Robust Tail Risk Estimation in Cryptocurrency Markets: Addressing GARCH Misspecification with Block Bootstrapping

Abstract

This study examines the use of Filtered Historical Simulation (FHS) to estimate tail risk in cryptocurrency markets for the optimization of robustness in this area under model misspecification. An ARMA-GARCH model is employed on the daily returns on Binance Coin and Litecoin in order to compare the performance of classical and block bootstrap procedures in residual risk. Diagnostic tests indicate that standardized residuals are dependent, contrary to the independent and identically distributed (i.i.d.) assumption of conventional FHS. Comparing the block and ordinary bootstrapping approaches, we find that block bootstrap produces wider, more conservative confidence intervals, particularly in extreme tails (e.g., 0.1% and 99.9% percentiles). The findings suggest that block bootstrapping can be employed as a correction instrument in risk modeling where the standard volatility filters do not work. The article highlights the necessity to account for remaining dependencies and offers practical recommendations for more robust tail risk estimation during volatile markets.

1. Introduction

Risk management of the financial markets heavily relies on reliable forecasting techniques for quantifying potential losses due to adverse price moves. Filtered Historical Simulation (FHS) is one technique widely used in density forecasting—a probability distribution estimate of future return—via the use of past records to develop scenarios of risk. FHS improves the Historical Simulation (HS) technique by filtering the data to make it more applicable, and it performs best when used for non-normal risks and out-of-the-ordinary market events. Nevertheless, one of the fundamental assumptions of FHS is that standardized residual returns—those normalized for volatility—ought to be independent and identically distributed (i.i.d.). In practice, financial time series tend to suffer from heteroscedasticity (time-variant variance), serial correlation (contagion between lagged and current values), and leptokurtosis (fat tails, indicating more likely large events). These stylized facts require more sophisticated modeling techniques to culminate in credible risk estimates.

One of the most prominent issues in financial risk management is tail risk, signifying the probability of extreme losses. Standard metric models require tail-based risk measures such as Value-at-Risk (VaR) (Christoffersen 2009; Kuester et al. 2006) and Expected Shortfall (ES) (Acereda et al. 2020) to quantify these risks. However, traditional models relying on normality assumptions underestimate serial correlation in returns and, thus, risks. To circumvent this, bootstrapping methods have gained use as non-parametric methods for inference that generate the distribution of financial returns under no constraining assumptions. Subsampling existing data with replacement using ordinary bootstrapping is just one of the frequent examples in risk measurement. However, if the starting sample is biased or filled with dependencies, the resampled data will also contain such issues, in turn, leading to false predictions.

To counteract such limitations, block bootstrapping offers a solution by resampling blocks in successive sequences, rather than individual observations. It preserves financial time series dependency, thus improving the accuracy of risk measures. However, the availability of the best block size is of critical concern as it determines the balance between preserving dependencies and statistical consistency. In this study, block bootstrapping is examined for its impact on risk prediction when compared to ordinary bootstrapping during use within the FHS context. Specifically, it examines how these techniques address embedded non-linearities and dependencies on cryptocurrency markets with volatile conditions and market imperfections that pose particular challenges to more conservative risk prediction models.

The objective of this study is to enhance the risk assessment of cryptocurrencies using an evolved version of the FHS method with block bootstrapping, as well as to convey a stronger evidence-based framework for tail risk forecasting. The study applies the method to Binance Coin (BNB) and Litecoin (LTC) and compares block bootstrapping performance in the detection of volatility clustering and long-memory. Due to the rapid growth of cryptocurrency markets and their appeal to retail, as well as institutional investors, understanding the limitations of traditional risk models is integral. The existing literature has indicated inefficiency in cryptocurrency markets, volatility spillovers, and abnormal risk behavior as a further justification of the significance of advanced risk assessment models. Previous studies (e.g., Kristjanpoller et al. 2024; Liu et al. 2023; Theiri et al. 2023; Chowdhury et al. 2023; Bouri et al. 2021; Ahelegbey et al. 2021; Baur et al. 2018) validate the use of tail risk estimation and other alternative volatility models, particularly for the majority of the most volatile instruments like cryptocurrencies.

Earlier research has resulted in excellent work recognizing sophisticated risk management methods like the FHS approach, which employs block bootstrap methods to yield tail risk and dependence on volatility estimation in high-frequency markets (Giannopoulos et al. 2024). One empirical research study quotes the potential of Bitcoin to predict US stock sector volatility and offers a plural list of recommendations while conveying specific modeling issues through the use of different distributions and HAR-RV models of cryptocurrency returns (Bouri et al. 2021). Additionally, numerous other studies on the inefficiency of cryptocurrency markets, prevailing extreme risks, and spillovers of volatility related to leading cryptocurrencies have been performed (Conlon et al. 2021; Nadarajah and Chu 2021; Pichl and Kaizoji 2021; Xu et al. 2021; Wang et al. 2021). Similarly, herding structure and confirmation bias of investors driving market actions are shown in other behavioral studies (Bouri et al. 2019; Xiong et al. 2021). Overall, these results emphasize the necessity of an advanced, new risk model such as the FHS to meet the multi-dimensionality of cryptocurrency risks and intricacies.

This study contributes to existing research on crypto-asset risk management that addresses essential knowledge gaps. To begin with, although FHS is central to capturing non-normal risk, the use of its practical application, notably, its biases when using bootstrapping approaches, necessitates more advanced analysis. Secondly, the study looks into how block bootstrapping and regular bootstrapping produce contrasting confidence intervals, which goes to further supports the implications that such differences would have for policy concerning risk management. Finally, it demonstrates the importance of out-of-sample backtesting in achieving exact risk forecasting models, most specifically where there is extreme market distress. With the long-term goal of facilitating better financial decision-making, this paper attempts to improve the accuracy and reliability of the estimation of Bitcoin risk by providing a detailed and empirical examination of bootstrapping procedures under the FHS model.

The rest of this study is organized as follows. Section 2 gives an overview of the theoretical and empirical literature on econometric modeling of cryptocurrency volatility using GARCH-type models and Filtered Historical Simulation (FHS), the limitations of classical bootstrapping in the presence of serial dependence, and the theory and application of block bootstrap techniques for tail risk estimation in financial markets. Section 3 outlines the methodology used, Section 4 discusses the empirical results, and Section 5 discusses and concludes with final remarks emphasizing the weaknesses of the proposed approach and providing ideas for the potential evolution of this subject’s study.

2. Literature Review

An understanding of cryptocurrency return tail risk dynamics has increased in significance in the last couple of years due to the unusual volatility, fat-tailed distributions, and complex dependency structures of digital asset markets. This literature review surveys three key streams: (1) econometric modeling of cryptocurrency volatility in GARCH-type models and Filtered Historical Simulation (FHS); (2) the limitations of the classical bootstrapping with serial dependence; and (3) theory and practice of block bootstrap techniques, specifically for tail risk estimation in finance.

2.1. Volatility Modeling and Filtered Historical Simulation in Cryptocurrency

Cryptocurrency returns are characterized by extreme volatility clustering and leptokurtosis, alongside evident regime shifts over time. GARCH-type models—particularly ARMA-GARCH(1,1)—have become the industry standard for capturing conditional heteroskedasticity in cryptocurrency markets. Empirical studies comparing model performance frequently find that GARCH-based volatility forecasts outperform simpler benchmark models such as ARMA alone or naïve historical volatility models when applied to Bitcoin, Ethereum, Litecoin, and other large-cap coins (Tunahan Akkuş and Çelik 2020) (ScienceDirect, SSRN).

Filtered Historical Simulation (FHS), first proposed by Barone Adesi, Giannopoulos, and Barone-Adesi et al. (1998, 1999), blends historical simulation with GARCH-filtered residuals, thereby accommodating time-varying volatility without sacriicing the empirical distribution of returns (Wikipedia Contributors 2024) (Wikipedia). More extensive empirical tests—such as Tian (2025)—testify that FHS significantly improves the accuracy of Value-at-Risk (VaR) forecasting, particularly at tail quantiles, and performs better than both standard Historical Simulation (HS) and GARCH with normal approximation (GARCH N) (arXiv). The findings point to the effectiveness of FHS for tail risk management in non-normal, volatile return distributions.

However, even FHS models rely on the assumption that the standardized residuals from a correctly specified GARCH model are independent and identically distributed (i.i.d.). In many cryptocurrency applications, residuals still exhibit hidden dependencies and heavy-tailed behavior, which may color VaR estimates—especially under temporal aggregation—leading to underestimation of extreme risk.

2.2. Limitations of Standard Bootstrap Under Dependence and Aggregation

The straightforward nonparametric bootstrap, which resamples individual i.i.d. observations, is found to be misleading where there is a financial time series with autocorrelation or heteroskedasticity (Ruiz and Pascual 2002) (e-archivo.uc3m.es, MDPI). It has been established that the treatment of dependency structures as something to be disregarded can result in bias and loss of precision in parameter estimation and prediction inference (Ruiz and Pascual 2002) (ResearchGate). Traditional bootstrap methods ruin serial dependence, thereby failing to capture tail behavior and risk measures correctly under financial situations, particularly if sampling is conducted at aggregated frequencies.

The issue is also compounded when observations are aggregated across time (e.g., from weekly or daily to a monthly series). Aggregation reduces extreme observations, dampens high-frequency variation, and smooths volatility, lowering the sensitivity of tail risk diagnostics. Even mild serial dependence heavily distorts tail quantile estimation since residual patterns are masked.

2.3. Block Bootstrap Methods for Dependent Time Series

Block bootstrap techniques were developed to preserve dependency by resampling contiguous blocks of observations rather than individual points. The Moving Block Bootstrap (MBB), non-overlapping block bootstrap, stationary (or Politis–Romano) bootstrap, and circular block bootstrap each maintain temporal structure to varying degrees (tsbootstrap documentation 2024) (arXiv). These methods help reduce bias in the estimation of variance, autocorrelation, and tail-related statistics when conventional bootstrapping fails.

A comprehensive examination by Cogneau and Zakamouline (2010) identifies that block bootstrap methods are particularly well-suited for financially dependent data, yet also identifies challenges with the choice of optimal block length and potential small-sample bias (Cogneau and Zakamouline 2010) (quantdevel.com). This bias happens particularly in moving block bootstrap implementations since overlapping blocks generate artificial negative serial correlation, and optimal block length choice has been conjectured to significantly influence estimation accuracy (Nordman and Lahiri 2014) (arXiv).

In addition, advanced theoretical frameworks—such as the fixed b asymptotic approach—enhance inference calibration of block bootstrap-based confidence sets with consideration of bandwidth/block-size choice uncertainty (Shao and Politis 2012) (arXiv).

2.4. Applications in Cryptocurrency Risk Contexts

Very few bodies of literature have studied the intersection of block bootstrapping and cryptocurrency tail risk estimation. In ultra-high-frequency environments, researchers employed standardized residuals from parameters like GARCH-type models and block bootstrap resampling to preserve concealed dependencies to enhance tail quantile forecasting within Bitcoin and Ethereum price series (Giannopoulos et al. 2024). These studies demonstrate that block-based Filtered Historical Simulation produces wider and more realistic forecast intervals, especially at extreme confidence levels (e.g., 0.1% and 99.9%), compared to standard bootstrap.

Despite these contributions, a systematic empirical comparison between ordinary bootstrap and block bootstrap within FHS frameworks remains limited, particularly in the context of mid-frequency cryptocurrency data (daily, weekly, monthly). Moreover, few studies examine the interaction between volatility persistence, leverage effects, and aggregation-induced dilution of tail detection capabilities.

2.5. Research Gaps and Value Added by This Study

Our study completes several important gaps in the extant literature. First, we present empirical evidence of i.i.d. assumption violation in ARMA–GARCH standardized residuals when applied to daily returns of Litecoin (LTC) and Binance Coin (BNB), which remain serially correlated despite filtering through GARCH. Our study bridges some of the most significant gaps in the literature thus far. To our knowledge, we empirically document deviations from i.i.d. for the first time in GARCH-filtered ARMA–GARCH residuals of BNB and LTC daily returns, and we document persistence of serial dependence after GARCH filtering. Second, by performing normal and block bootstrapping at weekly and monthly frequencies, we capture how temporal aggregation lowers power to signal tail events, as theories of information loss due to aggregation predict. Third, we compare outcomes of common and block bootstrap methods under the Filtered Historical Simulation (FHS) approach, demonstrating that common bootstrapping will tend to underestimate tail risk—especially at extreme quantiles—but block bootstrapping yields wider bands for confidence, which are closer to the true uncertainty. Fourth, our volatility model identifies significant persistence in BNB and a significant leverage effect in LTC, dynamics that will be undervalued in more mundane FHS applications. Fifth, our results are empirically validated on a sample of 6647 daily returns of two leading cryptocurrencies from 1999 to 2025, thus establishing the robustness of our evidence. Lastly, our evidence offers valuable and policy-relevant implications, warning risk managers and analysts against performing standard bootstrapping procedures on aggregate data in FHS models and demanding more stringent methodological requirements for tail risk estimation in digital asset markets.

In conclusion, while earlier research has shown the robustness of GARCH-Filtered Historical Simulation for measuring tail risk, and block bootstrap methods have been appreciated in the general time series literature, this study combines these methods within a single estimation of tail risk under aggregation in crypto markets. By highlighting the risk of spurious negatives and misspecified inference, our contribution is applicable to academic researchers as well as financial risk management practitioners with direct relevance.

3. Methodology

3.1. Filtered Historical Simulation Approach

In an effort to enhance the accuracy of risk estimation in the cryptocurrency markets, this paper applies the Filtered Historical Simulation (FHS) method through block bootstrap resampling. FHS—a model by Barone-Adesi et al. (1998, 1999)—builds on a simple historical simulation by incorporating an extra step of volatility filtering. It is a three-step process: (1) modeling asset return using an ARMA-GARCH model, (2) filtering previous residuals in order to shatter dependencies, and (3) resampling normalized residuals to generate future prices.

The first task is to model both the conditional mean equation using an Autoregressive Moving Average (ARMA) model, and the conditional variance equation using a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model. The ARMA model represents short-run price return dependence, while the GARCH model signifies volatility clustering—a highly observed characteristic of financial time series. The standard version of the ARMA-GARCH model is depicted as follows:

$$y_t={{\sum }_{i=1}^l{\delta }_i y}_{t-1}+{{\sum }_{i=1}^m{\theta }_i\epsilon }_{t-1}+{\epsilon }_t\hspace{1em}{\epsilon }_t~ \mathrm{N}(0, {\sigma }_t)$$

(1)

$${\sigma }_t^2=\omega +{\displaystyle {\sum }_{i=1}^p}{\alpha }_i{\displaystyle {\epsilon }_{t-i}^2}+{\displaystyle {\sum }_{j=1}^q}{\beta }_j{\sigma }_{t-j}^2$$

(2)

where y_t is the return at time t; δ_i captures serial correlation in asset prices; θ_i models the effect of past forecast errors; ε_t represents the error term, which can follow alternative distributions such as Student’s t or Generalized Error Distribution (GED) to better capture cryptocurrency return characteristics (Zhang et al. 2021); ${\sigma }_t^2$ represents the time-varying conditional variance.

After the estimation of the ARMA-GARCH parameters, standardized residuals are calculated, as each one is divided by its conditional standard deviation. The standardized residuals are innovations in the resampling procedure, taken randomly and multiplied by simulated conditional volatilities. Final price paths are built utilizing simulated returns, eventually producing density forecasts for future asset prices.

Residual diagnostics and volatility forecast performance are used as grounds for generating model choice; these are compared using criteria like Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and adjusted R. Should the best-fitting ARMA-GARCH model fail to remove dependency in residuals, other volatility models such as Asymmetric GARCH, IGARCH, TARCH, and GJR-GARCH may be considered.

3.2. Block Bootstrapping Approach

The shortcomings of classical bootstrapping for time series data include its use of independent observations which may be unrealistic when financial markets feature serially correlated returns. The block bootstrapping technique addresses this by ensuring that the financial time series’ dependency structure is retained. Instead of using single observations, the approach resamples adjoining blocks of observations to keep intact price returns’ temporal patterns.

The study employs the Moving Block Bootstrap (MBB) method, wherein a block is taken as s consecutive observations. The selection of the block size is significant: (1) Small blocks may miss dependencies. (2) Large blocks may introduce excessive overlap, conversely reducing variability in the resampled dataset.

To determine the most appropriate block size, the present study utilizes the method presented by Hall et al. (1995) in selecting the block lengths according to the Bootstrap Mean Squared Error (BMSE) criterion. The steps involve partitioning data into blocks of varying sizes (e.g., 2 to 50 observations).

• Resampling the blocks with replacement.
• Errors of estimation are calculated for every one of the block lengths.
• The block length with the lowest BMSE is selected.

Formally, let y_i be the observed time series, where i = 1, …, N. Define a block $B_i$ as $B_i=(y_i,\dots ,y_{i+s-1})$ for $1\le i\le N$, where N = n – s + 1 is the number of blocks in the dataset. The resampled time series $Y^{*}$ is constructed as follows:

$$Y^{*}=(B_1^{*},B_2^{*},\dots ,B_b^{*})=(y_{11}^{*},y_{12}^{*}, \dots , y_{1s}^{*},y_{21}^{*},y_{22}^{*}, \dots , y_{2s}^{*},{\dots ,y}_{b1}^{*},y_{b2}^{*}, \dots , y_{bs}^{*})$$

where each block $B_i^{\mathrm{*}}$ is randomly drawn from the original set of blocks (e.g., $B_i^{\mathrm{*}}=(y_{i1}^{\mathrm{*}},y_{i2}^{\mathrm{*}},\dots ,y_{is}^{\mathrm{*}}))$.

The application of block bootstrapping within the FHS setup enables the improvement in simulating price paths in terms of obtaining significant features of returns on cryptocurrencies like volatility clustering, fat tails, and serial dependence. The method offers a more realistic estimation of tail risk and Value-at-Risk (VaR) predictions than conventional risk modeling methods. By combining FHS with bootstrapping at the block level, this paper offers a consistent, reproducible framework for cryptocurrency risk analysis that guarantees correct inclusion of volatility dynamics and tail events in risk forecasts.

To determine the appropriate volatility model for each asset, we estimated a number of ARMA-GARCH-type specifications, such as GARCH(1,1), GARCH(1,2), GJR-GARCH(1,1), and EGARCH(1,1), for both LTC and BNB. Model selection was based on the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), with an inclination towards parsimony at the expense of goodness-of-fit. The alternative specifications’ AIC and BIC are presented in

Table 1. Model selection.

Model Type	AIC (BNB)	BIC (BNB)	AIC (LTC)	BIC (LTC)
GARCH(1,1)	−5.213	−5.186	−5.109	−5.083
GARCH(1,2)	−5.244	−5.211	−5.112	−5.078
GJR-GARCH(1,1)	−5.23	−5.198	−5.151	−5.116
EGARCH(1,1)	−5.215	−5.182	−5.14	−5.106

.

For BNB, the ARMA(0,0)-GARCH(1,2) was the optimal trade-off between fit and parsimony, while for LTC, introducing an asymmetric term (GJR-GARCH) greatly improved model performance since it picked up on the fact that there is a leverage effect in the series. This reflects the fact that negative shocks have a larger impact on LTC volatility than comparable positive shocks, a heavily reported empirical regularity in speculative assets. These specifications were subsequently used to estimate standardized residuals for the Filtered Historical Simulation procedure.

Although endogeneity is a central concern of econometric modeling, with causality being estimated from explanatory variables to asset returns, it is not the subject matter of the current study. Our model does not attempt to impose causality on explanatory variables on asset returns. Instead, we use an ARMA-GARCH model as a filter for volatility before tail risk estimation through Filtered Historical Simulation. In this context, the concern for endogeneity—typically arising from simultaneity, measurement error, or omitted variables in causal regression models—does not directly apply, as we are not estimating structural parameters or causal effects. The validity of our results hinges instead on the adequacy of the volatility model for filtering residuals and the appropriateness of the bootstrap techniques for resampling under temporal aggregation.

In this study, “Ordinary Bootstrap” is a standard name for the typical Filtered Historical Simulation (FHS) method, which uses standardized residuals resampled independently and with replacement from the filtered series (Barone-Adesi et al. 1999). This technique assumes residuals to be i.i.d., an assumption that typically will be challenged in practice, particularly in cryptocurrency markets with long-memory volatility and nonlinear behavior. As our diagnostic checks have shown, the ARMA-GARCH filtering process does not eliminate residual dependencies entirely. Rather than fixing the model structure per se, we investigate if alternative resampling techniques other than those employed above—block bootstrapping—can compensate for these dependencies. The exercise then reduces to quantifying the robustness of tail risk measures with less-than-perfect model assumptions, rather than selecting ideal models. This is a practical, application-driven exercise that is characteristic of empirical risk management.

4. Data and Results

4.1. Block Bootstrapping and Preliminary Analysis

Our dataset consists of the daily closing prices of two cryptocurrencies, Binance Coin (BNB) and Litecoin (LTC), collected from Investing.com (https://www.investing.com/) (accessed on 25 April 2025). All series are in US dollars. The sample period, depicted by the price availability of the two cryptocurrencies, starts on 9 November 2017 and concludes on 24 April 2024, hence including 2359 observations. The data points have been on a logarithmic scale for analysis. Logarithmic transformation maintains a plethora of advantages as related to the analysis of cryptocurrency prices. Firstly, it normalizes the distribution of such data and shifts it into a more appropriate set in light of the application of statistical analytical models. Secondly, it preserves variance stability, especially when considering that the processes of cryptocurrency prices are subject to change and may often be heteroscedastic. Moreover, logarithmic variations in prices reflect changes as a percentage, indicating that such measures are more interpretable and meaningful than absolute changes during financial audits. Finally, they allow for more evenly distributed data, as they reduce the issue of skewness and overall force a stronger performance of statistical tests and models. In summary, the utilization of logarithmic transformations improves both the analytical power and insights obtained from the cryptocurrency price audit.

Table 2. Descriptive statistics of daily log returns for BNB and LTC.

	BNB	LTC
Mean (%)	0.0040	0.0016
Standard Deviation (%)	0.0565	0.0543
Kurtosis (%)	25.5111	13.6180
Skewness (%)	1.9648	1.0458
Range	1.1407	0.9960
Minimum (%)	−0.4408	−0.3854
Maximum (%)	0.6999	0.6106
Largest (%)	0.6999	0.6106
Smallest (%)	−0.4408	−0.3854

presents descriptive statistics of log returns of BNB and LTC, including mean, standard deviation, skewness, kurtosis, range minimum, maximum, largest, and smallest, with respect to each cryptocurrency.

The average log return for BNB stands at 0.0040%, above that of LTC, at 0.0016%. Furthermore, BNB is more volatile, with a standard deviation of 0.0565% against LTC’s 0.0543%. In addition, log returns for BNB are heavily concentrated around the mean, depicted by its kurtosis, which stands at 25.5111% against LTC’s 13.6180%. Skewness is positive for both cryptocurrencies, with BNB at 1.9648% and LTC at 1.0458%. To expand, BNB has a bigger dispersion of returns: the range constitutes 1.1407% versus 0.9960% of LTC. The minimum log return for BNB equals −0.4408%, while for LTC, the figure stands at −0.3854%. The maximum log return for BNB is 0.6999%, and that of LTC is 0.6106%. While both cryptocurrencies are positively skewed with occasional large positive returns, descriptive statistics show that BNB is the highest in volatility and concentration of return distributions.

According to Baur et al. (2018), the return series of cryptocurrencies is intricate and non-standard. Many of these return series often feature extreme volatility, sudden shifts in market sentiment, as well as sudden price spikes, which traditional financial models often fail to elucidate. To capture such dynamics, the various orders of ARMA(l) have been estimated for both the mean and conditional variance equations by using the quasi-maximum likelihood (QML) method.1 Normal distribution, Student’s t-distribution, and Generalized Extreme Value (GEV) distribution were the three distributions that were taken into account in the likelihood function for conditional error probability. For the latter two distributions, the degrees of freedom were estimated together by leveraging the parameters in Equation (2). The filtering of a series of one-day price changes was analyzed by the conditional mean and variance equation specified as ARMA(0,0)-GARCH(1,2).

The Student’s t-distribution held the best distribution of errors that fit the log-likelihood function, allowing for a stronger reflection than that of the normal distribution. When both the first- and second-order ARCH terms are considered, the return series volatility is heteroskedastic with GARCH effects in spite of the fact that they represent a random walk.

Table 3. Model estimation of conditional mean and variance for BNB returns.

Parameter	Coefficient	Standard Error	t-Statistic
ω	0.0019	0.0028	7.7667
α	0.1223	0.0069	21.2889
β1	0.4416	0.0031	162.0032
β2	0.4398	0.0027	147.9689
d	4.3453	0.0716	62.11743

Note: A 1000 scale has been applied to the BNB return series. Heteroscedasticity-adjusted standard errors and the BFGS algorithm with inequalities were used for the estimation. The ML function has a value of −28,7988.7.

and

Table 4. Model estimation of conditional mean and variance for LTC returns.

Parameter	Coefficient	Standard Error	t-Statistic
ω	0.004422	0.000554	8.00214
α1	0.122109	0.005948	20.24143
β1	0.398986	0.003001	145.13233
β2	0.474213	0.003231	146.13321
γ	−0.02445	0.011232	−2.21024
d	4.033016	0.059132	68.20642

Note: A 1000 scale has been applied to the BNB return series. Heteroscedasticity-adjusted standard errors and the BFGS algorithm with inequalities were used for the estimation. The ML function has a value of −39.0031.

illustrate the estimates of the coefficients as well as their standard errors for BNB and LTC returns, respectively.

The conditional mean and variance model results for BNB returns are shown in Table 3. Estimation controls for robustness against a non-constant variance, while the key parameters are indicative of strong volatility clustering and persistence in the return series. The intercept term ω (0.0019) shows that this level of variance at baseline is highly negligible. Meanwhile, the α coefficient (0.1223) together with β coefficients (β₁ = 0.4416 and β₂ = 0.4398) shows a high effect on the present volatility of this commodity by lagged squared returns and past variances. Summing β₁ and β₂ yields 0.8814, less than 1, implying mean-reverting variance over time. The d (4.3453) parameter indicates possible nonlinear or long-memory effects and further supports the complexity of BNB return dynamics. The small number of degrees of freedom shows that there are fat tails in the BNB return data (it shows fat tails in the conditional distribution of ε), which are significantly heavier than would be expected under normality. The model captures high persistence in volatility, which is typical for financial returns. However, while the results provide actionable insights into risk management and portfolio optimization, further diagnostics are required beyond this to confirm adequacy. Results emphasize the importance of volatility modeling for comprehending the risks associated with BNB returns, given the pronounced patterns of clustering and persistence in the latter.

Similarly, for each of the three probability distributions of the error component in the likelihood function, parameters in Equation (1) with different orders in the ARMA and the conditional variance of the LTC return series were computed. The best form for the conditional variance was a GARCH(1,2) but with an additional term, γ, to account for asymmetric effects, given by

$${\sigma }_t^2=\omega +a{\left({\epsilon }_{t-1}+\gamma \right)}^2+{\beta }_1{\sigma }_{t-1}^2+{\beta }_2{\sigma }_{t-2}^2$$

(3)

where I_t−1 can take the value of 1 for ε_t−1 < 0; otherwise, it takes the value of 0. Table 4 reports on the coefficient estimates and the corresponding standard errors.

The key findings of the conditional mean and variance model for LTC returns are compiled in Table 4. The model estimated using the BFGS algorithm with heteroskedasticity-adjusted standard errors provides robust parameters of the scaled series. Significant volatility clustering and persistence are confirmed by key parameters: α₁ = 0.122109, while β₁ (0.398986) and β₂ (0.474213) confirm the contribution of lagged squared returns and past volatility to the current one. The sum of β₁ and β₂ (0.873199) is below 1, indicating that there is mean-reverting variance over time. In addition to the heteroskedastic long-memory, the variance of the rate of return for LTC has asymmetric effects, while the daily interval of the rate of return follows a random walk. The statistical significance of the γ coefficient then summarizes these effects, and its negative sign reflects the steep decline in prices compared to their total increase. The negative γ of −0.02445 conveys a leverage effect, where volatility rises higher after adverse price shocks than after favorable ones. Furthermore, the significant d (4.033016) may indicate nonlinear dynamics or long-memory effects in LTC returns. It also shows that the conditional distribution of ε exhibits fat tails. On its part, the minor number of degrees of freedom implies that the distribution of LTC returns is, indeed, highly fat-tailed-heavy compared to the normal distribution prediction. There is a considerable likelihood of extreme returns that deviate from the central tendency, as indicated by the reported low degrees of freedom in the returns’ t-distribution. This demonstrates that fat tails are present in the distribution, therefore allowing results to emphasize the complex nature of LTC’s volatility, which is crucial for risk management and trading strategies. Stability in the model adds to its reliability, but further confirmation by residual analysis and out-of-sample testing is recommended to confirm its forecasting ability. In summary, the findings are informative on LTC’s volatility dynamics to aid risk assessment and decision-making in cryptocurrency markets.

The squared standardized and standardized residuals are also checked for dependence in the present study. The GARCH residual and normalized residual returns are checked under the i.i.d. hypothesis. Five tests were chosen according to Brockwell and Davis (2016); the statistics and significance of results are shown in

Table 5. Diagnostic tests on standardized and squared standardized residuals of returns.

BNB	Standardized Residuals		Squared Standardized Residuals
Test	Statistic	p-Value	Statistic	p-Value
Ljung-Box Q(10)	94.96523	0.003	37.1351	0.002
McLeod-Li(10)	42.22342	0.001	0.075223	1.001
Turning Points	−2.9879	0.002	−0.2341	0.8097
Difference Sign	−0.0599	0.945	−2.31247	0.0298
Rank Test	−0.77211	0.523	−7.43872	0.004
LTC	Standardized Residuals		Squared Standardized Residuals
Test	Statistic	p-Value	Statistic	p-Value
Ljung-Box Q(10)	80.02712	0.002	30.03311	0.003
McLeod-Li(10)	34.45262	0.000	0.44456	0.9989
Turning Points	−0.69779	0.501	−0.69982	0.4862
Difference Sign	−0.29996	0.7498	0.121021	0.9103
Rank Test	−0.54327	0.6021	−10.29769	0.001

Note: to examine the hypothesis of i.i.d., several tests are performed on the GARCH residual and standardized residual returns.

.

The majority of tests conducted for various statistics, such as the Turning Points test, Difference Sign test, Ljung–Box test, and Rank Test, possess significantly low p-values that showcase unexplained dependencies and patterns in standardized residuals and squared standardized residuals. This represents the existence of patterns and relationships in data that the model failed to adequately capture. The tests were applied to the residual returns after passing them through all possible ARMA-GARCH specifications. With FHS for multistep forecasting, the best block length of the sequence of standardized residuals was then determined in a separate study.

The BMSE is employed for comparing the statistical performance of an estimator or a model. This method pre-processes the continuous data by breaking it down into blocks of lengths ranging from 2 to 50 consecutive observations. The required steps are data resampling, estimation of the parameter of interest for each resampled dataset, squaring the error, and averaging over all the resamples. The same process is repeated for different block lengths, and the resulting plot provides an idea regarding the behavior of the estimator against changing block lengths. This involves the random selection of data points with replacements within each block independently to supply several bootstrap samples. The findings of the block bootstrap approach of Hall et al. (1995) are showcased in

Table 6. Optimal block length determination for bootstrap methods.

	Optimal Block Length	Minimum Squared Error (MSE)
BNB	4	0.1685
LTC	3	0.1463

, alongside their corresponding minimal squared error values and ideal block length.

This optimum block length determination used by the bootstrapping on BNB and LTC returns supports the acquisition of knowledge concerning some type of dependence for their returns, as was reported in Table 5. That is four for BNB and three for LTC. It signifies the fact that the return series of the BNB has more complex dependencies or longer memory effects, which can be captured by a larger block length. Conversely, for LTC, the return series can be well proxied by small block lengths, reflecting quicker mean reversion or less dependence on past observations. The performance concerning the MSE values for the two cryptocurrencies is almost identical, having BNB at a value of 0.1685, and LTC at an MSE value of 0.1463. A lower MSE for LTC shows that the block bootstrapping method performs slightly better in modeling its return process compared to BNB. The smaller the MSE value, the better the model performs. These values convey the goodness of fit regarding block bootstrapping methods across the return series for every cryptocurrency. Lastly, Table 5 reiterates the importance of setting the block length appropriately during the application of block bootstrapping methods.

Variances in the optimal lengths for BNB and LTC establish a foundation for a more individualized treatment of each of the cryptocurrencies based on unequal dynamic structures. This minimal difference in MSE values results in the trade-off between model complexity and prediction accuracy, thus validating block length in serving to enhance the overall efficiency of the risk modeling process.

4.2. Simulating the Density of the Price Forecasts

Following Barone-Adesi et al. (1999), random innovations were selected from the sample set of standardized residuals and were input to generate multistep price and variance equations forecasts. The simulation process is as follows: the model estimate vector ε_t of the historical dataset (Equations (1) and (2)) and the corresponding coefficient estimates (Table 1 and Table 2) are built for BNB and LTC, respectively, by employing a five-member and two-member consecutive standardized residual return block, represented by e_i, i = 1, …, 60 days.

These estimated residuals are then standardized by the respective volatility estimates, defined as $h_t^{*}=\sqrt{{\widehat{\sigma }}_t^2}$, to obtain the innovations set, denoted as $z_t^{*}$; to simulate one step ahead, at $t+1$ value of the innovation, one random standardized residual return is drawn from the dataset, then scaled with the volatility of the period $t+1$. The innovations set, represented by $z_t^{*}$, is obtained by standardizing these estimated residuals through the corresponding volatility estimates, which are defined as $h_t^{*}=\sqrt{{\widehat{\sigma }}_t^2.}$ To simulate one step ahead, at t + 1 value of the innovation, one random standardized residual return is taken from the dataset and scaled with the volatility of the period t + 1. This is represented by $z_{t+1}^{*}=e_1^{*}\sqrt{h_{t+1}^{*}}$, with $\sqrt{h_{t+1}^{*}}=\sqrt{\widehat{\omega }+\widehat{\alpha } z_t^{*}+\widehat{\beta }{h}_t^{*}}$, and the coefficients ($\widehat{\omega }, \widehat{\alpha } , \widehat{\beta }$) are calculated by Equation (2). The asset return at the time $t+1$, represented by $y_{t+1}^{*}$, is finally simulated as follows: $y_{t+1}^{*}=\widehat{\delta }{y}_t+\widehat{\theta }{z}_t^{*}+z_{t+1}^{*}$, where Equation (2) is used to estimate the coefficients ($\widehat{\delta }, \widehat{\theta }$). We can calculate the corresponding simulated asset price using these simulated returns. For the other simulation paths, t + i, i = 2, 3, the volatility is not observable and has to be generated from the randomly chosen re-scaled residuals as $\sqrt{h_{t+i}^{*}}=\sqrt{\widehat{\omega }+\widehat{\alpha } z_{t+i-1}^{*}+\widehat{\beta }{h}_{t+i-1}^{*}}$, and simulated returns as $y_{t+i}^{*}=\widehat{\delta }{y}_{t+i-1}+\widehat{\theta }{z}_{t+i-1}^{*}+z_{t+i}^{*}$. These standardized residuals are then used in a multistep forecast with the optimum block length obtained. For each cryptocurrency, 1,000,000 paths for 60 one-day intervals were generated. The prediction intervals for 0.1% to 99.9% percentiles are shown in

Table 7. BNB prediction intervals for selected percentiles for 1, 2, …, 60-day horizon.

Day	OBNB 0.1%	BBNB 0.1%	OBNB 99.9%	BBNB 99.9%	OBNB 0.5%	BBNB 0.5%	OBNB 99.5%	BBNB 99.5%	OBNB 1%	OBNB 99%
1	99.56	99.56	100.35	100.35	99.74	99.75	100.24	100.24	99.8	100.19
2	99.39	99.33	100.51	100.54	99.63	99.61	100.33	100.36	99.71	100.28
3	99.25	99.06	100.6	100.68	99.55	99.52	100.41	100.45	99.64	100.34
4	99.11	98.85	100.7	100.8	99.48	99.43	100.47	100.52	99.58	100.39
5	99	98.71	100.79	100.9	99.42	99.34	100.52	100.59	99.53	100.44
6	98.89	98.59	100.85	100.99	99.36	99.24	100.57	100.65	99.49	100.48
7	98.79	98.37	100.93	101.08	99.3	99.14	100.62	100.7	99.44	100.52
8	98.71	98.19	101	101.15	99.25	99.08	100.67	100.75	99.4	100.56
9	98.63	98.08	101.05	101.24	99.21	99.02	100.71	100.8	99.36	100.59
10	98.46	97.88	101.18	101.35	99.11	98.91	100.79	100.89	99.29	100.66
15	98.17	97.49	101.4	101.62	98.93	98.7	100.93	101.05	99.15	100.78
20	97.89	97.1	101.61	101.84	98.77	98.46	101.06	101.2	99.02	100.89
25	97.48	96.47	101.92	102.22	98.54	98.19	101.26	101.41	98.84	101.04
30	97.23	96.09	102.16	102.45	98.39	97.99	101.38	101.55	98.73	101.14
35	96.81	95.51	102.46	102.84	98.17	97.72	101.56	101.74	98.57	101.28
40	96.52	95.09	102.68	103.08	98.04	97.52	101.68	101.88	98.46	101.36
45	96.16	94.53	103.01	103.47	97.83	97.24	101.85	102.06	98.32	101.49
50	95.91	94.15	103.2	103.71	97.69	97.06	101.96	102.19	98.21	101.58
55	95.5	93.58	103.51	104.12	97.48	96.79	102.12	102.38	98.07	101.7
60	95.18	93.12	103.78	104.38	97.34	96.6	102.24	102.51	97.98	101.78

Note: BBNB and OBNB denote block and ordinary bootstrap methods, respectively.

and

Table 8. LTC prediction intervals for selected percentiles 1, 2,…, 60-day horizons.

Day	OLT 0.1%	BLT 0.1%	OLT 99.9%	BLT 99.9%	OLT 0.5%	BLT 0.5%	OLT 99.5%	BLT 99.5%	OLT 1%	OLT 99%
1	99.57	99.57	100.37	100.39	99.76	99.76	100.22	100.22	99.81	100.18
2	99.39	99.35	100.51	100.52	99.66	99.64	100.32	100.35	99.73	100.26
3	99.26	99.16	100.6	100.66	99.58	99.54	100.39	100.43	99.66	100.32
4	99.13	99	100.69	100.79	99.51	99.47	100.45	100.5	99.61	100.37
5	99.01	98.74	100.77	100.88	99.45	99.37	100.51	100.57	99.56	100.42
6	98.92	98.55	100.85	100.93	99.39	99.3	100.56	100.61	99.51	100.46
7	98.83	98.31	100.91	101.08	99.34	99.22	100.6	100.67	99.47	100.49
8	98.74	98.22	100.98	101.13	99.29	99.14	100.64	100.7	99.43	100.53
9	98.68	98.08	101.03	101.25	99.24	99.06	100.69	100.76	99.4	100.57
10	98.52	97.96	101.16	101.35	99.15	98.96	100.76	100.84	99.33	100.63
15	98.23	97.62	101.38	101.58	98.98	98.76	100.91	101.01	99.2	100.75
20	97.97	97.29	101.61	101.83	98.82	98.52	101.03	101.16	99.08	100.85
25	97.57	96.79	101.9	102.16	98.6	98.29	101.23	101.36	98.91	101
30	97.32	96.47	102.13	102.38	98.46	98.09	101.35	101.49	98.8	101.1
35	96.89	96.03	102.44	102.77	98.24	97.85	101.53	101.68	98.65	101.24
40	96.64	95.67	102.65	103.01	98.12	97.66	101.65	101.8	98.55	101.32
45	96.29	95.14	102.95	103.4	97.92	97.43	101.82	101.99	98.41	101.45
50	96.07	94.78	103.16	103.66	97.79	97.27	101.92	102.1	98.32	101.53
55	95.73	94.33	103.5	104.03	97.6	97.03	102.09	102.3	98.17	101.65
60	95.43	93.94	103.74	104.32	97.48	96.84	102.21	102.42	98.08	101.73

Note: BLT and OBT denote block and ordinary bootstrap methods, respectively.

, using both the Block—represented by the symbol “B,”—and the standard bootstrap, represented by the letter “O.” Table 7 presents confidence bands for the BNB series while Table 8 depicts the LTC (LCN) series. The confidence intervals for specific percentiles of BNB and LTC returns, respectively, are also provided in these tables. These are extracted over a range of time periods spanning from 10 to 60 days, leveraging two different approaches—block and ordinary bootstrap—and share insights into potential price movements over a sixty-day (two-month) interval. To make forecast interval comparisons easier, each cryptocurrency’s starting price is set to USD 100.

Table 7 outlines a comparison of the block bootstrap and ordinary bootstrap methods for estimating the prediction intervals of BNB returns across several percentiles, 0.1%, 0.5%, 1%, and 99.9%, over horizons ranging from 1 to 60 days. Values of both methods are generally analogous across all time horizons, with only minor variations. For example, on Day 1, the two methods forecast the 0.1% percentile at 99.56 and the 99.9% percentile at 100.35, where the block bootstrap method forecasts lower for the 0.1% percentile and marginally higher for the 99.9% percentile. This shows that the block bootstrap method has the tendency to give wider confidence intervals, especially for extreme tail events, than the ordinary bootstrap. Both methods show, for the 0.1% and 0.5% percentiles, a decrease in the forecasted values as the forecast horizon increases; conversely, an increase is observed in the 99.9% percentile. For instance, the 0.1% percentile (BBNB) dips from 99.56 to 95.18, while the 99.9% percentile (BBNB) rises from 100.35 to 103.78 within the 60-day horizon, suggesting that the lower percentiles are more conservative estimates of BNB’s return, and that over time, predictions become less favorable. In contrast, higher percentiles indicate larger forecasted returns over time.

Notably, the difference between the 0.1% and 99.9% percentiles is large for all horizons, with the 99.9% percentiles always skewing higher. This leads to a sound conclusion and merely suggests that extreme positive tail events yield large returns, making them more important than extreme negative tail events, which culminate in large losses over a prolonged period of time. Commonly, a methodological difference signifies that the prediction intervals provided by block bootstrap will be wider as it captures more complex dependencies and volatility clustering over time. On the contrary, ordinary bootstrap grants narrower and stable intervals, suggesting a failure in accurately grasping these dependencies.

Overall, findings from Table 8 indicate that the block bootstrap method provides more risk-conservative tail risk estimates essential to risk management within the cryptocurrency market. Higher prediction intervals over time translate to higher uncertainty, and the fact that the block bootstrap is capable of capturing complex dependencies makes it a great tool to utilize when estimating long-horizon risk. Both methods, however, emphasize the importance of selecting the appropriate technique and percentiles in achieving the right determination of BNB’s risk and return pattern over different time frames.

Table 8 presents the predicted intervals of LTC returns at percentiles 0.1%, 0.5%, 1%, 99.5%, and 99.9% for the block bootstrap and ordinary bootstrap methods over horizons from 1 to 60 days. Similar to BNB, both methods result in values that are highly analogous for all percentiles and time horizons, with slight distinctions between the two approaches. Day 1 0.1% percentile predictions are 99.57 for OLT and 99.57 for BLT, while the 99.9% percentile predictions are 100.37 for OLT and 100.39 for BLT, respectively. Generally, both methods present almost identical overall prediction intervals. At the upper extreme percentiles—for instance, 99.9%—the block bootstrap method shows a slightly higher value, while granting lower values for the lower percentiles. For example, at Day 60, the estimate for OLT is 95.43, while that of BLT is 93.94 at the 0.1 percentile, despite the prediction for 103.74 at the 99.9% percentile by OLT, and 104.32 by BLT. In addition, with increased forecasting horizons, intervals across the 0.1%, 0.5%, and 1% percentiles show decreases, while those at 99.9% tend to rise. For instance, on Day 1, the 0.1% percentile for OLT was 99.57 and that of the 99.9% percentile was 100.37, while at Day 60 these shifted to 95.43 and 103.74, respectively. This suggests that the more prolonged a period of time, the more uncertainty exists, with 99.9% percentile predictions being more optimistic for higher returns in the long run. On the other hand, the 0.1 percentiles are more conservative estimates of possible losses. This distinction between the two methods can be seen at extreme percentiles, as the block bootstrap method tends to widen the ordinary bootstrap prediction intervals. This leads to a strong suggestion that the block bootstrap might be more effective in capturing complex dependencies and volatility clustering in the data. However, both methods indicate a similar trend where the prediction intervals tend to widen as the horizon grows—especially for higher percentiles—underscoring the increased uncertainty for long-term forecasts. These findings, therefore, support that the block bootstrap method provides estimates that are more conservative, in particular for tail events, while ordinary bootstrap produces relatively narrower intervals. Both methods demonstrate the importance of assessing tail risks when predicting LTC returns over different time horizons, with block bootstrap being a more robust method in capturing extreme market behavior.

In summary, Table 7 and Table 8 report on key information regarding the volatility of BNB and LTC returns at different time horizons and distributional quantiles. There is a precise assessment of various bootstrap methods for the estimation of these intervals, with results pointing to widening or narrowing width trends for larger time horizons that are concluded in confidence intervals across percentiles and time horizons. Moreover, there is also an expanding or shrinking interval with a growing time horizon perceived in the confidence intervals across the percentiles and horizons. Generally, it appears that FHS obtained by block bootstrap provides particularly wider prediction intervals than the standard bootstrap approach. The difference in width for the two methods is higher for extreme percentiles, such as 0.1% and 99.9%, compared with those closer to the median, 1–99%. Clearer intervals imply a better visualization of uncertainty and variability that can be captured in forecasted prices and are hence critical for risk management and decision-making.

5. Discussion and Conclusions

This study simulated the one-day movement of BNB and LTC prices using the FHS approach. The statistical tests suggest that the ARMA-GARCH model may not entirely capture the complex dependencies and heteroscedasticity in cryptocurrency return data. In an effort to address the remaining hidden dependencies during the simulation, the filtered residual returns were drawn as blocks of consecutive observations. The simulated confidence bands for the two cryptocurrencies result in wider bands for the block bootstrap, signifying greater uncertainty in the forecast. However, block bootstrap should be used with care and must consider factors including block size, data quality, and the assumptions underlying the technique.

Additionally, a significant d for both assets might suggest noteworthy nonlinear dynamics, hence requiring advanced modeling techniques, perhaps with implications for long-memory effects or other complex behaviors in the return series. For LTC, the negative γ further corroborates a leverage effect, indicating that negative price shocks tend to increase volatility more than positive price shocks do, depicting an important asymmetry for risk management under market stress.

The analysis of Table 3 and Table 4 is indicative of some significant implications of conditional means, variance, and volatility dynamics for BNB and LTC returns, respectively. The high coefficients of β₁ and β₂ of BNB are indicative of strong volatility persistence, as a high-volatility phase is likely to be followed by a longer phase of uncertainty. Similarly, the strong volatility clustering in LTC is significant for trustworthy risk forecasting. The fact that β₁ and β₂ are less than 1 in both cases reaffirms a fundamental characteristic of the mean-reverting nature of volatility—that in the long run, it indeed does lose steam even after periods of high, sustained variance. Also, a significant d for both securities would imply considerable nonlinear dynamics and thus the necessity for advanced model methods, potentially with implications regarding long-memory effects or other kinds of complex behavior in the return series. For LTC, the negative γ again implies a leverage effect, meaning that negative price shocks induce greater volatility than do positive price shocks, depicting an important risk management asymmetry in distress in the market.

The analysis of the prediction intervals for both cryptocurrencies shows that the block method tends to yield wider confidence bands than the ordinary bootstrap method. This is especially pronounced at extreme percentiles, such as 0.1% and 99.9%, compared to mid-range percentiles, such as 1% and 99%, which allow for a comprehensive understanding of the distribution of returns from extremely low to exceptionally high. It also reinforces the trend in confidence intervals over several horizons in time and percentiles, revealing that such intervals are likely to be expanded or contracted with added time at any point. These results facilitate our understanding of how to assess uncertainty for relatively high-frequency cryptocurrency markets and improve valuable insights supportive of enhanced risk management and decision-making.

Our empirical results are complementary to current contributions of recent research in highly volatile cryptocurrency markets. Wang et al.’s (2021) and Xu et al.’s (2021) research has emphasized the importance of tail risk measurement and volatility modeling in grasping cryptocurrency market dynamics. Conlon et al. (2021) and Xiong et al. (2021) are two new contributions that emphasize the relevance of behavioral traits and market efficiency in the analysis of trading risk for cryptocurrencies. Based on these foundations, and by providing a long data-driven risk analysis framework, the suggested FHS simulation approach contributes to the increasing debate over effective risk management techniques for comparatively high-frequency cryptocurrency markets.

The FHS can be useful in analyzing non-normal risks and tail occurrences in the financial markets. It allows for the risk assessment process to be more data-driven and realistic. However, its application needs to coincide with additional approaches to risk management, as we need to consider its limitations. Tail risk for speculative assets is key to having the ability to model tail behavior and to leveraging historical data to estimate the probability of a highly rare event. This study issues significant guidelines on how to address cryptocurrency market complexities by underlining deficiencies in traditional techniques of modeling and proposing alternative paths to improve accuracy in risk assessment. Furthermore, there is a plea for the recognition and reduction of the biases related to block bootstrap approaches, as it grants numerous practical strategies that risk management experts can further adopt to enhance decision-making. This report, therefore, stresses the importance of extended out-of-sample backtesting and calls for an increase in reliable and robust processes for risk analysis in daily cryptocurrency exchanges, especially during periods of market turmoil.

The research approach employed in this study is closely aligned with its key objective of enhancing risk analysis in crypto markets through the refinement of the Filtered Historical Simulation (FHS) and block bootstrapping techniques. Through the application of the ARMA-GARCH model in the analysis of Binance Coin (BNB) and Litecoin (LTC) daily log returns, the study is able to effectively obtain volatility dynamics and identify potential violations of the independent and identically distributed (i.i.d.) assumption—a critical component in deriving an accurate risk estimate. The implementation of block bootstrapping corrects for hidden dependencies in the return series with a more stable assessment of the tail risks when compared to the application of the standard bootstrap method. The comparison of confidence intervals at the tail percentiles exemplified in the research also supports its focus on improving the prediction of tail risk. The methodological choices are explicitly tied to the research objective by providing a data-driven and advanced methodology for assessing cryptocurrency market risks that aid in improved risk management decisions for high-frequency financial markets.

While the standard assumption of Filtered Historical Simulation (FHS) is that GARCH-filtered residuals are i.i.d., our diagnostic results indicate that this assumption does not hold in the case of both BNB and LTC. Rather than treating this as a disqualifying model flaw, we treat it as an empirical reality that reflects the limitations of volatility models in capturing all dependencies, particularly under high persistence and nonlinearity. This motivates our study of block bootstrap techniques, namely those intended to preserve local patterns of dependence on resampling. Here, therefore, the study reframes the application of FHS—not as a process that makes an assumption of perfect filtering of models, but as one that can be generalized to accommodate model misspecification. Our findings suggest that block bootstrapping is a corrective technique that improves tail risk estimation if perfect model assumptions cannot be guaranteed.

Although this study does not include an out-of-sample backtest process, it gives a robust in-sample diagnostic framework that shows how different bootstrapping procedures behave under actual model constraints. The emphasis here is on evaluating how the standard and block bootstrapping procedures respond to residual dependence and time aggregation, which are typically challenges to the operation of standard risk estimation procedures. We agree that backtesting VaR and ES predictions on hard lines through rolling windows and test statistics for point prediction (e.g., Kupiec’s test) is a valuable extension, and we identify this as one of the most worthwhile areas for future research. Even in the current in-sample context, however, cross-comparison per se generates decision-relevant evidence on limitations of traditional FHS and actual-world advantages of block bootstrap-based risk estimation.

5.1. Limitations and Future Research

Although this study provides useful information in line with cryptocurrency market risk analysis, certain limitations should be mentioned. The FHS method is appropriate in the estimation of non-normal risks and tail events, and therefore, it should be complemented with other methods to develop a more comprehensive risk assessment framework. Moreover, through widening confidence intervals and a reflection of higher uncertainty in projections, the implementation of the block bootstrap approach requires careful consideration as related to variables such as block size, data quality, and underlying assumptions. Prominently, model risk increases at tails, and further research is required to conduct out-of-sample backtesting for each methodology, especially during turmoil, to be able to fully understand their performance in realistic circumstances. Addressing these limitations could enhance the robustness and application of the proposed risk analysis framework in high-frequency cryptocurrency markets.

5.2. Concluding Remarks

In this study, the Filtered Historical Simulation (FHS) technique is blended with the block bootstrap (BB) method to improve risk estimation of cryptocurrency markets. Employing a volatility ARMA-GARCH model and incorporating bootstrap resampling techniques, we address some of the key challenges, including volatility clustering, fat-tailed distributions, and serial dependence, characteristics that are typically omitted from standard risk models.

The findings point out that Value-at-Risk (VaR) forecasting and tail risk estimation are both significantly enhanced as long as dependencies of financial time series are preserved, thereby validating the need for adaptive resampling techniques in risky asset classes. The results align with the overall aim of the paper to keep a concrete and data-driven environment for cryptocurrency risk measurement. Future research may explore additional non-parametric frameworks or machine learning-based volatility models for further improvement in risk forecasting of digital asset markets.

Rather than proposing an alternate approach to forecasting, this study is a contribution to the literature by empirically demonstrating the pitfalls of applying standard FHS with imperfect model filtering—a common occurrence in real financial data, especially within the cryptocurrency market. Our findings point toward the fact that if standardized ARMA-GARCH model residuals still exhibit dependencies, standard bootstrapping overestimates tail risk, while block bootstrapping yields more realistic and precise risk estimates. The crux of this paper is to explain that the block bootstrap is a valuable extension of FHS in contexts where perfect conditions are violated.