# Estimating Stochastic Volatility under the Assumption of Stochastic Volatility of Volatility

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## Abstract

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## 1. Introduction

## 2. Model Specification

## 3. Estimation Technique

## 4. Application

## 5. Financial Implications

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1 | Stochastic volatility is an essential component in an asset pricing model, in option pricing. See (Wong and Lo 2009; Mrázek et al. 2016; Cui et al. 2017; among others). |

2 | Heston (1993) assumed that the variance ${v}_{t}$ follows a square root process, ${dv}_{t}={\kappa}^{\ast}[{\theta}^{\ast}-{v}_{t}]dt+\sigma \sqrt{{v}_{t}}d{W}_{t}$, where σ controls the volatility of volatility. When σ is equal to zero, the volatility is deterministic. Following Heston (1993) and applying Ito’s rule (4), we are able to test whether volatility is stochastic via linear regression equations. We refer to the study implemented by Heston (1993) for more information regarding this issue. |

3 | Following Barndorff-Nielsen et al. (2009), (1) we deleted the entries with a timestamp outside the 9:30 a.m.–4 p.m. window when the market is open and (2) we deleted the entries with a bid, ask or a transaction price equal to zero. |

4 | For the simulation analysis, we considered a stochastic process of a single random variable based on the S&P 500 index assuming that the variance varies over time (i.e., exhibits serial correlated properties). To this end, we used a stochastic differential equation of the Ito type. The drift and the diffusion parameters are estimated from the original series by the method of maximum pseudo-likelihood (see Boukhetala 1996). |

5 | Following Bekaert and Hoerova (2014), the squared $VIX$ index can be decomposed into the variance of equity returns and the variance risk premium. For robustness purposes, in order to isolate the predictive power of past volatility and the volatility on volatility risk premium, we estimated (15) and (16) using the difference between the VIX indices and daily realized variance (estimated as the sum of 5-min intraday returns of the S&P 500) as dependent variables. In this regard, we also used in (15) and (16) the residuals obtained from ${VIX}_{t}={\beta}_{0}^{VIX}+{\beta}_{1}^{VIX}{v}_{t}+{u}_{3t}$ and their squared specifications as dependent variables. In all cases, the results remain qualitative and quantitatively similar. |

Panel A: 1-Min Sample | ||||

v_{t} | γ_{t} | v_{t} (annual) | γ_{t} (annual) | |

Mean | 1.59E-04 | 1.05E-03 | 2.4883E-02 | 1.65E-01 |

Median | 1.03E-04 | 1.73E-04 | 1.6142E-02 | 2.71E-02 |

Maximum | 1.15E-02 | 2.66E+00 | 1.8062E+00 | 4.16E+02 |

Minimum | 0.00E+00 | 4.16E-11 | 0.0000E+00 | 6.52E-09 |

Std. Dev. | 2.32E-04 | 1.91E-02 | 3.6384E-02 | 3.00E+00 |

Skewness | 1.37E+01 | 1.16E+02 | 1.3678E+01 | 1.16E+02 |

Kurtosis | 4.21E+02 | 1.55E+04 | 4.2115E+02 | 1.55E+04 |

Jarque–Bera | 1.80E+08 | 2.43E+11 | 1.8000E+08 | 2.43E+11 |

p-value | [0.00E+00] | [0.00E+00] | [0.00E+00] | [0.00E+00] |

No. of obs. | 24,224 | 24,224 | 24,224 | 24,224 |

Panel B: 5-Min Sample | ||||

v_{t} | γ_{t} | v_{t} (annual) | γ_{t} (annual) | |

Mean | 4.16E-04 | 2.96E-03 | 2.96E-02 | 2.11E-01 |

Median | 2.66E-04 | 4.43E-04 | 1.89E-02 | 3.15E-02 |

Maximum | 1.37E-02 | 1.33E+00 | 9.79E-01 | 9.48E+01 |

Minimum | 4.56E-06 | 1.06E-09 | 3.25E-04 | 7.52E-08 |

Std. Dev. | 5.67E-04 | 2.47E-02 | 4.04E-02 | 1.76E+00 |

Skewness | 6.98E+00 | 3.69E+01 | 6.98E+00 | 3.69E+01 |

Kurtosis | 1.02E+02 | 1.78E+03 | 1.02E+02 | 1.78E+03 |

Jarque–Bera | 2.08E+06 | 6.63E+08 | 2.08E+06 | 6.63E+08 |

Probability | [0.00E+00] | [0.00E+00] | [0.00E+00] | [0.00E+00] |

p-value | 5042 | 5042 | 5042 | 5042 |

Panel C: 30-Min Sample | ||||

v_{t} | γ_{t} | v_{t} (annual) | γ_{t} (annual) | |

Mean | 1.14E-03 | 3.31E-02 | 9.56E-03 | 2.78E-01 |

Median | 7.18E-04 | 2.09E-02 | 1.32E-03 | 3.83E-02 |

Maximum | 1.53E-02 | 4.46E-01 | 5.91E-01 | 1.72E+01 |

Minimum | 4.66E-06 | 1.35E-04 | 2.49E-07 | 7.25E-06 |

Std. Dev. | 1.38E-03 | 4.02E-02 | 3.90E-02 | 1.13E+00 |

Skewness | 3.63E+00 | 3.63E+00 | 9.34E+00 | 9.34E+00 |

Kurtosis | 2.48E+01 | 2.48E+01 | 1.13E+02 | 1.13E+02 |

Jarque–Bera | 1.84E+04 | 1.84E+04 | 4.38E+05 | 4.38E+05 |

p-value | [0.00E+00] | [0.00E+00] | [0.00E+00] | [0.00E+00] |

No. of obs. | 841 | 841 | 841 | 841 |

Panel D: Daily Sample | ||||

v_{t} | γ_{t} | v_{t} (annual) | γ_{t} (annual) | |

Mean | 6.60E-03 | 4.84E-02 | 2.34E-01 | 1.71E+00 |

Median | 3.77E-03 | 7.28E-03 | 1.33E-01 | 2.58E-01 |

Maximum | 1.36E-01 | 7.49E+00 | 4.83E+00 | 2.66E+02 |

Minimum | 6.87E-06 | 3.20E-05 | 2.43E-04 | 1.13E-03 |

Std. Dev. | 9.54E-03 | 2.87E-01 | 3.38E-01 | 1.02E+01 |

Skewness | 5.45E+00 | 1.93E+01 | 5.45E+00 | 1.93E+01 |

Kurtosis | 5.18E+01 | 4.48E+02 | 5.18E+01 | 4.48E+02 |

Jarque-Bera | 1.31E+05 | 1.04E+07 | 1.31E+05 | 1.04E+07 |

p-value | [0.00E+00] | [0.00E+00] | [0.00E+00] | [0.00E+00] |

No. of obs. | 1256 | 1256 | 1256 | 1256 |

Panel A: 1-Min Sample | |||||

Case 1: Returns | Case 2: Simulation | ||||

${\mathit{\delta}}_{\mathbf{1}}$ | ${\mathit{R}}^{\mathbf{2}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | ${\overline{\mathit{R}}}^{\mathbf{2}}$ | ||

${H}_{0}:{\delta}_{1}=0$ | 8.82E-05 *** | 0.3892 | ${H}_{0}:{\delta}_{1}=0$ | 8.49E-05 | 0.3789 |

(7.05E-07) | (7.15E-07) | ||||

Panel B: 5-Min Sample | |||||

Case 1: Returns | Case 2: Simulation | ||||

${\mathit{\delta}}_{\mathbf{1}}$ | ${\mathit{R}}^{\mathbf{2}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | ${\overline{\mathit{R}}}^{\mathbf{2}}$ | ||

${H}_{0}:{\delta}_{1}=0$ | 1.23E-04 *** | 0.3779 | ${H}_{0}:{\delta}_{1}=0$ | 1.16E-04 | 0.3688 |

(2.22E-06) | (2.28E-06) | ||||

Panel C: 30-Min Sample | |||||

Case 1: Returns | Case 2: Simulation | ||||

${\mathit{\delta}}_{\mathbf{1}}$ | ${\mathit{R}}^{\mathbf{2}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | ${\overline{\mathit{R}}}^{\mathbf{2}}$ | ||

${H}_{0}:{\delta}_{1}=0$ | 1.36E-04 *** | 0.341 | ${H}_{0}:{\delta}_{1}=0$ | 1.76E-05 | 0.3184 |

(6.40E-06) | (5.46E-06) | ||||

Panel D: Daily Sample | |||||

Case 1: Returns | Case 1: Returns | ||||

${\mathit{\delta}}_{\mathbf{2}}$ | ${\mathit{R}}^{\mathbf{2}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | ${\overline{\mathit{R}}}^{\mathbf{2}}$ | ||

${H}_{0}:{\delta}_{1}=0$ | 6.54E-03 *** | 0.3677 | ${H}_{0}:{\delta}_{1}=0$ | 3.21-E04 | 0.3810 |

(2.40E-04) | (1.37E-05) |

Panel A: 1-Min Sample | |||||

Case 1: Returns | Case 2: Simulation | ||||

${\mathit{\delta}}_{\mathbf{2}}$ | ${\mathit{R}}^{\mathbf{2}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | ${\overline{\mathit{R}}}^{\mathbf{2}}$ | ||

${H}_{0}:{\delta}_{2}=0$ | 6.95E+00 *** | 0.4937 | ${H}_{0}:{\delta}_{2}=0$ | 6.25E+00 | 0.4889 |

(4.56E-02) | (6.01E-02) | ||||

Panel B: 5-Min Sample | |||||

Case 1: Returns | Case 2: Simulation | ||||

${\mathit{\delta}}_{\mathbf{2}}$ | ${\mathit{R}}^{\mathbf{2}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | ${\overline{\mathit{R}}}^{\mathbf{2}}$ | ||

${H}_{0}:{\delta}_{2}=0$ | 1.68Ε+00 *** | 0.4770 | ${H}_{0}:{\delta}_{2}=0$ | 1.22E+00 | 0.4462 |

(2.48E-02) | (1.85E-02) | ||||

Panel C: 30-Min Sample | |||||

Case 1: Returns | Case 2: Simulation | ||||

${\mathit{\delta}}_{\mathbf{2}}$ | ${\mathit{R}}^{\mathbf{2}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | ${\overline{\mathit{R}}}^{\mathbf{2}}$ | ||

${H}_{0}:{\delta}_{2}=0$ | 2.87E-01 *** | 0.4491 | ${H}_{0}:{\delta}_{2}=0$ | 3.16E-01 | 0.4214 |

(1.09E-02) | (2.40E-02) | ||||

Panel D: Daily Sample | |||||

Case 1: Returns | Case 2: Simulation | ||||

${\mathit{\delta}}_{\mathbf{2}}$ | ${\mathit{R}}^{\mathbf{2}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | ${\overline{\mathit{R}}}^{\mathbf{2}}$ | ||

${H}_{0}:{\delta}_{2}=0$ | 5.02E+01 *** | 0.4710 | ${H}_{0}:{\delta}_{2}=0$ | 7.99E+01 | 0.3914 |

(1.50E+00) | (1.20E-01) |

**Note**: This table reports the estimates of the tests for the null hypothesis ${H}_{0}:{\delta}_{2}=0$, that is, if the volatility of volatility is stochastic, applying linear least squares regression. Two cases are considered. Case 1 is based on our real data estimated by using 1-min returns, 5-min returns, 30-min returns and daily returns for the S&P 500 equity index. Case 2 is based on a Monte Carlo analysis using 1000 replications with a sample size equal to the number of observations 24,224, 5042, 841 and 1259 for each panel, respectively. *** refer to significant levels of 1%.

Panel A: Units | |||

VIX Index | |||

${v}_{t-1}$ | ${\gamma}_{t-1}$ | $Adj.{R}^{2}$ | Wald test ${H}_{0}:{\beta}_{1}^{VIX}-{\beta}_{2}^{VIX}=0$ |

4.23E+02 *** | 1.33E+01 *** | 0.2381 | 46.2936 |

(8.86E+00) | (2.94E-01) | [0.00E+00] | |

VVIX Index | |||

${v}_{t-1}$ | ${\gamma}_{t-1}$ | $Adj.{R}^{2}$ | Wald test ${H}_{0}:{\beta}_{1}^{VIX}-{\beta}_{2}^{VIX}=0$ |

7.60E+02 *** | 7.63E+00 *** | 0.1175 | 25.6088 |

2.94E+01 | 9.76E-01 | [0.00E+00] | |

Panel B: Logarithm Units | |||

VIX Index | |||

${v}_{t-1}$ | ${\gamma}_{t-1}$ | $Adj.{R}^{2}$ | Wald test ${H}_{0}:{\beta}_{1}^{VVIX}-{\beta}_{2}^{VVIX}=0$ |

2.50E+01 *** | 6.19E-01 *** | 0.2953 | 39.8077 |

(6.47E-01) | (2.05E-02) | [0.00E+00] | |

VVIX Index | |||

${v}_{t-1}$ | ${\gamma}_{t-1}$ | $Adj.{R}^{2}$ | Wald test ${H}_{0}:{\beta}_{1}^{VIVX}-{\beta}_{2}^{VVIX}=0$ |

7.73E+00 *** | 8.16E-02 *** | 0.1414 | 23.9296 |

(8.16E-02) | (1.06E-02) | [0.00E+00] |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Alghalith, M.; Floros, C.; Gkillas, K.
Estimating Stochastic Volatility under the Assumption of Stochastic Volatility of Volatility. *Risks* **2020**, *8*, 35.
https://doi.org/10.3390/risks8020035

**AMA Style**

Alghalith M, Floros C, Gkillas K.
Estimating Stochastic Volatility under the Assumption of Stochastic Volatility of Volatility. *Risks*. 2020; 8(2):35.
https://doi.org/10.3390/risks8020035

**Chicago/Turabian Style**

Alghalith, Moawia, Christos Floros, and Konstantinos Gkillas.
2020. "Estimating Stochastic Volatility under the Assumption of Stochastic Volatility of Volatility" *Risks* 8, no. 2: 35.
https://doi.org/10.3390/risks8020035