# Realized Measures to Explain Volatility Changes over Time

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

Q1: “If volatility is priced, does an anticipated increase in volatility raise the required return on equity, leading to an immediate stock price decline?”

Q2: “Does a drop in the value of the stock (negative return) increase financial leverage, so that it makes the stock riskier and increases its volatility?”.

## 2. Data and Methodology

#### 2.1. Realized Measures

#### 2.1.1. Theoretical Considerations

#### 2.1.2. Classes of Realized Measures

#### 2.2. Data and Data Adjustments

## 3. Empirical Results

#### 3.1. Volatility Feedback Effect

#### 3.2. Leverage Effect Results

^{−3}. Moreover, Panel B shows that FTSE100 returns have a statistically significant negative impact on their past returns with coefficient $d$ being equal to −5.6477 × 10

^{−4}. Hence, in both cases, the hypothesis of leverage effect is accepted.

## 4. Practical Implications

#### 4.1. Out-of-Sample Analysis

^{−3}, and the mean absolute deviation was equal to 2.61 × 10

^{−6}with tracking signal equal to 107.3424. As for FTSE100, the mean forecast error was equal to −6.10 × 10

^{−3}, and the mean absolute deviation was equal to 2.03 × 10

^{−6}with tracking signal equal to −222.2500. Figure 3 depicts the forecast error both for of S&P500 and FTSE100 index.

#### 4.2. Portfolio Implications

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | The relation between asset returns and expected volatility motivates the research to produce volatility forecast models using several techniques (see Xu 1999; Bali and Theodossiou 2007; Wang and Zhu 2010; Kiliç and Ugur 2018; Shang et al. 2018; among others). |

2 |

**Figure 1.**S&P500 plots. Note: This set of figures depict the realized measures on the S&P500 index (from left to right): The first figure depicts the realized bi-power variation (5-min sub-sampled) estimator, the second figure depicts the realized bi-power variation (5-min) estimator, the third figure depicts the S&P500 index closing price, the fourth figure depicts the median realized variance (5-min) estimator, the fifth figure depicts the realized Kernel variance estimator (two-scale/Barlett), the sixth figure depicts the realized variance (5-min sub-sampled) estimator, the seventh figure depicts the realized variance (10-min) estimator, the eighth figure depicts the realized variance (10-min sub-sampled) estimator, the ninth figure depicts the realized semi-variance (5-min) estimator, the tenth figure depicts the realized semi-variance (5-min sub-sampled) estimator, the eleventh figure depicts the realized variance (5-min) estimator, and the twelfth figure depicts the return on the S&P500 index.

**Figure 2.**FTSE100 plots. Note: This set of figures depict the realized measures on the FTSE100 index (from left to right): The first figure depicts the realized bi-power variation (5-min sub-sampled) estimator, the second figure depicts the realized bi-power variation (5-min) estimator, the third figure depicts the FTSE100 index closing price, the fourth figure depicts the median realized variance (5-min) estimator, the fifth figure depicts the realized Kernel variance estimator (two-scale/Barlett), the sixth figure depicts the realized variance (5-min sub-sampled) estimator, the seventh figure depicts the realized variance (10-min) estimator, the eighth figure depicts the realized variance (10-min sub-sampled) estimator, the ninth figure depicts the realized semi-variance (5-min) estimator, the tenth figure depicts the realized semi-variance (5-min sub-sampled) estimator, the eleventh figure depicts the realized variance (5-min) estimator, and the twelfth figure depicts the return of the FTSE100 index.

**Figure 3.**Forecast errors plots. Note: This set of figures depicts the one-step-ahead forecast error of the expected returns in S&P500 and FTSE100 indices used in this study obtained from a rolling-estimation window of 1000 observations.

**Figure 4.**Volatility regimes. Note: This set figures depicts the volatility regimes in returns and equity prices of S&P500 and FTSE100 indices; the green color refers to high volatility regimes while the red color refers to low volatility regimes. Panel A refers to S&P500 index sample. Panel B refers to FTSE100 index sample.

**Figure 5.**Defined volatility regimes. Note: This set figures depicts the volatility regimes in returns (upper figure) and the probability of the high and low volatility regimes (bottom figure) of S&P500 and FTSE100 indices. On the bottom figure, the grey color refers to high volatility regimes while the black color refers to low volatility regimes. Panel A refers to S&P500 index sample. Panel B refers to FTSE100 index sample.

Code | Description |
---|---|

$BPV$ | Realized bi-power variation (5-min) |

$BP{V}^{SS}$ | Realized bi-power variation (5-min sub-sampled) |

$MedRV$ | Median realized variance (5-min) |

$R{K}^{Parzen}$ | Realized Kernel variance (non-flat Parzen) |

$R{K}^{TH2}$ | Realized Kernel variance (Tukey-Hanning(2)) |

$R{K}^{Barlet}$ | Realized Kernel variance (two-scale/Barlett) |

$RS{V}^{D}$ | Realized downside semi-variance (5-min) |

$RS{V}^{D,SS}$ | Realized downside semi-variance (5-min sub-sampled) |

$RV$ | Realized variance |

$R{V}_{10}$ | Realized variance (10-min) |

$R{V}_{10}^{SS}$ | Realized variance (10-min sub-sampled) |

$R{V}_{5}$ | Realized variance (5-min) |

$R{V}_{5}^{SS}$ | Realized variance (5-min sub-sampled) |

Variable | Coefficient | Marginal Impact | Coefficient | Marginal Impact |
---|---|---|---|---|

Panel A. S&P500 | Panel B. FTSE100 | |||

$BPV$ | 24.5036 *** | 0.0037 | 14.8874 | 0.0010 |

(5.9560) | (7.0660) | |||

$BP{V}^{SS}$ | 2.4522 | 0.0000 | 80.1180 *** | 0.0080 |

(13.1600) | (13.2200) | |||

$MedRV$ | 53.8661 *** | 0.0307 | 41.4710 *** | 0.0231 |

(4.493) | (4.003) | |||

$R{K}^{Barlet}$ | 20.8178 ** | 0.0012 | 1.9995 | 0.0000 |

(9.0300) | (7.0130) | |||

$RS{V}^{D}$ | 174.6970 *** | 0.2244 | 121.288 *** | 0.0409 |

(4.8200) | (8.712) | |||

$RS{V}^{D,SS}$ | 147.8110 *** | 0.1341 | 89.9744 *** | 0.0184 |

(5.5730) | (9.765) | |||

$RV$ | 138.2400 *** | 0.0896 | 85.2223 *** | 0.0154 |

(6.5380) | (10.1000) | |||

$R{V}_{10}$ | 13.5120 *** | 0.0026 | 10.6719 ** | 0.0011 |

(3.9240) | (4.7250) | |||

$R{V}_{10}^{SS}$ | 80.5904 *** | 0.0256 | 10.4318 | 0.0003 |

(7.383) | (9.0230) | |||

$R{V}_{5}^{SS}$ | 183.5250 *** | 0.0256 | 157.8510 *** | 0.0226 |

(16.8100) | (15.4000) | |||

c | 3.0330 × 10^{−4} * | 0.0007 | 5.9472 × 10^{−4} *** | 0.0034 |

(1.6720 × 10^{−4}) | (1.5020 × 10^{−4}) | |||

Obs. | 4552 | 4552 | ||

R adj | 0.7229 | 0.7436 |

Variable | Coefficient | Marginal Impact | Coefficient | Marginal Impact |
---|---|---|---|---|

Panel A. S&P500 | Panel B. FTSE100 | |||

${R}_{t-1,t}$ | −1.1040 × 10^{−3} *** | 0.0068 | −5.6477 × 10^{−}^{4} *** | 0.0037 |

(1.9790 × 10^{−4}) | (1.3730 × 10^{−}^{4}) | |||

c | 1.1636 × 10^{−}^{4} | 0.1755 | 8.5092 × 10^{−5} *** | 0.2158 |

(3.7350 × 10^{−6}) | (2.4060 × 10^{−6}) | |||

Obs. | 4550 | 4550 | ||

R-squared | 6.7900 × 10^{−}^{3} | 3.7000 × 10^{−3} |

Low Volatility Regimes | High Volatility Regimes |
---|---|

Panel A. S&P500 | |

0.8174 | 0.8809 |

0.8274 | 0.8101 |

0.7920 | 0.7452 |

0.7377 | 0.7849 |

0.7821 | 0.7921 |

Panel B. FTSE100 | |

0.6685 | 0.7423 |

0.6776 | 0.7564 |

0.7324 | 0.7762 |

0.6416 | 0.8731 |

0.7934 | 0.7697 |

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## Share and Cite

**MDPI and ACS Style**

Floros, C.; Gkillas, K.; Konstantatos, C.; Tsagkanos, A.
Realized Measures to Explain Volatility Changes over Time. *J. Risk Financial Manag.* **2020**, *13*, 125.
https://doi.org/10.3390/jrfm13060125

**AMA Style**

Floros C, Gkillas K, Konstantatos C, Tsagkanos A.
Realized Measures to Explain Volatility Changes over Time. *Journal of Risk and Financial Management*. 2020; 13(6):125.
https://doi.org/10.3390/jrfm13060125

**Chicago/Turabian Style**

Floros, Christos, Konstantinos Gkillas, Christoforos Konstantatos, and Athanasios Tsagkanos.
2020. "Realized Measures to Explain Volatility Changes over Time" *Journal of Risk and Financial Management* 13, no. 6: 125.
https://doi.org/10.3390/jrfm13060125