — gamma efficiency, vega efficiency, volatility regime, catalyst strength, cheapness, and liquidity — so there are no forward-looking signals, only outcomes. The framing question is whether the probability of winning this “OTM lottery” is better or worse than a classical 6-from-49 lotto.
The screener data (cayolargo.fi/options-otm-performance, BTC-focused with ETH, SOL, XRP, AVAX and TRX also tracked), captured here, covers 244 expired options with 51 winners of 2x or more, a 20.9% overall win rate, and a best return of 93.33x. The outcome distribution is the core of the lottery-ticket thesis: 30 JACKPOT outcomes (avg 45.17x), 4 BIG WINs (avg 5.71x), 17 WINNERs (avg 2.97x), 44 BREAKEVEN (avg 1.31x) and 149 LOSSes (avg 0.65x) — i.e. roughly 61% of trades lose money, but the rare jackpots are large enough to carry the aggregate payoff. The dealer-gamma regime is the key conditioning variable: in the entry-versus-exit regime transition matrix, near-flip (“FLIP”) entries win 21% (31/148), positive-gamma entries win 28% (11/40), and negative-gamma entries only 16% (9/56). The regime breakdown shows the dominant winning cohort is NEAR_FLIP / NORMAL-IV / FAIR-VRP (135 candidates, 21% win rate, 8.96x average return), while the NEGATIVE_GAMMA / NORMAL / FAIR cohort had a lower 15% win rate but a still-high 5.18x average return. Daily performance is heavily clustered: late-May to early-June produced the outsized payoffs (1 Jun: 50% win rate, 43.30x average, 93.33x best; 31 May: 64% win rate, 38.33x average; 30 May: 58% win rate, 25.04x average), whereas much of mid-June was flat-to-negative. The dashboard also charts score-versus-return, dimension importance, the days-to-expiry sweet spot, the optimal OTM-distance range, a rolling 7-day win rate, coin beta comparison, and touched-strike (gamma repricing versus intrinsic value).
These aggregate figures sit slightly below the more optimistic headline numbers in the launch post, which cited a win rate reaching 35% for options expiring within 10–15 days and BTC jackpot winners with a 46.3x factor — a reminder that the dataset is live and the read changes with the sample and the recent regime.
By Pawel Lachowicz, PhD.
Mentioned by Pawel Lachowicz, PhD in this LinkedIn discussion: cheap OTM crypto options on Deribit behave like lottery tickets, the win rate reaches 35% for options expiring within 10–15 days, the near-flip gamma regime powers most winning trades, and BTC records jackpot winners with a 46.3x factor.
Variance Swaps: Trading Commodity Volatility Directly#
Vitor Gaspar (commodity trader, derivatives and hedging) lays out a practitioner’s primer on variance swaps as a way to trade commodity volatility directly, without routing the view through options. Most commodity vol exposure is still expressed via options — buy a straddle, delta-hedge, and collect the gap between implied and realized volatility — but that path is noisy: strike selection, time decay, and delta-hedging costs all leak P&L away from the pure volatility view. A variance swap removes the option layer entirely: one leg pays the realized variance of the underlying over a period, the other pays a fixed strike, and the long-variance side profits when realized variance exceeds the strike.
The post is organised as “five things to understand.” (1) What it is — a direct swap on realized variance with no options and no delta hedging. (2) Why it is cleaner than a straddle — a straddle is a vega bet contaminated by gamma, theta, skew dynamics, and hedging cost, whereas the variance swap isolates the variance exposure exactly, with no strike to choose and no delta to hedge. (3) Where the liquidity is — concentrated in deep energy markets (Brent and WTI crude, Henry Hub and TTF natural gas), with limited but real liquidity in gold and copper and almost none in agricultural commodities, where vol exposure stays in straddle form. (4) The tail risk — because variance is the sum of squared returns, a single large move contributes disproportionately, so short-variance positions carry effectively unbounded exposure to one big move; the author cites the 2020 oil collapse and the 2022 Russia gas dislocation as events that drove variance-swap losses far beyond any reasonable strike. (5) Who uses them — institutional vol sellers (as an alternative to listed straddle strategies) and some energy-macro hedge funds, but rarely corporates, for whom the operational complexity outweighs the benefit (“the structure exists for the trading floor, not the treasury floor”).
His bottom line is that the variance swap is the cleanest instrument for a pure volatility view but also carries the most asymmetric tail: the clean vega exposure cuts both ways, because there is nothing else in the P&L to dilute a bad outcome. It is worth knowing for traders who run vol systematically, while those who trade vol opportunistically are usually better served by options. He closes by asking practitioners where they set the tail cap and how they size around it.
By Vitor Gaspar.
In the discussion, ex-Goldman Sachs exotics trader Eric Barthe adds an important precision: the variance swap is “the cleanest instrument to express a pure variance view, but not a pure vol view” — variance and volatility differ by a convexity term, so the payoff is linear in variance rather than in volatility itself. Taylor Lukof (ABR Dynamic Funds) cautions against the counterparty risk embedded in a tail-event product — “the worst thing would be to not get paid your insurance when you need it most.”
Normalizing volatility transform for Normal implied volatility#
The Normal (Bachelier) volatility model is the de facto market standard for quoting implied volatilities in fixed-income markets, where negative interest rates make the log-normal Black-Scholes quoting convention break down. Vladimir Lucic (Imperial College London) and Parviz Rakhmonov (Marex) revisit static arbitrage in this Normal implied volatility model and introduce the normalizing volatility transform (NVT), defined on a single maturity slice as f(k) = k / σ(k), where k = K - F_T is the moneyness (strike minus forward) and σ(k) is the Normal implied volatility. The construction is the Bachelier-model analogue of Masaaki Fukasawa’s (2012) normalizing transformation of the log-normal implied-volatility smile, and the paper links the two settings explicitly.
The authors restate call and put prices through the NVT and prove a sequence of arbitrage characterisations. First, a slice that is free of vertical-spread arbitrage has a monotonically increasing NVT f(k) (Theorem 2.1), mirroring Fukasawa’s log-normal result. Second, for large-strike (wing) arbitrage they derive an exact asymptotic condition — f(k)²/2 + 3·ln f(k) - ln k → +∞ as k → +∞ on the call wing, with a symmetric condition on the put wing as k → -∞ (Theorem 3.1). A finding the authors emphasise is that, unlike the log-normal case, the Normal model needs an extra condition on put option prices to rule out large-strike arbitrage. Third, they show a slice is free of strike arbitrage if and only if it is simultaneously free of butterfly and large-strike arbitrage (Theorem 3.2), constructing the implied risk-neutral distribution from the digital-put price, which serves as a CDF. Finally, at the surface level they prove that the absence of calendar arbitrage is equivalent to the total Normal implied volatility σ(k,T) being non-decreasing in maturity T for every moneyness — a notable contrast with the log-normal world, where it is total variance that must be non-decreasing. An arbitrage-free Normal volatility surface is then defined as one that is free of calendar arbitrage with every slice free of strike arbitrage, giving practitioners the exact conditions needed to build arbitrage-free Bachelier volatility surfaces.
This is a short (7-page) derivatives-pricing theory paper rather than an empirical trading study, so it reports no backtest performance statistics — there is no annualised return, Sharpe ratio, maximum drawdown or win rate. Its results are the four arbitrage theorems above, supported by appendix bounds on Normal call and put prices derived from the Komatsu and Gordon inequalities for the Mills’ ratio. There is no accompanying dataset or code release; the work is self-contained and reproducible from the proofs, building on Fukasawa (2012), Lucic (2021, SSRN abstract 3835233) and Choi, Kwak, Tee & Wang (2022) on the Bachelier model.
Mentioned by Parviz Rakhmonov in this LinkedIn discussion, where he explains that the paper bridges the gap by introducing the normalizing volatility transform and linking it naturally to the volatility transforms introduced earlier by Masaaki Fukasawa in the log-normal setting, investigating large-strike arbitrage conditions in detail and showing that the Normal case requires an extra condition on put option prices.
Detecting Structural Evolution of Implied Volatility Surface Using Gradient-Based Features#
This Aalto University master’s thesis (subtitle: A Machine Learning Approach to Market Regime Detection) develops a model-free methodology for detecting and interpreting structural changes in the implied volatility (IV) surface over time. Rather than fitting a parametric surface model, Shakhin Dzhafarov characterises the surface through local gradients — the partial derivatives of implied volatility with respect to moneyness and maturity — and quantifies how those gradients change from day to day at a grid of points on the surface. The daily gradient-change vectors, together with the concurrent move in the underlying index, become the features fed into an unsupervised clustering algorithm.
The clustering uses HDBSCAN (Hierarchical Density-Based Spatial Clustering of Applications with Noise), chosen because it does not require the number of clusters to be specified in advance and, crucially, explicitly labels points as noise. This matches the economic intuition of the problem: most days the volatility surface merely wiggles, and only occasionally does it undergo a genuine structural transformation. The clusters that survive correspond to recognizable, recurring dynamics — specific changes in volatility skew and in the term structure across different maturity and moneyness levels — giving practitioners a flexible, entire-surface framework for monitoring and interpreting regime shifts in options markets.
The study uses implied volatilities of S&P 500 index options across several moneyness and maturity levels spanning 2007 to 2025. It is an academic thesis, and no accompanying code repository is published.
As a detection and characterisation study rather than a trading backtest, the thesis reports no annualised return, Sharpe ratio, drawdown, or win rate. Its quantitative results are structural: the great majority of daily IV-surface changes are classified as noise, reflecting how infrequently true structural shifts occur, so the identified clusters are small — with sizes of up to 18 daily observations — yet each captures a distinct and interpretable skew or term-structure regime.
Mentioned by Nam Nguyen (Quantitative Strategist and Derivatives Specialist) in this LinkedIn discussion — later reposted by Sahand Haji Ali Ahmad — where he summarises the paper as a regime-classification approach built on the entire volatility surface, highlighting that most daily changes are noise while several recurring, meaningfully interpretable patterns can be detected.
A devilish question for option sellers: Which VRP is higher?#
Kris Abdelmessih (Moontower) poses a deceptively simple question to option sellers — between two names, which has the higher volatility risk premium (VRP)? — and uses it to expose how much the answer depends on how you measure the premium and, more fundamentally, on the difference between volatility and variance. The structural belief that implied volatility sits above realized volatility (so options are “overpriced” and the edge is in selling them) is granted, but the article argues that turning that belief into risk-adjusted, opportunity-cost-aware position sizing is far subtler than the usual “harvest the VRP” framing suggests.
Using HYG (a low-vol high-yield bond ETF) as a running example, the piece contrasts two ways to quote the premium: as a ratio (IV/RV), which eases cross-sectional comparison but flatters low-vol names, and as a spread (IV − RV), which compares edge more fairly. The deeper point is that a short option position’s P&L is governed by variance, not volatility: the net P&L is approximately ½·Γ·S²·(σ²_RV − σ²_IV) per day, so it is the gap in squared vols that pays. Combined with the at-the-forward approximations that theta scales with σ while gamma scales with 1/σ, this means a low-vol name simultaneously has less theta and more gamma — the two forces in the “gamma-theta tug of war”.
The consequence is that theta-neutral sizing does not deliver equal edge. To collect the same daily theta on a 10-vol name versus a 50-vol name you trade 5× the contracts, but to equalize the variance edge you would need 25× — the square of the theta ratio. Abdelmessih derives how rich a low-vol name must be to match: a 10-vol name would need to trade near 17.9 vol (a 79% ratio premium) to equal a 50-vol name priced at a 20% premium. The practical takeaways are that variance is the “V” that matters, that ratio screens mislead by making structurally disadvantaged low-vol names look rich, that vol-point spreads are a sounder ranking tool (though they too break down at very low vols), and that harvesting low-vol premium is costlier because it demands far more contracts and more gamma hedging. His HYG conclusion is counterintuitive: its options are genuinely rich versus realized vol, yet on a variance-edge basis a 2-3 point spread is actually cheap next to a 50-vol name trading at 60.
By Kris Abdelmessih (Moontower), published as an article on X (originally published April 2026).
Faster Volatility Estimators and the Gross-versus-Net Trap#
DeltaDog (@the_delta_dog), a discretionary trader working his way into systematic “going quant” territory, shares a progress note from building the volatility component of a risk model on daily data. He tests a family of estimators — a mix of slow and fast exponentially weighted moving averages (EWMAs) of squared log returns, plus high-low range combinations — and finds that the faster, shorter-lookback estimators consistently win. His representative example is sqrt(ewma(diff(log(close), 1)^2, 10)): the square root of an EWMA (span ~10) of one-day squared log returns, i.e. a short, highly responsive realized-volatility estimate. A fast estimator reacts quickly to changing conditions and adjusts position sizes sooner, whereas a slow estimator smooths volatility over a longer window and lags regime changes.
The important caveat is the author’s own follow-up about the difference between gross and net. In backtesting, gross refers to the strategy’s performance (returns, PnL, or the resulting equity-curve volatility) computed before trading frictions, while net is the same quantity computed after subtracting transaction costs — commissions, bid-ask spread, and slippage. Faster volatility estimators re-size positions more often, so they generate higher turnover and should therefore lose more to costs on a net basis. DeltaDog flags that in his chart the net line tracks the gross line almost perfectly — “my cost and execution assumptions is a joke when net is tracking gross like this. its just a chart” — meaning his modelled costs are so low that turnover is barely penalised, which is exactly why the high-churn fast estimators keep winning even after costs. When net stops tracking gross, the fast estimators’ turnover disadvantage would begin to bite.
This is the crux of the follow-up discussion and the “fast always wins” remark. pedma (@pedma7) replies that in his own work “faster always wins” too, but questions whether that will hold once costs are realistic; he is considering deliberately “taking the hit” and blending in slower estimators anyway for robustness, or adding a rolling adjustment as the strategy matures. DeltaDog agrees he is tempted to try “something fancier” but doubts it is worth it for now. The exchange captures a recurring quant dilemma: a more responsive signal that looks strictly better in-sample may owe its apparent edge to under-modelled trading costs, so a practitioner may deliberately choose a slower, more cost-tolerant estimator as insurance for the day realistic frictions are applied.
Shared by DeltaDog (@the_delta_dog) in this discussion on X.
Gammega at +3.5σ: When Spot and IV Pull Apart#
Cayo Largo Research documents a BTC options episode in which Gammega, a fourth-order Greek defined as the second derivative of Gamma with respect to implied volatility, rose sharply while spot rallied and 30-day ATM IV drifted lower. The note uses Cayo Largo’s 37-Greek heatmap and Gammega time-series for the BTC ATM 8-30d Deribit cohort to show that higher-order surface stress appeared before it became visible in the first-order Gamma exposure most desks monitor.
The key observation is the sequencing: Gammega began lighting up around 18:20 UTC on May 23, 2026, Gamma followed roughly five hours later, and the signal persisted into May 25 as Gammega moved beyond +3.5 sigma. The article argues that spot and IV moving in opposite directions can bend the gamma surface through cross-derivative effects even without an obvious volatility shock. Zomma, Zomanna, and Gamma are used to explain why the stress showed up in Gamma’s vol-sensitivity rather than in the delta-vol coupling, making Gammega an early warning signal for gamma instability in crypto options.
By Cayo Largo Research.
Realized Wavelet-Based Estimation of Integrated Variance and Jumps in the Presence of Noise#
Baruník and Vácha (Quantitative Finance, 2015; arXiv 1202.1854) build a realized-variance estimator using smooth wavelets and the Maximal Overlap Discrete Wavelet Transform (MODWT) inside the two-scale realized variance framework, so it uses all available ticks and stays feasible under microstructure noise. The decomposition simultaneously isolates jumps and splits realized variation into arbitrary investment horizons.
Our summary: this is the most rigorous existing recipe for scale-decomposed realized variance, and it is directly transplantable to perpetual-futures volatility estimation. The method is validated in simulation across noise levels, jump sizes, and long-memory fractional stochastic-volatility models, and applied to forex-futures volatility through the financial crisis. Its practical value for a fee-sensitive book is that it hands you jump-robust variance measured per horizon, which is the right input for scale-band volatility features rather than a single aggregate realized variance.
Forecasting Realized Volatility Using HAR Models and Wavelet Decomposition: A Volatility-Timing Perspective#
Adam Clements (Queensland University of Technology) and Puneet Vatsa (University of the Sunshine Coast; North American Journal of Economics and Finance 83, 102605, March 2026) wavelet-decompose realized volatility into short-, medium- and long-term components, fit a separate HAR model to each, and then judge the forecasts inside a volatility-timing portfolio rather than on statistical loss. The decomposition uses the Maximum Overlap Discrete Wavelet Transform with a Haar filter under reflecting boundary conditions — a detail that matters, since MODWT is shift-invariant and reflecting boundaries are the standard mitigation for the endpoint problem that invalidates many wavelet forecasting results.
Our summary: the headline is that the low-frequency component wins, and the reason it wins is more interesting than the fact. HAR-L does not produce lower portfolio volatility than plain HAR — it produces materially lower turnover at comparable risk, and that is where the utility gain comes from. Volatility forecasts drive portfolio weights, so a forecast that jitters forces the book to churn; stripping the high-frequency component removes revisions that were never worth acting on. This is the volatility-side analogue of the discipline this collection favours: measure the economic edge, not the error. The authors state the corollary explicitly — the forecasts delivering the greatest welfare gains may not minimise conventional statistical loss functions, so a horse race won on RMSE or QLIKE can select the wrong model for a fee-sensitive book. The finding is robust across two designs that usually break such results: forecasting portfolio weights directly rather than volatility, and substituting jump-robust volatility estimates. Attempting to exploit the medium-frequency component does not help, so this is not a general “more decomposition is better” claim — it is specifically the slow band that carries the actionable content, which matches Bandi et al. on the pricing side and the L1-filter result on the trend side.
Data and reproduction: realized volatility for the 20 largest equities in the S&P 500 from 31 January 2000 to 31 December 2024, giving 6,166 daily observations. Evaluation uses a utility-based framework in the manner of Fleming et al. (2001, 2003) and Bollerslev et al. (2018), assuming quadratic utility with constant relative risk aversion γ, with results reported at 1-day and 5-day horizons. Significance is assessed with a bootstrap using 999 samples and an average block length of 22 trading days. All inputs are standard published series; no code release is advertised. Published open access.
Key metrics: results are reported as pairwise utility gains from switching between forecasts rather than as strategy returns, so there is no annualised return, Sharpe ratio or drawdown figure. The substantive numbers are that HAR-L consistently delivers higher investor utility than benchmark HAR at the 5% significance level, achieves lower portfolio turnover relative to HAR, and produces comparable portfolio volatility — the gain is turnover reduction at unchanged risk, not risk reduction. Medium-frequency forecasts do not improve on HAR.
Enhancing Volatility Prediction: A Wavelet-Based Hierarchical Forecast Reconciliation Approach#
Clements et al. (Journal of Forecasting, 2026) apply hierarchical forecast reconciliation across a wavelet decomposition hierarchy, so that scale-level forecasts aggregate coherently up to the total.
Our summary: this addresses the missing piece whenever you forecast each scale band separately and then sum them — without reconciliation, the per-band forecasts can silently contradict the aggregate forecast, which is a real hazard for the “one model per scale” designs that recur throughout this collection. Reconciliation forces coherence between the band-level and total-level predictions. The entry is catalogued from the bibliographic record (Wiley full text was inaccessible), so the author list and detailed results are unverified.
Volatility Forecasting with Machine Learning: A Horse Race Across GARCH, HAR, and Tree-Based Models#
Akram Khan (2026) runs a controlled horse race between seven volatility-forecasting configurations on S&P 500 realized volatility from January 2004 through November 2025: three GARCH variants (GARCH, EGARCH, GJR-GARCH), the HAR-RV model of Corsi (2009), gradient-boosted trees (LightGBM, XGBoost), and an equal-weight ensemble of LightGBM, HAR-RV and GARCH(1,1). Every model sees the same data, the same 35-feature panel and the same expanding-window protocol, evaluated over a January 2022 to November 2025 test period (980 trading days) that spans the 2022 bear market, the 2023 rally, the 2024–2025 rate normalisation and a 19.4% drawdown year. QLIKE is the primary loss, alongside RMSE and Mincer–Zarnowitz R², with Diebold–Mariano two-sided p-values for all key pairs.
Our summary: this is a disciplined, reproducibility-first study whose main message is a null result — no model separates from the field at conventional significance. On the full sample the equal-weight ensemble has the lowest point-estimate QLIKE (0.3431) and GJR-GARCH is the best individual model (0.3447), but the ensemble’s edge over GJR-GARCH is not statistically significant (DM p = 0.90). The more interesting finding is a subperiod reversal: in the structurally out-of-distribution 2022 high-volatility regime the parametric GARCH family wins (EGARCH best at QLIKE 0.2346) while LightGBM ranks last, whereas in the calmer 2023–2025 period the tree models lead and HAR-RV is last — the GARCH variants adapt more gracefully to a regime their 2004–2021 training never saw, while the 35-feature trees do better once conditions normalise. LightGBM split-importance shows VIX is the dominant feature, the leverage effect enters through three return-based features (~19% of split count together), and long-horizon RV signals dominate short-horizon ones.
Data and reproduction: S&P 500 (^GSPC) and VIX daily OHLCV from Yahoo Finance, 2004–Nov 2025; a 35-feature engineered panel with forward targets. The paper ships full LaTeX source, data-collection and modelling scripts, per-day forecasts of all seven models, and an end-to-end audit that re-derives every paper number, released under permissive licences (CC BY 4.0 paper, CC0 data, MIT code) at the companion repository. It self-scores 2 on the Reproducibility Disclosure Score rubric of Khan (2026), “Machine Learning in Quantitative Finance: A Systematic Review” (SSRN 6562398).
Key metrics (full test sample, QLIKE): Ensemble 0.3431, GJR-GARCH 0.3447, XGBoost 0.3553, LightGBM 0.3632, EGARCH 0.3748, GARCH(1,1) 0.3806, HAR-RV 0.4198; GJR-GARCH has the best RMSE (0.0734) and MZ R² (0.3802). Subperiod QLIKE: 2022 EGARCH 0.2346 (best) vs LightGBM 0.3429 (worst); 2023–2025 XGBoost 0.3693 (best) vs HAR-RV 0.4568 (worst). Ensemble-vs-GJR-GARCH DM p = 0.90 (not significant).
Mentioned by Vivek V Rao (@VivekVRao1) in this discussion, who highlights the paper’s finding that “no model separates from the field at conventional significance levels” and adds that he has found NAGARCH — which shifts the news-impact curve — to outperform GJR-GARCH, which instead twists it so that squared returns from negative returns boost volatility more than from positive returns.
By Akram Khan.
The asymmetric relationship between returns and implied higher moments: Evidence from the crude oil market#
Xinxin Zhang, Elie Bouri, Yahua Xu, and Gongqiu Zhang (Energy Economics, 2022) investigate whether returns have an asymmetric relationship with changes in option-implied volatility, skewness, and kurtosis in the crude-oil market. Using United States Oil Fund (USO) option data, they estimate the implied higher moments and use quantile regressions to allow the relationship to vary with the state of the market and with investor heterogeneity.
The results distinguish the three moments rather than treating “the leverage effect” as a single phenomenon. Negative returns have a stronger effect than positive returns on implied volatility and kurtosis, particularly in their upper quantiles. For implied skewness, positive returns dominate at lower quantiles: contemporaneous positive returns are followed by greater skewness risk, whereas more negative returns are followed by less. This makes the paper a useful empirical benchmark for any option-surface workflow that recovers and studies higher risk-neutral moments.
Data and reproduction: the study uses USO ETF option data; the publisher page does not provide a data or code release, and the closed-access full text was not downloadable for this collection. PyDerivatives, a related research toolkit, is explicitly positioned by its maintainer as a replication and extension of this analysis, with option-implied density and moment tools alongside broader pricing-kernel functionality.
Key metrics: this is an econometric relationship study, not a tradable-strategy backtest, so it reports no annualised return, Sharpe ratio, maximum drawdown, or win rate. Its key empirical results are the signs and quantile-dependent strengths of the return–implied-moment relationships described above.
On Volatility-Managed Portfolios: The Role of Conditional Factor Correlations#
Pedro Barroso (Católica-Lisbon), Paulo F. Maio (Hanken School of Economics), and Haoxu Wang (ISEG Lisbon) reassess Moreira and Muir (2017) on an updated and unified sample. They rebuild the same nine volatility-managed equity factors — market, size, value, momentum, operating profitability, investment, and the q-factor investment and ROE factors, plus betting-against-beta — over a common 1967:2–2024:12 window, scaling each factor by the inverse of its realized variance and running spanning regressions of the scaled factor on the raw factor (and on the raw factor plus FF3). Three things differ from the original study: a single common sample period for all factors so they are directly comparable, three- and six-month realized variances alongside the one-month window, and a Bonferroni correction for testing nine factors at once. At the individual-factor level only three managed factors survive — momentum, ROE, and betting-against-beta. At the portfolio level the picture reverses: every mean-variance-efficient (MVE) combination of factors benefits from volatility scaling, including the Fama-French five-factor MVE portfolio whose individual constituents mostly do not. The authors trace this gap to time-varying correlations between factors.
Our summary: this is a targeted correction to the Moreira and Muir entry at the top of this page, and it relocates where a practitioner should expect volatility scaling to earn its keep. The single-factor breadth of the original result does not survive a unified sample with multiple-testing control and a no-leverage constraint: the scaled market and value factors, which were profitable in Moreira and Muir, are no longer significant, and the internet appendix shows this is not a post-2015 artefact — the same holds when the sample is truncated at Moreira and Muir’s own end date. Much of the remaining single-factor profitability comes from leverage. Capping the scaling weight at one, which matters because a weight above one implies heavy shorting of the loser leg, roughly halves the alphas for momentum, RMW, IA, ROE, and BAB, and kills RMW entirely under longer realized-variance windows. What survives is the portfolio result, and the mechanism the authors isolate is the paper’s real contribution. Decomposing MVE realized variance into individual factor variances and a correlation matrix, they rerun the strategy with the conditional correlation matrix replaced by the identity matrix (zero correlations) and by the full-sample unconditional matrix. Both ablations gut the FF5 result while leaving the shape of the strategy otherwise intact, which means the alpha is not coming from timing factor volatility at all — it comes from cutting exposure when the factors become more correlated with each other. That reframes volatility targeting on a multi-factor book as a correlation-management tool in disguise, and it matches how such books actually fail: not because one sleeve gets noisy, but because the sleeves stop diversifying each other at the worst moment. A useful diagnostic falls out of the FF6 result — FF6 keeps its alpha even with correlations shut down, and since FF6 is just FF5 plus momentum, its gains come from momentum rather than from the correlation channel.
Data and reproduction: all inputs are standard published series and freely available. Daily and monthly returns for ROE and IA come from Lu Zhang’s data library, BAB from AQR’s website, and the remaining factors from Kenneth French’s data library, over 1967:2–2024:12 (the start date is set by ROE and IA availability). Realized variances use 21, 63, and 126 daily returns for the one-, three-, and six-month windows; MVE weights are computed full-sample from the covariance matrix and mean returns, so the portfolio construction is in-sample by design and is not presented as a live trading rule. Statistical inference uses heteroskedasticity-robust (HC1) standard errors. No code release accompanies the paper, but the construction is fully specified in equations and the ablations are simple matrix substitutions, so replication is straightforward.
Key metrics: at the individual-factor level with unconstrained leverage and one-month realized variance, annualized alphas are momentum 10.78% (t = 6.33), BAB 9.34% (t = 7.91), ROE 5.25% (t = 5.69), RMW 2.76% (t = 3.23), and IA 1.74% (t = 2.59), while market (1.52%, t = 1.02), size (−1.01%, t = −1.00), value (1.02%, t = 0.93), and CMA (0.44%, t = 0.65) are insignificant. Imposing no leverage cuts these to momentum 3.94%, BAB 3.71%, ROE 2.30%, RMW 0.88%, IA 0.68%; against the Bonferroni critical value of 2.77 (5%/9), only momentum, ROE, and BAB clear the bar robustly. For MVE portfolios (unconstrained, one-month), all five are significant at the 1% level: C4 3.11% (t = 5.20), FF3 2.28% (t = 2.91), HXZ 2.22% (t = 6.02), FF6 2.09% (t = 5.37), FF5 1.23% (t = 3.49); under no leverage FF5 still gives 0.58% (t = 3.28) while FF3 falls to insignificance (0.71%, t = 1.73). The correlation ablation is the decisive number: FF5’s unconstrained alpha drops from 1.23% (t = 3.49) to 0.60% (t = 1.72) with zero correlations and 0.58% (t = 1.65) with unconditional correlations, and all six FF5 alphas become insignificant even before multiple-testing adjustment. Raw annualized Sharpe ratios for context: BAB 0.87, ROE 0.71, IA 0.58, momentum 0.48 among individual factors, versus 1.30 (HXZ), 1.20 (FF6), 1.04 (FF5), 0.93 (C4), 0.61 (FF3) for the MVE portfolios. The paper reports spanning-regression alphas rather than a tradable backtest, so no maximum drawdown, turnover, transaction-cost or win-rate figures are given; the authors explicitly note the case for timing is fragile before the estimation issues of Cederburg et al. (2020) and the trading-cost results of Barroso and Detzel (2021) and DeMiguel, Martín-Utrera and Uppal (2024) are even considered.
Mentioned by Ivan Blanco (Founder & CIO at Noax Capital, Associate Professor of Finance at CUNEF) in this discussion, who summarises it as “vol scaling barely helps most single factors — it helps the portfolio”, and highlights that much of the benefit comes from cutting exposure when factor correlations rise rather than when volatility alone rises: the risk that hurts a multi-factor book is factors moving together at the worst moment, not any single one getting noisy.
By Pedro Barroso, Paulo F. Maio, and Haoxu Wang. 21 pages, this version June 2026.
Volatility Salience: Evidence from the Options Market#
Yintai Ding, Gang Li, Shimeng Shi, and Jia Zhai introduce volatility salience theory (VST): a measure of how strongly a stock’s recent realised-volatility deviation stands out against a market benchmark. Using US equity-option data, they sort options by VST and evaluate delta-hedged call and put returns. Their stated interpretation is behavioural: investors overweight a salient volatility shock and bid implied volatility above fundamentals, so buying high-salience options subsequently earns lower delta-hedged returns.
The result is a cross-sectional pricing relation rather than a fully specified trading strategy. In the figure shared with the paper, the equal-weighted high-minus-low VST portfolio has roughly -0.5% monthly delta-hedged return at the one-month horizon for both calls and puts; the difference is statistically insignificant at months two and three, then is negative and significant again at later displayed horizons. The chart covers January 1996–August 2025. The abstract says the effect is distinct from the variance risk premium and other volatility measures, and stronger where arbitrage is limited, sentiment is high, or retail participation is greater.
Our summary: the useful hypothesis is not that high realised volatility alone makes an option expensive, but that the unusualness of its volatility relative to a benchmark can be priced. That distinction is exactly what the accompanying X discussion challenges: a deviation measure may simply proxy for the volatility level or variance-risk-premium richness, and a t+1 effect that weakens around t+2–t+3 may reflect ordinary volatility-shock decay. The authors claim controls that separate VST from those explanations, but the full SSRN text, regression specification, data description, and robustness tables were inaccessible in this run, so that claim cannot yet be independently assessed.
Data and reproduction: the abstract names US equity-option data; the available figure states the January 1996–August 2025 sample. No code or data release is identified in the accessible metadata. SSRN blocked direct download and the available open-access and curl retrieval paths did not provide a PDF.
Key metrics: the accessible material reports return spreads and statistical significance, not annualised return, Sharpe ratio, drawdown, turnover, or transaction costs. The clearest observed magnitude is the approximately -0.5% high-minus-low monthly delta-hedged return at t+1 for both calls and puts.
Mentioned by Ralph Sueppel (Macrosynergy) in this X discussion, which highlights the lower delta-hedged returns. The replies ask whether VST survives controls for volatility level and the variance risk premium, and whether the fast-decaying effect is simply volatility mean reversion.
Advanced Hedging Tactics in Volatility Strategies#
Menthor Q frames the management of a short-volatility oil-options book as part of the trade, not a mechanical afterthought. The first lever is hedging cadence: instead of delta-hedging futures every day, the article compares daily, three-day, weekly, and biweekly adjustments. The proposed intuition is that frequent re-hedging reacts to noise and locks in whipsaw losses, while a slower cadence can allow short-run oil-price mean reversion after a large move. Less frequent hedging deliberately accepts higher P&L variability in exchange for lower turnover and, potentially, a better hedge level.
The second lever is the volatility input to delta. If a trader believes realised volatility will be lower than the implied volatility used to price a sold option, the article proposes calculating delta from the lower realised-volatility estimate. In its $60 put / $70 spot illustration, a 20% rather than 30% volatility delta initially under-hedges the position, then the hedge is increased rapidly once spot approaches or breaches the strike. Economically, this is not a pure hedge: it retains directional and short-gamma exposure in exchange for an explicit view on realised volatility and a downside trend/threshold rule. The article also recommends avoiding near-expiry short gamma unless far from the strike, rolling before expiry week, and spreading strikes and maturities to avoid a concentrated gamma cliff.
Our reading: the useful distinction is between risk reduction and P&L optimisation. A hedge based on a discretionary realised-volatility forecast, a delayed cadence, or a decision to wait above strike can outperform a conventional delta rule only by accepting model, gap, and basis risk; it can therefore conflict with accounting and risk-system Greeks. The article says lower-volatility deltas improve P&L in backtests, but supplies no contracts, dates, return series, hedge-rule parameters, transaction costs, tail losses, or out-of-sample tests. It is best read as a set of hypotheses for a cost-aware, path-dependent hedging study rather than evidence that slower or under-hedged execution is generally superior.
By Menthor Q.
Published as an X Article shared in this X post.
Intraday Volatility Dynamics#
Carsten H. Chong, Marc Hoffmann, Mathieu Rosenbaum, and Grégoire Szymanski develop inference for the autocorrelation of intraday volatility changes that remains valid when spot-volatility estimates are distorted by sampling error, price jumps, and market-microstructure noise. Their approach combines the limiting autocorrelation structure of continuous-time increments at shrinking horizons—which must match fractional Gaussian noise—with infill asymptotics for realised autocovariances, yielding a feasible generalised-method-of-moments estimator.
Our summary: the contribution is a way to distinguish genuine short-horizon volatility dynamics from serial dependence introduced by the estimator itself. Applied to SPY transaction data, the authors find negative serial correlation in changes in both latent and realised intraday volatility. That makes the paper relevant for intraday volatility forecasts and execution-risk models, where an apparent mean-reversion signal may otherwise be an artefact of noisy high-frequency measurement.
Data and reproduction: the accessible abstract identifies SPY transaction data, but does not disclose the sample dates, cleaning rules, estimator parameters, code, or a downloadable data set. The SSRN preprint PDF was unavailable through the accessible public endpoint, so the detailed empirical design and numerical tables could not be independently checked.
Key metrics: this is an econometric inference paper rather than a trading-strategy backtest; it reports no annualised return, Sharpe ratio, drawdown, turnover, or transaction-cost result in the accessible material. Its primary empirical result is the negative serial correlation of latent and realised intraday-volatility changes in SPY; the accessible abstract does not provide the coefficient estimates or confidence intervals.
Mentioned by Piotr Pomorski (@PtrPomorski) in this X post.
Interactive Brokers IV Downloader#
A practical tool for downloading implied volatility data from Interactive Brokers. The post provides code and instructions for systematically collecting IV data across multiple symbols, which can be used for research, strategy development, and monitoring volatility levels. This is a companion piece to the IV Mean Reversion strategy, providing the data infrastructure needed to implement volatility-based trading systems.
By Peter, CrackingMarkets.
A Dirty Long Vol VIX Trade#
A Substack post by Robot James outlining a simple systematic long volatility trade in VIX futures and VIX ETPs. The core idea is that customer positioning in the VIX complex tends to be lopsided and sticky — retail traders and “tourists” systematically slam VIX futures, especially on upticks of volatility — which causes VIX futures to trade too cheap relative to fair value when unexpected bad news hits the market. The post argues that these positioning-driven dislocations create tradeable inefficiencies because VIX futures are volatile and positively skewed, and the crowded short side creates predictable under-reaction effects around shocks.
The proposed trade uses SPX options to determine whether VIX futures look cheap relative to underlying SPX volatility (since “it’s all SPX vol at the end of the day”) and takes long exposure to VIX futures or ETPs when they screen as significantly underpriced. The approach is framed as a dead simple rules-based long-vol setup that exploits sticky customer positioning rather than trying to forecast volatility directly. The full trade specification and backtest details are behind a paywall for paid subscribers.
By Robot James.
Mentioned by Robot James in this tweet, where he describes it as a way to trade the tendency of VIX futures to “get dead cheap when unexpected bad news hits” in an “incredibly simple systematic way”.
A GARCH-temporal fusion transformer model for the volatility prediction of exchange traded funds#
Lorenzo Petrosino, Luca Bacco, Giuliano Salvati, Mario Merone, and Marco Papi propose a hybrid volatility model that combines traditional GARCH econometrics with a Temporal Fusion Transformer. The study forecasts volatility for exchange traded funds composed of S&P 500 assets across sectors, using historical volatility and Garman-Klass volatility as target proxies.
Our summary: this is the strongest “econometrics plus TFT” volatility reference in the current batch. Instead of treating deep learning and GARCH as substitutes, it uses GARCH-derived structure alongside TFT’s multi-horizon covariate modeling. That makes it especially relevant for practitioners who already trust GARCH-style volatility pipelines but want to test whether attention-based neural forecasting can improve nonlinear and cross-feature behavior.
Key metrics: the paper reports that the hybrid GARCH-TFT model significantly outperforms alternative models when forecasting the Garman-Klass proxy and performs comparably to standalone TFT for historical volatility. The dataset covers January 1, 2002 to October 31, 2023 and is sourced from Yahoo Finance. The article evaluates forecast accuracy rather than a trading strategy, so it does not report Sharpe ratio, annualized return, or maximum drawdown.
Code and data: the authors cite a public companion repository at GitHub. The article page states that the market data can be obtained using Yahoo Finance.
Advances in forecasting realized volatility: a review of methodologies#
Radmir Mishelevich Leushuis and Nicolai Petkov survey all models used to forecast realized volatility in the academic literature from 2000 to the first half of 2024. The review covers classical linear models (HAR and extensions), machine learning approaches, and deep learning architectures, comparing their empirical performance across markets and evaluation metrics.
Our summary: this is the most comprehensive recent survey of the realized volatility forecasting literature and a useful orientation for practitioners deciding which model class to test. The headline finding is that a hybrid CNN-LSTM tops the out-of-sample accuracy leaderboard, but the survey is careful to note that deep models do not universally dominate—HAR and its extensions remain strong baselines. The paper catalogs HAR extensions that have been shown to improve forecasts in specific contexts: signed and negative semivariance, jumps, implied volatility, overnight returns, realized volatility-of-volatility, and asymmetry and leverage effects. Results vary significantly by market, evaluation horizon, and loss function, and no single extension wins uniformly.
Data and code: the survey covers published academic results across multiple asset classes and markets. No single dataset or code repository; individual papers cited within should be consulted for reproduction details. Published in Financial Innovation, 2025.
Key metrics: the top-performing hybrid CNN-LSTM model outperforms all standalone linear, ML, and deep learning benchmarks in out-of-sample forecast accuracy as measured in the reviewed studies. HAR extensions involving semivariance, jumps, and implied volatility show the most consistent gains; overnight return and vol-of-vol extensions show more mixed results.
Assessing the Risk of Bitcoin Futures Market: New Evidence#
Anupam Dutta proposes an augmented HAR model for forecasting the realized volatility of Bitcoin futures that incorporates time-varying jump intensity estimated through a GARCH-jump process. The paper tests whether jump-induced volatility carries information about future realized variance beyond what is already captured by the continuous path component and the Bitcoin implied volatility index.
Our summary: the paper is notable for two reasons. First, it provides one of the cleaner demonstrations that jump volatility and leverage effects add predictive information to HAR-RV for Bitcoin futures specifically—a crypto context where jumps are frequent and large. Second, the novel finding that jump-induced volatility is informative incrementally to Bitcoin’s own IV index is practically important: it means that even after conditioning on the market’s forward-looking volatility expectation, high-frequency jump realizations still contain residual information. For HAR practitioners building Bitcoin volatility models, this supports including a jump component in addition to IV.
Data and code: Bitcoin futures high-frequency price data and the Bitcoin implied volatility index. Published in Annals of Data Science, 2024 (DOI: 10.1007/s40745-024-00517-4).
Key metrics: the augmented HAR model with GARCH-jump-derived volatility and leverage effects outperforms standard HAR-type models in both in-sample and out-of-sample analyses. Jump-induced volatility provides incremental out-of-sample predictive power relative to the Bitcoin IV index. Results are reported in forecast accuracy terms; no annualized return or Sharpe metrics are reported.
Forecasting Bitcoin realized volatility by measuring the spillover effect among cryptocurrencies#
Yue Qiu, Yifan Wang, and Tian Xie study whether cross-cryptocurrency spillover variables improve HAR-type realized volatility forecasts for Bitcoin. The paper constructs spillover measures from Ethereum, Litecoin, and other major cryptocurrencies and embeds them as exogenous regressors in an augmented HAR framework.
Our summary: the paper addresses a feature that is specific to crypto markets: Bitcoin’s realized volatility is not independent of the broader crypto ecosystem. When major altcoins move sharply, Bitcoin volatility often follows, and the authors show that this spillover information is statistically significant and improves out-of-sample forecasts. For practitioners running Bitcoin volatility models, this suggests that cross-asset realized volatility from correlated cryptocurrencies is a more natural exogenous input than general macro or equity spillovers, because the timing and magnitude of crypto co-movements are tightly coupled.
Data and code: high-frequency price data for Bitcoin and multiple altcoins. Published in Economics Letters, Volume 208, November 2021 (DOI: 10.1016/j.econlet.2021.110092).
Key metrics: the HAR model augmented with cross-cryptocurrency spillover variables outperforms the standard HAR benchmark in out-of-sample Bitcoin realized volatility forecasting. Results are reported using mean squared error and related accuracy metrics; no trading strategy performance metrics are reported.
GNAR-HARX Models for Realised Volatility: Incorporating Exogenous Predictors and Network Effects#
Tom Ó Nualláin introduces a hybrid modelling framework that combines Generalised Network Autoregressive (GNAR) structure with Heterogeneous Autoregressive (HAR) dynamics and exogenous predictors. The model forecasts realized volatility across ten international stock indices using approximately 16 years of daily data from 2005 to 2020, evaluating performance under both QLIKE loss and mean squared error.
Our summary: this paper is a direct empirical warning about the metric-dependency of exogenous variable results in HAR-type models—a pattern that maps closely onto practitioner experience. The best QLIKE model is a local GNAR-HAR without any exogenous variables, while the best MSE model is a GNAR-HARX that includes implied volatility. Exogenous variables help on mean squared error while hurting or not improving on QLIKE, which is exactly the split seen when broader feature packs are added to HAR-type models in crypto and equity contexts. The network structure finding—that fully connected cross-market graphs outperform dynamically estimated graphical lasso networks—is also useful for anyone attempting cross-asset spillover modelling.
Data and code: daily realized volatility for ten international stock indices, 2005–2020. The paper is available on arXiv (arXiv:2510.24443). Code availability not confirmed at time of writing.
Key metrics: the local GNAR-HAR without exogenous variables achieves the lowest QLIKE score. The GNAR-HARX with implied volatility achieves the lowest MSE. GNAR-HAR(X) specifications outperform univariate HAR(X) benchmarks across both metrics. Fully connected networks consistently outperform graphical lasso networks. No trading strategy performance metrics are reported.
The role of implied volatility in forecasting future realized volatility and jumps in foreign exchange, stock, and bond markets#
Thomas Busch, Bent Jesper Christensen, and Morten Ørregaard Nielsen examine whether model-free implied volatility derived from options prices adds forecasting information beyond the realized continuous and jump variation components already used in HAR-type models. The study covers three major asset classes—foreign exchange (EUR/USD, JPY/USD, GBP/USD), equity (S&P 500), and fixed income (30-year US Treasury bond)—and uses high-frequency intraday data to construct realized volatility, realized bipower variation, and the jump component.
Our summary: this is the canonical reference for HAR-X models that incorporate implied volatility. The central result is that implied volatility contains information about future realized volatility that is incremental to the continuous and jump components of past realized volatility, and this holds consistently across FX, equity, and bond markets. The paper also shows that implied volatility is informative about future jump activity, not just total realized variance. For practitioners building HAR-based volatility forecasters, this paper provides the direct empirical justification for including an IV term alongside standard HAR lags.
Data and code: the study uses high-frequency tick data for three FX rates, the S&P 500, and 30-year Treasury futures. No public code repository was identified. Published in Journal of Econometrics, Volume 160, Issue 1, 2011, pages 48–57.
Key metrics: implied volatility adds statistically significant incremental forecasting power for realized volatility across all five assets. The gains are strongest for the continuous variation component and for future jump activity. Results are reported in terms of out-of-sample R-squared and Mincer-Zarnowitz regression coefficients rather than trading performance metrics.
Transaction activity and bitcoin realized volatility#
Konstantinos Gkillas, Maria Tantoula, and Manolis Tzagarakis investigate whether Bitcoin transaction activity improves out-of-sample realized volatility forecasts beyond standard HAR and random forest models. The paper decomposes Bitcoin’s high-frequency realized volatility into its continuous and jump variation components and tests whether on-chain transaction flow data carries additional predictive information.
Our summary: this is a short but directly relevant paper for anyone building HAR-type Bitcoin volatility models with exogenous inputs. The main result is that transaction activity, combined with the jump component, improves out-of-sample forecasts in both the HAR and random forest frameworks. On-chain activity is a natural candidate because Bitcoin trading volume and transaction counts are publicly available at high frequency and capture information about market participation that is not embedded in price-based realized volatility measures alone.
Data and code: Bitcoin high-frequency price data and on-chain transaction data. Published in Operations Research Letters, Volume 49, Issue 5, 2021, pages 715–719 (DOI: 10.1016/j.orl.2021.06.016).
Key metrics: incorporating transaction activity and jump variation improves out-of-sample forecast accuracy for Bitcoin realized volatility relative to baseline HAR and RF models. Results are reported using standard forecast evaluation metrics; no Sharpe ratio or drawdown metrics are reported as the paper focuses on volatility forecasting accuracy rather than a trading strategy.