# Hurst Exponent Analysis: Evidence from Volatility Indices and the Volatility of Volatility Indices

^{*}

## Abstract

**:**

_{gk,t}, V

_{p,t}, V

_{rs,t}, and V

_{s,t}, and then estimate the volatility of volatility indices through the GARCH(1, 1) model. Based on the values of the Hurst exponent, we analyze the trace of the behavior of three trading strategies, i.e., the momentum-based strategy, the random walk, and the mean-reversion strategy. The results are highly recommended for financial analysts dealing with volatility indices as well as for financial researchers.

## 1. Introduction

_{α}, has the same law as the process a

^{H}X

_{t}, where 0 ≤ t ≤ 1. An example of a self-similar stochastic process is a fractional Brownian motion. When a fractional Brownian motion has a Hurst exponent larger than 0.5, it means that its increments (the increments of this motion) are positively autocorrelated (Garcin 2019). Given that H belongs to the interval (0, 1), a fractional Brownian motion X

_{t}is a centered Gaussian process with covariance function:

^{2}. For more details about the fractional Brownian motion, see (Garcin 2019). Many studies focus on the Hurst exponent and fractal dimension analysis; it measures how rough a fractal object is (Rehman 2009).

_{s,t}, V

_{p,t}, V

_{gk,t}, and V

_{rs,t}. Finally, our study follows a graphical representation of the previous volatility measures. In Section 5, we discuss the main results analytically, the limitations of this work, and give the main conclusion from our research.

## 2. Hypotheses

_{1}) is as follows: Do the values of the Hurst exponent on the volatility of volatility indices better reflect the period’s changes than the values of the Hurst exponent on the volatility indices? The answer of this question requires the following steps and corresponding hypotheses.

_{gk,t}, V

_{p,t}, V

_{rs,t}, and V

_{s,t}by using the open, high, low, and close prices of five volatility indices during the period of 2001–2021. This estimation follows the procedure of (Chan and Lien 2003). The mathematical background is available in detail in the next section. According to (Chan and Lien 2003), the quantity V

_{s}, as the first logarithmic difference between the high and low prices, overestimates the other three measures, V

_{p,t}, V

_{rs,t}, and V

_{gk,t}. We extend this result by examining the GFC and COVID-19 periods. Even during intense periods, V

_{s}overestimates the other three measures. This is the reason we isolated the V

_{s}-figure; see for instance Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5.

_{2}) is: Do the values of the Hurst exponent on the volatility indices reflect the well-defined behavior of the financial market in the sense of three financial strategies or do the risk managers, investors, and modelers trust the behavior of financial market based on the values of the Hurst exponent on the volatility indices? We continue our analysis of the time series in the sense of the volatility of volatility. More precise, applying GARCH(1, 1) to the volatility measures V

_{s,t}, V

_{p,t}, V

_{rs,t}, and V

_{gk,t}, we obtain the volatility of volatility indices for the period 2001–2021 and the sub-periods 2001–2007 (before the GFC), 2008–2010 (GFC), 2008–2021 (after the GFC), and 2020–2021 (COVID-19). Then, by using the Hurst exponent on these volatility of volatility indices, we test how persistent (or not) our time series data is.

_{3}) is: Do the values of the Hurst exponent on the volatility of volatility indices reflect the well-defined behavior of the financial market in the sense of three financial strategies, or do the risk managers, investors, and modelers trust the momentum of the financial market based on the values of the Hurst exponent on the volatility of volatility indices? This last hypothesis relates to the case of rough (or not) volatility. The term “rough” is associated with the Hurst exponent through R/S analysis and the fact that the volatility model is a function of fractional Brownian motion. According to the works of (Cont and Das 2022) and (Fukasawa et al. 2019), if H is closer to 1 (0.5 < H < 1), then the increments of the time series are smoother than Brownian motion. Conversely, if H approaches 0 (0.5 to 0), then the increments of the time series become rougher than Brownian motion. Consequently, the term “rough” in our case is related to the distance of the values of the Hurst exponent from the value of 0.5 (Brownian motion). Thus, our final hypothesis (H

_{4}) is: How rough is the volatility or the volatility of volatility with respect to the given financial data?

## 3. Methodology & Data Description

_{s,t}, V

_{p,t}, V

_{gk,t}, and V

_{rs,t}. Then, a GARCH model is applied to the vectors V

_{s,t}, V

_{p,t}, V

_{gk,t}, and V

_{rs,t}; after that, we proceed with the estimation of volatility of volatility. The computation of the Hurst exponent is based on (Szóstakowski 2018) and (Ceballos and Largo 2018).

_{t}represents the opening price on day t, and let C

_{t}, H

_{t}, and L

_{t}be the closing, high, and low prices, respectively. Staring with the case of the first logarithmic difference between H

_{t}and L

_{t}, i.e., the high and low prices, we have:

_{s,t}, V

_{p,t}, V

_{gk,t}, and V

_{rs,t}based on the volatility indices considered. In general, GARCH (Bollerslev 1986) is a method for estimating volatility measure. Here, we follow the notation of (Jafari et al. 2007) in order to describe some of the basic components of the GARCH model. Suppose that r

_{t}expresses a return on an asset. Then:

^{2}, and ε

_{t}∼iid N(0, 1). GARCH implies an estimation for the variance through the constant variables α

_{0}, …, α

_{p}and β

_{1}, …, β

_{q}such that:

_{1}+ β

_{1}< 1 with α

_{1}≥ 0, β

_{1}≥ 0 and α

_{0}> 0. The corresponding numerical results are available in Section 4. Finally, we apply Hurst exponent analysis in order to examine if the time series is persistent or not. According to (García and Requena 2019), the latter case indicates that our data satisfy the behavior of ordinary Brownian motion instead of the former case where there is memory associated with the data. For this reason, we use the R/S analysis (Mandelbrot 1967), which represents one of the statistical measures of the variability of the time series. For this process, we follow the references of (Szóstakowski 2018) and (Ceballos and Largo 2018).

_{n}:

_{s,t}, V

_{p,t}, V

_{gk,t}, and V

_{rs,t}by using Equations (1)–(4), respectively (see Table 1 and Table 2). For all cases, the normality test is rejected.

## 4. Numerical Results

_{gk,t}, V

_{p,t}, V

_{rs,t}, and V

_{s,t}by using the volatility indices of five international markets for all periods (2001–2021 and all sub-periods). Then we estimate the Hurst exponent on the volatility indices for the whole period of 2001–2021 and the respective sub-periods (Table 3). Further, we apply GARCH(1, 1) on the aforementioned range-based estimators in order to compute the volatility of volatility indices during the same period and sub-periods, and finally, we apply the Hurst exponent to the volatility of volatility indices. This is the desired result, because in this way we test how persistent our time series data are. The volatility of the volatility indices is estimated from an AR(1)-GARCH(1, 1)-model. The results for the variance in Equation (6) show that the p-value is equal to zero (or less than 0.05), and as a consequence, the model’s parameters are highly significant. Beyond that, the sum of parameters α and β is less than 1; this indicates that we have a stationary solution with finite expected value. In addition, the sum of the model’s coefficients is very close to one, meaning that the volatility shocks are highly persistent and in equilibrium with the extracted results of this work related to the Hurst exponent. More precisely, the general result across the whole time-period is that the Hurst exponent indicates a time series with a long-term positive autocorrelation (see Table 4). Mathematically, this is the time interval of H between the values (0.5, 1).

_{1}) is well-defined. However, the values of the Hurst exponent on the volatility indices do not reflect the well-defined behavior of the financial market in the sense of three financial strategies for the aforementioned reasons, and thus, hypothesis (H

_{2}) is rejected. According to Table 3 and Table 4, the values of the Hurst exponent are between 0.5 and 1; this implies that both the volatility and the volatility of volatility with respect to the given financial data are not rough (hypothesis (H

_{4})). In this paper, we also examined if the range of the time interval affects the values of the Hurst exponent; we obtained a negative result. Further, and regardless of the potential financial strategies of our time period, the values of the Hurst exponent are greater than 0.5 for the whole period of 2001–2021 (Table 4). This indicates that there is no anti-persistent time series, i.e., the future values do not tend to return to a long-term mean. This implies that there is no mean-reverting financial strategy. This type of strategy helps investors to trade despite difficult circumstances (long-term variations), given that the future values will eventually return to a long-term mean. On the other hand, and according to the Hurst exponent results (Table 4), in most cases, the market encourages a momentum-based strategy, whereby investors have the information that securities will maintain their price dynamics as a function of time (momentum), either upward or vice-versa. In this case, risk managers, investors, and modelers might go towards a specific stock market under the momentum-based strategy. There is also the option for risk managers, investors, and modelers to diversify their portfolios accordingly, considering a mix of momentum-based and mean-reverse strategies and avoiding risk investments with a value of H close to the values of a random walk. Then we could claim that the hypothesis (H

_{4}) is well-defined; that is, risk managers, investors, and modelers can rely on the values of the Hurst exponent on the volatility of volatility indices reflecting the well-defined behavior of the financial market in the sense of three financial strategies.

## 5. Discussion and Conclusions

_{1})). The results are highly recommended for financial analysts dealing with volatility indices as well as for financial researchers.

_{1}). In particular, investors, modelers, and risk managers may follow Hurst exponent values on the volatility of volatility indices for their decisions. Finally, high-frequency data and rough volatility should be considered, to determine if realized variance measures show a high/low Hurst parameter.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

VIX (CBOE) | Chicago Board Options Exchange Volatility index |

VXN (CBOE Nasdaq 100) | CBOE Nasdaq Volatility Index |

VXD (DJIA) | Dow Jones Industrial Average Volatility Index |

VHSI (HSI) | Hang Seng index Volatility Index |

KSVKOSPI (KOSPI) | Korea Composite Stock Price Index |

GFC | Global financial crisis |

H | Hurst exponent |

R/S | Rescaled Range Analysis |

AR | Autoregressive Process |

GARCH | Generalized AutoRegressive Conditional Heteroskedasticity |

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**Figure 1.**Time-varying dynamics of volatility measures of the CBOE Market Volatility Index. Range-based estimators (

**a**) V

_{s,t}and (

**b**) V

_{gk,t}, V

_{p,t}, V

_{rs,t}[Author’s own processing].

**Figure 2.**Time-varying dynamics of volatility measures of the CBOE NASDAQ 100 Volatility Index. Range-based estimators (

**a**) V

_{s,t}and (

**b**) V

_{gk,t}, V

_{p,t}, V

_{rs,t}[Author’s own processing].

**Figure 3.**Time-varying dynamics of volatility measures of the DJIA Volatility Index. Range-based estimators (

**a**) V

_{s,t}and (

**b**) V

_{gk,t}, V

_{p,t}, V

_{rs,t}[Author’s own processing].

**Figure 4.**Time-varying dynamics of volatility measures of the HSI Volatility Index. Range-based estimators (

**a**) V

_{s,t}and (

**b**) V

_{gk,t}, V

_{p,t}, V

_{rs,t}[Author’s own processing].

**Figure 5.**Time-varying dynamics of volatility measures of the KSVKOSPI Volatility Index. Range-based estimators (

**a**) V

_{s,t}and (

**b**) V

_{gk,t}, V

_{p,t}, V

_{rs,t}[Author’s own processing].

**Table 1.**Descriptive statistics of volatility indices from 5 international markets [Author’s own processing].

CBOE Market Volatility Index|Price History | Close | High | Low | Open |

Μean | 19.681 | 20.707 | 18.905 | 19.812 |

Μedian | 17.250 | 18.080 | 16.520 | 17.340 |

Μaximum | 82.690 | 89.530 | 72.760 | 82.690 |

Μinimum | 9.1400 | 9.3100 | 8.5600 | 9.0100 |

Std. Dev. | 8.9360 | 9.5939 | 8.3575 | 8.9923 |

Skewness | 2.2932 | 2.3856 | 2.1518 | 2.2773 |

Κurtosis | 7.9056 | 8.5988 | 6.6939 | 7.6987 |

Jarque-Bera | 18.110 | 20.964 | 13.729 | 17.346 |

Probability | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

Observations | 5203 | 5203 | 5203 | 5203 |

CBOE NASDAQ 100 Volatility Index|Price History | Close | High | Low | Open |

Μean | 24.340 | 22.880 | 21.061 | 21.968 |

Μedian | 20.560 | 20.300 | 18.860 | 19.605 |

Μaximum | 80.640 | 86.520 | 73.860 | 80.570 |

Μinimum | 10.310 | 10.970 | 9.6600 | 10.310 |

Std. Dev. | 11.067 | 9.0678 | 7.9105 | 8.4678 |

Skewness | 1.6283 | 2.4683 | 2.2681 | 2.3566 |

Κurtosis | 2.5228 | 9.3483 | 7.7727 | 8.4064 |

Jarque-Bera | 3679.05 | 21.477 | 15.564 | 17.849 |

Probability | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

Observations | 5203 | 5203 | 5203 | 5203 |

DJIA Volatility Index|Price History | Close | High | Low | Open |

Μean | 18.650 | 20.024 | 17.572 | 18.656 |

Μedian | 16.120 | 17.465 | 15.170 | 16.120 |

Μaximum | 74.600 | 80.240 | 63.250 | 74.600 |

Μinimum | 2.7100 | 9.5600 | 2.4700 | 2.7100 |

Std. Dev. | 8.2446 | 9.0035 | 7.7995 | 8.2493 |

Skewness | 2.1898 | 2.1293 | 2.0826 | 2.2062 |

Κurtosis | 6.8074 | 6.5143 | 5.9404 | 6.9090 |

Jarque-Bera | 14.202 | 13.129 | 11.409 | 14.566 |

Probability | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

Observations | 5202 | 5202 | 5202 | 5202 |

HSI Volatility Index|Price History | Close | High | Low | Open |

Μean | 21.678 | 18.931 | 17.729 | 18.341 |

Μedian | 19.660 | 18.465 | 17.500 | 18.020 |

Μaximum | 104.29 | 68.640 | 58.610 | 58.870 |

Μinimum | 2.0294 | 2.0707 | 1.9985 | 2.0402 |

Std. Dev. | 10.254 | 8.1025 | 7.1187 | 7.5525 |

Skewness | 1.7779 | 0.68914 | 0.19054 | 0.39290 |

Κurtosis | 6.3893 | 3.7278 | 2.1159 | 2.6870 |

Jarque-Bera | 11.364 | 1761.31 | 515.37 | 873.569 |

Probability | 0.000000 | 0.000000 | 0.000000 | 2.03 × 10^{−185} |

Observations | 2426 | 2676 | 2676 | 2675 |

KSVKOSPI Volatility Index|Price History | Close | High | Low | Open |

Μean | 16.346 | 16.912 | 15.999 | 16.465 |

Μedian | 15.240 | 15.670 | 14.920 | 15.340 |

Μaximum | 69.240 | 71.750 | 62.080 | 71.290 |

Μinimum | 0.010914 | 0.011174 | 0.010879 | 0.011155 |

Std. Dev. | 7.9787 | 8.4060 | 7.6776 | 8.0174 |

Skewness | 0.89333 | 1.0452 | 0.76862 | 0.90655 |

Κurtosis | 4.3562 | 5.1291 | 3.8254 | 4.5845 |

Jarque-Bera | 2906.89 | 4022.5 | 2228.7 | 11.752 |

Probability | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

Observations | 3147 | 3147 | 3147 | 3187.04 |

**Table 2.**Volatility estimates. Descriptive statistics on range-based estimators. [Author’s own processing].

CBOE Market Volatility Index|Price History | V_{gk} | V_{p} | V_{rs} | V_{s} |

Μean | 0.0038030 | 0.0037981 | 0.0039638 | 0.086164 |

Μedian | 0.0018780 | 0.0019078 | 0.0018067 | 0.072696 |

Μaximum | 0.30795 | 0.25293 | 0.30252 | 0.83704 |

Μinimum | 0.000000 | 0.000000 | 0.000000 | 0.0000 |

Std. Dev. | 0.0082953 | 0.0079785 | 0.0094253 | 0.055655 |

Skewness | 15.579 | 14.463 | 14.593 | 3.0640 |

Κurtosis | 427.86 | 359.09 | 345.82 | 20.355 |

Jarque-Bera | 3.98973 × 10^{+07} | 2.81359 × 10^{+07} | 2.61108 × 10^{+07} | 97.966 |

Probability | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

Observations | 5203 | 5203 | 5203 | 5203 |

CBOE NASDAQ 100 Volatility Index|Price History | V_{gk} | V_{p} | V_{rs} | V_{s} |

Μean | 0.0029664 | 0.0030914 | 0.0031269 | 0.078422 |

Μedian | 0.0015422 | 0.0016296 | 0.0015031 | 0.067187 |

Μaximum | 0.49927 | 0.36051 | 0.74883 | 0.99932 |

Μinimum | 0.0000 | 0.0000 | 0.0000 | 0.000000 |

Std. Dev. | 0.0089028 | 0.0075764 | 0.012429 | 0.049134 |

Skewness | 39.489 | 26.260 | 47.917 | 3.8547 |

Κurtosis | 2111.0 | 1109.9 | 2816.0 | 38.789 |

Jarque-Bera | 8.57543 × 10^{+08} | 2.37258 × 10^{+08} | 1.52557 × 10^{+09} | 300.56 |

Probability | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

Observations | 4612 | 4612 | 4612 | 4612 |

DJIA Volatility Index|Price History | V_{gk} | V_{p} | V_{rs} | V_{s} |

Μean | 0.015907 | 0.012977 | 0.021436 | 0.12924 |

Μedian | 0.0022831 | 0.0023729 | 0.0022821 | 0.081075 |

Μaximum | 1.6846 | 1.3000 | 2.6153 | 1.8977 |

Μinimum | −0.00013469 | 0.0000 | −0.00046366 | 0.0000 |

Std. Dev. | 0.055918 | 0.046756 | 0.083584 | 0.13874 |

Skewness | 13.467 | 15.625 | 15.595 | 3.7924 |

Κurtosis | 300.86 | 355.86 | 385.30 | 25.684 |

Jarque-Bera | 1.97765 × 10^{+07} | 2.76597 × 10^{+07} | 3.23887 × 10^{+07} | 155.450 |

Probability | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

Observations | 5202 | 5202 | 5202 | 5202 |

HSI Volatility Index|Price History | V_{gk} | V_{p} | V_{rs} | V_{s} |

Μean | 0.0020828 | 0.0019981 | 0.0023188 | 0.059696 |

Μedian | 0.00085300 | 0.00083030 | 0.00085529 | 0.047958 |

Μaximum | 0.10961 | 0.086440 | 0.18831 | 0.48933 |

Μinimum | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Std. Dev. | 0.0054811 | 0.0048891 | 0.0075498 | 0.044407 |

Skewness | 9.1365 | 8.8341 | 11.524 | 3.4983 |

Κurtosis | 111.45 | 105.41 | 189.48 | 19.140 |

Jarque-Bera | 1.4216 × 10^{+06} | 1.27379 × 10^{+06} | 4.06097 × 10^{+06} | 46.304 |

Probability | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

Observations | 2426 | 2676 | 2675 | 2676 |

KSVKOSPI Volatility Index|Price History | V_{gk} | V_{p} | V_{rs} | V_{s} |

Μean | 0.0015916 | 0.0016425 | 0.0017795 | 0.051696 |

Μedian | 0.00057376 | 0.00062004 | 0.00054095 | 0.041444 |

Μaximum | 0.16515 | 0.27974 | 0.32143 | 0.88028 |

Μinimum | 2.8075 × 10^{−05} | 3.1385 × 10^{−05} | 0.0000 | 0.0093241 |

Std. Dev. | 0.0064552 | 0.0071245 | 0.0096723 | 0.043337 |

Skewness | 15.196 | 23.862 | 21.189 | 6.2728 |

Κurtosis | 290.69 | 796.66 | 573.95 | 72.830 |

Jarque-Bera | 1.1201 × 10^{+07} | 8.35198 × 10^{+07} | 4.34308 × 10^{+07} | 716.154 |

Probability | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

Observations | 3147 | 3147 | 3147 | 3147 |

**Table 3.**Values of the Hurst exponent on the volatility indices of 5 international markets. [Author’s own processing].

Volatility Indices | Period 2001–2021 | Period before GFC 2001–2007 | GFC 2008–2010 | Period after GFC 2008–2021 | COVID-19 2020–2021 |
---|---|---|---|---|---|

CBOE Volatility Index | 0.923132 | 0.990811 | 0.97901 | 0.949799 | 0.937536 |

CBOE NASDAQ 100 Volatility Index | 0.944038 | 1.01016 | 0.967554 | 0.951156 | 0.958043 |

DJIA Volatility Index | 0.916349 | 0.991099 | 0.977751 | 0.943301 | 0.976587 |

HSI Volatility Index | 0.938568 | 0.945965 | 0.978547 | 0.948096 | 1.02819 |

KSVKOSPI Volatility Index * | 0.92346 | - | 0.972173 | 0.92346 | 1.02054 |

_{gk,t}, V

_{p,t}, V

_{rs,t}, V

_{s,t}are available upon request).

**Table 4.**Values of the Hurst exponent on the volatility of volatility indices of 5 international markets. [Author’s own processing].

Volatility Indices | Period 2001–2021 | Period before GFC 2001–2007 | GFC 2008–2010 | Period after GFC 2008–2021 | COVID-19 2020–2021 |
---|---|---|---|---|---|

CBOE Volatility Index | V_{s}: 0.738147 | V_{s}: 0.816659 | V_{gk}: 0.67335V _{p}: 0.682425V _{s}: 0.761437 | V_{s}: 0.679823 | V_{gk}: 0.706384V _{s}: 0.748151 |

CBOE NASDAQ 100 Volatility Index | - | V_{p}: 0.559094V _{s}: 0.630445 | V_{s}: 0.63375 | V_{s}: 0.598392 | V_{gk}: 0.779025V _{p}: 0.705984V _{rs}: 0.775285V _{s}: 0.777791 |

DJIA Volatility Index | - | V_{s}: 0.700755 | V_{s}: 0.702298 | V_{s}: 0.731611 | - |

HSI Volatility Index | V_{s}: 0.71753 | - | - | V_{s}: 0.71753 | V_{p}: 0.630344V _{s}: 0.721309 |

KSVKOSPI Volatility Index * | V_{p}: 0.530579V _{s}: 0.58489 | - | V_{p}: 0.574546V _{s}: 0.584908 | V_{p}: 0.530579V _{s}: 0.58489 | V_{s}: 0.750765 |

_{gk,t}, V

_{p,t}, V

_{rs,t}, V

_{s,t}are available upon request).

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## Share and Cite

**MDPI and ACS Style**

Zournatzidou, G.; Floros, C.
Hurst Exponent Analysis: Evidence from Volatility Indices and the Volatility of Volatility Indices. *J. Risk Financial Manag.* **2023**, *16*, 272.
https://doi.org/10.3390/jrfm16050272

**AMA Style**

Zournatzidou G, Floros C.
Hurst Exponent Analysis: Evidence from Volatility Indices and the Volatility of Volatility Indices. *Journal of Risk and Financial Management*. 2023; 16(5):272.
https://doi.org/10.3390/jrfm16050272

**Chicago/Turabian Style**

Zournatzidou, Georgia, and Christos Floros.
2023. "Hurst Exponent Analysis: Evidence from Volatility Indices and the Volatility of Volatility Indices" *Journal of Risk and Financial Management* 16, no. 5: 272.
https://doi.org/10.3390/jrfm16050272