# Asymmetric Realized Volatility Risk

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## Abstract

**:**

## 1. Introduction

## 2. Volatility Risk and the Conditional Distribution of Asset Returns

**Figure 1.**The kurtosis of the simulated distribution under the assumption that shocks to realized volatilityhave the normal inverse Gaussian (NIG) (upper line) and normal distributions.

**ex ante**distribution of returns must come from the negative dependence between ${\epsilon}_{t}$ and ${\eta}_{t}$. Writing the expression for the third moment,

#### 2.1. Volatility Risk: Empirical Regularities

Mean | SD | Skewness | Kurtosis | ${Q}_{0.75}$ | ${Q}_{0.9}$ | ${Q}_{0.95}$ | ${Q}_{0.99}$ | |
---|---|---|---|---|---|---|---|---|

S&P 500 | 1.00 | 0.25 | 1.86 | 12.17 | 1.11 | 1.28 | 1.43 | 1.84 |

DJIA | 1.01 | 0.30 | 1.94 | 15.52 | 1.15 | 1.35 | 1.53 | 1.95 |

FTSE | 1.00 | 0.33 | 2.97 | 27.12 | 1.13 | 1.37 | 1.55 | 2.14 |

CAC | 1.00 | 0.25 | 2.14 | 21.84 | 1.12 | 1.29 | 1.41 | 1.76 |

Nikkei | 1.00 | 0.27 | 1.19 | 6.72 | 1.14 | 1.33 | 1.47 | 1.86 |

IBM | 1.00 | 0.22 | 1.13 | 6.96 | 1.11 | 1.25 | 1.38 | 1.72 |

GE | 1.00 | 0.20 | 0.85 | 5.30 | 1.11 | 1.25 | 1.37 | 1.63 |

WMT | 1.00 | 0.24 | 1.25 | 7.82 | 1.12 | 1.29 | 1.41 | 1.74 |

AT&T | 1.00 | 0.27 | 1.67 | 10.25 | 1.12 | 1.32 | 1.48 | 1.87 |

**Figure 3.**In-sample percentage errors for the HAR (heterogeneous autoregressive) model with leverage effects.

**Table 2.**Descriptive statistics for returns standardized by in-sample realized volatility fitted values.

Mean | SD | Skewness | Kurtosis | ${Q}_{0.01}$ | |
---|---|---|---|---|---|

S&P 500 | −0.019 | 1.061 | −0.392 | 4.305 | −2.780 |

DJIA | 0.019 | 1.090 | −0.340 | 3.944 | −2.867 |

FTSE | −0.009 | 1.147 | −0.177 | 3.694 | −2.905 |

CAC | −0.011 | 1.053 | −0.182 | 3.346 | −2.654 |

Nikkei | −0.054 | 1.083 | −0.152 | 3.715 | −2.832 |

IBM | 0.035 | 1.033 | −0.076 | 4.505 | −2.539 |

GE | −0.025 | 1.012 | 0.026 | 3.897 | −2.479 |

WMT | −0.039 | 1.001 | 0.066 | 3.909 | −2.432 |

AT&T | −0.037 | 1.028 | 0.024 | 4.397 | −2.523 |

**Figure 5.**Sample autocorrelations for the squared (

**left**) and absolute (

**right**) residuals of the HAR model with leverage effects.

**Figure 6.**GARCHstandard deviation series (

**top**) and realized volatility fitted values (

**bottom**) for the HAR model with leverage effects.

## 3. The Dually Asymmetric Realized Volatility Model

#### 3.1. Model Details

#### 3.1.1. Long Memory Specification

^{1}Statistical tests for distinguishing between those alternatives, such as the one proposed Ohanissian et al. [30], have been hampered by low power. Finally, Granger and Ding [31] and Scharth and Medeiros [5] discuss how estimates of the fractional differencing parameter are subject to excessive variation over time.

#### 3.1.2. Extended Leverage Effects

#### 3.1.3. The Distribution of the Volatility Disturbances

#### 3.1.4. Days-of-the-Week and Holiday Effects

#### 3.2. The Impact of Microstructure Noise and Other Issues

Statistic | ${r}_{t}$ | ${RV}_{t}$ | ${r}_{t}/{RV}_{t}$ | $\Delta {RV}_{t}$ |
---|---|---|---|---|

Mean | 0.012 | 0.984 | 0.042 | 0.000 |

SD | 1.343 | 0.605 | 1.013 | 0.359 |

Skewness | −0.186 | 3.417 | 0.040 | −0.913 |

Kurtosis | 10.570 | 26.342 | 2.743 | 65.184 |

Min | −9.470 | 0.212 | −3.296 | −6.840 |

${Q}_{0.1}$ | −1.443 | 0.478 | −1.282 | −0.308 |

${Q}_{0.25}$ | −0.610 | 0.606 | −0.658 | −0.135 |

${Q}_{0.75}$ | 0.654 | 1.153 | 0.711 | 0.127 |

${Q}_{0.9}$ | 1.368 | 1.584 | 1.382 | 0.301 |

Max | 10.957 | 9.673 | 3.230 | 5.280 |

^{2}The use of jumps does not seem to bring important forecasting advantages in our framework. On the other hand, the inclusion of a jump equation would substantially increase the complexity of the model, requiring us to model and estimate the joint distribution of return, volatility and jump shocks. For predicting and simulating the model multiple periods ahead, this is a substantial burden. Since the ultimate interest lies in the conditional distribution of returns, a parsimonious alternative is to ignore the distinction between continuous and jump components in realized volatility and to carefully model the distribution of returns given realized volatility (considering the possible impact of jumps on it). Corsi and Reno (2012) provides some evidence that the impact of jumps on volatility is quite transitory.

#### 3.3. Estimation and Density Forecasting

^{3}The use of this approximate estimator does not impact, in any away, the main arguments of this paper. For reference, the log-likelihood function is given by:

- In the first step, the functional form of the model is used for the evaluation of forecasts ${\widehat{RV}}_{t+1}$ and ${\widehat{h}}_{t+1}$ conditional on past realized volatility observations, returns and other variables.
- Using the estimated copula, we randomly generate S pairs of return (${\tilde{\epsilon}}_{t+1,j}$, $j=1,..,S$) and volatility (${\tilde{\eta}}_{t+1,j}$, $j=1,..,S$) shocks with the according marginal distributions. Antithetic variables are used to balance the return innovations for location and scale.
- We obtain S simulated volatilities through ${\tilde{RV}}_{t+1,j}={\widehat{RV}}_{t+1}+{h}_{t}{\tilde{\eta}}_{t+1,j}$, $j=1,...,S$. Each of these volatilities generate a returns ${\tilde{r}}_{t+1,j}={\widehat{\mu}}_{t}+{\tilde{RV}}_{t+1,j}{\tilde{\epsilon}}_{t+1,j}$.
- This procedure can be iterated in the natural way to generate multiple paths for returns and realized volatility.

## 4. Empirical Analysis

#### 4.1. Realized Volatility Measurement and Data

^{4}Following the results of Hansen and Lunde [44], we adopt the previous tick method for determining prices at time marks where a quote is missing.

**Figure 8.**Time Series of returns (

**top**), realized volatility (

**middle**) and log realized volatility (

**bottom**) for the S&P 500 index.

#### 4.2. Full Sample Parameter Estimates and Diagnostics

**Table 4.**Estimated parameters (S&P 500): ARFIMAmodels. DARV, dually asymmetric realized volatility; AE, asymmetric effects.

Parameter | ARFIMA | ARFIMA + AE | ARFIMA-GARCH | ARFIMA + AE-GARCH | DARV-FI | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ψ | 0.989 | (0.029) | 0.653 | (0.028) | 0.970 | (0.053) | 0.546 | (0.028) | 0.400 | (0.025) | ||||

d | 0.340 | (0.011) | 0.261 | (0.008) | 0.464 | (0.012) | 0.352 | (0.013) | 0.367 | (0.015) | ||||

${\varphi}_{1}$ | 0.064 | (0.017) | 0.033 | (0.019) | −0.075 | (0.017) | −0.047 | (0.020) | −0.074 | (0.019) | ||||

${\lambda}_{1}$ | − | − | −0.055 | (0.006) | − | − | −0.049 | (0.005) | −0.072 | (0.006) | ||||

${\lambda}_{2}$ | − | − | −0.018 | (0.003) | − | − | −0.012 | (0.003) | −0.023 | (0.004) | ||||

${\lambda}_{3}$ | − | − | −0.014 | (0.002) | − | − | −0.011 | (0.002) | −0.013 | (0.002) | ||||

${\theta}_{0}$ | 0.105 | (0.005) | 0.089 | (0.006) | 0.002 | (0.000) | 0.001 | (0.000) | 0.013 | (0.001) | ||||

${\theta}_{1}$ | − | − | − | − | − | − | − | − | 0.101 | (0.007) | ||||

${\theta}_{2}$ | − | − | − | − | 0.845 | (0.018) | 0.852 | (0.015) | − | − | ||||

${\theta}_{3}$ | − | − | − | − | 0.127 | (0.017) | 0.121 | (0.015) | − | − | ||||

α | 0.906 | (0.046) | 0.855 | (0.052) | 1.841 | (0.125) | 1.663 | (0.117) | 1.800 | (0.168) | ||||

β | 0.548 | (0.046) | 0.479 | (0.049) | 1.075 | (0.106) | 0.910 | (0.095) | 1.037 | (0.140) | ||||

κ | 0.169 | (0.024) | 0.157 | (0.023) | 0.207 | (0.024) | 0.228 | (0.024) | 0.254 | (0.025) |

Parameter | HAR/AE-GARCH | DARV (HAR) | |||
---|---|---|---|---|---|

${\varphi}_{0}$ | 0.090 | (0.008) | 0.087 | (0.009) | |

${\varphi}_{1}$ | 0.231 | (0.016) | 0.250 | (0.016) | |

${\varphi}_{2}$ | 0.357 | (0.017) | 0.362 | (0.022) | |

${\varphi}_{3}$ | 0.273 | (0.016) | 0.244 | (0.018) | |

${\lambda}_{1}$ | −0.052 | (0.005) | −0.072 | (0.006) | |

${\lambda}_{2}$ | −0.016 | (0.003) | −0.024 | (0.004) | |

${\lambda}_{3}$ | −0.007 | (0.002) | −0.009 | (0.002) | |

${\theta}_{0}$ | 0.001 | (0.000) | 0.001 | (0.001) | |

${\theta}_{1}$ | − | − | 0.054 | (0.003) | |

${\theta}_{2}$ | 0.853 | (0.014) | − | − | |

${\theta}_{3}$ | 0.122 | (0.014) | − | − | |

α | 1.752 | (0.136) | 1.669 | (0.149) | |

β | 1.014 | (0.116) | 0.931 | (0.121) | |

κ | 0.242 | (0.024) | 0.272 | (0.025) |

ARFIMA | ARFIMA + AE | ARFIMA GARCH | ARFIMA + AE GARCH | DARV (FI) | HAR + AE GARCH | DARV (HAR) | |
---|---|---|---|---|---|---|---|

Log-Likelihood | 292.34 | 455.29 | 757.00 | 882.31 | 955.65 | 870.12 | 915.38 |

${R}^{2}$ | 0.688 | 0.723 | 0.739 | 0.778 | 0.789 | 0.778 | 0.787 |

BIC | −455.59 | −749.20 | −1,360.69 | −1,579.05 | −1,717.64 | −1546.58 | −1629.04 |

SD (${\widehat{\nu}}_{t}$) | 0.323 | 0.295 | 0.312 | 0.288 | 0.281 | 0.288 | 0.283 |

Ljung–Box (1) (${\widehat{\nu}}_{t}$) | 0.000 | 0.000 | 0.000 | 0.021 | 0.009 | 0.325 | 0.000 |

Ljung–Box (5) (${\widehat{\nu}}_{t}$) | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

Ljung–Box (10) (${\widehat{\nu}}_{t}$) | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

Skewness (${\widehat{\eta}}_{t}$) | 4.58 | 3.86 | 1.63 | 1.68 | 2.02 | 1.64 | 1.95 |

Kurtosis (${\widehat{\eta}}_{t}$) | 65.02 | 52.03 | 9.82 | 10.47 | 14.24 | 9.88 | 13.74 |

ARCH(1) (${\widehat{\eta}}_{t}$) | 0.000 | 0.000 | 0.306 | 0.397 | 0.693 | 0.422 | 0.692 |

ARCH (5) (${\widehat{\eta}}_{t}$) | 0.000 | 0.000 | 0.910 | 0.876 | 0.315 | 0.843 | 0.340 |

ARCH (10) (${\widehat{\eta}}_{t}$) | 0.000 | 0.000 | 0.293 | 0.510 | 0.158 | 0.636 | 0.194 |

K-S Test (${\widehat{\eta}}_{t}$) | 0.000 | 0.000 | 0.051 | 0.076 | 0.716 | 0.015 | 0.913 |

Ljung–Box (1) $(z-\overline{z})$ | 0.073 | 0.362 | 0.000 | 0.039 | 0.927 | 0.251 | 0.181 |

Ljung–Box (1) ${(z-\overline{z})}^{2}$ | 0.000 | 0.000 | 0.199 | 0.250 | 0.994 | 0.283 | 0.716 |

Ljung–Box (1) ${(z-\overline{z})}^{3}$ | 0.003 | 0.011 | 0.002 | 0.115 | 0.998 | 0.619 | 0.357 |

Ljung–Box (1) ${(z-\overline{z})}^{4}$ | 0.000 | 0.000 | 0.274 | 0.208 | 0.965 | 0.479 | 0.857 |

#### 4.3. Point Forecasts

Model | 1 Day | ||||||
---|---|---|---|---|---|---|---|

${R}^{2}$ | RMSE | MAE | MAPE | ${R}^{2}(\Delta )$ | SPA in MSE | SPA in ${R}^{2}$ | |

ARFIMA | 0.767 | 0.342 | 0.189 | 0.205 | 0.178 | 0.001 | 0.002 |

ARFIMA + AE | 0.817 | 0.301 | 0.174 | 0.191 | 0.317 | 0.001 | 0.000 |

ARFIMA-GARCH | 0.790 | 0.316 | 0.167 | 0.166 | 0.244 | 0.034 | 0.033 |

ARFIMA + AE-GARCH | 0.829 | 0.283 | 0.157 | 0.160 | 0.380 | 0.573 | 0.585 |

DARV (FI) | 0.834 | 0.277 | 0.156 | 0.161 | 0.407 | 0.719 | 0.727 |

HAR + AE-GARCH | 0.827 | 0.285 | 0.159 | 0.166 | 0.374 | 0.405 | 0.416 |

DARV (HAR) | 0.831 | 0.280 | 0.158 | 0.165 | 0.395 | 0.400 | 0.382 |

Model | 5 Days (Cumulated) | ||||||

${\mathit{R}}^{\mathbf{2}}$ | RMSE | MAE | MAPE | ${\mathit{R}}^{\mathbf{2}}(\Delta )$ | SPA in MSE | SPA in ${\mathit{R}}^{\mathbf{2}}$ | |

ARFIMA | 0.790 | 1.593 | 0.936 | 0.195 | 0.089 | 0.008 | 0.003 |

ARFIMA + AE | 0.819 | 1.494 | 0.875 | 0.183 | 0.132 | 0.009 | 0.001 |

ARFIMA-GARCH | 0.820 | 1.348 | 0.789 | 0.148 | 0.123 | 0.005 | 0.007 |

ARFIMA + AE-GARCH | 0.834 | 1.296 | 0.749 | 0.141 | 0.165 | 0.009 | 0.012 |

DARV (FI) | 0.841 | 1.269 | 0.733 | 0.139 | 0.201 | 0.538 | 0.536 |

HAR + AE-GARCH | 0.842 | 1.265 | 0.740 | 0.143 | 0.173 | 0.014 | 0.015 |

DARV (HAR) | 0.847 | 1.244 | 0.728 | 0.141 | 0.213 | 0.787 | 0.761 |

Model | 22 Days (Cumulated) | ||||||

${\mathit{R}}^{\mathbf{2}}$ | RMSE | MAE | MAPE | ${\mathit{R}}^{\mathbf{2}}(\Delta )$ | SPA in MSE | SPA in ${\mathit{R}}^{\mathbf{2}}$ | |

ARFIMA | 0.651 | 8.246 | 5.062 | 0.235 | 0.137 | 0.008 | 0.002 |

ARFIMA + AE | 0.683 | 7.862 | 5.528 | 0.283 | 0.202 | 0.000 | 0.000 |

ARFIMA-GARCH | 0.701 | 7.183 | 4.238 | 0.178 | 0.220 | 0.043 | 0.043 |

ARFIMA + AE-GARCH | 0.714 | 7.031 | 4.066 | 0.170 | 0.272 | 0.010 | 0.011 |

DARV (FI) | 0.720 | 6.953 | 3.994 | 0.167 | 0.283 | 0.252 | 0.290 |

HAR + AE-GARCH | 0.732 | 6.806 | 4.059 | 0.178 | 0.309 | 0.010 | 0.007 |

DARV (HAR) | 0.736 | 6.752 | 4.020 | 0.176 | 0.332 | 0.576 | 0.573 |

ARFIMA | ARFIMA + AE | ARFIMA GARCH | ARFIMA + AE GARCH | DARV (FI) | HAR + AE GARCH | DARV (HAR) | |
---|---|---|---|---|---|---|---|

DJIA | 0.207 | 0.269 | 0.262 | 0.345 | 0.381 | 0.346 | 0.377 |

(0.000) | (0.004) | (0.018) | (0.142) | (0.802) | (0.128) | (0.659) | |

FTSE | 0.236 | 0.314 | 0.259 | 0.347 | 0.368 | 0.334 | 0.346 |

(0.001) | (0.001) | (0.001) | (0.023) | (0.819) | (0.002) | (0.004) | |

CAC | 0.199 | 0.265 | 0.232 | 0.278 | 0.301 | 0.269 | 0.283 |

(0.001) | (0.005) | (0.008) | (0.167) | (0.862) | (0.013) | (0.026) | |

Nikkei | 0.213 | 0.256 | 0.223 | 0.266 | 0.270 | 0.258 | 0.259 |

(0.037) | (0.096) | (0.040) | (0.510) | (0.834) | (0.120) | (0.091) | |

IBM | 0.193 | 0.232 | 0.254 | 0.290 | 0.296 | 0.294 | 0.301 |

(0.000) | (0.000) | (0.000) | (0.423) | (0.604) | (0.354) | (0.809) | |

GE | 0.161 | 0.199 | 0.206 | 0.259 | 0.281 | 0.251 | 0.275 |

(0.004) | (0.007) | (0.007) | (0.111) | (0.851) | (0.051) | (0.576) | |

WMT | 0.269 | 0.287 | 0.296 | 0.321 | 0.334 | 0.316 | 0.325 |

(0.002) | (0.001) | (0.078) | (0.051) | (0.789) | (0.130) | (0.340) | |

AT&T | 0.211 | 0.221 | 0.237 | 0.252 | 0.259 | 0.251 | 0.259 |

(0.000) | (0.000) | (0.002) | (0.019) | (0.896) | (0.005) | (0.720) |

#### 4.4. Volatility Risk

$\Delta {RV}_{t}$ | >80th Percentile | >90th Percentile | >95th Percentile | >99th Percentile | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Model | ${R}^{2}$ | MSE | ${R}^{2}$ | RMSE | ${R}^{2}$ | RMSE | ${R}^{2}$ | RMSE | |||

ARFIMA | 0.092 | 0.399 | 0.080 | 0.504 | 0.070 | 0.637 | 0.000 | 1.068 | |||

ARFIMA + AE | 0.035 | 0.402 | 0.013 | 0.513 | 0.000 | 0.658 | 0.034 | 1.114 | |||

ARFIMA-GARCH | 0.254 | 0.353 | 0.255 | 0.446 | 0.231 | 0.569 | 0.059 | 1.002 | |||

ARFIMA + AE-GARCH | 0.246 | 0.354 | 0.248 | 0.448 | 0.214 | 0.577 | 0.039 | 1.015 | |||

DARV (FI) | 0.300 | 0.343 | 0.317 | 0.428 | 0.285 | 0.549 | 0.198 | 0.900 | |||

HAR + AE-GARCH | 0.242 | 0.355 | 0.244 | 0.449 | 0.209 | 0.578 | 0.037 | 1.016 | |||

DARV (HAR) | 0.309 | 0.343 | 0.324 | 0.426 | 0.286 | 0.549 | 0.166 | 0.909 |

${RV}_{t}$ | >80th Percentile | >90th Percentile | >95th Percentile | >99th Percentile | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Model | ${R}^{2}$ | MSE | ${R}^{2}$ | RMSE | ${R}^{2}$ | RMSE | ${R}^{2}$ | RMSE | |||

ARFIMA | 0.542 | 0.633 | 0.397 | 0.818 | 0.198 | 1.044 | 0.000 | 1.452 | |||

ARFIMA + AE | 0.663 | 0.544 | 0.563 | 0.700 | 0.473 | 0.874 | 0.134 | 1.403 | |||

ARFIMA-GARCH | 0.584 | 0.582 | 0.455 | 0.744 | 0.299 | 0.919 | 0.004 | 1.305 | |||

ARFIMA + AE-GARCH | 0.672 | 0.515 | 0.574 | 0.658 | 0.493 | 0.789 | 0.069 | 1.227 | |||

DARV (FI) | 0.685 | 0.499 | 0.592 | 0.639 | 0.520 | 0.754 | 0.360 | 1.055 | |||

HAR + AE-GARCH | 0.669 | 0.515 | 0.568 | 0.658 | 0.486 | 0.784 | 0.067 | 1.223 | |||

DARV (HAR) | 0.682 | 0.502 | 0.589 | 0.646 | 0.517 | 0.758 | 0.347 | 1.049 |

ARFIMA | ARFIMA + AE | ARFIMA GARCH | ARFIMA + AE GARCH | DARV (FI) | HAR + AE GARCH | DARV (HAR) | |
---|---|---|---|---|---|---|---|

${R}^{2}$ of $E(\Delta R{V}_{t}|\Delta R{V}_{t}>{Q}_{0.9}):$ | |||||||

DJIA | 0.100 | 0.074 | 0.147 | 0.136 | 0.224 | 0.141 | 0.210 |

FTSE | 0.084 | 0.106 | 0.138 | 0.135 | 0.211 | 0.124 | 0.191 |

CAC | 0.036 | 0.073 | 0.245 | 0.219 | 0.169 | 0.206 | 0.192 |

Nikkei | 0.038 | 0.029 | 0.123 | 0.108 | 0.153 | 0.105 | 0.158 |

IBM | 0.008 | 0.014 | 0.219 | 0.226 | 0.287 | 0.229 | 0.291 |

GE | 0.170 | 0.120 | 0.270 | 0.264 | 0.285 | 0.253 | 0.294 |

WMT | 0.059 | 0.082 | 0.105 | 0.109 | 0.170 | 0.116 | 0.165 |

AT&T | 0.016 | 0.020 | 0.021 | 0.019 | 0.026 | 0.023 | 0.033 |

${R}^{2}$ of $E\left(R{V}_{t}\right|R{V}_{t}>{Q}_{0.9}):$ | |||||||

DJIA | 0.333 | 0.463 | 0.389 | 0.506 | 0.542 | 0.494 | 0.540 |

FTSE | 0.114 | 0.234 | 0.210 | 0.278 | 0.311 | 0.266 | 0.300 |

CAC | 0.078 | 0.162 | 0.213 | 0.202 | 0.220 | 0.188 | 0.218 |

Nikkei | 0.466 | 0.490 | 0.506 | 0.526 | 0.526 | 0.503 | 0.512 |

IBM | 0.431 | 0.466 | 0.462 | 0.485 | 0.498 | 0.486 | 0.497 |

GE | 0.435 | 0.478 | 0.463 | 0.499 | 0.520 | 0.480 | 0.511 |

WMT | 0.135 | 0.186 | 0.186 | 0.216 | 0.241 | 0.206 | 0.228 |

AT&T | 0.288 | 0.307 | 0.310 | 0.317 | 0.337 | 0.339 | 0.357 |

#### 4.5. Value-at-Risk

1% VaR | ||||||||
---|---|---|---|---|---|---|---|---|

Monte Carlo | Forecast VaR | |||||||

Failures | UC | IND | ES | Failures | UC | IND | ||

ARFIMA | 0.007 | 0.162 | 0.643 | 0.006 | 0.021 | 0.000 | 0.951 | |

ARFIMA + AE | 0.008 | 0.358 | 0.599 | 0.006 | 0.023 | 0.000 | 0.943 | |

ARFIMA-GARCH | 0.009 | 0.815 | 0.536 | 0.005 | 0.025 | 0.000 | 0.806 | |

ARFIMA + AE-GARCH | 0.009 | 0.646 | 0.556 | 0.005 | 0.025 | 0.000 | 0.773 | |

DARV (FI) | 0.009 | 0.815 | 0.536 | 0.005 | 0.025 | 0.000 | 0.806 | |

HAR + AE-GARCH | 0.010 | 0.990 | 0.515 | 0.006 | 0.025 | 0.000 | 0.773 | |

DARV (HAR) | 0.009 | 0.646 | 0.556 | 0.004 | 0.026 | 0.000 | 0.740 | |

2.5% VaR | ||||||||

Monte Carlo | Forecast VaR | |||||||

Failures | UC | IND | ES | Failures | UC | IND | ||

ARFIMA | 0.023 | 0.510 | 0.943 | 0.015 | 0.044 | 0.000 | 0.238 | |

ARFIMA + AE | 0.024 | 0.817 | 0.839 | 0.014 | 0.042 | 0.000 | 0.310 | |

ARFIMA-GARCH | 0.030 | 0.161 | 0.480 | 0.014 | 0.047 | 0.000 | 0.143 | |

ARFIMA + AE-GARCH | 0.030 | 0.125 | 0.950 | 0.014 | 0.046 | 0.000 | 0.193 | |

DARV (FI) | 0.030 | 0.161 | 0.932 | 0.013 | 0.046 | 0.000 | 0.193 | |

HAR + AE-GARCH | 0.029 | 0.204 | 0.506 | 0.013 | 0.047 | 0.000 | 0.155 | |

DARV (HAR) | 0.030 | 0.125 | 0.455 | 0.013 | 0.045 | 0.000 | 0.207 | |

5% VaR | ||||||||

Monte Carlo | Forecast VaR | |||||||

Failures | UC | IND | ES | Failures | UC | IND | ||

ARFIMA | 0.054 | 0.447 | 0.654 | 0.027 | 0.071 | 0.000 | 0.837 | |

ARFIMA + AE | 0.056 | 0.250 | 0.032 | 0.027 | 0.072 | 0.000 | 0.011 | |

ARFIMA-GARCH | 0.060 | 0.035 | 0.130 | 0.026 | 0.076 | 0.000 | 0.941 | |

ARFIMA + AE-GARCH | 0.062 | 0.013 | 0.007 | 0.026 | 0.073 | 0.000 | 0.027 | |

DARV (FI) | 0.061 | 0.022 | 0.009 | 0.025 | 0.072 | 0.000 | 0.034 | |

HAR + AE-GARCH | 0.060 | 0.044 | 0.013 | 0.025 | 0.076 | 0.000 | 0.004 | |

DARV (HAR) | 0.059 | 0.055 | 0.014 | 0.025 | 0.075 | 0.000 | 0.001 |

Date | Return | ${RV}_{t}$ | ARFIMA | ARFIMA + AE | ARFIMA GARCH | ARFIMA + AE GARCH | DARV (FI) | HAR + AE GARCH | DARV (HAR) |
---|---|---|---|---|---|---|---|---|---|

September 29, 2008 | −9.219 | 4.845 | 0.002 | 0.001 | 0.003 | 0.002 | 0.002 | 0.003 | 0.002 |

October 7, 2008 | −5.911 | 4.017 | 0.007 | 0.009 | 0.019 | 0.024 | 0.026 | 0.023 | 0.027 |

October 9, 2008 | −7.922 | 4.393 | 0.008 | 0.009 | 0.021 | 0.022 | 0.025 | 0.022 | 0.027 |

October 15, 2008 | −9.470 | 3.665 | 0.018 | 0.013 | 0.044 | 0.039 | 0.031 | 0.047 | 0.043 |

October 22, 2008 | −6.295 | 3.678 | 0.113 | 0.145 | 0.129 | 0.149 | 0.164 | 0.161 | 0.174 |

November 5, 2008 | −5.412 | 2.499 | 0.057 | 0.053 | 0.066 | 0.067 | 0.065 | 0.086 | 0.084 |

November 19, 2008 | −6.311 | 3.532 | 0.033 | 0.027 | 0.043 | 0.038 | 0.042 | 0.042 | 0.048 |

November 20, 2008 | −6.948 | 5.858 | 0.035 | 0.068 | 0.045 | 0.077 | 0.100 | 0.083 | 0.106 |

December 1, 2008 | −9.354 | 2.562 | 0.005 | 0.004 | 0.014 | 0.011 | 0.010 | 0.015 | 0.013 |

January 20, 2009 | −5.426 | 2.505 | 0.019 | 0.016 | 0.023 | 0.019 | 0.025 | 0.014 | 0.019 |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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^{3}${(1-L)}^{d}=1-dL+\frac{d(d-1){L}^{2}}{2!}-\frac{d(d-1)(d-2){L}^{3}}{3!}+...$^{4}The fully electronic E-Mini S&P500 futures contracts feature among the most liquid derivative contracts in the world, therefore closely tracking price movements of the S&P 500 index. The index prices used for the other series are unfortunately less frequently quoted. The volatility measurements for the DJIA, FTSE 100, CAC 40 and Nikkei 225 indexes are therefore of somewhat inferior quality compared to the S&P 500 index and the stocks.

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Allen, D.E.; McAleer, M.; Scharth, M. Asymmetric Realized Volatility Risk. *J. Risk Financial Manag.* **2014**, *7*, 80-109.
https://doi.org/10.3390/jrfm7020080

**AMA Style**

Allen DE, McAleer M, Scharth M. Asymmetric Realized Volatility Risk. *Journal of Risk and Financial Management*. 2014; 7(2):80-109.
https://doi.org/10.3390/jrfm7020080

**Chicago/Turabian Style**

Allen, David E., Michael McAleer, and Marcel Scharth. 2014. "Asymmetric Realized Volatility Risk" *Journal of Risk and Financial Management* 7, no. 2: 80-109.
https://doi.org/10.3390/jrfm7020080