Heterogeneity, Jumps and Co-Movements in Transmission of Volatility Spillovers Among Cryptocurrencies

3.1 Realized Volatility Estimation

A realized volatility estimator affects the quality of all HAR models. Log price p _t of an asset is based on the general stochastic volatility jumps-diffusion model:

μ _t stands for the drift term with a continuous variation sample path, σ _t represents the stochastic volatility process and is positive, W _t represents the driving standard Brownian motion and k _t dq _t is the random jump size, while 0 ≤ t ≤ T.

For our analysis, we estimate daily realized volatility series with the realized volatility measure (Andersen and Bollerslev 1998), given by:

where R _i,t is the daily log return within day t and i = 1, …, N is the total number of intraday observations within a day t. We rely on the use of quadratic variation to estimate volatility as the best estimator of integrated volatility considering that volatility includes jumps:

where ∫ t − 1 t σ s 2ds stands for the continuous sample path variation and ∑ t − 1 < s < t k s 2 represents the discontinuous jump variation. The daily realized volatility estimator of quadratic variation can be used when N → ∞ as a consistent estimator of quadratic variation.

3.2 Jumps

Volatility at a given daily point might include jump variation. To take this into consideration, we detect jumps components from realized volatility. We rely on the methodology of Aït-Sahalia and Jacod (2012) and Duong and Swanson (2015), to detect and quantify small, large, upside, downside, and asymmetric jumps in all univariate and multivariate HAR models, as follows:

3.3 Realized Skewness and Realized Kurtosis Risks

Following Amaya et al. (2015), we define realized skewness R S t that measures the asymmetry of the daily return distribution. We standardize RS _t , by the RV _t . RS _t in daily frequency, is given as follows:

whereR _i,t stands for the intra-day log-return within day t and i = 1, …, N and N is the total number of intra-day log-returns within a day. The interpretation can be given as follows: for a zero value the tails of the daily return distribution on both sides of the mean value balance out overall (symmetric distribution). When the value is negative, the left tail is longer than right side tail (left-skewed distribution). When the value is positive, the right tail is longer than the left side tail (right-skewed distribution).

Subsequently, we estimate the kurtosis risk for a univariate price process (Barndorff-Nielsen and Shephard 2004). Following Amaya et al. (2015), we define realized kurtosis as R K t as we consider the existence of extreme deviations in daily return distribution. RK _t is standardized by the RV _t . RK _t in daily frequency can be defined as:

(where R _i,t is the intra-day log-return within day t and i = 1, …, N and N is the total number of intra-day log-returns within a day. The interpretation can be given as follows: it shows extreme deviations from the Gaussian distribution. When extreme deviations are similar to the Gaussian distribution is called mesokurtic distribution). When there are fewer and less extreme deviations than the Gaussian distribution (kurtosis is called platykurtic distribution). When there are more extreme deviations than the Gaussian distribution (kurtosis is called leptokurtic distribution).

3.4 Realized Covariance Estimation

We use realized covariance estimator introduced by Barndorff-Nielsen and Shephard (2004) to estimate realized covariances RCov _t for each pair of cryptocurrencies, as follows:

where R _i,t represent the intra-day return series of cryptocurrencies a and b, respectively and i = 1, …, N is the number of intraday returns.

3.5 HAR Models

We employ a family of univariate HAR models (Duong and Swanson 2015) for in-sample realized volatility predictions of the four cryptocurrencies. These models incorporate various jump components and include: HAR-RV-C (continuous only), HAR-RV-CJ (continuous and jumps), HAR-RV-C-PV (bipower variation), HAR-RV-C-UJ (upside jumps), HAR-RV-DJ (downside jumps), HAR-RV-C-UDJ (upside and downside jumps), HAR-RV-C-LJ (large jumps), HAR-RV-C-APJ (asymmetric jumps) and HAR-RV-C- RSK (Realized Skewness and Kurtosis).

We further extend these models to a multivariate framework (MHAR) to investigate spillover transmission between cryptocurrencies. All MHAR models incorporate jumps and are categorized into two versions: MHAR-RCov (including realized variance covariances) and MHAR-RV (including variances only).

Each of the MHAR-RV models incorporates the jump components of the univariate models (listed above). Additionally, we incorporate realized skewness and kurtosis risks into both univariate and multivariate HAR models, resulting in a total of 18 models for analysis.

HAR-RV-C (continuous)

The benchmark model (Duong and Swanson 2015).

where RVC _t stands for the continuous component of realized volatility.

HAR-RV-CJ (continuous and jumps)

where RVJ _t stands for the jumps component of realized volatility.

HAR-RV-C-TBV (bipower variation)

where TBPV _t stands for the threshold bipower variation.

HAR-RV-C-UJ (upside jumps)

where RJ t + is the upside jump.

HAR-RV-DJ (downside jumps)

where RJ t − stands for the downside jump.

HAR-RV-C-UDJ (upside and downside jumps)

Is the combination of both upside and downside jumps.

HAR-RV-C-LJ (truncated large jumps)

where RVLJ _t stands for the large jump.

HAR-RV-C-APJ (asymmetric jumps)

where RJA _t is the asymmetric jump.

We employ the following HAR model including realized skewness and realized kurtosis in both a univariate and multivariate framework, also:

HAR-RV-C-RSK (Skewness and Kurtosis)

is the combination of both Realized Skewness and Realized Kurtosis.

In each of the above HAR-RV models we examine forecasting horizons of h = 1, 7, 30 for one day, week and month, respectively.

3.6 Model Estimation

Based on the univariate HAR models, we extend the two versions of MHAR models, with the inclusion of jumps measures. A multivariate version of HAR model with the inclusion of daily realized covariances is defined as (Fengler and Gisler 2015):

y t = vec h I Co v t ̂ is the half-vectorized time t daily realized covariance matrix with size N. β ₀ is a K × 1 vector of intercepts, β _h represents the K × K coefficient matrices, ɛ _t represents a K × 1 vector of zero mean and finite variance innovations and y _h,t stands for a normalized h-period aggregate of y _t .

The restricted model that does not include covariances can be defined as:

y ̃ t is the N × 1 vector of the N realized variances, β ̃ 0 stands for a N × 1 intercept and β ̃ h is the N × N coefficient matrices. ε ̃ t represents a N × 1 vector of innovations.

Following Fengler and Gisler (2015), the two versions of MHAR models are constrained VAR (p) models. The first equation can be then re-defined as follows:

where φ ₀ = β ₀, φ _i represent autoregressive coefficient matrices and ɛ _t is the error term vector, assuming serial uncorrelatedness.

In this study, to address the curse of dimensionality we use the least absolute shrinkage and selection operator (lasso) (Tibshirani 1996), to estimate and select variables for the MHAR-RV and MHAR-RCov models. Subsequently, the new model is written as:

y t = vec y 1 , … , y T is the KT × 1 vector with the observations for the variables of the left-hand side. X = Z ⊗ I _k where ⊗ stands for the Kronecker product. I _k represents the unit matrix of size K and Z = Z 1 , … , Z T ⊺ is the T × 3K matrix of the explanatory variables and Z t = y 1 , t − 1 ⊺ , y 7 , t − 1 ⊺ , y 30 , t − 1 ⊺ ⊺ . ε = vec ε 1 , … , ε T , stands for the KT × 1 vector of innovations and B = vec β 1 , β 7 , β 30 is the 3K ² × 1 vector of parameters. We estimate elements of B with the lasso:

λ ≥ 0 represents a tuning parameter to control the amount of shrinkage. B _i are the elements of the vector B and L ₁ is the penalty of the lasso. The less important coefficients are set to zero.

3.7 Spillover Index

To quantify volatility spillovers between cryptocurrencies, this study employs the forecast error variance decomposition (FEVD) approach for Vector Autoregressive (VAR) models, as proposed by Diebold and Yilmaz (2012, 2014). Fengler and Gisler (2015) employed a restricted VAR model with a moving average representation for spillover analysis:

The coefficient matrices A _i then derive from A _i  = φ ₁ A _i−1 + φ ₂ A _i−2 + ⋯ + φ _p A _i−p .

Pesaran and Shin (1998), proposed the generalized variance decomposition approach to analyze the directions of spillovers, as follows:

where A _h represents the vector of the coefficients for each of the HAR model with h days forecasting horizon, Σ stands for the covariance matrix of the error vector ɛ, σ _ij represents the variance of the error term for the jth equation. e _i denotes the binary selection vector with 1 for the ith entry and zero for others. Following Diebold and Yilmaz (2012), the normalization of the error variances before employing in the spillover index is required. The normalized error variances are expressed as follows:

The spillover index allows us to quantify the contribution of shocks from one cryptocurrency to the forecast error variance of another. Specifically, it estimates the proportion of the one-step-ahead error variance in forecasting cryptocurrency j’s realized variance that can be attributed to shocks originating from cryptocurrency i. The spillover index is calculated as the ratio of the sum of the off-diagonal elements divided by the total number of cryptocurrencies included in the HAR model.

4.1 Data

This study focuses on four major cryptocurrencies based on their market capitalization as of July 2019. The selected cryptocurrencies are Bitcoin (BTC), Litecoin (LTC), Ripple (XRP), and Ethereum (ETH). The empirical analysis covers a period from July 1, 2017, to February 28, 2021. Daily realized volatility and jump series are estimated using intraday logarithmic returns. Additionally, daily variances and covariances are calculated. Data are obtained at a one-hour sampling frequency from CryptoCompare.com.^[1] Table 1 provides the average daily contributions to the total realized variance for the four cryptocurrencies of the continuous component, jumps (CJ), large jumps (LJ), small jumps (SJ), upside jumps (UJ), and downside jumps (DJ).

4.2 Univariate HAR Models Predictions

This section explores the effectiveness of univariate HAR models with jump components in forecasting daily realized volatility for the four cryptocurrencies considered (Bitcoin [BTC], Litecoin [LTC], Ripple [XRP], and Ethereum [ETH]). Tables 2 and 3 present the corresponding results.

Specifically, Table 2 reports the coefficients and adjusted R-squared (adj. R ²) values for the HAR-RV-C (continuous component), HAR-RV-CJ (continuous and jumps), HAR-RV-C-TBV (bipower variation), and HAR-RV-C-UJ (upside jumps) models. Table 3 presents the same statistics for the HAR-RV-C-DJ (downside jumps), HAR-RV-C-UDJ (upside and downside jumps), HAR-RV-C-LJ (large jumps), HAR-RV-C-APJ (asymmetric jumps), and HAR-RV-C-RSK (realized skewness and kurtosis) models.

The in-sample forecasting results demonstrate that the daily predictability coefficients (β _CD and β _JD) have a statistically significant impact on realized volatility predictability for Litecoin, Ripple, and Ethereum in many cases. When examining the HAR-RV-C and HAR-RV-CJ models (Table 2), we observe significant heterogeneity and a stronger influence from jumps compared to the in-frequency continuous effect.

Furthermore, Table 3 highlights the statistical significance of most daily and many weekly coefficients associated with the jump components in the models. In most cases, the daily jump effect exhibits a higher absolute value compared to the weekly and monthly effects. Additionally, models incorporating downside jumps, asymmetric jumps, and the combination of both upside and downside jumps achieve the highest adj. R ² values.

For Bitcoin, the HAR-RV-C-UDJ, HAR-RV-C-DJ (downside jumps), and HAR-RV-C-LJ (large jumps) models generate statistically significant predictions for both daily and weekly realized volatility. For Ethereum, the HAR-RV-C-APJ, HAR-RV-C-DJ, and HAR-RV-C-UDJ models demonstrate significant predictability on a daily and weekly basis. Litecoin and Ripple exhibit similar results, with strong daily predictability impact from models incorporating jumps (HAR-RV-CJ, HAR-RV-C-UDJ, HAR-RV-C-DJ, and HAR-RV-C-APJ).

Tables 2 and 3 reveal that models including downside jumps, the combination of upside and downside jumps, and asymmetric jumps yield statistically significant predictability for daily and weekly realized volatility of Bitcoin and Ethereum, with higher adj. R ² values. Overall, the HAR models display similar predictability power for Litecoin and Ripple, with slight differences observed in their adj. R ² values. The in-sample predictions further suggest evidence of short memory, as shown by the higher absolute values of daily and weekly jump effects across all HAR models and cryptocurrencies.

4.3 Multivariate HAR Models – Static Analysis (Spillovers)

This subsection discusses the results from multivariate HAR (MHAR) models with jumps inclusion to measure volatility spillovers between four cryptocurrencies (Bitcoin [BTC], Litecoin [LTC], Ripple [XRP], and Ethereum [ETH]). Two versions of MHAR models are estimated using lasso regression following Fengler and Gisler (2015). Specifically, MHAR-RCov incorporate realized covariances, while MHAR-RV focus on variances only. The forecast horizon (h) is set to 14 days, and the number of lags (p) p = 2.^[2] Spillover transmission is measured using the frequency Connectedness package in R (Krehlik 2023).

Tables 6 and 7 present the spillover results for the MHAR-RCov models including covariances. More particular, Table 6 reports results for the MHAR-RCov-C (continuous component only), MHAR-RCov-CJ (continuous and jumps), and MHAR-RCov-C-TBV (bipower variation) models. Table 7 summarizes results for models incorporating various jump properties (upside, downside, combined, large, asymmetric, skewness, and kurtosis). Directional spillovers (from and to others) and spillover indices (in bold) are included, following the approach of Diebold and Yilmaz (2012, 2014).

Overall, the inclusion of jumps in the MHAR models significantly impacts the spillover analysis. As previously observed in Tables 2 and 3, models with jumps exhibit higher statistical significance (based on coefficient p-values and R-squared values) compared to models without jumps. This enhanced predictability of realized volatility translates to a larger magnitude of spillovers between the cryptocurrencies considered. Furthermore, the inclusion of covariances in the MHAR-RCov models (Tables 6 and 7) leads to higher level of spillover indices compared to models with variances only (Tables 4 and 5).

Table 6 highlights the importance of jumps to estimate volatility spillovers. The spillovers from the MHAR-RCov-CJ model are higher than those from the MHAR-RCov-C model (continuous component only). Table 7 further emphasizes the role of jump properties in spillover analysis. Large jumps, the combination of upside and downside jumps, and upside jumps exhibit the strongest impact, based on the sum of directional spillovers (from and to others). Overall, the spillover index values for the MHAR-RCov models range from 71.40 % to 88.12 % (Tables 6 and 7), indicating a high degree of interconnectedness among the cryptocurrencies considered.

Tables 4–7 also provide insights into the transmitters and recipients of spillovers. The last row in each table identifies the transmitters, while the last column identifies the recipients of spillovers. Across all models, Bitcoin and Ethereum exhibit the highest spillover transmission to other cryptocurrencies (based on the “TO” row). In contrast, directional spillovers from others (i.e. received by each cryptocurrency) exhibit slight differences across the four cryptocurrencies (based on the “FROM” column). Overall, directional spillovers appear to be relatively low across all models for the full sample period. Tables 6 and 7 further show that Bitcoin and Ethereum are the primary transmitters of spillovers. In most cases, directional spillovers from others involving covariances are slightly higher than those involving variances only.

Additionally, we examine the MHAR-RV models (variances only) presented in Tables 4 and 5. Specifically, a comparison of the spillover indices between these tables and Tables 6 and 7 (MHAR-RCov models with covariances) reveals consistently higher indices when covariances are included. Similar to the previous case, Table 4 highlights the importance of jumps inclusion to estimate spillovers. The MHAR-RV-CJ model (with jumps) exhibits higher spillovers compared to the MHAR-RV-C continuous model (without jumps), which is further reflected in the higher spillover index for the MHAR-RV-CJ model. In Table 5 we observe that as in the MHAR-RCov case, the combination of upside and downside jumps, large jumps, upside jumps, and downside jumps that exhibit the strongest influence. Overall, the inclusion of both jumps and covariances significantly enhances the measurement of spillover levels between the cryptocurrencies considered in the analysis.

4.4 Dynamic Analysis – Frequency Domain Analysis

While the static analysis provides valuable insights into the spillover behavior within the cryptocurrency market (2017–2021), it may overlook significant temporal variations. This period witnessed important events such as the 2018 price crash and the COVID-19 pandemic. To capture these dynamic aspects, we employ a rolling window analysis that decomposes connectedness over time. Following the frameworks of Diebold and Yilmaz (2012, 2014) for time-domain analysis and Barunik and Krehlik (2018) for frequency domain analysis, we conduct a rolling window analysis for both MHAR-RCov and MHAR-RV models. A window of 100 days and a forecast horizon of 14 days are used throughout the dynamic analysis, with a lag order of p = 2.

Figure 1 presents the rolling sample plot for total volatility spillovers. Figures 2–4 depict the spillover indices across three distinct time-frequency ranges: short-term (1–7 days), medium-term (7–30 days), and long-term (more than 30 days). Frequency connectedness is computed for all MHAR models (with and without covariances) using lasso regression estimation. In the corresponding figures, red lines represent MHAR-RCov spillovers, while black lines represent MHAR-RV spillovers.

Figure 1:

Figure 1 depicts the rolling estimates of spillover indices for a 14-day forecast horizon across both MHAR model versions. Red lines represent the dynamic connectedness for MHAR-RCov models (with covariances), while black lines represent the connectedness for MHAR-RV models (variances only). The rolling window used for estimation is 100 days. Jump series are estimated as described in Section 3.2, and the MHAR models incorporate various jump specifications as detailed in Section 3.5 (MHAR-RV-C-UJ, MHAR-RV-C-DJ, MHAR-RV-C-UDJ, MHAR-RV-C-LJ, MHAR-RV-C-APJ, and MHAR-RV-C-RSK).

Figure 2:

Frequency connectedness within the cryptocurrency market using a 14-day forecast horizon and a rolling window of 100 days. Frequency connectedness is estimated for three bands: short-term (1–7 days), medium-term (7–30 days), and long-term (over 30 days). Red lines represent MHAR-RCov models (with covariances), while black lines represent MHAR-RV models (variances only). Jumps are estimated as described in Section 3.2, and MHAR-RCov models are based on MHAR-RV-C (continuous component), MHAR-RV-CJ (continuous and jumps), and MHAR-RV-C-TBV (bipower variation) models, as detailed in Section 3.5.

Figure 3:

Frequency connectedness within the cryptocurrency market using a 14-day forecast horizon and a rolling window of 100 days. Frequency connectedness is estimated for three bands: short-term (1–7 days), medium-term (7–30 days), and long-term (over 30 days). Red lines represent MHAR-RCov models (with covariances), while black lines represent MHAR-RV models (variances only). Jumps are estimated as described in Section 3.2, and MHAR-RCov models are based on the MHAR-RV-C-UJ, MHAR-RV-C-DJ, MHAR-RV-C-UDJ models, as estimated in sub-Section 3.5.

Figure 4:

Frequency connectedness within the cryptocurrency market using a 14-day forecast horizon and a rolling window of 100 days. Frequency connectedness is estimated for three bands: short-term (1–7 days), medium-term (7–30 days), and long-term (over 30 days). Red lines represent MHAR-RCov models (with covariances), while black lines represent MHAR-RV models (variances only). Jumps are estimated as described in Section 3.2, and MHAR-RCov models are based on the MHAR-RV-C-LJ, MHAR-RV-C-APJ, MHAR-RV-C-RSK models, as estimated in sub-Section 3.5.

All figures reveal a clear response of both MHAR-RCov and MHAR-RV models to external events, particularly during July 2017-February 2021. These events include the 2018 cryptocurrency price crash and the COVID-19 pandemic (first and second waves). Sudden upward and downward movements in spillovers are observed throughout the sample period, highlighting the dynamic nature of interconnectedness within the cryptocurrency market. Furthermore, the spillover dynamics exhibit wave-like patterns with peaks (Asiri, Alnemer, and Bhatti 2023; Iyer and Popescu 2023; Mensi et al. 2023), suggesting a shared dynamics and response by both MHAR models to external events.

We can further observe the higher level of spillovers for MHAR-RCov models compared to MHAR-RV models. This difference is evident across all models and points out the importance of incorporating jumps and covariances when analyzing spillover dynamics. Specifically, a significant increase in spillovers is observed beginning in late 2017 and early 2018, with the spillover index exceeding 85 % for most models. The index then exhibits cyclical behavior throughout 2018, followed by another upward surge in the second half of the year (reaching 90 %). Subsequent declines are observed before another rise in late 2018 and early 2019 (ranging from 80 % to 90 % for most of the year, particularly for MHAR-RCov-DJ, MHAR-RCov-UDJ, and MHAR-RCov-LJ models). These findings support the inclusion of jumps when modeling spillover dynamics between cryptocurrencies, as jumps capture the impact of both the continuous component and discrete events.

In line with previous findings regarding the importance of jumps inclusion in forecasting returns and volatility in traditional financial markets (Andersen et al. 2012; Duong and Swanson 2015), the inclusion of jumps in spillover models is crucial for cryptocurrency markets. These jumps capture the impact of real-world events, as highlighted by Hu et al. (2019) who link Bitcoin price jumps to major economic occurrences. We further support this notion, demonstrating that jumps significantly contribute to spillover dynamics during periods of market stress. The COVID-19 pandemic (first wave) is an example of this effect. The resulting uncertainty led to a surge in the spillover index (Harb et al. 2024) exceeding 95 % for most models in early 2020. Models incorporating jumps (MHAR-RCov-DJ, MHAR-RCov-UDJ, MHAR-RCov-LJ, and MHAR-RCov-APJ) captured this more effectively, with spillover indices remaining above 90 %. A subsequent decline followed in mid-2020, with spillover indices ranging from 80 % to 95 % for the aforementioned models (MHAR-RCov-DJ, MHAR-RCov-UDJ, and MHAR-RCov-LJ).

Overall, the inclusion of covariances and jumps leads to a higher level of spillovers compared to models with variances only. This difference in spillover levels highlights the importance of focusing on the dynamics of MHAR-RCov based connectedness analysis. Additionally, certain jump effects (downside jumps, combined upside-downside jumps, and large jumps) appear to be earlier indicators of the impact of external events (such as the COVID-19 outbreak) on realized volatility spillovers and the level of interdependence within the cryptocurrency market. The dynamics of the total spillover index observed in this study, are in line with previous findings (Antonakakis, Chatziantoniou, and Gabauer 2019; Ji et al. 2018; Shahzad et al. 2021).

Turning now our attention to the frequency-domain analysis it reveals similar patterns across all models. As previously noted, the inclusion of covariances in MHAR models generally leads to higher spillover levels compared to models with variances only. Frequency decomposition of connectedness shows the time-frequency dynamics, with spillover plots confirming that covariances and specific jump effects contribute to an overall increase in connectedness, particularly at short-term (1–7 days) and medium-term (7–30 days) horizons.

Across all models, long-term connectedness remains relatively low, indicating weaker interdependence between cryptocurrencies at longer frequencies. While long-term spillover evidence is very little, short-term and medium-term dynamics reveal distinct patterns. Figures 2–4 illustrate the similar evolution of medium-term and long-term connectedness, with the latter consistently lower than short-term connectedness.

We further notice that medium-frequency components (one week to one month) influence the total connectedness observed in Figures 1–4. For example, the period encompassing the first and second waves of the COVID-19 pandemic (early and late 2020), which witnessed peak total connectedness, was primarily driven by the increased risk stemming from these medium-term events. Furthermore, when short-term connectedness declines, medium-term connectedness tends to rise, and vice versa. These short- and medium-term dynamics characterize the total connectedness within the cryptocurrency market. This observed pattern suggests positive expectations regarding future uncertainty in the cryptocurrency market, with shorter-term effects to influence more heavily the dynamics in this specific market. Market participants appear to anticipate that events leading to heightened uncertainty will not have a lasting impact. For instance, the period following the 2018 cryptocurrency crash exhibited greater influence from short-term factors, a similar pattern also observed after the initial wave of the COVID-19 pandemic.

In contrast, the heightened interconnectedness observed during the first and second waves of the COVID-19 pandemic appears to be primarily driven by medium- and long-term components. This suggests that cryptocurrency market participants anticipated the pandemic’s impact to extend over a longer period. Finally, in Figures 2–4, we again observe that MHAR-RCov-DJ, MHAR-RCov-UDJ, and MHAR-RCov-LJ models seem to better capture the total connectedness within the market. This further highlights the importance of incorporating jumps when measuring the connectedness in the cryptocurrency market.

4.5 Practical Implications

In this sub-section, we explore the practical implications of jumps for cryptocurrency asset management. Investors exposed to downside risk are likely interested in portfolio diversification beyond Bitcoin. However, diversification during periods of heightened uncertainty (e.g. COVID-19 pandemic) can be constrained, as highlighted by Asiri, Alnemer, and Bhatti (2023). Therefore, we examine the use of jump measures to investigate portfolio performance for four cryptocurrencies (Bitcoin, Litecoin, Ripple, and Ethereum).

The findings discussed in the previous sub-sections indicate that the jumps distinction between good and bad component of volatility, are important to model linkages between cryptocurrencies in an attempt to accurately assess risk. For this reason, we further exploit jumps measures, to construct an optimal portfolio minimizing daily downside jumps for this specific case. Specifically, the optimal weights for each cryptocurrency are calculated based on intra-day portfolio returns. If no downside jump is observed for a particular cryptocurrency on a given day, it receives a 100 % portfolio weight. However, if multiple cryptocurrencies experience no downside jumps, equal weights are assigned to those cryptocurrencies. The portfolio return is then constructed using these estimated optimal weights.

Figure 5 presents the daily optimal weights for the four cryptocurrencies, minimizing the downside jump effect within the portfolio. A total of 1,771 weight combinations were considered, ranging from 0 % to 100 % with 5 % increments. For example, on days where multiple cryptocurrencies exhibit downside jumps, the optimal portfolio might consist of equal weights (50 %) for each cryptocurrency. Conversely, if only one cryptocurrency experiences no downside jump, it would receive a 100 % weight in the portfolio on that day. Finally, the weight combination minimizing the overall portfolio’s downside jump is selected for each day.

Figure 5:

This figure presents the weights of the optimal portfolio of the four cryptocurrencies including bitcoin (BTC), litecoin (LTC), ethereum (ETH), and ripple (XRP), in a daily basis covering the period from 1st July 2017 to 27th February 2021.