Short-Term Stock Price Forecasting Using Geometric Brownian Motion

In these days of big data, machine learning, and AI, many researchers are showing growing interest in sophisticated models for stock price prediction or to refine basic models of stock dynamics. Reference [1] takes the opposite approach. It uses a classical model for stock price dynamics—the Geometric Brownian Motion (GBM)—and examines whether it can still be used to forecast stock prices. Specifically, the study applies four volatility measures to large-cap stocks in an emerging market to estimate volatility, then incorporates these estimates into the GBM to generate price forecasts.

The authors pointed out,

One effective method for forecasting short-term investment involves models like GBM. This study specifically applied GBM over a two-week period, focusing on the crucial aspect of volatility measurement. By examining four distinct volatility measurements, simple volatility (S), log volatility (L), high-low volatility (HL) and high-low-closed volatility (HLC), the findings indicate that simple volatility (S) yielded the closest forecast to actual stock prices, as evidenced in Table 3 and Figure 1.

Furthermore, the overall high accuracy of the forecasts generated by GBM, with most MSE, MAPE, and MAD values falling below 10% as shown in Table 4, confirms its potential as a valuable tool for short-term stock market forecasting. These results suggest that for investors and analysts focusing on short-term investment in the Malaysian stock market, utilizing GBM with a simple volatility measurement can provide a reasonably accurate basis for making timely trading decisions.

In short, and somewhat surprisingly, the simple GBM model combined with a basic volatility measure delivers the most accurate forecasts over short horizons of up to two weeks.

We note the following,

1. The forecast accuracy is limited to the short term,
2. Although four volatility measures are tested, the simplest performs best,
3. The analysis is conducted in an emerging market, and
4. The sample size is small.

Overall, this study runs counter to the current trend and suggests that simple models—both in volatility measurement and price dynamics—can still be effective. This is an interesting study and worth further examination.

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

References

[1]  FS Fauzi, SM Sahrudin, NA Abdullah, SN Zainol Abidin, SM Md Zain, Forecasting stock market prices using Geometric Brownian Motion by applying the Optimal Volatility measurement, Mathematical Sciences and Informatics Journal (2025) Vol. 6, No. 2

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Option Pricing with Quantum Mechanical Methods

It is well known that put options are often overpriced, especially in equities. The literature is filled with papers explaining this phenomenon. However, most research still relies on the Black-Scholes-Merton framework, where the underlying asset follows a Geometric Brownian Motion (GBM).

Reference [1] also addresses this question, but it departs from the usual framework by casting the problem into a model rooted in quantum mechanics. Essentially, the new approach proceeds as follows:

1. Start with a general stochastic process and solve it by converting the Fokker–Planck (FP) equation into the Schrödinger equation.
2. Introduce the delta potential and the Laplace distribution for the stock price.
3. Derive a closed-form solution for European put options within the context of quantum mechanics.

The authors pointed out,

To resolve the well-known overpriced put puzzle, we propose an option pricing model inspired by quantum mechanics. Starting from an SDE of stock returns, we convert the FP equation into the Schrödinger equation. We then obtain the PDF of stock returns and a closed-form solution of European options. Our model indicates that S&P 500 index returns follow a Laplace distribution with power-law decay in the tail. We demonstrate that our QM outperforms GBM-based models in describing S&P 500 index returns and their corresponding put option prices. Our results indicate that high put option prices in the market are close to fairness and can be accurately modeled via quantum approaches.

In short, the paper proposes a quantum-mechanics–inspired option pricing model that converts the Fokker–Planck equation into the Schrödinger equation, yielding both the return distribution and a closed-form solution for European options. The model shows that S&P 500 returns follow a Laplace distribution with power-law tails and that quantum methods outperform GBM-based models in explaining return dynamics and put option prices.

This is an interesting formulation of option pricing theory. We note that the framework operates in the physical world rather than under the risk-neutral measure. We believe it may have practical applications in option trading, particularly for traders who rehedge less frequently.

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

References

[1] Minhyuk Jeong, Biao Yang, Xingjia Zhang, Taeyoung Park & Kwangwon Ahn, A quantum model for the overpriced put puzzle, Financial Innovation (2025) 11:130

Originally Published Here: Option Pricing with Quantum Mechanical Methods

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Enhancing the Wheel Strategy with Bayesian Networks

The option wheel strategy is a systematic approach that combines selling cash-secured puts and covered calls. The process begins by selling puts on a stock the investor is willing to own; if assigned, the investor acquires the shares and then sells covered calls against the position to collect additional premium. The cycle repeats, though returns depend heavily on underlying volatility, assignment risk, and disciplined position management.

This is another popular options strategy among investors and was widely promoted by trading educators. However, experienced investors recognize that it suffers from the same drawback as the covered call strategy.

Reference [1] revisits the wheel strategy, but with a twist: it applies an LLM-based Bayesian network on top of the wheel framework. Essentially, this Bayesian network is used to characterize market regimes and guide position sizing and strike selection. The authors pointed out,

This paper introduces a novel model-first hybrid AI architecture that overcomes key limitations of using LLMs directly for quantitative financial decision-making, specifically in options wheel strategy decisions. Instead of employing LLMs as decision-makers, we use them as intelligent model constructors. This approach yields strong and stable returns with enhanced downside protection, achieving a Sharpe ratio of 1.08 and a maximum drawdown of -8.2%. The strategy delivers 15.3% annualized returns over 18.75 years (2007–September 2025), including volatile periods such as 2020–2022. Additionally, the model provides full transparency through 27 decision factors per trade… Our comprehensive baseline comparisons demonstrate the effectiveness of the model-first architecture. Pure LLM approaches yield 8.7% returns with a 0.45 Sharpe ratio. Static Bayesian networks achieve 11.2% returns and a 0.67 Sharpe ratio. Rules-based systems produce 9.8% returns with a 0.52 Sharpe ratio. In contrast, our hybrid approach attains 15.3% returns and a 1.08 Sharpe ratio, while maintaining superior risk management.

In short, using the LLM-based Bayesian network, the performance of the wheel strategy improved significantly.

We find the results appear unusually impressive and [glossary_exclude]warrant [/glossary_exclude]caution, but the underlying design and architecture are worth examining.

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

References

[1] Xiaoting Kuang, Boken Lin, A Hybrid Architecture for Options Wheel Strategy Decisions: LLM-Generated Bayesian Networks for Transparent Trading, arXiv:2512.01123

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Numerical Methods for Implied Volatility Surface Construction in Crypto Markets

The implied volatility surface is a fundamental building block in modern financial markets, as it underpins the pricing of both vanilla and exotic instruments and supports key risk-management functions such as hedging and scenario analysis. It has been modeled extensively in traditional finance; in crypto, however, few studies exist. Given the volatile nature of the crypto market, it is important to examine this area.

Reference [1] proposes a numerical method for reconstructing the implied vol surface of major cryptocurrencies: Bitcoin, Ethereum, Solana, and Ripple. The main steps are as follows:

1. Apply the Black-Scholes-Merton (BSM) equation,
2. Convert it into a discretized framework using the Finite Difference Method, and
3. Fit the resulting implied volatilities into bivariate polynomials.

The authors pointed out,

In this study, we developed and implemented a method for reconstructing smooth local volatility surfaces for cryptocurrency options by extending the generalized BS model with a bivariate polynomial volatility function. The proposed approach combines a finite difference method with an optimization routine to calibrate volatility surfaces from observed option prices. The resulting local volatility functions are smooth, flexible, and differentiable with respect to both the underlying asset price and time, which enhances their analytical tractability and practical usability…

Through extensive computational tests using [glossary_exclude]real option[/glossary_exclude] data from BTC, ETH, SOL, and XRP, we confirmed that the reconstructed local volatility surfaces successfully reproduce observed market prices across different maturities and strike ranges. In particular, the method captures the market phenomenon that volatility tends to increase as the underlying asset deviates from the spot level. Numerical comparisons showed that the model-generated prices closely matched the actual market prices, which highlights the effectiveness of the proposed calibration procedure. Our algorithm provides a tractable and robust methodology for approximating volatility surfaces in highly volatile crypto markets…

In short, the authors successfully develop a numerical procedure that accurately reconstructs the implied volatility surface for major cryptocurrencies.

This paper carries important practical relevance. However, we note the confusing terminology: the authors refer to their volatility surface as “local volatility,” which may be misleading, as it can be mistaken for the classical local volatility surface introduced by Derman, Kani, and Dupire. These two concepts are distinct.

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

References

[1] Yunjae Nam, Youngjin Hwang & Junseok Kim, Reconstructing Smooth Local Volatility Surfaces for Cryptocurrency Options, Int. J. Appl. Comput. Math (2025) 11:242

Article Source Here: Numerical Methods for Implied Volatility Surface Construction in Crypto Markets

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ChatGPT as a Personal Financial Advisor: Capabilities and Limitations

Artificial intelligence (AI) is advancing rapidly, and traders and investors are finding ways to leverage this progress to gain an additional edge. Reference [1] examines the effectiveness of AI—ChatGPT, in particular—in personal finance.

Unlike previous studies that focus on quantitative aspects, the paper evaluates AI performance in a qualitative way. Specifically, it prompts ChatGPT with 21 personal finance cases and assesses not only the accuracy of its suggestions but also their contextual appropriateness, emotional intelligence, and attention to detail. These dimensions are critical for real-world impact, especially when users make decisions based on AI-generated advice. The authors pointed out,

We see clear improvements in ChatGPT-4o compared to ChatGPT-3.5, such as more detailed suggestions and alternative solutions (out-of-box thinking). However, the newer model, in its current form, does not appear to be capable of replacing human financial advisors. This is because it tends to provide generalized advice, overlooks important aspects of the financial planning process, such as determining client goals and expectations, and makes mathematical errors in retirement problems. Moreover, ChatGPT sometimes lacks a moral or legal compass…

We find that the quality of financial advice improves (but not always) with prompt engineering. However, the issue is that through prompt engineering, ChatGPT appears to mirror the focus of the user’s attention. If a user is not thinking about taxes, ChatGPT may still provide useful financial advice, but it may omit any considerations of taxes….However, we suggest that this tool be used with great caution as its omissions of important details, such as taxes and legal issues could create problems for users. Finally, we believe that the benefits of using ChatGPT outweigh its drawbacks in the personal finance domain.

In short, the paper finds that ChatGPT-4o shows meaningful improvements over earlier versions in handling personal finance cases, but it still cannot replace human advisors due to its generalized advice, omissions of key details, and occasional mathematical, legal, or moral oversights. This study concludes that ChatGPT is useful for initial guidance, yet must be used with caution, and that in personal finance, the benefits of using ChatGPT outweigh its drawbacks.

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

References

[1] Minh Tam Tammy Schlosky, and Sterling Raskie, ChatGPT as a Financial Advisor: A Re-Examination, Journal of Risk and Financial Management, 18(12), 664.

Originally Published Here: ChatGPT as a Personal Financial Advisor: Capabilities and Limitations

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Option-Implied Information as a Predictor of Stock Returns

Options are powerful tools. They serve not only as an effective instrument for risk management and speculation, but they also provide valuable information about the underlying asset. That is, we can use option‐implied signals to predict the underlying. For example, when the correlation between the S&P 500 and its implied volatility turns positive, it often precedes a market correction.

Reference [1] conducts similar research. It uses implied volatility and the option Greeks of U.S. stocks to predict the direction of the underlying using a machine learning technique. The authors constructed a long/short portfolio and achieved superior risk-adjusted returns. By studying call and put options of exchange-listed U.S. stocks—including end-of-day bid and ask prices, volume, and implied volatility—from 1996 to December 2022, the authors pointed out,

This study provides empirical evidence that option‐implied volatility and Greeks are valuable predictors of extreme stock returns. The results demonstrate that incorporating these option‐related variables via machine learning significantly enhances predictive accuracy compared with traditional logistic regression models. The long–short portfolio constructed by a model utilizing option variables delivers strong financial performance, significantly outperforming a benchmark portfolio using only stock characteristics. Implied volatility and delta emerge as the most important option‐related variables, while the other Greeks, gamma, theta, and vega, also add value when considered through nonlinear interactions. The results also reveal that put options are more informative than call options in predicting extreme returns and that crashes are easier to predict than jumps. This paper contributes to the asset pricing literature by showing that options provide leading information about extreme stock returns, beyond what can be inferred from stock characteristics.

In short, this study shows that option-implied volatility and Greeks are powerful predictors of extreme stock returns. A long–short portfolio built on option signals delivers strong performance, with implied volatility and delta as the most influential predictors and put options proving more informative than calls.

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

References

[1] Panayiotis C. Andreou, Chulwoo Han, Nan Li, Predicting Stock Jumps and Crashes Using Options, Journal of Futures Markets, 2025; 1–20

Originally Published Here: Option-Implied Information as a Predictor of Stock Returns

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Calendar Anomalies in Digital Assets: A Study of Major Cryptocurrencies

In the financial market, seasonality refers to systematic return patterns that recur at specific calendar intervals. It has been studied extensively in the equity space, but little work has been carried out on cryptocurrencies.

Reference [1] addresses this gap. Specifically, it examines seasonality patterns of 10 major cryptocurrencies using data from 2013 to 2024. The paper investigates the (i) Monday effect, (ii) weekend effect, (iii) January effect, and (iv) Halloween effect. The author pointed out,

Our study revisits seasonalities in crypto markets following Kaiser (2019). We do not identify robust return anomalies in the original sample, especially in more mature markets. We thereby show that the few anomalies, that count as “well-established” in prior literature, e.g., the Monday effect in BTC, do not persist in data after 2015 and should therefore be interpreted as a statistical artifact rather than an anomaly. In the cross-section of crypto currencies, the typical Monday effect, if any, appears negative. However, we find that trading activity is significantly and substantially lower at weekends and that this effect is robust across assets and time.

In brief, regarding Bitcoin, the paper concluded,

• It finds no robust or persistent seasonality in Bitcoin returns.
• The historically “well-known” positive Monday effect in Bitcoin disappears after 2015.
• January and Halloween effects are not reliable for Bitcoin—any significance is inconsistent and not stable across time windows.
• Across the full sample and the post-2018 subsample, Bitcoin shows no statistically significant calendar anomalies in returns when tested properly (including bootstrap tests).
• In the cross-section of 500 coins, the typical Monday effect is negative, aligning with equity market patterns, meaning Bitcoin’s earlier positive Monday effect was likely a statistical artifact, not a robust pattern.
• The only stable pattern the paper identifies is lower weekend trading activity.

Seasonality is an important component of trading and risk management. As the crypto market matures, it will be interesting to stay vigilant and observe how these seasonal patterns evolve, if at all. This is also suggested by the author.

Our study, rather descriptive in nature, offers some interesting findings. The entry of institutional investors could lead to increased trading during regular trading hours, thereby increasingly creating inefficiencies around the weekend. Low trading activity on the weekend could also delay the speed of reaction to news. Hence, our insights may be of interest to researchers and traders continuing to challenge the EMH in crypto markets. However, only time will tell whether the negative cross-sectional Monday effect will persist in the long term.

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

References

[1] Mueller, L. (2024). Revisiting seasonality in cryptocurrencies. Finance Research Letters, 64, 105429

Article Source Here: Calendar Anomalies in Digital Assets: A Study of Major Cryptocurrencies

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A Recent Review of Pairs Trading and Statistical Arbitrage

Pairs trading, or statistical arbitrage, is one of the oldest quantitative trading strategies, and it is still employed today. Over the years, it has expanded from classical distance methods to more sophisticated approaches, and practitioners have increasingly questioned its profitability.

Reference [1] provides a thorough review of the pairs trading literature between 2016 and 2023. The findings are as follows.

1. Distance Methods

Distance-based approaches focus on selecting trading pairs using measures such as the sum of squared errors (SSE) or absolute errors (SAE) of normalized price differences. These methods provide simple and intuitive frameworks for identifying co-moving assets and have shown consistent profitability across global markets, including during downturns. Research cited in the document highlights strong market-neutral properties and robustness even after transaction costs. Future work may extend distance methods using richer optimization frameworks, alternative similarity metrics, and broader datasets.

2. Cointegration Methods

Cointegration techniques rely on long-run equilibrium relationships between asset prices, providing a theoretically grounded basis for pairs trading. The document notes extensive evidence supporting their validity across equity and bond markets. Advances involve adaptive modeling, regime-switching structures, and incorporating external variables such as macroeconomic or ESG data. Future work aims to strengthen resilience by integrating alternative datasets and improving modeling flexibility under complex market conditions.

3. Stochastic Control Methods

Stochastic control frameworks treat pairs trading as a continuous-time optimization problem, dynamically adjusting positions based on spread levels, horizon risk, and divergence risk. These methods extend the classical OU process to include jump-diffusions, regime switching, and stochastic volatility, improving realism and adaptability. The document emphasizes strong empirical performance across various markets, while also noting practical challenges such as transaction costs and liquidity constraints. Future research includes integrating AI/ML for improved adaptability and explicitly modeling trading frictions.

4. Time Series Methods

Time series techniques—including GARCH models, OU processes, and fractional OU extensions—focus on short-term dynamics, volatility clustering, and mean reversion. They allow adaptive trading thresholds based on volatility forecasts and have demonstrated improved returns even after accounting for transaction costs. The document highlights opportunities for hybrid models, combining time series techniques with machine learning, copulas, or stochastic control, as well as incorporating slippage, liquidity constraints, and application to emerging markets and high-frequency settings.

5. Other Methods (Copula, Hurst Exponent, Entropic Approaches)

The document identifies several alternative approaches designed to capture features that traditional statistical methods miss. Copula methods model complex joint distributions and tail dependencies; Hurst exponent approaches capture long-memory effects; and entropic methods account for model uncertainty. These techniques enhance robustness by addressing nonlinear dependence structures and heavy-tail behavior in spreads. Future research may refine these methods, integrate them with machine learning, and test them across diverse asset classes and market regimes.

This comprehensive review of statistical arbitrage strategies will assist practitioners in research, particularly in adapting these methods to new asset classes such as crypto.

Regarding the profitability, we believe that while simple methods have historically been profitable, increasing competition and market efficiency mean that more sophisticated approaches are often required to maintain or enhance profitability. However, sophistication alone is not sufficient; effectiveness depends on model design, data quality, and market conditions.

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

References

[1] Sun, Y. (2025). A Review of Pairs Trading: Methods, Performance, and Future Directions. WNE Working Papers, 19/2025 (482). Faculty of Economic Sciences, University of Warsaw.

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Fair Volatility: A Multifractional Model for Realized Volatility

Volatility is an important measure of market uncertainty and risk. For decades, realized volatility has been computed from the squared returns. Recent research, however, has highlighted several deficiencies in traditional volatility measures.

Reference [1] continues this line of inquiry, identifying three key inefficiencies in conventional volatility estimation,

• Volatility is path-independent and blind to temporal dependence and non-stationarity,
• Its relevance collapses in derivative-intensive strategies, where volatility often represents opportunity rather than risk,
• It lacks an absolute benchmark, providing no guidance on what level of volatility is economically fair in efficient markets.

To address these issues, the author introduces the Hurst–Hölder exponent within the Multifractional Process with Random Exponent (MPRE) framework, incorporating it into the stochastic equation describing asset dynamics. This relationship leads to a formal definition of fair volatility—the level of volatility implied under market efficiency, where prices follow semi-martingale dynamics.

The authors pointed out,

This work establishes the Hurst-Hölder exponent as a superior, informationally equivalent substitute for volatility in financial risk measurement, provided price dynamics are locally fractional. Its adoption offers three principal advantages:

• Path-Dependent Risk. It directly quantifies path roughness, capturing deviations from semi-martingale behavior that volatility alone cannot. It moves beyond measuring mere variability to diagnosing the type and intensity of randomness.
• Absolute Benchmarking. Its value is intrinsically meaningful. Unlike volatility, which requires relative comparison, the exponent provides an absolute scale anchored by the martingale benchmark of H(t) =1/2.
• Theoretical Synthesis. It provides a contribution to resolve the apparent dichotomy between market efficiency and behavioral finance. These are not opposing models but alternating market phases, dynamically captured by the exponent’s fluctuation around its efficient equilibrium.

The convertibility of Hurst-Hölder exponent into realized volatility is established by Proposition 1. This enables the determination of a confidence interval around the volatility level that prevails under conditions of informational market efficiency, that is, when prices exhibit submartingale behavior. This benchmark is the level that we termed fair volatility.

Theoretically, the results are noteworthy, as the paper contributes to the literature addressing inefficiencies in traditional volatility measures and extends the Black–Scholes–Merton framework by [glossary_exclude]accounting [/glossary_exclude]for the behavior (mean-reverting or trending) of the underlying asset.

However, practical applications of this approach remain to be seen. Given that the authors have already defined fair volatility, it would be valuable to see trading strategies developed around this concept and their performance evaluated in real markets.

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

References

[1] Sergio Bianchi, Daniele Angelini, Fair Volatility: A Framework for Reconceptualizing Financial Risk, 2025, arXiv:2509.18837

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Volatility, Skewness, and Kurtosis in Bitcoin Returns: An Empirical Analysis

As cryptocurrencies become mainstream, researchers have begun examining their statistical properties, particularly volatility, which represents the second moment of the return distribution. However, limited attention has been given to higher-order moments, specifically skewness and kurtosis. Given that cryptocurrencies are highly volatile and exhibit heavy-tail risks, their return distributions are not log-normal, making the study of skewness and kurtosis essential.

Reference [1] effectively analyzes the volatility, skewness, and kurtosis of Bitcoin and their relationships with Bitcoin returns. The authors use 5-minute high-frequency trading data from 2013 to 2024 to study these properties. They pointed out,

This paper employs 5-minute high-frequency data and quantile regression to examine the relationships between returns and higher-order moments in the Bitcoin market. These findings reveal significant asymmetric relationships between returns and higher-order moments in the Bitcoin market. Specifically: First, Bitcoin returns exhibit significant impacts on higher-order moments (namely volatility, skewness, and kurtosis), with contemporaneous returns demonstrating stronger effects than lagged returns. Second, negative returns show significantly negative correlations with changes in volatility and kurtosis, but significantly positive correlations with skewness changes. Third, at the upper quantiles of volatility and kurtosis changes, as well as the lower quantiles of skewness changes, the impact of negative returns on higher-order moments exceeds that of positive returns. Behavioural finance theories help explain these mechanisms.

The paper also provides insights for both investors and regulators

… investors should enhance risk awareness and optimize asset allocation. Investors must fully recognize Bitcoin’s unique risk structure, particularly the tail risks reflected by higher-order moments. When making investment decisions, they should consider not only volatility but also skewness and kurtosis to comprehensively assess risks. Diversification across Bitcoin and traditional assets can mitigate portfolio risks. Investors should also develop science-based strategies aligned with their risk tolerance and investment objectives, avoiding herd behaviour and excessive speculation.

We find this article particularly interesting and important, especially the conclusion that the correlation between Bitcoin returns and volatility is negative. However, we have observed that the options market has not yet priced in this negative correlation. Further research is warranted.

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

References

[1] Can Yang and Zhen Fang, The asymmetric relationships between returns and higher-order moments: evidence from the Bitcoin market, Applied Economics, 2025

Article Source Here: Volatility, Skewness, and Kurtosis in Bitcoin Returns: An Empirical Analysis

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