Extension of the Black-Scholes-Merton Model to Include Supply Change Rate for Ethereum Options

Ethereum (ETH) is a cryptocurrency that resembles a combination of a currency, a stock, and a commodity. It is a non-dividend-paying crypto asset with a dynamic supply change parameter. Ethereum options have been traded since 2019. Deribit is the largest ETH options exchange by volume, with a market share of approximately 80%.

A particularity of ETH is the changing nature of its supply. Specifically, the supply change rate can be expressed as follows,

Supply change rate = Net issuance rate = Issuance rate − Burn rate

Reference [1] generalized the Black-Scholes-Merton (BSM) formalism to include ETH’s supply change rate. The authors pointed out,

The net issuance rate introduces skewness in volatility structures. This skewness is weighted for deep-in-the-money options, which is consistent with the behavior of option prices. As strike prices increase significantly, the implied volatilities asymptotically approach each other.

Moreover, the sensitivity of the results to even minor changes in the net issuance rate parameter is noteworthy. Figures 12 - 13 demonstrate this effect. This indicates that the inclusion of the net issuance rate can cause significant changes in option pricing. Consequently, options could be fundamentally mispriced if the parameter is completely ignored.

The main contribution of the thesis is the identification of a deterministic factor in the pricing of crypto asset options, the supply change rate, which is not taken into account in the traditional BSM model. The extended BSM model, or alternatively the crypto asset BSM, presented in this thesis includes this rate in the model. The supply change rate can take both positive and negative values within its mathematically defined limits. Moreover, the crypto asset BSM can be used for any other crypto asset that has a supply change parameter, preferably with low block times.

Basically, the author employed the formula used for incorporating stock dilution effects and extended it to ETH options.

Another interesting insight from the paper is that it shows the volatility smirk of ETH options, where, unlike equity options, out-of-the-money call options have higher implied volatility than at-the-money calls.

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

References

[1] Teemu Laurikainen, An extension of the Black-Scholes-Merton options pricing model to Ethereum, Aalto University, 2025

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Time Series vs. Machine Learning: A Systematic Evaluation

Forecasting is important in finance, as it helps investors, analysts, and institutions make informed decisions under uncertainty. Up to now, most forecasting techniques have relied on traditional time series methods, such as ARIMA, GARCH, and exponential smoothing. However, with recent advancements in machine learning and artificial intelligence, these technologies have increasingly found applications in financial forecasting. Their ability to capture complex, nonlinear relationships and process large volumes of data has opened new possibilities for improving prediction accuracy in areas such as asset prices, volatility, and risk.

Reference [1] presents a systematic comparison of traditional time series techniques with newer AI/ML approaches. It highlights the weaknesses of traditional time series methods, notably they assume stationarity and linear relationships, which often do not hold in financial markets. These models struggle with non-stationary data, non-linear dynamics, and large datasets, limiting their ability to capture the full complexity of market behavior.

The paper also discusses the advantages of AI-driven methods, particularly that they excel at capturing complex, non-linear relationships in financial data, and adapting to changing market conditions without manual intervention. They also handle large, high-dimensional datasets effectively, uncovering hidden patterns and making more accurate predictions than traditional models.

The authors made several comparisons using criteria such as,

• Accuracy
• Computational Complexity
• Flexibility and Adaptability
• Interpretability

The authors pointed out,

The comparison of both approaches revealed that while traditional methods are more interpretable and computationally efficient, AI-driven techniques provide greater accuracy and adaptability, especially when dealing with the dynamic and volatile nature of modern financial markets. However, the challenge of obtaining high-quality, reliable data and avoiding overfitting remains for both types of models.

In practice, the decision to use traditional methods versus AI-driven approaches depends largely on the nature of the financial data and the specific forecasting needs. Traditional methods may still be the preferred choice for simpler, well-behaved datasets where linearity and stationarity are present, or when computational resources are limited. They are also suitable for scenarios where interpretability is essential, such as regulatory environments or when model transparency is required. Conversely, AI-driven models should be considered when forecasting complex, non-linear, or high-dimensional financial data, such as stock prices or forex rates, where traditional models struggle. These models are particularly useful when predictive accuracy is paramount, and sufficient computational resources are available to handle the increased complexity.

In short, the new AI/ML techniques offer advantages but also come with disadvantages. However, nothing prevents us from combining these two approaches and leveraging their respective strengths.

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

References

[1] Gwokkwan Sun, and Shuhan Deng, Financial Time Series Forecasting: A Comparison Between Traditional Methods and AI-Driven Techniques, Journal of Computer, Signal, and System Research, Vol. 2 No. 2 (2025)

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Tail Risk Hedging with Corporate Bond ETFs

Tail risk hedging is a strategy designed to protect portfolios against extreme market moves that occur infrequently but have a significant impact when they do. These “tail events” lie at the far ends of a return distribution and often coincide with financial crises, sharp market crashes, or systemic shocks. A well-structured tail risk hedge, typically involving options or volatility instruments, can provide substantial value during periods of heightened uncertainty.

Reference [1] proposed a tail risk hedging scheme by shorting corporate bonds. Specifically, it constructed three signals—Momentum, Liquidity, and Credit—that can be used in combination to signal entries and exits into short high-yield ETF positions to hedge a bond portfolio.  The authors pointed out,

The research above constructed signals on the Investment Grade bond market to inform a dynamic hedge that deploys liquid bond ETFs as hedges to effectively and quickly protect high carry bond funds. It succeeded in lowering absolute and relative risk, increasing annualised returns, and improving Sortino for PIMIX and avoiding drawdowns for DODIX, in a realistic framework that incorporates trading costs, funding costs, and volume sized hedge positions.

Credit Risk, Liquidity, and Momentum signals derived from options, duration times spread, and cumulative duration-neutral returns respectively, each seemed to capture some orthogonal information about the IG bond market. Hedge performance considering individual signals, followed by their combination, proves this point - with an optimal improvement in Sortino of ≥ 0.7 using the joint signals. When searching the hedge model’s parameter space, results remain strong and consistent over a wide array of tested parameters.

Hedging is cost effective as the research has focused on establishing short positions in IG (LQD) and HY (HYG) bond ETFs rather than shorting individual IG corporate bonds. IG bond ETFs are liquid and have low bid-ask spreads, and establishing shorts in the IG bond ETF space via LQD & HYG provides great downside convexity which benefits the efficacy of the hedge. While IG and HY CDXs have far larger traded volumes than LQD & HYG, they do not have the same downside convexity and prove to be not as effective as ETFs

In short, it's possible to develop an effective tail risk hedging strategy using corporate bond ETFs.

An interesting insight from this paper is that it points out how using corporate ETFs benefits from downside convexity while using credit default swaps such as IG CDXs does not.

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

References

[1] Travis Cable, Amir Mani, Wei Qi, Georgios Sotiropoulos and Yiyuan Xiong, On the Efficacy of Shorting Corporate Bonds as a Tail Risk Hedging Solution, arXiv:2504.06289

Originally Published Here: Tail Risk Hedging with Corporate Bond ETFs

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Machine Learning for Algorithmic Trading: A Comprehensive Review

Thanks to the advancement in computing technologies, we’re seeing more widespread use of machine learning, especially deep learning, in the financial services sector. It’s no longer just a theoretical tool; it's showing up in everything from credit risk models to algorithmic trading strategies.

Reference [1] provides a comprehensive review of deep learning techniques used in the financial sector, with a focus on algorithmic trading. It offers a structured analysis of deep learning’s applications across various areas of trading, aiming to identify key trends, challenges, and emerging opportunities by critically evaluating existing research.

The paper provides detailed insights into methodologies applied in different sub-areas of trading such as,

• Stock price prediction
• Market volatility prediction
• Portfolio optimization
• Sentiment analysis for trading
• Risk management
• Anomaly detection and fraud detection
• Supply chain forecasting

Specifically, in volatility forecasting, it highlights,

Recent studies have emphasized the significance of incorporating multiple data streams, including macroeconomic indicators, sentiment analysis, and high-frequency trading data, to enhance the predictive accuracy of volatility models [129,130]. The findings suggest that hybrid models outperform single-model approaches, but data noise and overfitting remain challenges. As shown in Table 8, a variety of models have been applied to different datasets, each with specific contributions and limitations.

Overall, the authors concluded,

This review has highlighted the transformative potential of deep learning in algorithmic trading, where models such as LSTM, CNN, and Reinforcement Learning have shown substantial improvements in predicting financial markets and optimizing trading strategies. However, significant challenges remain, particularly related to data quality, overfitting, and the interpretability of complex DL models. Financial markets are noisy, volatile, and influenced by a multitude of factors, making it difficult for models to generalize well. Additionally, the black-box nature of DL models raises concerns for traders and regulators who require transparency in decision-making. Emerging trends such as attention mechanisms, transformer architectures, and hybrid models offer promising solutions to these challenges, alongside integrating alternative data sources like social media sentiment and news. Future research must focus on improving model robustness, developing explainable AI techniques, and addressing computational efficiency to unlock the full potential of DL in real-world trading environments. By overcoming these hurdles, DL can significantly enhance the accuracy and effectiveness of algorithmic trading, providing traders with more powerful tools for navigating complex financial markets.

In short, deep learning is useful but still has its limitations.

In our experience, being able to leverage advances in computing is definitely an edge, but domain knowledge remains essential.

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

References

[1] MD Shahriar Mahmud Bhuiyan, MD AL Rafi, Gourab Nicholas Rodrigues, MD Nazmul Hossain Mir, Adit Ishraq, M.F. Mridha, Jungpil Shin, Deep learning for algorithmic trading: A systematic review of predictive models and optimization strategies, Array, Volume 26, 2025, 100390,

Post Source Here: Machine Learning for Algorithmic Trading: A Comprehensive Review

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Interest Rate Sensitivity in Low-Volatility Investing

Low-volatility investing is a strategy that focuses on stocks with historically lower price fluctuations, aiming to achieve strong risk-adjusted returns. Despite conventional finance theory suggesting that higher risk should lead to higher returns, research has shown that low-volatility stocks often outperform their high-volatility counterparts on a risk-adjusted basis. By reducing drawdowns and offering a smoother return profile, low-volatility investing appeals to risk-conscious investors, particularly in uncertain market environments.

Given the appeal of low-volatility investing, there are, however, some concerns about its sensitivity to changes in interest rates, particularly its viability in a higher-yield environment. Reference [1] investigates this issue. The authors pointed out,

The results confirm that low-volatility stock deciles do indeed exhibit positive, statistically significant bond betas (that become negative for high-volatility deciles), and this exposure carries over to the most popular low-volatility indexes such as the S&P 500 Low Volatility Index and the MSCI USA Minimum Volatility Index, even after accounting for their exposures to value, quality, and investment factors. The estimated bond betas roughly correspond to a duration of a two-year Treasury bond, but—as our robustness tests show—this sensitivity does not appear to be very stable over time. It can be quite effectively mitigated by applying leverage (especially within the context of long–short strategies) or by carefully avoiding excessive industry tilts, such as overallocating to companies from the utilities or consumer staples sector.

… Even in 2022, one of the worst years on record for US Treasuries, exposure to interest rates failed to materially affect the performance of low-volatility strategies. The negative bond contribution was more than offset by high positive returns on undervalued, high-quality, and conservative stocks overrepresented in low-volatility portfolios. On a more pessimistic note, however, equity style exposures of our sample low-volatility strategies seem to account for much of their raw excess returns generated in the past 30 years, suggesting some skepticism as to how much value added these strategies can bring to an already diversified and quality-tilted portfolio.

In short, the results confirm that long-only low-volatility strategies exhibit positive, statistically significant bond betas, even after controlling for exposures to value, quality, and investment factors. However, this sensitivity is not stable over time and can be effectively mitigated through leverage or by avoiding excessive industry concentrations.

This article sheds new light and provides insights into low-volatility investing. Let us know what you think in the comments below or in the discussion forum.

References

[1] Juliusz Jabłecki, Low-Volatility Equity Strategies and Interest Rates: A Bittersweet Perspective, The Journal of Beta Investment Strategies, Volume 16, Issue 1 Spring 2025

Article Source Here: Interest Rate Sensitivity in Low-Volatility Investing

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Variational Autoencoders for Arbitrage-Free Volatility Modeling

Machine learning and AI are transforming investing by enabling data-driven decision-making, uncovering hidden patterns, and automating complex strategies. From algorithmic trading and portfolio optimization to risk management and sentiment analysis, AI-driven models process vast amounts of data with speed and precision, identifying opportunities that traditional methods might miss.

Most ML and AI approaches have been applied to building predictive models. Reference [1], however, suggests using ML techniques in risk management. Specifically, it explores the use of Variational Autoencoders for generating synthetic volatility surfaces for stress testing and scenario analysis.

The paper develops a robust synthetic data generation framework using parameterized Heston models. It implements comprehensive arbitrage validation, ensuring critical no-arbitrage conditions, including calendar spread and butterfly arbitrage constraints, are preserved. The authors pointed out,

First, we have demonstrated that synthetic data generation using carefully parameterized Heston models can effectively overcome the traditional barriers of limited market data in illiquid markets. By generating over 13,500 synthetic surfaces—compared to the typical constraint of fewer than 100 market-observable surfaces, we have significantly enhanced the robustness and reliability of our VAE training process. Our methodology succeeds in preserving critical no-arbitrage conditions, specifically, both calendar spread and butterfly arbitrage constraints validate its practical applicability in real-world trading environments. .. A key innovation of our approach is expanding the idea of latent space optimization which was alluded to by Bergeron et al [2] and its independence from historical market data for training purposes. This characteristic makes our framework particularly valuable for emerging markets, newly introduced derivatives, and other scenarios where historical data is scarce or non-existent. The ability to generate realistic, arbitrage-free synthetic surfaces provides practitioners with a powerful tool for price simulation and risk assessment in illiquid markets...The successful reconstruction of surfaces with significant missing data points (demonstrated through our test case with 100 randomly removed points) showcases the model’s robustness and practical utility. Extending the framework to examine the model’s performance under various market stress scenarios could constitute further research directions.

This is a significant contribution to the advancement of ML and AI in finance, particularly in risk management—an area with much yet to be explored.

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

References

[1] Nteumagne,  B. F.; Donfack,  H. A.; Wafo Soh,  C. Variational Autoencoders for Completing the Volatility Surfaces. Preprints 2025, 2025021482. https://ift.tt/BxHoiVu

Originally Published Here: Variational Autoencoders for Arbitrage-Free Volatility Modeling

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Improving Portfolio Management with Volatility of Volatility

Managing portfolios using volatility as a filter has proven effective. Reference [1] builds on this research by proposing the use of volatility of volatility for portfolio management. The rationale behind using volatility of volatility is that it represents uncertainty.

Unlike risk, which refers to situations where future returns are unknown but follow a known distribution, uncertainty means that both the outcome and the distribution are unknown. Stocks may exhibit uncertainty when volatility or other return distribution characteristics vary unpredictably over time.

Practically, the author used a stock’s daily high and low prices to derive its volatility of volatility. They pointed out,

Specifically, we hypothesise that the benefits of volatility management are more pronounced for low uncertainty stocks and during periods of low aggregate uncertainty. To test these hypotheses, we use a measure of uncertainty based on the realised volatility-of-volatility (vol-of-vol) derived from intraday high and low prices. We first examine the relation between this measure of uncertainty and future returns. Consistent with the extant literature, we find that uncertainty is positively related to returns, and that it contains unique information about future returns not captured by other stock characteristics. We then explore the role of uncertainty in the performance of volatility management, across individual stocks and over time. We show that volatility management yields a significantly larger improvement in risk-adjusted performance for stocks with low uncertainty compared to those with high uncertainty and, for the market portfolio, it yields better performance during periods of low aggregate uncertainty compared to periods of high uncertainty. We also show that uncertainty potentially explains the performance of volatility management when applied to different asset pricing factor portfolios. Furthermore, our findings complement the sentiment-driven explanation of Barroso and Detzel (2021), revealing that the effect of sentiment on volatility management crucially depends on the level of uncertainty.

In short, using the volatility of volatility as a filter proves to be effective, particularly for low-uncertainty stocks.

We find it insightful that the author distinguishes between risk and uncertainty and utilizes the volatility of volatility to represent uncertainty.

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

References

[1] Harris, Richard D. F. and Li, Nan and Taylor, Nicholas, The Impact of Uncertainty on Volatility-Managed Investment Strategies (2024). https://ift.tt/rVJf7iS

Originally Published Here: Improving Portfolio Management with Volatility of Volatility

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Incorporating Liquidity into Option Pricing: a Stochastic Approach

Liquidity is an overlooked research area, yet it plays a crucial role in financial markets. Trading system developers often use the bid-ask spread as a proxy for liquidity, but this approach is less effective in the options market.

Reference [1] proposes a method for integrating liquidity into the option pricing model. Essentially, it introduces market liquidity as a variable that can change randomly and affects stock prices through a discounting factor and market liquidity levels.

The paper begins by incorporating a stochastic liquidity variable into the SDE for stock price under the P measure. Here the liquidity process is modeled as an Ornstein-Uhlenbeck process. The SDE is then transformed into the Q measure in which European options are analytically evaluated using the derived closed-form characteristic function. The authors pointed out,

We incorporate three main factors, that is, stochastic volatility, economic cycles, and liquidity risks, into one model used for option pricing. A combination of Heston stochastic volatility and regime switching is selected for modeling the price of the underlying stock when there are no liquidity risks. The stock price is then discounted based on the level of market liquidity levels described by a mean reverting stochastic process. The employment of regime switching Esscher transform provides a risk‐neutral measure as well as the corresponding model dynamics, yielding a European option pricing formula in closed form. Significant impacts of the three factors can be seen through the performed numerical experiments. Our analysis with real data also confirm the necessity to consider stochastic liquidity, which has greatly improved model performance. By leveraging the stochastic liquidity component, our proposed model can help investors refine their hedging positions, better responding to liquidity shocks, and thus mitigate risks more effectively.

In short, liquidity is integrated as a discount factor, and the study demonstrates its impact on option prices.

This research provides a framework for incorporating liquidity into options trading, however, we found it less intuitive. Let us know what you think in the comments below or in the discussion forum.

References

[1] Xin-Jiang He, Hang Chen, Sha Lin, A Closed-Form Formula for Pricing European Options With Stochastic Volatility, Regime Switching, and Stochastic Market Liquidity, Journal of Futures Markets, 2025; 1–12

Originally Published Here: Incorporating Liquidity into Option Pricing: a Stochastic Approach

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Stock and Volatility Simulation: A Comparative Study of Stochastic Models

Stress testing and scenario analysis are essential tools in portfolio management, helping portfolio and risk managers assess potential vulnerabilities under extreme market conditions. By simulating adverse scenarios such as financial crises, interest rate shocks, or geopolitical events, these techniques provide insights into how a portfolio might behave under stress and identify potential weaknesses.

Reference [1] investigates several stochastic models for simulating stock and volatility paths that can be used in stress testing and scenario analysis. It also proposes a method for evaluating these stochastic models. The models studied include

• Geometric Brownian Motion (GBM),
• Generalized Autoregressive Conditional Heteroskedasticity (GARCH),
• Heston stochastic volatility,
• Stochastic Volatility with Jumps (SVJD), and a novel
• Multi-Scale Volatility with Jumps (MSVJ).

The authors pointed out,

When the objective is to evaluate and simulate scenarios that reflect market crashes, both short-term events and long-term crises, models such as GBM and the Heston model have been shown to be more effective. These models are better equipped to capture the sudden and severe price movements associated with market crashes, as demonstrated by their performance in reproducing historical drawdowns and their ability to capture tail risk…

If the objective is to generate future scenario simulations for option pricing, the MSVJ model has proven to be the most suitable choice. The MSVJ model’s superior performance in capturing the range of the actual TQQQ price, as evidenced by its highest WMCR for both price and volatility, makes it particularly valuable for option pricing…

When the primary goal is to simulate the most realistic price path and volatility paths for TQQQ, the SVJD model has demonstrated superior performance. By capturing both stochastic volatility and jump processes, the SVJD model can generate price and volatility trajectories that closely resemble the observed dynamics of TQQQ. Portfolio managers can utilize this model for more accurate backtesting of trading strategies and better assessment of portfolio risk under various market conditions.

In short, each model has its strengths and weaknesses and serves a particular purpose.

This study is an important contribution to the advancement of portfolio risk management. Let us know what you think in the comments below or in the discussion forum.

References

[1]  Kartikay Goyle, Comparative analysis of stochastic models for simulating leveraged ETF price paths, Journal of Mathematics and Modeling in Finance (JMMF) Vol. 5, No. 1, Winter & Spring 2025

Post Source Here: Stock and Volatility Simulation: A Comparative Study of Stochastic Models

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Are Weekend Gaps Always Filled? A Look at Stock Indices

The weekend price gap is a well-known phenomenon in financial markets, particularly in assets that trade continuously during the week but pause over the weekend, such as stocks, futures, and options. When markets reopen after the weekend, prices sometimes experience a gap up or down due to news, geopolitical events, or macroeconomic developments that occurred while trading was halted. For investors, weekend gaps present both risks and opportunities.

The prevailing belief among traders is that gaps are usually filled. Reference [1] examines this assumption, specifically studying weekend gap dynamics in the DJIA, NASDAQ  100, and the DAX. The authors pointed out,

While our findings do not support a universal tendency for markets to revert to the prior closing price at short distances, they reveal that more pronounced movements at larger thresholds may indicate a partial gap-filling mechanism…Although our descriptive statistics and Chi-square tests show minimal evidence of a predictable “fill-the-gap” bias within ranges closer to the Monday open, there is suggestive evidence for directional price movements further away from the gap, reflecting possible longer-horizon effects…

Our regression and correlation analyses reveal that larger gaps typically coincide with higher short-term volatility, reinforcing the argument that weekend price discontinuities signal an increased uncertainty or risk (Hull & Basu, 2016; Mandelbrot, 1972; Plastun et al., 2020). This effect is particularly pronounced for the DJIA and NASDAQ, where an expanded gap size correlates with a greater likelihood of hitting the Take Profit and Stop Loss thresholds alike…Meanwhile, the DAX, though hinting at a moderate positive association between gap size and Take Profit rates, presents less robust evidence—highlighting how regional factors, sectoral composition, or liquidity conditions may temper volatility responses to weekend gaps.

In short, small to medium-sized gaps are not necessarily filled; rather, they are indicative of increased volatility. Larger gaps, however, exhibit some directional predictability and can be used to design trading strategies. Additionally, European market dynamics differ from those of the U.S.

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

References

[1] Marnus Janse van Rensburg, and Terence Van Zyl, Price Gaps and Volatility: Do Weekend Gaps Tend to Close?, J. Risk Financial Manag. 2025, 18, 132

Originally Published Here: Are Weekend Gaps Always Filled? A Look at Stock Indices

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