Closed-Form Option Pricing Under the Physical Measure

The Black–Scholes–Merton (BSM) model is one of the most celebrated option pricing models and remains widely used by academics and practitioners alike. At its core, it states that the value of an option equals the cost of replicating its payoff through continuous dynamic hedging. In practice, however, continuous rebalancing is neither feasible nor costless. Traders and risk managers hedge discretely, making the real-world distribution of the underlying asset increasingly important.

Despite its practical relevance, relatively little research has focused on option pricing under the physical measure. We previously discussed one such paper, and Reference [1] is another noteworthy contribution to this area.

The paper derives closed-form pricing formulas for European call and put options under the physical measure (P) by solving the Feynman–Kac partial differential equation under P. The author pointed out,

Furthermore, this physical measure perspective can be elegantly extended to practical pricing and capital allocation. The core idea is straightforward: by calculating the expected real-world payout u0 at maturity T, an option seller can determine the exact present value required to cover this expected liability by simply discounting it at the risk-free rate, yielding e^−rT u₀. This specific amount can be securely allocated today into risk-free bonds or a bank account, providing a robust, expectation-based capital reserve without relying on the flawed continuous hedging assumption. Moreover, this framework allows for a novel calibration approach. By inverting the closed-form physical expectation formula, we can extract the implied physical parameters—specifically the implied real-world drift µ (or m) and volatility σ—directly from the observed market prices of call and put options. This shifts the calibration paradigm from fitting unobservable risk-neutral parameters to backing out the market’s implied real-world expectations, grounding the pricing model in actual market consensus rather than theoretical arbitrage constraints.

So basically, under this framework, the risk-free rate in the BSM formula is replaced by the underlying's real-world drift, µ, resulting in the options prices under the physical measure.

Other interesting derivations in this paper are:

• The variance of the call payoff and a measure for economic capital as the discounted expected payoff plus a multiple of the payoff's standard deviation.
• A drift rho, a new sensitivity measuring the dependence of the expected option payoff on µ, showing that higher real-world drift increases the expected liability of a written call.

This paper explores an important research direction that traders and portfolio managers should be aware of. Although the author emphasizes applications for option sellers, the framework is equally relevant to option buyers and risk managers.

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

References

[1] Halidias, N. (2026), Beyond Risk-Neutral Pricing: The Practical Case for Physical Measure Expectations in Option Selling, Working Paper

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A Statistical Mechanical Model of Trend, Volatility, and Correlation

Trend and mean reversion have been studied extensively. However, Reference [1] takes a refreshing approach by applying a lattice model borrowed from statistical mechanics to analyze market trends. In a follow-up paper [2], the authors extend the model and examine the relationship between trend, future volatility, and correlations.

The study utilizes 33 years of daily data across 24 futures markets, spanning equities, interest rates, foreign exchange, and commodities. Trend horizons range from 2 to 1,024 trading days, with trend strength measured using the t-statistic, as in the previous paper. The authors pointed out,

The e and f terms refine such models by also taking current trends into account. The positive value of f implies that the variance tends to grow day after day in times of strong trends, which explains why it is high after trends have built up, as shown in fig. 1, right. The negative value of e shows that the variance grows faster in times of strong down-trends as opposed to up-trends. We have found that this asymmetry is particularly strong for equities and mostly stems from short-term trends. Since trends measure cumulative recent returns, this can be interpreted as the “leverage effect”: strong negative returns tend to be followed by an increase of the variance.

… Since g < 1, the correlation tends to revert to the long-term correlation. The o-term refines such models by taking trends into account. It implies, e.g., that the next-day correlation of two assets is about 0.1 − 0.2 higher (lower) than their average correlation, when both trends are strong (ϕ, ψ ≥ 2) and point in the same (opposite) direction.

In short, the paper finds that,

• Future volatility depends on current trend strength, not just current volatility,
• Future cross-asset correlations depend on the trend strengths of both assets. Strong common trends imply higher future correlations, especially in downtrends,
• The leverage effect is strongest for equities.

This study provides another interesting perspective on market trends and volatility through the lens of statistical mechanics. It has implications not only in trading but also in risk management.

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

References

[1] Sara A. Safari, Christof Schmidhuber, Trends and Reversion in Financial Markets on Time Scales from Minutes to Decades, arXiv:2501.16772

[2] Sara A. Safari, Christoph Schmidhuber, Trends, Volatility, Correlations, and Critical Phenomena in Financial Markets, arXiv:2606.20145

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How 0DTE Options Are Reshaping Market Dynamics

Market dynamics have changed since the COVID-19 pandemic, driven by several structural developments. One of the most notable has been the rapid growth of retail trading, particularly in short-dated options.

Reference [1] examines the impact of zero-days-to-expiration (0DTE) options on market dynamics using trade-level Cboe data for SPY, QQQ, and IWM options from 2012 to 2025. The study investigates how the introduction and expansion of 0DTE trading have altered option market behavior. The authors pointed out,

.. We find that the expansion of daily expirations is associated with a significant increase in ultra-short-dated (0–1DTE) trading, accompanied by evidence of maturity substitution away from both short-term (2–5DTE) and medium-term (6–30DTE) options. This shift is associated with a marked increase in directional trading intensity, particularly following Tuesday and Thursday expirations, as well as a rise in small trade activity consistent with increased retail participation. We also document a reflexive relationship between ultra-short-dated trading and directional behavior, suggesting that the growth of 0DTE options both reflects and reinforces speculative trading dynamics. We also find that the expansion of short-dated trading is associated with increased market maker intermediation, indicating greater hedging and liquidity provision demands in these markets. Additionally, we use ordinary least squares and instrumental variables regression strategies to estimate the relationship between expected volatility and increased trading volume in short-term (0–6DTE) options relative to long-term (≥7DTE) options. We find evidence that a higher percentage of short-term options volume is associated with a higher level of volatility expectations at a 30-day time horizon over our data period (2012–2024)...

In short, the paper concludes that weekly expirations have significantly increased 0–1DTE trading, shifting activity away from longer-dated options through a maturity-substitution effect. It also finds that directional trading and retail participation have increased, with 0DTE activity and directional trading reinforcing each other through a reflexive feedback mechanism. The paper shows that greater short-dated trading is associated with higher 30-day implied volatility and a flatter 7-day/30-day term structure.

These findings provide valuable insights into the evolving options market. The analysis of option trading activity by day of the week is particularly interesting as it may help explain the day-of-the-week seasonality observed in the options market.

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

References

[1] Dalvi, S., and LaFond, H., Effects of Weekly Option Introduction on Market Participant Behavior and Volatility Expectations. Working paper, 2026.

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Extending the Dealer Gamma Exposure Framework

Dealer Gamma Exposure (GEX) is an interesting concept that has gained attention in recent years and is now used by both retail and institutional traders. Reference [1], however, estimates that GEX calculations can have an error margin of 30% to 50%. The paper identifies several sources of GEX estimation error, notably,

• Dealer-position sign assumptions,
• Using a single ATM IV instead of the full volatility surface,
• Ignoring intraday (especially 0DTE) flows.

It also discusses situations where GEX may fail, including sovereign crises, physical commodity squeezes, geopolitical commodity shocks, and currency-peg or intervention regimes.

The author pointed out,

Dealer gamma exposure (GEX) has become the dominant retail and semi-institutional framework for interpreting equity index dynamics. This paper argues that GEX, while analytically valuable in stationary regimes, carries a quantifiable 30--50% error margin and fails systematically in the four market configurations that generate the largest dislocations: sovereign crises, physical commodity squeezes, geopolitical commodity shocks, and currency peg breaks. We propose a four-lens framework that retains GEX as a baseline (Lens 1: Gamma) and augments it with three additional dimensions: (ii) Vega Exposure, which identifies the structural short-volatility positioning that precedes regime ruptures through the five-step short-vol unwind mechanism; (iii) Risk Reversal (25-delta), which reads directional fear and greed through the options skew and produces the framework's single most important operational rule --- when RR contradicts GEX, follow RR; and (iv) Term Structure and Physical Signals, which integrates VIX forward curve dynamics, commodity lease rates, inventory drawdowns, and backwardation patterns to detect dislocations invisible to listed options data. The framework's decision rule is confluence: when three of four lenses align, the signal is actionable. Sizing follows a Bayesian Kelly criterion (Sukhov, 2026). The framework is validated against five publicly documented, time-stamped calls made by the author's research desk on the social platform X (formerly Twitter) prior to major market dislocations…

In short, the paper proposes an extension of the GEX framework. The original gamma exposure calculation is retained, but the framework is expanded to incorporate vega exposure, 25-delta risk reversal, term structure, and physical-market signals.

Under this broader framework, the author presents five documented market calls from August 2024 to February 2026, each posted publicly before the relevant event, including the August 2024 VIX spike, DeepSeek/NVDA, Iran-Hormuz oil, the dispersion break, and the silver squeeze.

GEX remains an interesting concept, and this paper provides a useful refinement of the framework, although the sample size is small. Let us know what you think in the comments below or in the discussion forum.

References

[1] Djouad, D., Beyond Dealer Gamma: A Four-Lens Framework for Reading Options Market Dislocations. CrossVol Research Working Paper. MPRA Paper No. 129365, 2026.

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Multi-Agent LLM Systems for Trading

Large language models (LLMs) are evolving rapidly, from early text-generation systems to the latest models capable of sophisticated reasoning and multi-step decision-making. Reference [1] applies LLMs to the development of a gold trading system. Unlike traditional quantitative models, which are typically built around forecasting future prices or returns, the authors use LLMs as a committee of analysts rather than as predictive models.

The objective is to test whether structured LLM reasoning can generate better trading decisions than conventional indicator-based approaches. Specifically, the framework consists of three agents responsible for data analysis, risk assessment, and trade decision-making. The authors employ Chain-of-Thought prompting and evaluate model performance across temperature settings ranging from 0.0 to 1.0. They pointed out,

In this study, we analyzed whether LLM-assisted trading strategies could achieve improved performance compared to traditional indicator-based strategies when applied to historical gold market data. More specifically, we examined how LLM-assisted strategies performed compared to baseline strategies under the same market conditions.

…The results suggest that LLM-assisted strategies outperformed the baseline strategies in terms of total return. The LLM-based strategies achieved returns ranging from 33.07% to 61.17%, while the best-performing baseline strategy reached 17.53%. Additionally, all LLM-assisted strategies achieved a Sharpe ratio above 1.0, whereas none of the baseline strategies surpassed that threshold. The temperature setting t = 0.5 resulted in the best overall performance and achieved the highest values across most performance metrics. This indicates that a moderate level of randomness in the LLM output may contribute positively to trading performance.

In short, the paper finds that all LLM-based strategies outperform the traditional technical-indicator benchmarks in terms of total return, while achieving Sharpe ratios above 1.0.

Although the study has several limitations, including a single asset, only 294 trading days of data, the absence of transaction costs, no incorporation of news or sentiment inputs, and no out-of-sample tests, it nevertheless offers an interesting direction for trading-system development. Rather than serving solely as forecasting engines, LLMs may be used as autonomous agents or as specialized decision-making modules within a broader trading framework.

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

References

[1] Phillips, A., & Emilsson, V. (2026), LLM-Assisted Gold Trading: Evaluating Reasoning-Based Strategies Against Technical Indicator Baselines, Bachelor's thesis, Department of Computer and Systems Sciences, Stockholm University.

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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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