# An Empirical Study of Volatility in Cryptocurrency Market

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data

#### Descriptive Statistics

- R
_{it}= Daily log return of cryptocurrency at day t; - P
_{it}= Closing price of crypto at day t; - P
_{i}_{,t−1}= Closing price of crypto at day t − 1.

## 3. Data Analysis

#### 3.1. Unit Root Test

_{o}is the constant, and ‘e’ is an error term.

#### 3.2. ARCH Effect Test

#### 3.3. Granger Causality

#### 3.4. GARCH Model

_{1}and β

_{1}are coefficients of the ARCH and GARCH terms, respectively, where ‘${\alpha}_{1}$’ represents the ARCH effect which estimates the response to any shock or news in the crypto currency market. ‘${\beta}_{1}$’ represents the GARCH effect which identifies the persistency of the volatility. A high ARCH coefficient (${\alpha}_{1}$) indicates a greater sensitivity of volatility to the news coming and a high GARCH (${\beta}_{1}$) value depicts the presence of a high persistency of volatility and the volatility taking more time to die out (Chaudhary et al. 2020; Rastogi 2014). The GARCH model will be stable only if the sum of ${\alpha}_{1}$ and ${\beta}_{1}$ will be less than one or else the data will reflect an explosive nature. As seen in Table 9, there is a strong presence of the ARCH and GARCH effects across all the cryptos and there is a strong persistency factor across all the cryptos. A high volatility across all currencies can be observed from the GARCH graphs (Figure 2).

#### 3.5. EGARCH Model

#### 3.6. GARCH in Mean (GARCH-M) Model

_{t}which is the return of the cryptocurrency, is taken as a function of its past volatility (${h}_{t-1}$) and if the coefficient of ${h}_{t-1}$ denoted as δ is significant and also positive, then it can be concluded that the increase in the risk is compensated by high returns and δ can be attributed to the risk premium.

#### 3.7. DCC(1,1) Model

_{i}

_{,}

_{t}and r

_{j}

_{,}

_{t}, and then deploying the AR(1) models, there are two residual time variables, a

_{i}

_{,}

_{t}and a

_{j}

_{,}

_{t}. For these two variables, H

_{t}represents the dynamic conditional covariance matrix of two-time series r

_{i}

_{,}

_{t}and r

_{j}

_{,}

_{t}.

- R
_{t}: - represents the dynamic conditional correlation (DCC) matrix.
- D
_{t}: - represents the diagonal matrix from the covariance matrix H
_{t}. - D
_{t}^{−1}: - the inverse of the D
_{t}Matrix.

_{t}, R

_{t}, D

_{t}, and D

_{t}

^{−1}is:

_{t}

^{−1}HtD

_{t}

^{−1}

_{i,t}and ε

_{j,t}; the following relationship is obtained by defining the following variables

- Q
_{t} - indicates Covariance Matrix
- G
_{t} - indicates the diagonal matrix of the covariance matrix Q
_{t}, - Qt
^{−1} - indicates the inverse matrix of the matrix Q
_{t}. - C
_{t} - indicates Correlation Matrix

_{t}, C

_{t}, G

_{t}, and D

_{t}

^{−1}are:

_{t}= G

_{t}C

_{t}G

_{t}

_{t}= G

_{t}

^{−1}Q

_{t}G

_{t}

^{−1}

_{t}, H

_{t}, and Q

_{t}, assume:

_{ij}

_{,t}= σ

_{i}

_{,t}ρ

_{ij}

_{,t}σ

_{j}

_{,t}, σ

_{ji}

_{,t}= σ

_{i}

_{,t}ρ

_{ji}

_{,t}σ

_{j}

_{,t}

_{i}

_{,t}= σ

_{i}

_{,t}ε

_{i}

_{,t}and a

_{j}

_{,t}= σ

_{j}

_{,t}ε

_{j}

_{,t}from AR(1) and GARCH(1,1) (Engle 1982, 2001).

_{ij}

_{,t}and ρ

_{ji}

_{,t}are the varying correlation.

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Currency | Market Cap |
---|---|

Bitcoin | USD 377.53B |

Ether | USD 129.53B |

Litecoin | USD 3.79B |

XRP | USD 16.02B |

BITCOINRT | ETHERRT | LITECOINRT | XRPRT | |

Mean | 0.20% | 0.32% | 0.18% | 0.26% |

Median | 0.23% | 0.22% | −0.03% | 0.00% |

Minimum | −0.4973 | −0.5896 | −0.4868 | −0.653 |

Maximum | 0.2276 | 0.2586 | 0.607 | 1.028 |

Std.Dev. | 4% | 6% | 6% | 8% |

Skewness | −0.8444 | −0.6056 | 0.6600 | 1.8551 |

Kurtosis | 15.0827 | 11.9877 | 15.5543 | 32.6549 |

Observations | 1913 | 1913 | 1913 | 1913 |

Jarque-Bera | 11,864.2 | 6555.72 | 12,701.7 | 71,193.7 |

Probability | 0 | 0 | 0 | 0 |

BITCOINRT | ETHERRT | LITECOINRT | XRPRT | |

BITCOINRT | 1.0000 | |||

ETHERRT | 0.7210 | 1.0000 | ||

LITECOINRT | 0.6825 | 0.6931 | 1.0000 | |

XRPRT | 0.4676 | 0.5083 | 0.5235 | 1.0000 |

One-Year Return | ||||

BITCOIN | ETHER | LITECOIN | XRP | |

Mean | 175% | 595% | 1092% | 334% |

Med | 61% | 65% | 24% | 22% |

Max | 1832% | 14,171% | 44,380% | 6921% |

Min | −83% | −92% | −87% | −93% |

SD | 277% | 1733% | 4630% | 1075% |

Three-Year Return | ||||

Mean | 78% | 86% | 36% | 31% |

Med | 79% | 66% | 20% | 18% |

Max | 149% | 267% | 280% | 178% |

Min | −0.12% | −15.52% | −56.74% | −40.37% |

SD | 34% | 74% | 67% | 45% |

Five-Year Return | ||||

Mean | 94% | 133% | 96% | 63% |

Med | 102% | 132% | 99% | 73% |

Max | 121% | 239% | 171% | 108% |

Min | 49% | 23% | 1% | −1% |

SD | 22% | 63% | 66% | 37% |

BITCOINRT | ETHERRT | LITECOINRT | XRPRT | |

T Stat | −47.051 | −46.831 | −46.98 | −30.04 |

Prob. | 0.0001 | 0.0001 | 0.0001 | 0.0001 |

BITCOINRT | ETHERRT | LITECOINRT | XRPRT | |

F-Stats | 9.7978 | 25.1836 | 36.7409 | 133.5966 |

Prob. | 0.00 | 0.00 | 0.00 | 0.00 |

Null Hypothesis: Indicates Does Not Granger Cause | F-Statistic | Prob. | Type of Causality |
---|---|---|---|

ETHERRT BITCOINRT | 8.5385 | 0.0002 * | Unidirectional |

BITCOINRT ETHERRT | 0.0623 | 0.9396 | No causality |

LITECOINRT BITCOINRT | 1.0697 | 0.3433 | No causality |

BITCOINRT LITECOINRT | 2.2198 | 0.1089 | No causality |

XRPRT BITCOINRT | 4.6675 | 0.0095 * | Unidirectional |

BITCOINRT XRPRT | 0.6506 | 0.5218 | No causality |

LITECOINRT ETHERRT | 0.2374 | 0.7887 | No causality |

ETHERRT LITECOINRT | 0.9367 | 0.3921 | No causality |

XRPRT ETHERRT | 2.9438 | 0.0429 ** | Unidirectional |

ETHERRT XRPRT | 0.3821 | 0.6825 | No causality |

XRPRT LITECOINRT | 0.5818 | 0.559 | No causality |

LITECOINRT XRPRT | 1.1575 | 0.3145 | No causality |

BITCOIN RT | ETHER RT | LITECOIN RT | XRP RT | |||||

Coeff. | Prob. | Coeff. | Prob. | Coeff. | Prob. | Coeff. | Prob. | |

c | 0.0015 | 0.1076 | 0.0024 | 0.0478 | 0.0012 | 0.3621 | 0.0020 | 0.2155 |

AR(1) | −0.7508 | 0.0000 | −0.8053 | 0.0000 | −0.7799 | 0.0000 | −0.4235 | 0.1371 |

MA(1) | 0.7131 | 0.0000 | 0.7637 | 0.0000 | 0.7419 | 0.0000 | 0.3634 | 0.2148 |

BITCOINRT | ETHERRT | LITECOINRT | XRPRT | |||||

Coeff. | Prob. | Coeff. | Prob. | Coeff. | Prob. | Coeff. | Prob. | |

C | 0.0001 | 0.0010 | 0.0003 | 0.0010 | 0.0002 | 0.0010 | 0.0004 | 0.0010 |

ARCH (${\alpha}_{1}$) | 0.1008 | 0.0010 | 0.0927 | 0.0010 | 0.0673 | 0.0010 | 0.3923 | 0.0010 |

GARCH (${\beta}_{1}$) | 0.8376 | 0.0010 | 0.7961 | 0.0010 | 0.8737 | 0.0010 | 0.6304 | 0.0010 |

BITCOINRT | ETHERRT | LITECOINRT | XRPRT | |||||

Coeff. | Prob. | Coeff. | Prob. | Coeff. | Prob. | Coeff. | Prob. | |

${\alpha}_{0}$ | −0.6223 | 0.001 | −0.5334 | 0.001 | −0.438 | 0.001 | −0.7695 | 0.001 |

${\alpha}_{1}$ | 0.1701 | 0.001 | 0.2075 | 0.001 | 0.1557 | 0.001 | 0.4378 | 0.001 |

λ | −0.0649 | 0.001 | −0.0226 | 0.0031 | 0.0274 | 0.001 | 0.0697 | 0.001 |

β | 0.9214 | 0.001 | 0.9343 | 0.001 | 0.9411 | 0.001 | 0.9159 | 0.001 |

BITCOINRT | ETHERRT | LITECOINRT | XRPRT | |||||

Coeff. | Prob. | Coeff. | Prob. | Coeff. | Prob. | Coeff. | Prob. | |

Δ | 0.0413 | 0.7360 | 0.1186 | 0.3143 | 0.1363 | 0.3360 | 0.0182 | 0.7593 |

C | 0.0001 | 0.0010 | 0.0002 | 0.0010 | 0.0002 | 0.0010 | 0.0003 | 0.0010 |

A | 0.1009 | 0.0010 | 0.1128 | 0.0010 | 0.0678 | 0.0010 | 0.3626 | 0.0010 |

Β | 0.8373 | 0.0010 | 0.8223 | 0.0010 | 0.8725 | 0.0010 | 0.6538 | 0.0010 |

BITCOINRT | ETHERRT | LITECOINRT | XRPRT | |

BITCOINRT | 1.0000 | |||

ETHERRT | 0.7210 | 1.0000 | ||

LITECOINRT | 0.7379 | 0.7456 | 1.0000 | |

XRPRT | 0.6143 | 0.6869 | 0.6818 | 1.0000 |

Estimate | t-Value | Pr(>|t|) | ||
---|---|---|---|---|

All Currencies | α | 0.033 | 7.815 | 0.000 |

β | 0.963 | 186.198 | 0.000 | |

Bitcoin–Ether | α | 0.058 | 2.330 | 0.020 |

β | 0.934 | 27.697 | 0.000 | |

Bitcoin–XRP | α | 0.037 | 3.337 | 0.001 |

β | 0.956 | 66.737 | 0.000 | |

Bitcoin–Litcoin | α | 0.048 | 3.615 | 0.000 |

β | 0.948 | 62.468 | 0.000 | |

Ether–Litcoin | α | 0.038 | 5.163 | 0.000 |

β | 0.962 | 128.784 | 0.000 | |

Ether–XRP | α | 0.034 | 2.882 | 0.004 |

β | 0.962 | 64.787 | 0.000 | |

Litcoin–XRP | α | 0.034 | 3.421 | 0.001 |

β | 0.965 | 86.364 | 0.000 |

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## Share and Cite

**MDPI and ACS Style**

Gupta, H.; Chaudhary, R.
An Empirical Study of Volatility in Cryptocurrency Market. *J. Risk Financial Manag.* **2022**, *15*, 513.
https://doi.org/10.3390/jrfm15110513

**AMA Style**

Gupta H, Chaudhary R.
An Empirical Study of Volatility in Cryptocurrency Market. *Journal of Risk and Financial Management*. 2022; 15(11):513.
https://doi.org/10.3390/jrfm15110513

**Chicago/Turabian Style**

Gupta, Hemendra, and Rashmi Chaudhary.
2022. "An Empirical Study of Volatility in Cryptocurrency Market" *Journal of Risk and Financial Management* 15, no. 11: 513.
https://doi.org/10.3390/jrfm15110513