Impact of Spot-Volatility Correlation on Option Returns

It is well known that equity indices tend to exhibit a negative correlation with their volatility. There is some research on this relationship, often linking it to the leverage effect. Recently, we observed that the equity market has been behaving unusually, with the spot-volatility correlation turning positive. It is therefore timely to revisit research on this relationship.

Reference [1] studies how changes in the spot-volatility correlation affect expected option returns. The study uses SPX options from 1996 to 2023, realized variance calculated from 5-minute S&P 500 futures data, and a 30-day rolling return-variance correlation. The authors formulate the research problems using the Heston option pricing model under the physical measure. They pointed out,

This paper examines how the leverage effect affects expected index option returns. We rely on the Heston (1993) stochastic volatility framework and the exponential-affine pricing kernel in Heston et al. (2024b) to characterize the relation between the return-variance correlation and expected index call and put returns…

The theoretical analysis predicts a negative (positive) relation between the market leverage effect and call (put) expected returns, and the magnitudes of these relations increase for out-of-the-money contracts for calls. We confirm these theoretical predictions using a long sample of weekly S&P 500 index option returns. The estimated signs are robust across alternative formation days, the exclusion of expiration weeks, longer maturities, and a monthly holding period aligned with the option expiration cycle, and the results remain statistically significant.

In short, the paper concludes that,

• When correlation increases, i.e., becomes less negative, expected call returns decrease and expected put returns increase. Equivalently, a more negative correlation raises expected call returns and lowers expected put returns;
• For calls, the effect becomes substantially stronger OTM; for puts, it becomes weaker OTM.

This article tackles a relatively underexplored but important topic. An interesting irony is that the correlation can become positive when investors aggressively buy out-of-the-money call options as the equity market rallies, thus increasing volatility and causing the correlation to turn positive. However, by doing so, they also reduce the expected returns on their calls.

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

References

[1] Driessen, J., Jeon, J., and Sun, Y. (2026), The Leverage Effect and Expected Option Returns, Working Paper

Article Source Here: Impact of Spot-Volatility Correlation on Option Returns

source https://harbourfronts.com/impact-spot-volatility-correlation-option-returns/

Effectiveness of Volatility-Based Exit Rules

In the trading literature, we often see discussions about entry rules, but much less about exits. Some practitioners claim that exits are more important than entries. Is this really the case?

We have discussed how the effectiveness of exit rules depends on market conditions. Reference [1] continues this line of research by examining whether volatility-based take-profit/stop-loss rules improve a technical trading system in USD/JPY.

The author uses daily USD/JPY data from 2022 to 2025, with 2022–2023 treated as in-sample and 2024–2025 as out-of-sample. The system uses MACD crossovers to determine long/short entries and reversals, while ATR thresholds determine take-profit and stop-loss exits.

The paper pointed out,

This study examined ATR-based take-profit and stop-loss rules as components of adaptive risk-management design in algorithmic trading systems. The objective was not simply to evaluate the effectiveness of volatility-adaptive exit rules, but to identify the conditions under which interactions between exit-rule design, trading-model structure, and market conditions contribute to robust system performance. The findings demonstrate that the effectiveness of such rules depends on the interaction between exit-rule design, trading-model structure, and market conditions…

In conclusion, the findings demonstrate that the effectiveness of ATR-based exit rules depends critically on their compatibility with underlying model structures and market conditions. Rather than generating universally positive effects, ATR-based exit rules improve performance only under specific conditions characterized by compatibility between appropriately specified trading models, suitable TP/SL multiplier combinations, and prevailing market dynamics. In this context, volatility-adaptive exit rules function primarily as a complementary mechanism that reinforces trading structures rather than independently generating superior outcomes.

In short, the paper concludes that,

• The effectiveness of ATR-based exit rules depends on the trading model and market conditions,
• Volatility-based exit rules complement and reinforce existing trading structures rather than independently generating superior performance.

This paper emphasizes again that profit targets and stop-loss exits are not always beneficial; their effectiveness depends on the market regime.

Last but not least, do not simply accept claims that "exits are more important than entries." As always, test and verify with data.

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

References

[1] Kang, B.-K. (2026), Conditional effectiveness of volatility-adaptive exit rules in algorithmic trading systems: Evidence from the USD/JPY market. Journal of Risk and Financial Management, 19, 554.

Article Source Here: Effectiveness of Volatility-Based Exit Rules

source https://harbourfronts.com/effectiveness-volatility-based-exit-rules/

Retail Participation in the 0DTE Options Market

Retail options trading has become an important force in today's financial markets, particularly with the rapid growth of short-dated options trading. As a result, researchers have increasingly focused on understanding retail trading behavior. A previous study we discussed found that retail traders tend to buy short-dated options, especially out-of-the-money contracts, while frequently selling long-dated options.

Along this line of research, Reference [1] examines retail trading behavior in SPX 0DTE options. The authors utilize transaction-level OPRA data for SPX/SPXW options from May 19, 2022, to January 31, 2026, primarily from CBOE. They pointed out,

This paper documents a new and economically important feature of retail participation in options markets: synchronization driven by algorithmic execution. In the market for SPX 0DTE options, retail trading is not only large but highly structured in time, strategy, and execution. The resulting intraday patterns are sharp, stable, and predictable, and they differ qualitatively from the behavior emphasized in much of the existing retail trading literature.

Our central empirical finding is a pronounced intraday periodicity in SPX 0DTE trading activity. Trading volume and trade counts spike at fixed clock times, with bursts arriving abruptly at the second level and dissipating quickly thereafter. These patterns intensify over time and are overwhelmingly concentrated in complex, ultra-short-maturity option strategies. Simple trades and longer-dated options display little comparable periodicity.

We show that these spikes are associated with a distinct composition of trades. At deterministic times, trading shifts toward small, complex, short-premium strategies with standardized geometry.. . The evidence suggests that many independent traders deploy similar strategy templates and timing rules supplied or encouraged by retail-facing fintech platforms.

In short, the paper finds that,

• Trade counts and volume spike sharply at exact hour and half-hour marks, including 10:00, 10:30, 13:00, 13:30, and 14:00;
• Activity jumps almost instantaneously and largely dissipates within 30–60 seconds;
• The spikes are concentrated in 0DTE contracts and driven by complex multi-leg orders;
• The dominant strategies are short put verticals, short call verticals, and short iron condors, rather than long-call lottery bets.

The authors interpret these findings as evidence that fintech platforms, APIs, no-code bots, templates, defaults, and margin constraints synchronize otherwise independent retail accounts.

The paper also emphasizes the importance of incorporating retail order flow into market analysis, given its growing influence on market dynamics,

Taken together, the evidence suggests that the rise of retail algorithmic trading represents a structural change in options markets. As retail traders increasingly operate through rules and templates, understanding market behavior requires attention not only to trader characteristics but also to the technological and institutional features that govern how trades are generated and executed.

It is interesting to note that, while previous research found that retail traders tend to buy short-dated options, this paper points in the opposite direction: retail traders tend to sell options, but only at specific times during the trading day.

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

References

[1] Garcia-Ares, P. A., Amaya, D., Pearson, N. D., & Vasquez, A. (2026), The rise of algorithmic retail option traders, SSRN 6480379

Article Source Here: Retail Participation in the 0DTE Options Market

source https://harbourfronts.com/retail-participation-0dte-options-market/

Optimization in the Indicator and Parameter Space

In today's era of supercomputers, artificial intelligence (AI), and machine learning (ML), many tasks that once required hours or even days can now be completed in a fraction of a second.

Parameter optimization is a common technique in systematic trading in which the parameters of a trading indicator or strategy are varied to improve its historical performance. Reference [1] extends this idea by optimizing not only the parameters of individual indicators, but also the selection of the indicators themselves. In other words, the optimization is performed in both the indicator space and the parameter space.

Specifically, the paper proposes MADTIP, a memetic algorithm that combines a genetic algorithm to select indicator combinations with Simulated Annealing (SA), the Firefly Algorithm (FA), and Particle Swarm Optimization (PSO) to optimize indicator parameters. The framework uses indicators from four categories: trend, momentum, volatility, and volume.

Active indicators vote on the trading signal, and a trade is executed when more than 50% generate the same buy or sell signal. The method is evaluated on daily data from 2015 to 2023 covering Taiwanese equities, Apple, Nvidia, SPY, Bitcoin, and Ethereum.

The authors pointed out,

This study presented and validated the Memetic Algorithm with Diverse Technical Indicator Pool (MADTIP), a novel framework designed to enhance the robustness of automated trading strategies. By integrating a diversity preservation mechanism into the trading strategy optimization process, the proposed framework addresses the critical limitation of premature convergence and unadaptable often observed in static-pool trading strategy optimization methods. The experimental results, conducted across a rigorous walk-forward validation period covering the 2020 COVID-19 market crash, support three key findings. Firstly, the ablation study confirmed that the inclusion of indicator diversity is not merely additive but produces resilient trading strategies… Lastly, the proposed MADTIP-SA framework consistently outperformed both the passive Buy-and-Hold strategy and the Random Forest baseline. Most notably, in high-volatility assets such as Bitcoin, the framework achieved superior capital preservation, reducing volatility by over 90% compared to the ML baseline while maintaining competitive profitability.

In short, the paper concludes that the MADTIP framework produces superior trading performance, primarily because indicator diversity improves adaptability and risk control.

This paper is another example of how modern computing has greatly expanded the scope and speed of quantitative strategy development. However, the increased flexibility also substantially increases the risk of overfitting.

In this study, the validation framework is not sufficiently rigorous.  For example,

• The walk-forward validation, which uses two-year rolling training windows followed by one-year out-of-sample tests, is relatively simplistic;
• The paper provides little discussion of data-snooping bias, multiple-testing issues, or the robustness of the selected hyperparameters.

Overall, the use of advanced optimization techniques to assist strategy design is promising and has great potential, but stronger validation frameworks are needed before such methods can be considered robust.

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

References

[1] Chen, C.-H., Chideme, K., Huang, Y.-L., & Hong, T.-P. (2027). A hybrid memetic optimization framework with the diverse technical indicator pool for finding adaptive and profitable trading strategies. Expert Systems with Applications, 332, 133666.

Article Source Here: Optimization in the Indicator and Parameter Space

source https://harbourfronts.com/optimization-indicator-parameter-space/

Evaluating the Performance of AI-Powered ETFs and AI/ML Stocks

Artificial intelligence has become a major focus in finance and trading. It is often portrayed as a transformative technology capable of consistently generating trading profits.

But is this really the case?

Reference [1] takes a critical look at the performance of AI-powered ETFs, AI-focused ETFs, and AI/machine learning (AI/ML) stocks. The authors examine monthly total returns of 21 AI-powered ETFs, 44 AI-focused ETFs, and 42 AI/ML stocks from January 2005 to March 2025. They pointed out,

This study compares the performance of AIPs, AI-focused thematic ETFs, AIML, and broad technology stocks using monthly data from 2005 to 2025. Combining asset-pricing models, volatility estimation, rolling performance measures, quantile regressions, and multivariate panel analysis, we examine how these portfolios differ in return generation, risk-adjusted efficiency, tail behavior, and their association with an investor-attention proxy.

Across methods, a consistent contrast emerges. AIMLs deliver the strongest cumulative performance and the largest standalone abnormal returns; however, they also exhibit substantially higher volatility, deeper drawdowns, and less stable exposure patterns. AIPs, in turn, exhibit lower market exposure, smoother volatility dynamics, and higher Sharpe-type efficiency, positioning them as a stability-oriented channel of exposure to the AI theme...

Overall, this paper shows that AIPs, as an observed investment class, exhibit more efficient risk conversion than comparator portfolios. Their comparative strength is visible in lower beta, smoother volatility dynamics, milder downside deterioration, and higher return per unit of realized volatility. The evidence does not identify a clean causal AI implementation effect or a distinct AI-generated alpha premium. Rather, it shows that AIPs provide a stability-oriented route to AI exposure. In contrast, direct AI/ML equity exposure offers higher upside but is less stable, and AI-focused thematic ETFs occupy an intermediate, more state-sensitive position.

In short, the paper concludes that AI/ML stocks offer the highest upside and abnormal returns but also the greatest volatility and drawdown risk. AI-powered ETFs' main advantage lies in risk reduction, including lower beta, smoother volatility, smaller downside deterioration, and higher returns per unit of realized risk, rather than in a proven causal effect of AI or persistent AI-generated alpha.

One interesting, and important, conclusion is that AI itself is not a direct alpha-generating engine. Instead, it contributes to performance indirectly through improved risk management, volatility targeting, and portfolio construction.

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

References

[1] Sovbetov, I., Hatipoglu, Y. Z., & Can, E. N. (2026). Risk-adjusted performance of AI-powered portfolios: Evidence on volatility, beta, and returns. Borsa Istanbul Review.

Article Source Here: Evaluating the Performance of AI-Powered ETFs and AI/ML Stocks

source https://harbourfronts.com/evaluating-performance-ai-powered-etfs-ai-ml-stocks/

Improving Momentum with a Volatility Regime Filter

Regime classification plays an important role in portfolio management. There are several approaches to detecting market regimes, ranging from simple to highly sophisticated methods. Reference [1] proposes a simple regime classification framework for momentum trading.

The paper utilizes an expanding-window percentile rule applied to realized market volatility. Realized volatility is computed from daily market returns as the rolling standard deviation over a 21-trading-day window. On each trading day, the prevailing volatility is compared with the 75th percentile of the expanding distribution of all realized volatility observations through the prior day. Days above the threshold are classified as turbulent, while the remainder are classified as calm.

In calm regimes, the strategy maintains full exposure to the momentum factor with a weight of one. In turbulent regimes, it reduces momentum exposure to zero and holds cash earning the risk-free rate on the full notional. There is no fractional exposure or interpolation between states.

The author pointed out,

The evidence points in a specific direction. On risk-adjusted return, the conditioned strategy holds an edge that is economically meaningful but statistically marginal, with an excess-return Sharpe ratio of 0.837 against 0.592 for the unconditional strategy, significant only at the 10% level. On tail risk, the evidence is far clearer. The conditioned strategy reduced maximum drawdowns sharply and consistently across five crisis episodes spanning five decades, with the widest gap during the Global Financial Crisis, where it lost 5.1% against the unconditional strategy's 57.1%. A subperiod decomposition locates the advantage almost entirely within crisis-containing periods. We read these findings together as showing that regime-conditioning behaves as a form of crash insurance. It carries a small cost during calm markets and pays off during turbulent ones, which explains why a full-sample average understates its value and why a measure as direct as maximum drawdown reveals it more clearly than the Sharpe ratio.

In short, the paper concludes that implementing this regime filter improves risk-adjusted returns, with a particularly significant reduction in drawdowns.

An interesting aspect of the paper is that the author interprets the volatility-managed momentum strategy as a form of crash insurance: it carries a small cost during calm periods and pays off during turbulent ones, rather than as an alpha-generating strategy, in contrast to some broad claims in the volatility-managed literature.

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

References

[1]  Fabian, S. W. (2026). Statistical Evaluation of Momentum, Mean Reversion, and Regime Dynamics in U.S. Equity Markets (Working paper). Eastern Illinois University.

Originally Published Here: Improving Momentum with a Volatility Regime Filter

source https://harbourfronts.com/improving-momentum-volatility-regime-filter/

Do Short-Dated Options Lead the Underlying Market?

Options volume has increased dramatically in recent years, particularly in very short-dated options (0DTE and 1DTE). One theory suggests that information extracted from option trading activity can predict future stock returns.

Along these lines, Reference [1] examines whether short-dated option prices lead stock prices. It analyzes 2025 data covering 30 U.S. underlyings and 77.44 million consecutive short-dated option trade pairs. The authors define an event as an empirical delta that violates the Black-Scholes-Merton delta bounds and treat these violations as a screen for option-stock desynchronization. The paper pointed out,

Short-dated option trade pairs that violate the frictionless delta bound contain information about where the underlying stock moves next, but the relevant horizon is measured in seconds. The evidence survives independent WRDS TAQ NBBO reconstruction and matched controls, appears in both 0DTE and non-0DTE contracts, and is consistent with hedge-pressure effects. A continuous version of the screen shows that the result is not an artifact of the discrete bound: the same-direction response grows smoothly with the distance between the empirical and model delta, and is already present for in-bound deviations that never cross the bound. The effect is far smaller than stock-side crossing costs, so it should be interpreted as price discovery rather than an outside trading strategy. The contribution is to show that option-to-stock price discovery can occur at the first-second horizon.

In short, the article concludes that short-dated option prices can lead stock quotes by about one second, but the economic effect reflects price discovery rather than exploitable profits after transaction costs. The lead is explained by hedging pressure, i.e., when inferred hedge pressure aligns with the event, continuation is positive, and when it is opposed to the event, continuation is negative.

The findings debunk the claim that short-dated option prices provide a tradable lead over the underlying stocks.

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

References

[1] Willeboordse, F. H. (2026). Seconds of price discovery: Evidence from short-dated options. Economics Letters

Originally Published Here: Do Short-Dated Options Lead the Underlying Market?

source https://harbourfronts.com/short-dated-options-lead-underlying-market/

Volatility Timing: Does It Really Add Value?

Volatility timing is the practice of adjusting portfolio exposure in response to changes in market volatility. Some practitioners regard volatility timing as one of the most effective tools in portfolio management. However, not everyone shares this view.

We have previously discussed how the effectiveness of volatility timing varies across industries and depends on several factors. Reference [1] goes deeper into this topic by examining 153 U.S. equity long-short factors over the 1972–2024 period. The paper investigates whether volatility-managed factor portfolios outperform their static counterparts and identifies the factor characteristics associated with any performance improvements. It examines several cross-sectional drivers, including volatility, skewness, excess kurtosis, maximum drawdown, downside volatility, and volatility persistence.

The authors pointed out,

This paper has examined whether volatility timing improves risk-adjusted returns across factor portfolios and sought factor characteristics which could explain its effectiveness. The results provide a nuanced answer to both questions. While volatility timing delivers modest improvements in performance on average, these gains are not consistently statistically significant, suggesting that it is not a universally reliable enhancement to factor investing.

At the same time, the analysis reveals substantial and systematic variation in timing performance across factors. This variation is not random, but closely linked to underlying risk characteristics, particularly downside risk and return asymmetry. Factors with greater exposure to adverse states such as momentum benefit more consistently from volatility management, indicating that the effectiveness of volatility timing depends on how risk is distributed across different market conditions rather than on its overall level.

The main contribution of the paper is therefore to clarify when volatility timing is most effective, rather than whether it works in general. By showing that timing gains are driven by downside and tail-related risks, the results provide a coherent explanation for the mixed evidence in the existing literature. It suggests that volatility timing should be applied selectively, focusing on factors whose risk is concentrated in adverse states, rather than uniformly across all factor portfolios.

In short, the paper concludes that volatility timing provides,

• Modest Sharpe improvement; however, the improvements are generally not statistically significant across factors;
• More consistently positive alpha. Alpha improvements are greater but still not universal;
• Lower realized volatility, not higher returns; however, the benefits differ substantially across factor themes.

The findings suggest that the benefits of volatility management are not universal but depend on the characteristics of the underlying factors. It works best for factors whose risk is concentrated in adverse downside states.

This article reminds us once again to question all claims in finance, and of volatility timing in particular.

References

[1] Sannerholm, F., & Jogdal, J. (2026). Is there an edge in volatility-managed portfolios? If so, where is it? University of Gothenburg, Graduate School, School of Business, Economics and Law.

Article Source Here: Volatility Timing: Does It Really Add Value?

source https://harbourfronts.com/volatility-timing-really-add-value/

Explaining the Decline of Trend-Following CTAs

Regime change has become a recurring theme across financial markets. One of the most recognizable examples is the deterioration in the performance of trend-following Commodity Trading Advisors (CTAs) after the Global Financial Crisis.

Reference [1] investigates the reasons behind this shift using data from approximately 100 liquid futures contracts spanning 1995–2025, together with a CTA proxy. The paper evaluates four competing hypotheses,

1. Capacity constraints;
2. The electronification of futures markets;
3. A structural shift in the interaction between CTA trades and aggregate order flow; and
4. A microstructural mechanism whereby liquidity offered to trend followers has dried up.

The authors pointed out,

The interpretive frame we propose takes the self-fulfilling impact loop – signal → trade → impact → reinforced signal – as the mechanism through which trend exists in the first place. Trend followers do not merely harvest a pre-existing anomaly: their aggressive directional flow, mediated by impact, sustains the very price patterns they trade on. The loop has two preconditions: that aggressive execution be feasible at reasonable cost, and that the relationship between aggressive flow and price remain intact. Both held until roughly 2010 across the futures universe; since then, both have been compromised on small-tick contracts and preserved on large-tick ones.

The mechanism behind this asymmetry is the post-2008 transition to HFT-dominated market making, which replaced a generation of inventory-tolerant liquidity providers with one whose business model is structurally incompatible with absorbing predictable directional flow. The resulting liquidity withdrawal in front of CTA orders operates in both tick-size tiers, but its consequences are asymmetric. On dense large-tick books, residual depth at the best quotes and at deeper levels remains sufficient for execution to proceed largely unperturbed; the loop continues to turn, and both the signal and the PnL survive. On sparse small-tick books, withdrawal removes the residual depth that previously supported fast trend execution, forcing trend followers either to “walk the book” aggressively or to retreat from the contract altogether. As they retreated, the loop broke on its input side: the impact-mediated reinforcement of nascent trends disappeared along with the flow that produced it.

In short, the authors examine trend following through the lens of a momentum feedback loop. They reject the first three hypotheses and conclude that volatility-normalized tick size is the key factor distinguishing markets where trends persist from those where they collapse.

The paper argues that HFT-dominated market making disrupted this feedback loop, primarily in small-tick markets, by withdrawing liquidity in the presence of predictable directional order flow.

This study provides valuable insights into the mechanisms underlying trend following and the structural factors driving its changing performance.

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

References

[1] Kurth, J. G., Eisler, Z., Rej, A., & Bouchaud, J.-P. (2026). Is Trend Still Your Friend? A Microstructural Account of the Demise of Short-Term Trend-Following. arXiv:2607.01550

Article Source Here: Explaining the Decline of Trend-Following CTAs

source https://harbourfronts.com/explaining-decline-trend-following-ctas/

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

Post Source Here: Closed-Form Option Pricing Under the Physical Measure

source https://harbourfronts.com/closed-form-option-pricing-physical-measure/