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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Probabilistic AI in Finance: A Comprehensive Literature Review

Probabilistic AI is a branch of artificial intelligence that models uncertainty explicitly, allowing systems to reason and make predictions even when data is incomplete or noisy. Instead of producing single-point estimates, it generates probability distributions over possible outcomes, capturing both what is known and how confident the model is.

Reference [1] reviewed the research literature on probabilistic AI as applied to finance. Specifically, it followed a rigorous article selection process and ultimately analyzed 62 papers published between 2004 and 2024. The authors pointed out,

In this review, we perform a systematic literature review following a SLR approach to review 62 papers on the topic of probabilistic AI in finance. We examine these papers across dimensions such as model type, output, asset class, and uncertainty type. Additionally, we provide insights into the geographical distribution of research, contributor backgrounds, and the historical development of the field. Our findings suggest that most articles on probabilistic AI claim to enhance point predictions, and few articles have an explicit focus on improving uncertainty estimation within finance. Moreover, probabilistic AI offers valuable capabilities for financial modeling, including non- parametric distribution estimation, separation of uncertainty types, and capturing non-linear dynamics. However, the lack of comprehensive benchmarking and robust evaluations, especially in comparison to traditional models, makes it difficult to assess their true performance.

An important implication of our findings is the need for more interdisciplinary collaboration. Analysis of author backgrounds indicates that research in this area is largely dominated by computer scientists, with relatively limited participation from financial experts. As a result, computer scientists often lack the domain-specific knowledge needed to effectively model financial problems, while financial researchers, despite being better positioned to address such challenges, have seldom adopted probabilistic AI techniques, likely due to technical barriers. This review serves as a starting point for bridging these divides, guiding financial researchers in adopting these methods and helping computer scientists better frame their approaches within the financial context.

In short, the review highlights both the promise and the current limitations of probabilistic AI in finance, particularly the lack of robust benchmarking and systematic evaluation against traditional models. Advancing this field will require stronger interdisciplinary collaboration, where domain expertise from finance and innovation from computer science are combined to produce models that are both technically sound and economically meaningful.

We believe that the conclusion regarding domain knowledge and collaboration applies not only to probabilistic AI but also to deterministic, traditional AI and machine learning in finance.

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

References

[1] Sivert Eggen, Tord Johan Espe, Kristoffer Grude, Morten Risstad, Rickard Sandberg, Financial Time Series Uncertainty: A Review of Probabilistic AI Applications, Journal of Economic Surveys, 2025; 00:1–39

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Expiration Effects and Return Anomalies in Option Markets

A growing body of research has recently investigated anomalies in option returns, such as option return momentum, and these anomalies are often attributed to market inefficiencies. Reference [1], however, proposed and tested a different hypothesis: these anomalies originate from option returns around expiration days.

Specifically, the author isolated the return of delta-hedged call options on the option expiration day—i.e., the third Friday of the month—and the following Monday to examine how these returns contribute to overall monthly option performance. They pointed out,

This paper identifies expiration-driven liquidity effects as a key driver of option return anomalies. We show that predictable option returns are largely concentrated around expiration, when investors rolling over positions create large order imbalances that overwhelm market makers’ risk-bearing capacity. These frictions lead to significant price distortions, explaining much of the observed anomaly predictability.

Our findings reveal that more than half of the monthly anomaly returns occur during the two-day expiration window, while returns outside this period are significantly weaker. This pattern holds across a broad set of stock, fundamental, and option characteristics and is particularly pronounced for S&P 500 stocks, where expiration fully accounts for anomaly returns.

These results challenge the view that option anomalies reflect behavioral biases or inefficiencies. Instead, they highlight the role of intermediary constraints and systematic liquidity demands in shaping option prices. Future research could further explore the impact of expiration dynamics on market participants and whether similar patterns exist in other derivative markets.

In short, the paper demonstrates that expiration effects are a major determinant of option return patterns. Many well-documented anomalies weaken or disappear when the expiration window is excluded, while return predictability is concentrated around expiration.

The authors also provide an explanation for the negative option returns observed near expiration, attributing them to the rollover of covered call positions. Finally, the paper highlights the role of market maker constraints and systematic liquidity demands in shaping option prices.

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

References

[1] Pedro A. Garcia-Ares and Dmitriy Muravyev, Option returns: a tale of the expiration rollover day, 2025, fma.org

Article Source Here: Expiration Effects and Return Anomalies in Option Markets

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Dynamic Correlations Between Bitcoin and the NASDAQ-100 Index

As cryptocurrencies, particularly Bitcoin (BTC), become more mainstream, investors have noticed an increasing correlation between BTC and the stock market. Some investors even argue that BTC now functions as a substitute risk asset for the technology sector.

Reference [1] presents a quantitative study on the correlation between BTC and the NASDAQ-100 (NDX) index, the causal relationships between them, and the influence of market sentiment on this relationship. Specifically, the study first employs univariate GARCH(1,1) models to capture volatility persistence in BTC and NDX returns. Second, it applies the DCC-GARCH model to measure time-varying correlations. Finally, it examines the influence of investor sentiment through regression and dynamic correlation analysis based on VIX levels.

The author pointed out,

This study aimed to quantitatively assess volatility spillovers, dynamic conditional correlations, and the moderating role of investor sentiment between Bitcoin (BTC) and the Nasdaq-100 index (NDX). The research specifically examined whether Bitcoin functions as a diversification asset and how market fear, indicated by the V IX, influences the correlation dynamics between BTC and NDX.

The key findings can be summarized as follows. First, both BTC and NDX exhibited significant volatility persistence, with BTC showing higher volatility magnitude and longer persistence (aBTC + [iBTC = 0.9322, (INDX + [iNDX = 0.9839). Second, the dynamic conditional correlation (DCC) between BTC and NDX was found to be varying over time and was significantly intensified during systematic stress events, such as the COVID-19 crisis and periods of monetary tightening. Third, the evidence suggested a weak yet significant bidirectional volatility spillover, particularly indicating that BTC volatility notably influenced NDX volatility during market stress periods. Fourth, investor sentiment (VIX) was identified as a strong moderator, with the regression results (y = 0.0106, p < 0.001) clearly showing that increases in market fear significantly amplified the correlation between BTC and NDX.

In short, the article concluded that,

• Both BTC and NDX showed strong volatility persistence, with BTC exhibiting higher volatility and longer persistence.
• The conditional correlation between BTC and NDX varied over time and intensified during market stress events such as the COVID-19 crisis and monetary tightening periods.
• A weak but significant bidirectional volatility spillover existed, with BTC volatility notably influencing NDX volatility during stress periods.
• Investor sentiment, measured by the VIX, acted as a strong moderator—rising market fear significantly increased the correlation between BTC and NDX.

This paper provides empirical evidence supporting investors’ observations and offers insights that can be integrated into trading and portfolio management frameworks.

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

References

[1] Boyu Zhang, From Diversifier to Amplifier? Investigating the BTC—NDX Linkage and the Modulating Role of VIX, Dean&Francis, 2025

Originally Published Here: Dynamic Correlations Between Bitcoin and the NASDAQ-100 Index

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Market Timing Through Volatility Clustering and Causal Structure Identification

Volatility plays a crucial role in trading and risk management. These days, more portfolio managers are aware of and utilize volatility measures, even if they do not trade options. For example, we have previously discussed volatility-managed portfolios, where asset volatilities are used to size positions.

Reference [1] contributes to this body of research by classifying stocks according to their volatilities and then trading them. Specifically, the authors first employ the Gaussian Mixture Model (GMM) to group stocks with similar volatility characteristics. They then apply a causal inference framework based on the Granger Causality Test (GCT) and augmented with Effective Transfer Entropy (ETE) to identify lead-lag relationships, ultimately trading based on market timing signals generated by the lead stock.

The article pointed out,

This paper has put forward a novel methodology for volatility-driven statistical arbitrage in the context of stock market forecasting, which integrates sophisticated statistical techniques with machine learning models. By employing the GMM clustering algorithm to classify stocks according to mid-range volatility, we have demonstrated the potential of such clusters to act as indicators for predicting price movements. This method, combined with a robust causality analysis framework involving Granger Causality Tests, Peter-Clarke Momentary Conditional Independence, and Effective Transfer Entropy, identifies significant predictive relationships between stocks. Furthermore, the integration of DTW and KNN enhances predictive accuracy by aligning and classifying time series data, thereby enabling the anticipation of profitable trading opportunities.

The effectiveness of this integrated approach is demonstrated by the backtesting results. The trading strategies based on identified volatility clusters and causal relationships consistently outperformed the Buy & Hold benchmark across multiple performance metrics, including total returns, Sharpe Ratio, and maximum drawdown. Notably, the volatility-driven strategy yielded substantial returns with controlled risk exposure, as evidenced by the superior Sortino and Calmar ratios in comparison to conventional strategies.

In short, by using GMM to cluster stocks with comparable volatility characteristics and applying causal inference techniques to determine lead-lag pairs, the authors developed a trading strategy that achieved superior risk-adjusted returns.

This is an interesting development. However, we note two limitations:

• The framework involves multiple models, which increases the number of parameters and the risk of overfitting, and
• It appears that only in-sample tests were conducted.

Despite these limitations, there are valuable insights to be gained from this approach.

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

References

[1] Ivan Letteri, Statistical Arbitrage Volatility-Driven with Statistics and Machine Learning Models for Stock Market Forecasting, SN COMPUT. SCI. 6, 918 (2025).

Article Source Here: Market Timing Through Volatility Clustering and Causal Structure Identification

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Tail Risk Hedging and Trend Following: A Combined Framework

Investing in equities offers strong growth potential, but a key drawback is exposure to periodic drawdowns, which many investors find psychologically difficult to endure. To address this, some have proposed tail risk hedging strategies. However, portfolio managers are often reluctant to adopt them due to high carry costs, and researchers continue to debate their effectiveness.

Reference [1] implemented a particular tail risk hedging strategy and overlaid it on a trend-following approach, referred to as the Portable Alpha Portfolio.

The Portable Alpha Portfolio consists of two components: 100% exposure to the MSCI ACWI Index as the beta source, while alpha is generated through a tail risk hedging overlay and a 50% exposure to a trend-following strategy.

• The tail hedge is constructed by systematically purchasing three tranches of 10-delta SPX put options with one year to expiration, rolled quarterly, and notionally sized.
• The trend-following component includes 79 futures contracts, with normalized returns computed over four lookback periods: 3, 6, 9, and 12 months. Positions are taken long when the lookback return is positive and short otherwise.

The authors pointed out,

This study examines the performance enhancements from applying a Portable Alpha framework to a global equity portfolio by overlaying both trend-following and tail risk hedging strategies. The Portable Alpha portfolio maintains 100% exposure to the MSCI ACWI Index as its beta source, while alpha is introduced through a tail risk hedging overlay and a 50% exposure to a trend-following strategy…

The resulting Portable Alpha portfolio generated a large, positive, and statistically significant alpha of 0.25% per month after controlling for traditional equity factors, global government bond, and commodity returns. Absolute outperformance was concentrated during periods of crisis, particularly in the first half of the sample, with more recent periods showing broadly comparable returns to ACWI. However, the Portable Alpha portfolio consistently delivered superior risk-adjusted performance across the full sample, with notable improvements in downside protection. These findings are consistent with those in Schwalbach and Auret (2023). Performance attribution confirms that the Portable Alpha portfolio effectively captured ACWI excess returns, demonstrating that convex return streams can be overlaid to enhance performance without diluting core equity exposure.

In summary, this study finds that combining tail risk hedging with trend following adds value to a long-only equity portfolio, supporting the case for incorporating tail risk hedging into portfolio management.

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

References

[1] Bruno Schwalbach & Christo Auret, Enhancing global equity returns with trend-following and tail risk hedging overlays, Investment Analysts Journal, 2025

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The Role of Investor Attention Index in Explaining Bitcoin Volatility

Modeling and forecasting volatility is essential in trading and risk management. Extensive research has been conducted on volatility modeling in traditional financial markets, and recently, attention has increasingly been directed toward cryptocurrency volatility. The standard approach often relies on econometric models.

Reference [1] applied the GARCH-MIDAS model to study Bitcoin volatility. However, unlike traditional approaches, it introduced two new types of explanatory variables:

• The first is investor attention, measured through Google Trends, social media activity, and news coverage, which reflects market participants’ interest and captures short-term market dynamics in speculative environments.
• The second is uncertainty indices, such as GEPU and GPR, which represent broader macroeconomic and geopolitical risks influencing market stability over longer horizons.

The authors pointed out,

This study highlights the significant role of investor attention and uncertainty indices in modelling Bitcoin volatility using the GARCH-MIDAS framework. Although uncertainty indices are more commonly applied in cryptocurrency volatility analysis, our findings demonstrate that attention indices, such as GTCA, outperform uncertainty indices like GEPU, GPR, and WUI in both explanatory power and forecasting performance. Notably, GTCA is easier to construct and, when developed comprehensively, provides superior insights into Bitcoin volatility dynamics.

Furthermore, an extended analysis of Ethereum, XRP, and BNB (Table 12) confirms that GTCA’s predictive power extends to other major cryptocurrencies, albeit with lower volatility levels compared to Bitcoin.

The two-variable models combining GTCA with GEPU, GPR, or WUI were identified as the most explanatory based on in-sample data and the best-performing according to out-of-sample results. These findings underscore the benefits of integrating GTCA with general uncertainty indices, as the complementary information they provide enhances the accuracy and reliability of Bitcoin volatility predictions.

This study contributes to the literature by being the first to integrate investor attention and uncertainty indices within a unified model for Bitcoin volatility. The construction of the GTCA index using a comprehensive set of search terms ensures robustness and distinguishes this work from prior studies. Additionally, by evaluating both in-sample and out-of-sample performance, this research addresses a critical gap in cryptocurrency volatility forecasting.

In short, the study found that the Google Trends Cryptocurrency Attention (GTCA) index is a significant driver of Bitcoin volatility and successfully integrated it, along with other uncertainty measures, into an effective econometric framework.

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

References

[1] Serkan Aras, Mehmet Ozan Ozdemir, Cihan Çılgın, Uncertainty or investor attention: Which has more impact on Bitcoin volatility?, Research in International Business and Finance 77 (2025) 103002

Article Source Here: The Role of Investor Attention Index in Explaining Bitcoin Volatility

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Impact of Artificial Intelligence on Financial Markets: a Quantitative and Qualitative Analysis

Artificial intelligence (AI) has become an integral part of modern finance, transforming how institutions analyze data, manage risk, and execute trades. By leveraging machine learning algorithms and natural language processing, AI systems can identify complex patterns in large financial datasets, forecast market movements, and detect anomalies that might signal fraud or structural inefficiencies.

As financial data grows in both volume and complexity, AI enables more adaptive, data-driven approaches, bridging the gap between quantitative modeling and real-time decision processes. Consequently, AI has a measurable impact on the market. Reference [1] examined the impact of AI on trading and risk management, performing both quantitative and qualitative analyses. The author pointed out,

As we have previously discussed, the core footprint of algorithmic trading and the entire spectrum of the financial markets, in one way or another, has been influenced by Artificial Intelligence (AI) technology. AI has profoundly changed the efficiency of the market, the execution of trading orders and the accuracy of trading decisions during the real-time analysis of available data, which has improved to an unprecedented level. Due to the advances in technology like AI, liquidity and the overall operations of the market have vastly improved because unlike humans, AI systems are able to outperform in trade execution speed and precision (Agarwal et al., 2021). AI also improves the accuracy of prediction that assists traders and investors in strategising and managing risks more effectively. Nevertheless, the integration of AI systems within trading structures has some consequences. Even if the implementation of AI technology comes with a lot of advantages, the stability of the market adds a new layer of risk. Arguments regarding flash crashes, categorised as a sudden drastic price decline due to an algorithm blunder, and systemic risk—the collapse of one AI system triggering an avalanche effect within a network of interlinked AI systems—pose a concern (Agarwal et al., 2021).

In short, AI has an overall positive impact on the market, particularly from the perspectives of execution and liquidity. However, it can also increase risks, as evidenced by the higher volatility associated with AI.

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

References

[1] Neeraj Kahol Sharma, Assessing the AI Impact on Financial Markets through Algorithmic Trading, Journal of Basic Science and Engineering, Vol. 22, No. 1, (2025)

Originally Published Here: Impact of Artificial Intelligence on Financial Markets: a Quantitative and Qualitative Analysis

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Return and Variance Risk Premia in the Bitcoin Market

The volatility risk premium (VRP) has been studied extensively in the literature, especially in equities. However, little work has been done in the crypto space. Reference [1] fills this gap by investigating the Bitcoin return premium (BP) and the Bitcoin variance risk premium (BVRP).

The authors utilized Bitcoin options data traded on Deribit from July 2017 to December 2022 to calculate the BVRP. The implied variance is computed using the same methodology as the VIX index. Additionally, they derived risk-neutral densities (RNDs) from the option volatility surface and subsequently used RNDs to classify Bitcoin market states as high volatility or low volatility.

The authors pointed out,

The BP is significantly higher than that of traditional investment assets such as currencies, commodities, and stocks, averaging around 66% per year….The annualized risk-neutral and physical monthly variances, proxied by the squared Bitcoin Volatility Index (BVIX) and the realized variance (BRV), are significantly high: 0.72 and 0.58, respectively. The corresponding variance risk premium is 0.14, much higher than that of the S&P 500 Index—approximately 2%, according to Bollerslev, Tauchen, and Zhou (2009).

We further analyze conditional estimates across market regimes. Our results indicate that BP is higher in the HV regime... The overall increased volatility leads to higher option price premia on average across all moneyness levels. The variance under the two probability measures is quite different and introduces a substantial VRP in both market regimes. Surprisingly, the low volatility cluster is characterized by a higher VRP of 0.17 compared to the high volatility cluster of 0.12, suggesting a potential disconnect between variance and VRP.  By relying on the average values of the premia within each cluster, we observe a negative relationship between the BP and BVRP in the Bitcoin market. This contrasts with the findings in S&P 500 Index market, where a positive relationship between variance risk premium and future returns has been reported by Bollerslev, Tauchen, and Zhou (2009) in a regression setup. Simultaneously, our clustering results indicate a positive relationship between returns and variance, supporting the inverse leverage effect in the Bitcoin market documented by Hou et al. (2020).

In short, the paper found that Bitcoin is far more volatile and exhibits a higher variance risk premium than the S&P 500, and that the Bitcoin return premium (BP) is also elevated. Risk premia vary over time as a function of two distinct volatility regimes.

This is an important contribution to the understanding of the crypto options market. However, the dataset extends only through 2022. Since then, Bitcoin has reached a broader user base, and its volatility dynamics have evolved. It would be interesting to examine whether the findings hold in the current market.

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

References

[1] Caio Almeida, Maria Grith, Ratmir Miftachov, Zijin Wang, Risk Premia in the Bitcoin Market, arXiv:2410.15195

Originally Published Here: Return and Variance Risk Premia in the Bitcoin Market

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