Does Gamma P&L Really Measure Convexity?

Decomposing option P&L into individual Greek contributions is a useful practice. The decomposition is typically derived from a Taylor expansion and serves both risk-management and research purposes. For example, some studies use gamma P&L as a measure of convexity and as a tool for analyzing the variance risk premium.

Reference [1] challenges this conventional interpretation. The authors argue that the gamma term, ½Γ(ΔS)², does not necessarily measure pure convexity. To support their argument, they compute the theoretical Black-Scholes gamma and compare it with an empirical gamma estimated from actual option price changes after removing delta and vega effects.

The authors pointed out,

…At the daily horizon, the ½*Γ_BS(∆S)² term loads primarily on vega contamination from negative spot–IV comovement rather than on convexity. The consequences are most direct for the variance risk premium literature that decomposes delta-hedged returns and attributes the ½*Γ_BS(∆S)² term to convexity exposure. If that term is dominated by vega contamination at the daily horizon, such attributions rest on a channel that is not identified….

…The broader implication is that Greek identification is horizon-specific. The Taylor channels do not scale uniformly with ∆t: delta, theta, vega, and gamma become observable at different rates, and cross-channel comovement can dominate the residual used to isolate a coefficient. A decomposition that is useful at one horizon can therefore fail at another unless the implied channel coefficients are checked directly.

The paper shows that correlations between theoretical and empirical gamma are close to zero at intraday horizons and become negative at daily horizons, ranging from -0.25 to -0.47.

In summary, the authors conclude that while the daily ½Γ(ΔS)² term can be calculated, it may largely reflect vega contamination arising from the leverage effect rather than pure convexity. Because spot and implied volatility are negatively correlated, volatility changes can be absorbed into the gamma term, distorting its interpretation.

Our experience is that the Greek P&L decomposition remains a useful approximation, particularly under normal market conditions. However, the paper’s findings reframe the conventional understanding of option convexity and the interpretation of gamma P&L.

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

References

[1] Willeboordse, F. H. (2026), Does Gamma Survive the Close? Finance Research Letters, Article 110281.

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Walk-Forward Analysis for Cryptocurrency Forecasting

In today's age of AI and machine learning, developing trading strategies from historical data is becoming increasingly accessible. As a result, a growing portion of the research process is shifting from model development to model validation. However, unlike the sell-side, where model validation guidelines are well established, the buy-side literature still lacks a coherent and unified framework for validating trading systems. Regardless of the modeling approach, walk-forward analysis remains an essential component of the validation process.

A walk-forward analysis repeatedly trains and tests a model on sequential time periods, providing a more realistic assessment of out-of-sample performance than the traditional single train-test split commonly used in the industry. Reference [1] applies a walk-forward testing framework to an ensemble of traditional statistical methods alongside modern neural approaches for cryptocurrency forecasting. The authors employ expanding-window walk-forward validation to avoid look-ahead bias and better mimic real-world deployment. They pointed out,

We introduced a reproducible walk-forward benchmark for cryptocurrency return and volatility forecasting. In the full benchmark, ARIMA remains strongest for one-day returns, Chronos is marginally strongest for seven-day returns, and PatchTST shows clear gains on one-day realized volatility, with a smaller and less statistically distinguished lead at the seven-day horizon. The broader lesson is methodological: in crypto forecasting, careful validation and strong baselines matter as much as model class. Our scope is intentionally narrow: daily data on five major assets, point forecasts only, and zero-shot use of one foundation model.

In short, the paper finds that neural and foundation models provide the greatest benefit when forecasting persistent signals such as volatility, but offer little advantage for noisy daily return prediction. More importantly, the results suggest that validation quality matters more than model novelty.

This paper once again emphasizes the importance of a robust model validation framework in systematic trading research.

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

References

[1] Korir, G., Mbalu, N., Aranotu, C., & Tekenah, H. (2026), When Do Neural Forecasters Help? A Walk-Forward Benchmark on Cryptocurrency Returns and Volatility, Working paper

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Is the VRP Still the Same Risk Premium?

The volatility risk premium (VRP) is the difference between implied volatility and subsequently realized volatility, reflecting the compensation investors pay for volatility insurance. It has long been held that the VRP is generally negative for option buyers and positive for option sellers. However, market dynamics and regimes evolve over time, and the VRP should evolve as well. Recent studies have begun examining how changes in market structure, such as the growth of overnight trading, have affected the VRP.

Reference [1] takes this line of research further by conducting a broad examination of the VRP and its evolution over time. The study analyzes S&P 500 option returns from 1987 to 2025, including 95% out-of-the-money puts, ATM straddles, delta-hedged strategies, and variance-swap/VIX-based strategies.

The authors pointed out,

The conventional wisdom in asset pricing is that there is a large variance risk premium for the S&P 500 – larger than can be accounted for by exposure to the market alone, and that options earn large negative alphas because they are exposed to volatility and jump risk. This paper replicates past findings that options historically earned negative returns and CAPM alphas, but since around 2012 there is no longer evidence for those negative premia.

Though the paper does not go into great depth on the theory side, it does offer an explanation, which is that the decline in SPX options premia can be explained by declining asymmetry in dealer portfolios and the hedging costs they face.

…the results documented here should perhaps not be puzzling at all – they are just one more example of the performance of a trading strategy decaying after it is announced in a finance journal. That is a good thing: markets are getting more efficient. At a deep level, the implication of the results is that now it is much less expensive for investors to hedge deep losses in the aggregate stock market than it used to be.

In short, the paper identifies a structural break around 2012 using multiple break-detection tests. The authors argue that this shift coincided with important changes in market structure, such as the fact that retail investors became increasingly able to sell options, reducing persistent option overpricing, while dealer positioning moved from net short-option exposure toward a more neutral stance. Furthermore, declining dealer hedging frictions, such as bid-ask spreads and basis risk, are also consistent with lower volatility risk premia.

This is a timely and insightful paper that reinforces a broader theme: market and volatility dynamics are changing, and market participants must adapt accordingly.

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

References

[1] Dew-Becker, I., & Stefano Giglio. The Decline of the S&P 500 Variance Risk Premium. Working paper, June 2026.

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Narrative Risk Premia and the Pricing of Market Stories

Sentiment analysis has become an important tool in financial markets, helping researchers and practitioners extract information from news articles, social media, and other textual sources. Reference [1] introduces a new concept called Narrative Risk Premia. Like sentiment analysis, it analyzes market discourse using news, analyst reports, and institutional communications, but it is fundamentally different in scope.

While sentiment measures the level of optimism or pessimism that is sometimes noise, narratives capture changes in the perceived probability distribution of future outcomes, particularly tail risks. Moreover, narratives are well-structured stories linking facts and their causal implications to future outcomes, whereas sentiment primarily measures the degree of optimism or pessimism in the market without specifying the underlying causal story.

The author constructs a Market Narrative Intensity Index (MNII) based on four components: salience, emotional intensity, persistence, and narrative-attention volatility. The MNII index is then utilized as a factor in asset-pricing tests. The paper pointed out,

This paper has investigated the role of persistent market narratives in driving volatility, mispricing, and risk premia in financial markets. While going beyond the sentiment/information approaches to the study of market narratives, the paper defined the narrative risk premia as the narrative-driven uncertainty risky to the concerned investors and introduced the concept of narrative risk premia as the cost of the narrative-driven uncertainty that requires estimation. Via the application of the Market Narrative Intensity Index (MNII) within the time series and asset pricing models and the related regime-switching approach, it was demonstrated that market narratives systematically drive market risk.

In short, the results show that narrative intensity predicts future volatility spikes, assets associated with stronger narratives earn higher expected returns, and narrative effects are nonlinear and regime-dependent. Moreover, fear-based narratives have a stronger impact than optimistic narratives. The author also argues that the framework can be extended to other asset classes, including bonds, commodities, and cryptocurrencies.

This is an interesting new concept that offers a different perspective on market behavior and may help explain phenomena such as bubbles, crashes, and other episodes of extreme market reactions,

It has implications, especially for understanding bubbles and crashes. Bubbles do not appear because of optimism, but because of narratives that compress risk perceptions and push prices high. Crashes happen because narratives fall apart, causing sudden price and volatility adjustments. By acknowledging narratives as slow-motion mispricing corrections, it is understood why the markets stay so far apart for such extended durations and why corrections turn out so harsh and sudden once they happen.

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

References

[1] Christodoulou-Volos, C. (2026), Narrative Risk Premia: How Persistent Market Narratives Generate Volatility and Mispricing, Journal of Cultural Analysis and Social Change, 11(1), 2520–2534.

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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

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