Interaction Between P and Q Measures: A Put-Call Parity Perspective

Option pricing is typically conducted under the risk-neutral (Q) measure, whereas the drift of the underlying asset in the real world is specified by the physical (P) measure. The interaction between these two measures is subtle and has been the subject of several studies, including research on delta hedging and the volatility risk premium.

Reference [1] contributes to this literature by examining put-call parity. Specifically, the author argues that while put-call parity is a terminal payoff condition operating under the Q measure, enforcing it in practice is not costless, and these costs are influenced by the real-world drift of the underlying asset.

The author pointed out,

This paper studies the carry-space wedge in put–call parity from the perspective of drift-sensitive implementation risk. Put–call parity is a static no-arbitrage relation over terminal payoffs and plays a central role in Black–Scholes–Merton-type risk-neutral pricing. In actual markets, however, the position that enforces parity is not held as a frictionless static identity. It must be maintained until maturity while being exposed to interim price paths, variation margin, trading frictions, funding conditions, and finite-capital constraints. The observed carry-space wedge may therefore reflect the path-dependent capital burden of parity enforcement rather than a terminal pricing error.

The main contribution of the paper is to add a drift-preserving component to the standard diffusion-based GBM path-risk term. The baseline support-capital argument implies an rσ√τ burden under zero-drift Brownian motion. When drift is preserved, a first-order approximation produces an additional directional margin-burden component of the form rµτ. I derive this term and test its empirical counterpart in SPX and RUT index options. The empirical µ used in the regression is not an observed future expected return. Because future physical drift is unobservable, I proxy for it using the rolling OLS slope of past log total-return paths.

In short, using SPX and RUT index options, the paper finds that the drift term adds explanatory power for the carry gap beyond volatility-based path-risk measures, bid-ask spreads, and financial-condition controls. The author interprets this as evidence that the capital burden of parity enforcement may be influenced by physical-market trends, even though option pricing itself remains risk-neutral.

The results bring me to the thought that hedgers and arbitrageurs have to assume and manage margin requirements, funding costs, and capital constraints. This is not always convenient. Hence, their returns reflect, at least in part, these costs.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Shin, U. (2026). The P behind Q: Empirical Evidence from Physical Drift in Put–Call Parity, arXiv:2605.12250v2.

 

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The Information Content of the Spot VIX Term Structure

Considerable research has been devoted to the study of the VIX futures term structure, showing that the shape of the curve contains valuable information about the future direction of volatility. Trading strategies have been developed based on the informational content of the VIX futures curve.

However, much less attention has been paid to the term structure of spot VIX indices. Reference [1] helps fill this gap by examining the spot VIX term structure, including the VIX, VIX9D, and VIX3M indices. The objective is to determine whether inversions in the spot VIX term structure contain predictive information about future realized volatility.

The author pointed out,

The VIX term structure contains forecasting information about future realised volatility beyond what the VIX level itself implies. I document this in four steps. First, inversion depth has substantial incremental explanatory power above the VIX level, adding between 2.4 and 6.9 percentage points of R-squared in sample. Second, the predictive content is concentrated at the front end of the curve. Measures using the VIX9D index dominate the conventional spot-minus-three-month measure across every robustness check. Third, the signal carries information about the volatility risk premium and is associated with the gap between realised volatility and the VIX at horizons of 5 and 10 days. Fourth, the result survives augmentation with Corsi-style HAR-RV controls and improves out-of-sample forecasts in a recursive 2019 to 2026 test. The Clark-West nested-model statistic rejects equal predictive accuracy at the 1 percent level in every measure-by-horizon cell tested.

In short, the paper concludes,

• The spot VIX term structure contains predictive information about future realized volatility beyond the information embedded in the VIX level alone.
• The predictive power is strongest at the front end of the curve, with VIX9D-based measures consistently outperforming traditional spot-versus-VIX3M measures.
• The signal is related to the volatility risk premium and helps explain the gap between implied and subsequently realized volatility over 5- and 10-day horizons.
• The results remain robust after controlling for realized volatility dynamics and improve out-of-sample volatility forecasts during the 2019–2026 test period.

The findings of this paper are insightful. Notably, the study shows that VIX9D contains the strongest predictive information among the term-structure measures examined. As a result, we will pay more attention to short-dated options and short-term volatility indicators going forward.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Lim Boon Chuan (2026), The Front End of the VIX Term Structure and Forward Realised Volatility, SSRN 6752518

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How Effective Are LLM Trading Agents?

AI is evolving rapidly across many areas of life, and trading is no exception. Large Language Models (LLMs) have increasingly been used as tools to assist financial analysts with research, data synthesis, and decision-making.

More recently, researchers and practitioners have moved beyond analyst assistance and begun exploring LLMs as autonomous end-to-end trading agents. As of this writing, even brokerage platforms such as Robinhood have introduced services that allow investors to trade using AI agents.

But how effective are these systems?

Reference [1] reviews recently proposed and developed end-to-end LLM trading agents and evaluates their performance and limitations. The authors pointed out,

The two-year arc of end-to-end LLM trading agents has been productive on its own terms: it forced concrete questions about how language models behave when asked to produce orderable decisions, and the resulting catalogue of architectures has real research value. The trouble starts when this exploratory progress is reported as deployment progress—headline Sharpe numbers from short, in-cutoff windows carried into abstracts, talks, and follow-up surveys as if they had already answered the deployment question.

Our argument has been narrow but specific. Public evidence cannot yet distinguish robust predictive ability from temporal contamination, unmodeled friction, short-sample noise, narrative fitting, and parametric prior; and even were every evaluation step cleaned up, three structural gaps would remain-language confidence is not tradable probability, narrative ability is not numerical execution, and model priors can become undisclosed implicit factor exposures. P1–P6 and the modular alternative are not a new framework but the minimum bookkeeping that lets a reader tell which kind of evidence a paper is offering—historical backtest, prototype, deployable claim, or autonomous trading ability—and lets reviewers hold each level of claim to its matching evidence floor.

So basically, the paper concludes that reported alpha from trading agents should not be interpreted as evidence of deployable trading capability. It shows evidence that reported performance can collapse once evaluation moves beyond the model's knowledge cutoff.

The authors argue that current LLM trading results are confounded by three major issues:

• Temporal contamination (training-data leakage),
• Unmodeled trading frictions,
• Short-sample and multiple-testing effects.

They propose six minimum reporting protocols and recommend a modular architecture instead of end-to-end LLM trading,

• LLMs extract information from news, filings, and transcripts,
• Independent quantitative models perform forecasting,
• Separate modules handle calibration, risk management, sizing, and execution.

This paper provides a transparent, scientific assessment of the current state of LLM-based trading agents, separating evidence-based findings from the hype often found in popular media.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Ye, Y., Han, J., Hu, A., Bu, J., Chen, Y., Wen, L., Mandic, D., Sun, D. D., Yinghui, X., & Xu, Z. (2026). The Alpha Illusion: Reported Alpha from LLM Trading Agents Should Not Be Treated as Deployment Evidence. arXiv:2605.16895v1, 16 May 2026.

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Decomposing the Variance Risk Premium, Part 2

The volatility risk premium (VRP) is the difference between implied volatility and subsequently realized volatility, and is one of the most extensively studied phenomena in options markets. We previously discussed Reference [1], which decomposes the VRP into upside and downside components and studies their dynamics separately. Reference [2] applies a similar framework to the same index, the S&P 500, but using a more recent dataset.

The author pointed out,

We examine four main points. First, we test whether investors pay a higher premium for volatility associated with equity price declines than for volatility associated with price increases. By decomposing the variance risk premium into upside and downside components using option prices and high-frequency equity return data, we find that the downside variance risk premium is statistically more pronounced than the aggregate variance risk premium.

Second, we examine whether risk premium associated with downside variance and skewness are related to the required return on equities. Empirically, these premium predict future returns, suggesting that investors view rare, large drawdowns and volatility during market declines as risk, and that compensation for bearing such risks is linked to expected equity returns.

Third, we investigate the relationship between the prediction horizon and predictive power (adjusted R2). Consistent with prior findings, predictive power for variance-related premium peaks around three to five months, while skewness-related premium exhibit relatively stronger predictive power at longer horizons.

Fourth, we evaluate whether the term structure (the difference between longer- and shorter-maturity premium) improves return forecasting. While we find limited evidence of improvement for the variance risk premium, the term structure of the skewness risk premium is statistically significant and suggests that when the longer-maturity skewness risk premium is lower (more negative) than the shorter-maturity premium, long-horizon equity returns subsequently rise, and vice versa. This is consistent with the interpretation that when investors anticipate longer-horizon tail risk, required long-run equity returns increase.

In short, the paper finds that downside VRP is substantially more pronounced than aggregate VRP. It also shows that downside variance and skewness risk premiums predict future equity returns, with variance-related predictive power strongest at medium horizons and skewness-related predictive power stronger at longer horizons. Finally, the study finds that the term structure of skewness risk premium contains forecasting information, suggesting that expectations of longer-horizon tail risk are linked to higher future required equity returns.

We believe this paper largely revisits the framework of the earlier study [1], albeit using more recent data and more extensive robustness tests, while ultimately reaching very similar conclusions.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Feunou, B., Jahan-Parvar, M. R., & Okou, C. (2016), Downside Variance Risk Premium, Journal of Financial Econometrics 16 (3), 341-383

[2] Akio Ito, Variance Risk Premium, Skewness Risk Premium and Equity Expected Returns, SSRN Working Paper 6712647, 2025

Originally Published Here: Decomposing the Variance Risk Premium, Part 2

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Volatility Measures for Regime Classification

Regime detection and classification are important in portfolio management and asset allocation. One of the key inputs into regime detection models is volatility. Reference [1] examines which volatility measure is most effective for regime classification. The authors study three volatility measures,

1. Implied volatility (VIX/VIXBR),
2. GARCH conditional volatility,
3. Historical volatility.

They then incorporate them into a Hidden Markov Model framework. The paper pointed out,

The empirical evidence supports conditional volatility (GARCH as the superior proxy for regime identification in both the Brazilian and U.S. markets). Unlike implied volatility, which exhibited excessive sensitivity to short-term noise and threshold variations, the GARCH- based specification provided the greatest parameter stability and classification robustness, essential attributes for operationalizing dynamic portfolios.

A key finding of this research is the structural identification of three volatility regimes (low, medium, and high). Contrary to binary specifications often assumed in the literature for interpretability, the Bayesian Information Criterion (BIC) results demonstrated that a three-state model better captures the complex dynamics of financial markets, specifically identifying transitional phases that binary models fail to detect. This granular identification allowed for a more precise assessment of international risk transmission…

From an investment perspective, the results highlight the distinct roles of regime-based strategies. The Dynamic Regime strategy consistently outperformed the traditional Static Mean-Variance (Single Regime) strategy in risk-adjusted metrics, successfully mitigating severe drawdowns, most notably in the U.S. market, where the static strategy suffered structural losses (−18.49%) while the dynamic strategy preserved capital (+1.56%). However, the dynamic strategy did not outperform the Naive (1/N) benchmark in terms of total cumulative return…

In short, the authors conclude that GARCH conditional volatility provides the most stable and operationally reliable regime classification. Implied volatility reacts faster to market changes but produces noisier regime switching. The study also finds that a three-regime framework is superior to a simple low/high volatility classification, as the intermediate regime captures transition periods and uncertainty normalization phases.

Another important point emphasized in the paper is that regime-based strategies are best viewed as risk-management tools rather than universal return-enhancing solutions. Regime-based allocation improves drawdown control and risk-adjusted performance relative to static mean-variance optimization.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Bitencourt, W. A., & Iquiapaza, R. A. (2026), Comparative analysis of volatility proxies and regime-based asset allocation, International Review of Economics and Finance, 109, 105366.

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Delta Hedging Performance Under Different Measures

Managing an option book is not trivial. There is considerable research on hedging errors, optimal hedging frequency, and the choice of volatility inputs used in hedging algorithms. Reference [1] continues this line of research by examining the effectiveness of hedging strategies under five different volatility measures:

1. Flat ATM implied volatility
2. Stochastic Volatility Inspired (SVI) implied volatility
3. Close-to-close realized volatility
4. Parkinson realized volatility
5. Yang–Zhang realized volatility.

The study uses OptionMetrics SPX options data from 2019 to 2024, including the COVID crash and the 2022 hiking cycle, and analyzes 2000 stratified options across four VIX regimes. The author pointed out,

This thesis provides the first empirical evaluation of SVI-calibrated implied volatility as a delta-hedging input on real S&P 500 index options. The results challenge three common assumptions.

First, more information does not automatically improve hedging. The SVI surface, despite encoding the full implied volatility smile with high calibration quality (median RMSE of 19.5 bps), produces 9.4% higher hedging error variance than flat ATM implied volatility. The calibration noise embedded in the five-parameter SVI fit outweighs the informational benefit of strike-specific volatilities for most option types.

Second, statistical efficiency in volatility estimation does not translate into hedging efficiency. The simple close-to-close realized volatility estimator, which uses only closing prices, outperforms both the Parkinson and Yang–Zhang estimators, which incorporate intraday range information. For hedging purposes, smoothness (low sensitivity to intraday noise) appears more valuable than efficiency (low estimation variance under GBM).

Third, and most importantly, the optimal volatility input is strongly dependent on option moneyness and market regime. SVI surface IV reduces hedging error for out-of-the-money calls by 6–12%, where the smile slope carries genuine hedgeable information. Realized volatility dominates for out-of-the-money puts, where the skew premium makes implied-based deltas biased. No single input wins everywhere, suggesting that a moneyness-conditional hedging strategy, using different volatility inputs for different regions of the option space, would outperform any static approach.

In short, the results show that, contrary to intuition, SVI surface volatility does not improve aggregate delta hedging performance. In fact, SVI increases the standard deviation of hedging errors by 9.4% relative to flat ATM implied volatility. Close-to-close realized volatility performs best overall, reducing hedging error standard deviation by 5.8%, while the Parkinson estimator performs worst. The effectiveness of volatility measures, however, is also found to depend on regime and moneyness.

Another important finding is that higher-order effects, including vanna, volga, jumps, and model misspecification, dominate hedging errors.

This paper provides valuable insights for both traders and risk managers. Let us know what you think in the comments below or in the discussion forum.

References

[1] Annigeri, Z. (2026), Regime-Dependent Delta Hedging with SVI-Calibrated Volatility Surfaces: An Empirical Analysis of SPX Index Options, Rutgers Business School, SSRN 6465741

Article Source Here: Delta Hedging Performance Under Different Measures

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Network Effects in Social Media Sentiment

Social media sentiment has become increasingly important in modern portfolio and risk management. Most studies on social media rely on aggregate sentiment measures, such as average bullishness scores or overall positive-versus-negative comment ratios. Reference [1] introduces an innovative approach to analyzing social media sentiment by investigating network effects, specifically how high-centrality users, i.e., “influencers,” affect the behavior and sentiment of regular users. The study utilizes data from the r/stocks subreddit from January 2019 to June 2022, covering approximately 3.5 million comments.

To study network effects, the authors construct a daily Reddit interaction graph in which nodes represent users and edges represent direct comment replies. They then compute eigenvector centrality to identify influential users, divide users into centrality quintiles, measure sentiment within each group, and test whether lagged sentiment from high-centrality users predicts future sentiment among lower-centrality users and the broader network. They pointed out,

In this study, we examine the relationship between online social interactions and financial markets, specifically focusing on the sentiment dissemination within a stock market community on Reddit. Our findings demonstrate that highly active users can spread their sentiments to a broader audience. This influence becomes more pronounced under two conditions: (1) when there is reduced disagreement among high-centrality nodes and (2) during periods of high market volatility. Additionally, we find that the COVID-19 pandemic represents a structural shift that enhances the influence of high-centrality nodes as increased online activity and uncertainty reshaped network dynamics…

The practical implications of our findings are twofold. For market participants, sentiment-based trading strategies can provide increased profitability, especially in commission-free trading environments. In addition, network sentiment can be an effective tool for market timing and creating downside protection. From a policy standpoint, while online networks can enhance information dissemination, the ability of a few highly active users to stimulate the beliefs of others can be exploited or, to a certain extent, can inflate the prices of specific assets in the market; one example being the GameStop short squeeze case.

In short, the results show that sentiment from influential users significantly predicts sentiment among regular users, with dissemination effects becoming stronger during the COVID period, high-volatility environments, and periods of low disagreement among influential users. The authors also develop a trading strategy based on these findings. The sentiment-timing strategy materially reduces drawdowns, while the long-only version outperforms buy-and-hold before transaction costs.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Akarsu, S., & Yılmaz, N. (2026), The dynamics of online social interactions and implications on stock market returns, Journal of Economic Interaction and Coordination.

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From Pinning to Amplification: Evidence from S&P500 Options

Options pinning, the tendency of underlying prices to gravitate toward strikes with concentrated open interest near expiration, is well documented. However, given the rapidly changing options landscape, it is worth reassessing whether this effect still holds.

Reference [1] examines options pinning using data from 2016 to 2025, a period marked by the proliferation of weekly and zero-days-to-expiration (0DTE) options, based on 1.6 million near-expiry SPY contracts. The author pointed out,

We find no evidence of options pinning in S&P 500 expiration dynamics across five tests on 2,294 trading days (2016–2025). The significant finding (p < 0.001) shows the opposite: high near-expiry ATM open interest is associated with wider, not narrower, daily ranges. This is consistent with the gamma amplification mechanism documented by Barbon & Buraschi (2021) and the theoretical predictions of Jeannin et al. (2008) under net short dealer gamma.

We interpret these results as evidence of a possible regime shift in S&P 500 expiration dynamics—from pinning in earlier periods (as documented by Golez & Jackwerth through 2009) to amplification in the modern era, potentially driven by the structural growth of short-dated options and changes in dealer positioning. We offer this interpretation as a hypothesis for further investigation, not as a definitive conclusion.

In short, the paper finds no evidence of pinning. Instead, on days with high open interest, prices exhibit approximately 16% wider ranges, indicating a shift from pinning to amplification. This change is likely driven by the growth of short-dated options and increased retail demand for long options, which leaves dealers short gamma and leads to larger price moves.

An interesting point discussed in the paper is that open interest is more informative than implied volatility, signifying that dealer positioning is important information.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Elms, N. (2026), From Pinning to Amplification: Evidence of a Regime Shift in S&P 500 Options Expiration Dynamics, 2016–2025, SSRN 6564078

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VIX Forecasting Using Crypto Overnight Returns

Prediction is central in finance. A growing line of research uses cross-asset signals to forecast market movements. A recent example showed that Bitcoin can serve as a strong leading indicator in a machine learning-based trading system.

Along similar lines, Reference [1] examines whether cryptocurrency overnight returns, defined as price changes during U.S. equity market closures, can predict the VIX. For this study, the authors use five-minute data of Bitcoin and Ethereum from 2018 to 2025, motivated by the idea that crypto markets are highly sensitive to sentiment, and that overnight returns capture this information. They pointed out,

This study examines the informational content of cryptocurrency returns through a novel temporal decomposition framework that aligns cryptocurrency trading activity with U.S. equity market hours. Our analysis shows that the overnight returns of both Bitcoin and Ethereum capture a distinct dimension of investor sentiment, which significantly improves the predictability of equity market volatility…

We then explore the predictive power of cryptocurrency overnight returns for VIX dynamics. In-sample analysis reveals a significantly negative relationship between cryptocurrency overnight returns and subsequent trading-hour VIX changes, indicating that positive overnight sentiment predicts a reduction in equity market uncertainty during the following trading session. Out-of-sample analysis demonstrates that models incorporating cryptocurrency overnight returns consistently outperform baseline models, with results remaining robust across subperiods and extending to other U.S. implied volatility indices. The economic significance of these findings is further validated through long-short trading strategies in VIX derivatives, where overnight-return-augmented models generate superior performance across different risk-aversion levels.

In short, the results show that crypto overnight returns have a negative predictive relationship with the VIX, with strong in-sample and out-of-sample performance. A trading strategy based on this signal is found to be profitable, and the results hold across different periods, including COVID and non-COVID regimes, and extend to other volatility indices.

Once again, the study reinforces the role of cryptocurrencies as leading indicators for broader market dynamics.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Gu, M., Lin, J., & Liu, S. (2026), Beyond Conventional Sentiment Indicators: Cryptocurrency’s Hidden Potential in VIX Forecasting, Economic Modelling.

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Regime-Aware Trading Strategies with Machine Learning

Regime detection is important in portfolio management and remains an active area of research, particularly in the age of machine learning and AI. Reference [1] proposes a trading strategy based on machine learning, combined with regime detection using a Hidden Markov Model.

Specifically, the machine learning technique used is LightGBM, a gradient boosting framework that employs histogram-based split finding for efficient training on high-dimensional tabular data. The study utilizes 63 features spanning technical, macro, and cross-asset data. The resulting strategy is compared against baseline approaches, including XGBoost classifier, logistic regression, SMA 50/200 crossover, and time-series momentum.

The author pointed out,

This study addresses the question of whether machine learning can generate statistically validated alpha in equity markets while adapting to changing conditions. The main contribution is a regime-aware LightGBM framework that provides three advances over prior work…

2. Feature importance hierarchy. The ablation study reveals that cross-asset features (Bitcoin) contribute more predictive value than traditional technical indicators, while SHAP analysis shows that macroeconomic features (yield curve, gold/equity ratio) dominate over stock-specific patterns for high-beta technology stocks. This challenges the conventional emphasis on technical indicators in equity forecasting.
3. Regime-adaptive decision logic. The model autonomously learns different strategies for different market conditions: mean reversion logic in bear markets (prioritizing distance from 200-day SMA) versus risk appetite monitoring in bull markets (prioritizing market beta and gold/equity flows). This adaptive behavior, revealed through regime-specific SHAP analysis, demonstrates that ML models can internalize economically sound reasoning.

The framework achieves a portfolio Sharpe ratio of 1.18 (95% CI: [0.53, 1.84]) and outperforms four baseline models (XGBoost, Logistic Regression, SMA crossover, momentum) under identical walk-forward evaluation. The consistent ∼17% alpha-positive rate across both NASDAQ-100 and S&P 500 universes suggests the approach generalizes beyond the specific training universe.

In short, the paper finds that cross-asset features, particularly Bitcoin as a leading indicator, provide the strongest predictive value; macroeconomic indicators outperform traditional technical indicators for high-beta stocks; and the model adapts its decision logic across regimes, shifting from mean reversion in bear markets to risk appetite monitoring in bull markets.

It is somewhat surprising to us that Bitcoin emerges as the strongest leading indicator, while it is less surprising that macro indicators outperform technical indicators.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Antonio Pagliaro (2026), Regime-Aware LightGBM for Stock Market Forecasting: A Validated Walk-Forward Framework with Statistical Rigor and Explainable AI Analysis, Electronics, 15(6), 1334.

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