Delta Hedging Under Fractional Brownian Motion

The Black–Scholes–Merton (BSM) model is the most frequently used option pricing framework in finance. However, it relies on simplifying assumptions, some of which are not realistic. Ongoing efforts aim to extend and generalize the BSM model, and Reference [1] represents a recent contribution in this direction.

The paper proposes an option pricing model in which the underlying asset follows Fractional Brownian Motion (FBM) rather than Geometric Brownian Motion. It not only derives pricing formulas for call and put options, but also introduces a hedging strategy based on FBM. Specifically, the strategy consists of buying a call option and dynamically delta-hedging it using deltas computed from the proposed fractional model.

The authors pointed out,

This study provides a comprehensive empirical evaluation of dynamic delta hedging strategies under fBm compared to standard Bm, focusing on two major financial crises: the 2007–2008 global financial crisis and the COVID-19 recession. While Bm assumes memoryless price movements, fBm incorporates long-range dependence, making it a potentially more robust framework for modeling asset dynamics during turbulent market conditions.

Using historical data from the S&P 500 and NYSE indices, the research estimates the Hurst exponent to calibrate fBm models and assess their predictive and hedging performance. The findings reveal that fBm generally improves prediction accuracy and significantly enhances hedging efficiency, especially when the Hurst exponent is close to or above 0.5. Notably, fBm-based strategies reduce downside risk, eliminate negative P&L outliers, and offer more consistent returns during and after crisis periods.

However, the study also uncovers that when the Hurst exponent falls below 0.5-as observed in the NYSE post-COVID—fBm may introduce greater volatility and tail risk, underscoring the importance of careful calibration. A key insight is the distinction between forecasting accuracy and hedging effectiveness: even when predictive gains are marginal, fBm’s memory-sensitive structure can lead to superior risk mitigation.

In short, the paper finds that FBM-based hedging improves prediction accuracy and, more importantly, enhances hedging efficiency and downside risk control when the underlying asset is trending (Hurst exponent is near or above 0.5), but volatility and tail risk may increase when the asset exhibits mean-reverting behavior (H < 0.5).

The results are somewhat counterintuitive. In practice, delta hedging is generally understood to add value primarily in mean-reverting regimes, yet the study suggests that delta hedging is effective when the underlying asset exhibits trending behavior.

We also note that the analysis is conducted using simulated option prices. It would be useful to evaluate this hedging strategy using real, empirical data. Nonetheless, this is an important study and [glossary_exclude]warrants [/glossary_exclude]further investigation.

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

References

[1]  Dufera, T.T., Kesto, D.A. & Legesse, T.S. Dynamic Delta Hedging During Crises: Fractional Brownian Motion in Action. Comput Econ (2025).

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When More Information Hurts: Social Media and Investor Underperformance

In today’s digital era, social media is ubiquitous, and market participants are exposed to an unprecedented volume of financial information. While investors are arguably more informed, it remains an open question whether this translates into better and more disciplined decision-making.

A recent incident illustrates this concern: a group of online traders, including both professionals and retail investors, followed an online influencer known as “Captain Condor” and collectively lost approximately USD 50 million around Christmas, with many losing their entire life savings; compounding these losses, members also paid annual fees exceeding USD 5,000 to gain access to the social media guru’s group.

This episode again raises a fundamental question: does more information necessarily lead to better judgment, and how is social media reshaping the investment landscape?

Reference [1] examines this issue formally, using StockTwits data from 2013 to 2022 to analyze the impact of social media activity around earnings announcements.

The authors pointed out,

Focusing on earnings announcements, I find that social media information is overly optimistic and fails to predict fundamentals. This social media attention is associated with systematic noise trading, specifically retail buying pressure from retail investors in the equity market, which generates a 58 bps price drift ahead of earnings announcements. Such systematic noise trading worsens price revelation, undermining price informativeness ahead of earnings announcements…

Overall, these findings reveal that social media-driven optimism can lead retail investors to trade systematically in the equity and options market, leading to reduced price efficiency and potentially harming their wealth.

In short, the paper finds that social media information around earnings announcements is overly optimistic and fails to predict fundamentals. This behavior worsens price discovery, reduces price efficiency in both equity and options markets, and can ultimately harm retail investors’ wealth.

In conclusion, increased information flow does not automatically result in better decision-making, underscoring the importance of critical thinking, judgment, and disciplined analysis in the presence of abundant information.

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

References

[1]  Edna Lopez Avila, Charles Martineau, and Jordi Mondria, Social Media and the Distortion of Price Revelation, SSRN 4439793, 2025

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Multifractality and Market Efficiency Across Asset Classes

The Fractal Market Hypothesis (FMH) is increasingly studied and applied by both finance academics and practitioners. We previously discussed the use of Detrended Fluctuation Analysis to estimate the Hurst exponent for major cryptocurrencies.

Continuing this line of research, Reference [1] applies Multifractal Detrended Fluctuation Analysis (MFDFA) to examine cryptocurrency, commodity, foreign exchange, and equity markets, specifically Bitcoin, Ethereum, crude oil, gold, EUR/USD, USD/JPY, the S&P 500, and the BIST 100 using data from January 1, 2018 to December 19, 2022.

In addition to the standard Hurst exponent used in prior studies, the paper also incorporates the q-th order fluctuation function to generalize the Hurst exponent. The scaling behavior of the resulting h(q) function serves as a measure of multifractality.

The authors pointed out,

The findings reveal that BTC, ETH, crude oil, and BIST 100 exhibit h (2) > 0.5, indicating long memory and persistent structure. These time series deviate from the mean and exhibit autocorrelations with trends and cycles, suggesting that increases or decreases in past periods are likely to continue in the future. In contrast, gold, EUR/USD, USD/JPY and S&P 500 have h (2) less than 0.5, implying short memory and mean-reverting behavior. These variables show negative autocorrelation, where increases or decreases in the past are often followed by opposite movements in the future. Importantly, none of the analyzed time series follow a random walk, indicating that past returns could be used to predict future returns…

The degree of multifractality varies significantly across the variables. EUR/USD has the lowest degree of multifractality, followed by USD/JPY, S&P 500, crude oil, ETH, gold, BTC, and BIST 100. Among the markets, the foreign exchange market demonstrates the lowest degree of multifractality, while the cryptocurrency and commodity markets exhibit higher degrees. Within the stock market, S&P 500 displays low multifractality, but BIST 100 stands out as the variable with the highest degree of multifractality among all analyzed. The findings further reveal that as the degree of multifractality increases, so does the associated risk.

In short, the paper finds that none of the analyzed assets follow a random walk, implying a predictable structure in returns. It shows that cryptocurrencies and commodities exhibit stronger persistence and higher multifractality than foreign exchange and developed equity markets, with BTC, ETH, crude oil, and BIST 100 displaying long memory, while gold, major FX pairs, and the S&P 500 exhibit mean-reverting behavior.

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

References

[1] Temel, F., Tuğay, O. Testing the Fractal Market Hypothesis Using MFDFA Across Multiple Asset Classes. Comput Econ (2025).

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Incorporating Momentum into Option Pricing Models

The Black–Scholes–Merton (BSM) model is a cornerstone of derivative pricing; however, it is not without limitations, and researchers continue to extend it. Reference [1] proposes an extension by incorporating intraday momentum into the BSM framework. This is achieved by introducing a drift term that represents intraday momentum, measured using a simple moving average of returns.

The model also adopts a modified Heston-type structure in which volatility follows a mean-reverting square-root process, allowing it to capture volatility clustering and remain consistent with empirical features such as volatility smiles. The momentum-driven drift adjustment influences the expected price path, while the stochastic volatility process models uncertainty around that path.

The authors pointed out,

In this study, we compute time-varying volatility using a Heston-type stochastic volatility model and incorporate the resulting volatility path into a modified Black-Scholes option pricing model with an additional momentum term in the drift component. This momentum term, derived from recent relative price changes, introduces a dynamic correction to the drift rate, reflecting short-term market sentiment and directional tendencies. By capturing intraday effects that classical BS models overlook, the framework enhances the realism of derivative pricing under rapidly changing conditions.

Our numerical results demonstrate that momentum significantly impacts option price trajectories. Specifically, under high positive or negative momentum values (e.g., Mt = ±2), deviations from the classical BS model become substantial. Positive momentum amplifies option prices over time, while negative momentum tends to attenuate them. These findings indicate that momentum is a non-negligible factor in option pricing and suggest that incorporating momentum can improve modeling fidelity in high-frequency or sentiment-sensitive trading environments.

In short, the paper introduces a momentum term based on recent price changes to dynamically adjust the drift, capturing short-term intraday effects. Numerical results show that strong positive or negative momentum leads to substantial deviations from standard BSM prices, indicating that momentum is an important factor in option pricing.

This represents an interesting and potentially useful extension of the BSM model for traders and risk managers. However, as noted by the authors, the findings are based on simulated results rather than empirical data, and it would be valuable to see the model tested on real market data.

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

References

[1] Hossain, M.S., Yuan, X. & Sultan, S. Momentum-Driven Option Pricing: Integrating Intraday Trends into Financial Derivative Models. Comput Econ (2025).

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Quantifying Recency Bias in Investor Volatility Expectations

Investors and traders often suffer from behavioral biases, which is where behavioral finance originates. Among these biases, recency bias is probably the most detrimental, yet it has infrequently been studied in a comprehensive quantitative manner.

Reference [1] addresses this gap by investigating recency bias in stocks with high idiosyncratic volatility (IVOL). The authors hypothesize that investors excessively extrapolate recent changes in volatility, particularly when high-IVOL stocks have become more volatile in recent periods relative to earlier ones, and they propose using changes in IVOL as a measure of recency bias.

The paper further develops a trading strategy to exploit this bias by buying low-IVOL stocks with declining volatility and short-selling high-IVOL stocks with increasing volatility. The authors pointed out,

We bring in the role of investors’ excess extrapolation associated with the recency bias as an explanation of the IVOL anomaly. We hypothesize that investors have higher tendency to excessively extrapolate past return volatilities of high IVOL stocks whose returns became more volatile in recent periods. The extrapolation bias accelerates investors’ preference for such stocks and further enhance the magnitude of overvaluation.

Accordingly, we form a recency-enhanced IVOL strategy to capture investors’ excess extrapolation to emphasize more on recent IVOL. We show that it generates significant and robust profitability. The other components of the standard IVOL strategy, which is referred to as the non-recency IVOL strategy, is mostly unprofitable. An implication to practitioners is that considering the recency effect is important when trading against idiosyncratic volatility. Our study also adds to the vast literature on the understanding of the IVOL anomaly. The implication to future studies examining the anomaly is that the role of recency biases should be considered when examining overvaluation of high IVOL stocks.

In short, the paper attributes the IVOL anomaly to investors’ recency bias, and by incorporating this recency effect, a recency-enhanced IVOL strategy generates significant profitability.

This is an important topic, with implications not only for stocks and volatilities but also for trading strategies themselves, as traders often abandon sound strategies due to recent poor performance. Further research in this area would be highly valuable.

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

References

[1] Wen-Chi Lo, Kuan-Cheng Ko, Recency biases and the idiosyncratic volatility puzzle, Finance Research Letters, Volume 91, March 2026, 109468

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Forecasting Market Crashes with Machine Learning Techniques

Predicting market direction is challenging, and forecasting market crashes is even more difficult, yet this remains a growing area of research. We previously discussed market correction prediction, and Reference [1] continues this line of inquiry by examining how machine learning can be used to predict market crashes within the Adaptive Market Hypothesis framework.

The study considers three categories of factors:

1. Internal factors, such as technical indicators designed to capture endogenous market dynamics, including momentum, trend strength, and money flow arising from investor behavior and adaptive learning;
2. External factors, including macroeconomic and commodity variables that proxy for systematic, exogenous risks affecting fundamental valuations; and
3. Volatility features that quantify market fear and uncertainty.

The authors evaluate the performance of three predictive models—logistic regression, random forest, and a long short-term memory (LSTM) network. They pointed out,

The findings of this thesis suggest that while market crashes remain inherently difficult to forecast with perfect accuracy, they are not entirely random events. Meaningful predictive signals do exist, but their detection requires a careful consideration of model choice and complexity. The primary conclusion is not that one model is universally superior, but that different models reveal different facets of predictability, presenting a practical trade-off for risk managers and investors.

The Logistic Regression model, with its high recall, serves as an excellent "earlywarning system." Its strength lies in its sensitivity; it is highly effective at flagging periods of potential danger, making it suitable for risk monitoring applications where the cost of a missed event is catastrophic. Its primary drawback is the high rate of false positives, which would make it costly to use as a direct trading signal.

The LSTM network, conversely, represents a more refined and balanced predictor. By matching the high recall of the logistic model while offering improved precision, it provides a more reliable signal. This suggests that incorporating the temporal dimension of financial data is a key avenue for enhancing predictive power. The practical implication is that while linear relationships capture the brute force of market panic, sequence modeling is required to understand the more subtle, evolving patterns that precede it. The choice between these models is therefore a strategic one, contingent on the specific application and the user’s tolerance for different types of error.

In short, the study concludes that market crashes are difficult to forecast but not entirely random, and different models capture different aspects of predictability. Logistic regression functions well as a high-recall early warning tool, while LSTM models provide more balanced signals.

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

References

[1] Michele Della Mura, Predicting Stock Market Crashes, A Comparative Analysis of Econometric and Machine Learning Models, Politecnico di Torino, 2025

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