# Information-Theoretic Measures and Modeling Stock Market Volatility: A Comparative Approach

^{*}

## Abstract

**:**

## 1. Introduction

`.`These models can be classified into different categories. Some examples of these models include historical volatility models based on the computation of the standard deviation of the historical return series, the random walk model, the historical average model, the simple moving average model, exponential smoothing, and the exponentially weighted moving average model. On the other hand, some examples of the regime-switching models accommodate the threshold-autoregressive model by Cao and Tsay (1992) and the smooth transition exponential smoothing model by Taylor (2004). Few examples of research work that used historical volatility models include Taylor (1986, 1987); Figlewski (1997); Figlewski and Green (1999) and Andersen et al. (2001). In literature, these models have shown good forecasting performance as compared to other class of volatility models. Poon and Granger (2005) discussed several practical problems involved in forecasting volatility.

**Hypothesis**

**1**

**(H1).**

**Hypothesis**

**2**

**(H2).**

## 2. Methodology

#### 2.1. Volatility

#### 2.2. GARCH Models

- The process ${\left\{{\epsilon}_{t}\right\}}_{t\in \mathbb{Z}}$ is called GARCH (p, q) process if $E\left({\epsilon}_{t}|{\mathcal{F}}_{t-1}\right)=0\text{},$ and $Var\left({\epsilon}_{t}|{\mathcal{F}}_{t-1}\right)={\sigma}_{t}^{2}$;
- The random variables ${Z}_{t}$ are identical and independent, ${\epsilon}_{t}$ is the residual series and ${\sigma}_{t}^{2}$ its conditional variance;
- $\omega >0,\text{}{\alpha}_{i}\text{}\ge 0$, ${\beta}_{j}\ge 0$ are real parameters and ensures that ${\sigma}_{t}^{2}>0$ at all times.

- where$\delta 0,{\alpha}_{1}\ge 0,{\beta}_{1}\ge 0,\left|{\gamma}_{1}\right|1,$ and$\omega 0$. ${\alpha}_{1}$ and ${\gamma}_{1}$ represent sign effect and leverage effect;
- For leverage effect ${\gamma}_{1}$ must be statistically significant and negative;
- Returns are stationary if $0<{\beta}_{1}<1$ and EGARCH captures serial dependence and leverage effects in returns;
- If $\left|{Z}_{t-1}\right|$ is small, then ${\sigma}_{t}$ decreases. For large $\left|{Z}_{t-1}\right|$ , the value of ${\sigma}_{t}$ increases;
- Due to log-transformation of variance, it guarantees positivity of variance without any restriction on parameters.

- where ${\alpha}_{1}>0,{\beta}_{1}0,{\gamma}_{1}0,\omega 0$ and${\gamma}_{1}$indicates asymmetry of returns;
- ${I}_{t-1}$ assumes value equals to 1 for ${\epsilon}_{t-1}^{2}<0$ (negative-shock), and zero otherwise;
- For positive and significant ${\gamma}_{1}$, leverage effect exists.

- where ${\alpha}_{1,+}\ge 0,{\alpha}_{1,-}\ge 0,{\beta}_{1}\ge 0,\text{}\mathrm{and}\text{}\omega 0$ are real parameters;
- The variable ${\sigma}_{t}$ is strictly positive and denotes the conditional standard deviation of ${\epsilon}_{t}$;
- The current volatility depends on both the modulus and the sign of the past returns through ${\alpha}_{1,+}$ and ${\alpha}_{1,-}$;
- For ${\alpha}_{1,+}>$0, the effect of the bad news is greater than those of the good news;
- The GJR-GARCH model due to Glosten et al. (1993) is a version of TGARCH, which corresponds to squaring the variables involved in Equation (8).

#### Conditional Distributions and GARCH Modeling

- where$x\in \left[-\infty ,\infty \right],$and $\mu ,\lambda ,\upsilon $ are location, scale, and shape parameters, respectively, and $\mathsf{\Gamma}$ is a Gamma function.

- where $\mu \in \left(-\infty ,\infty \right),\lambda \in \left(0,\infty \right)$ are location and scale parameters, respectively, and $\upsilon $ represents the tail-thickness parameter;
- For $\upsilon =2$ the distribution converges to the standard normal distribution, and for $\upsilon <2$, it has thicker tails than the normal distribution.

- where $\lambda ,\upsilon $ denoted skewness and shape parameters, respectively. For negative skewness $\lambda <0$, and for positive skewness $\lambda >0$. The parameter $\lambda \in \left[-1,1\right]$;
- For $\lambda =0,$the skewed-GED distribution converges to the GED;
- The sign function equals to −1 for negative values of its argument and equals to 1 for positive values;
- The values of $k=\frac{{2}^{\frac{2}{\upsilon}}\sigma m\lambda \mathsf{\Gamma}\left(0.5+{\nu}^{-1}\right)}{\sqrt{\pi}}$ , and $m=\frac{\pi \left(1+3{\lambda}^{2}\right)\mathsf{\Gamma}\left(\frac{3}{\upsilon}\right)-{16}^{\frac{1}{\upsilon}}{\lambda}^{2}\mathsf{\Gamma}\left(0.5+{\nu}^{-1}\right)\mathsf{\Gamma}\left(\frac{1}{\upsilon}\right)}{\pi \mathsf{\Gamma}\left(\frac{1}{\upsilon}\right)}$

- where $\xi $ denotes the asymmetry coefficient, $\mu ,\sigma $ are mean and the standard deviation, respectively.
- The density is skewed to the right if $\mathrm{log}\left(\xi \right)>0$ and skewed to the left if $\mathrm{log}\left(\xi \right)<0$.

#### 2.3. Information Theoretic Measures and Volatility Modeling

#### 2.3.1. Shannon Entropy Measure

- where, $X=\left({x}_{1},{x}_{2},\dots ,{x}_{m}\right)$ the convention $0\mathrm{ln}0=0$ holds, and $p=\left({p}_{1},{p}_{2},\dots ,{p}_{m}\right)$, ${p}_{i}$ represents the probability of ${x}_{i}$ , for $i=1,2,\text{}\dots ,m$. therefore,$\text{}{p}_{i}0$ $\forall \text{}i$ and ${{\displaystyle \sum}}_{i=1}^{m}{p}_{i}=1.$
- The entropy reaches to its maximum value if all events follow the equally likely assumption;
- The entropy corresponding to an event with probability less than one has a positive sign.

- (1)
- Maximum Likelihood (ML);
- (2)
- effreys: Bayesian estimate with a = 1/2;
- (3)
- Laplace: Bayesian estimate with a = 1;
- (4)
- Schurmann–Grassberger (SG): Bayesian estimate with a = 1/(length of underlying asset prices series);
- (5)
- Minimax with a = sqrt (sum (underlying asset prices series))/(length of underlying asset prices series);
- (6)
- Shrink entropy: Uses James–Stein-type shrinkage at the level of cell frequencies.

#### 2.3.2. Tsallis Entropy

- where$X=\left({x}_{1},{x}_{2},\dots ,{x}_{m}\right),$ $0<q<\infty $ and $q\ne 1.$ Rare events of interests denote $q<1$ , and frequently encountered have $q>1$;
- The q-exponential function ${e}_{q}^{x}={\left(1+\left(1-q\right)x\right)}^{\frac{1}{1-q}}$ , whose inverse is the q-logarithmic function ${\mathrm{ln}}_{q}^{x}=\frac{{x}^{1-q}-1}{1-q},q\ne 1$;
- The value of the parameter q decreases to 1 as the frequency of the data decreases. The values$1\le q\le 2$ emphasize highly volatile signals.

#### 2.3.3. Approximate and Sample Entropy

- Suppose the underlying time series, $X=\left({x}_{1},{x}_{2},\dots ,{x}_{L}\right)$ of length $L$;
- For $l<L$, let ${a}_{l}\left(i\right)$ and ${b}_{l}\left(j\right)$ be two vectors of length $l$ and $d\left({a}_{l}\left(i\right),{b}_{l}\left(j\right)\text{}\right)$ denotes distance between the two vectors. Therefore,$$d\left({a}_{l}\left(i\right),{b}_{l}\left(j\right)\text{}\right)=\mathrm{max}\left\{\left|{x}_{i+k}-{x}_{j+k}\right|:0\le k\le l-1\right\}.$$
- Two vectors ${a}_{l}\left(i\right)$ and ${b}_{l}\left(j\right)$ are called similar if $d\le r$, where $r>0$ denotes the specified tolerance. Now compute the relative frequency ${f}_{i}\left(r\right),$ for each of the $L-\left(l-1\right)$.$${f}_{i}\left(r\right)=\frac{{s}_{l}\left(i\right)}{L-\left(l-1\right)}$$
- where ${s}_{l}\left(i\right)$ denote number of vectors ${b}_{l}\left(j\right)$ similar to ${a}_{l}\left(i\right)$ for a fixed $i$and $l.$ Now computing the average frequency ${\eta}_{l}\left(r\right),$$${\eta}_{l}\left(r\right)=\frac{{\sum}_{i}ln{f}_{i}\left(r\right)}{L-\left(l-1\right)}$$
- Finally, ApEn can be computed by using the following statistics.$${\mathrm{ApEn}}_{l}\left(r,L\right)={\eta}_{l}\left(r\right)-{\eta}_{\left(l+1\right)}\left(r\right)$$

- where${E}_{1}$ denotes number of vectors pairs of length $l+1$ and $d\left[\left({X}_{l}\left(i\right),{X}_{l}\left(j\right)\right)\le r\right]$ with $i\ne j$;
- ${E}_{2}$ denotes total number of templates equals to length $l$ with $i\ne j.$

## 3. Empirical Analysis

#### 3.1. Stationarity and Normality Tests

#### 3.2. Testing the ARCH Effect

#### 3.3. Volatility Estimation Using GARCH Models

^{−12}. We conclude that the volatility dynamics of PSX-100 closed returns during the COVID-19 pandemic is significantly higher. Figure 2 shows a clear picture of the variation between the volatilities of the two periods.

#### 3.4. Volatility Assessment Using Shannon and Tsallis Entropy

#### 3.5. Regularity and Randomness Using Approximate Entropy and Sample Entropy

## 4. Discussion and Conclusions

- Both best-fitted models suggested the remarkable existence of the leverage effect in the Pakistan Stock Exchange;
- A high variation in the estimated volatility of the pandemic period has observed, and extreme downturns in prices are expected;
- The behavior of the variance is asymmetric for PSX-100 closing returns of both examined periods;
- The GARCH volatility modeling shows a usual behavior of the Pakistani market response towards bad news.

- We have pointed out that no remarkable difference between the randomness of volatility series before and during the COVID-19 pandemic, based on Shannon and Tsallis entropy estimates;
- All values of entropies are positive on both periods, and volatility shows non-linear dynamics;
- The overall relative difference of estimated entropies is not significant;
- The overall relative difference of estimated entropies is not significant under both the Shannon and Tsallis entropies;
- In the case of closing prices of the PSX-100, the Shannon entropy shows almost similar behavior under all estimation methods;
- In the case of ApEn and SampEn, the market shows mixed behavior, and the results are more sensitive. The sample entropy results reported more randomness in the pre-pandemic period for the Pakistani Stock Market. We detected both entropies are very sensitive to the selection of parameters.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Model | LOGLIK. | BIC | SIC | HQIC | AIC |
---|---|---|---|---|---|

Student T | |||||

GJRGARCH | 745.2045 | −5.9935 | −5.8795 | −5.9956 | −5.9476 |

EGARCH | 741.4358 | −5.9629 | −5.8489 | −5.9649 | −5.917 |

TGARCH | 740.9011 | −5.9585 | −5.8446 | −5.9606 | −5.9126 |

Asymmetric Student T | |||||

GJRGARCH | 746.4149 | −5.9952 | −5.867 | −5.9978 | −5.9436 |

EGARCH | 742.1684 | −5.9607 | −5.8325 | −5.9633 | −5.9091 |

TGARCH | 741.6314 | −5.9564 | −5.8281 | −5.9589 | −5.9047 |

Generalized Error | |||||

GJRGARCH | 745.4326 | −5.9954 | −5.8814 | −5.9974 | −5.9495 |

EGARCH | 742.1894 | −5.969 | −5.855 | −5.971 | −5.9231 |

TGARCH | 741.8362 | −5.9661 | −5.8522 | −5.9682 | −5.9202 |

Skewed GED | |||||

GJRGARCH | 746.4959 | −5.9959 | −5.8677 | −5.9985 | −5.9443 |

EGARCH | 742.675 | −5.9648 | −5.8366 | −5.9674 | −5.9132 |

TGARCH | 742.3509 | −5.9622 | −5.834 | −5.9648 | −5.9106 |

Model | LOGLIK. | BIC | SIC | HQIC | AIC |
---|---|---|---|---|---|

Student T | |||||

GJRGARCH | 611.1641 | −6.1547 | −6.0209 | −6.1579 | −6.1006 |

EGARCH | 611.0953 | −6.1438 | −5.9933 | −6.1478 | −6.0829 |

TGARCH | 611.8439 | −6.1617 | −6.0279 | −6.1648 | −6.1075 |

Asymmetric Student T | |||||

GJRGARCH | 612.1173 | −6.1543 | −6.0037 | −6.1582 | −6.6851 |

EGARCH | 615.0001 | −6.1939 | −6.0601 | −6.197 | −6.1075 |

TGARCH | 614.4997 | −6.1786 | −6.028 | −6.1825 | −6.0933 |

Generalized Error | |||||

GJRGARCH | 616.2783 | −6.2069 | −6.0731 | −6.2101 | −6.1075 |

EGARCH | 616.9883 | −6.204 | −6.0534 | −6.2079 | −6.0933 |

TGARCH | 612.136 | −6.1647 | −6.0309 | −6.1678 | −6.1397 |

Skewed GED | |||||

GJRGARCH | 612.2701 | −6.1558 | −6.0053 | −6.1598 | −6.0933 |

EGARCH | 612.6194 | −6.1696 | −6.0358 | −6.1727 | −6.1397 |

TGARCH | 613.2739 | −6.1661 | −6.0155 | −6.1700 | −6.1176 |

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**Figure 2.**Comparison of estimated volatilities of PSX-100 returns. The pre-pandemic period spans from 1 January 2019 to 31 December 2019, and pandemic period spans from 1 January 2020 to 24 December 2020. This suggest that the COVID-19 affected significantly the PSX-100 volatility.

**Figure 5.**The plot of the approximate entropy and sample entropy estimates for estimated volatility series, closing prices and closing returns of PSX-100.

Returns | Before COVID-19 | COVID-19 | ||||||
---|---|---|---|---|---|---|---|---|

Closing | Opening | High | Low | Closing | Opening | High | Low | |

Minimum | −0.0268 | −0.0268 | −0.0335 | −0.0367 | −0.071 | −0.071 | −0.0666 | −0.0655 |

1st Quartile | −0.0071 | −0.0078 | −0.0068 | −0.0065 | −0.0054 | −0.0054 | −0.0064 | −0.0054 |

Median | 0.0001 | 0.0004 | 0.0001 | 0.0012 | 0.0012 | 0.0012 | 0.0011 | 0.0022 |

Mean | 0.0003 | 0.0004 | 0.0003 | 0.0004 | 0.0002 | 0.0002 | 0.0002 | 0.0002 |

3rd Quartile | 0.0075 | 0.0079 | 0.007 | 0.0075 | 0.0085 | 0.0086 | 0.0075 | 0.0082 |

Maximum | 0.0351 | 0.0427 | 0.0337 | 0.0374 | 0.0468 | 0.0468 | 0.056 | 0.0498 |

Skewness | 0.1326 | 0.1736 | 0.1744 | −0.0276 | −1.151 | −1.1479 | −0.9461 | −1.1668 |

Kurtosis | 3.1178 | 3.206 | 3.3096 | 3.4331 | 8.0603 | 8.0203 | 9.1404 | 7.7247 |

SD | 0.0116 | 0.0122 | 0.0107 | 0.0118 | 0.0155 | 0.0156 | 0.0138 | 0.0152 |

CV | 0.0243 | 0.0327 | 0.0282 | 0.0323 | 0.0125 | 0.0147 | 0.0144 | 0.015 |

Range | 0.0619 | 0.0696 | 0.0672 | 0.0741 | 0.1179 | 0.1179 | 0.1226 | 0.1153 |

Inter−Quartile range | 0.0145 | 0.0157 | 0.0138 | 0.0141 | 0.0139 | 0.014 | 0.0139 | 0.0136 |

JB Stats | 0.7971 | 1.5441 | 2.0296 | 1.6715 | 303.1570 | 298.9304 | 404.2784 | 272.5008 |

Lilliefors Stats | 0.0479 | 0.0336 | 0.0298 | 0.0375 | 0.1281 | 0.1276 | 0.0986 | 0.1193 |

Pearson Chi Sq. Test | 19.8455 | 10.5772 | 13.3577 | 17.6829 | 66.4959 | 66.1870 | 43.9431 | 70.3577 |

Period of Variables | Stationary Tests | ARCH Effect | |||
---|---|---|---|---|---|

Variable | ADF Stats | Philip Perron Stats | KPSS Stats | LM Stats | LB Stats |

Closing Returns—Before COVID-19 | −12.9559 (0.0001) | −12.9191 (0.0001) | 0.4042 (0.0753) | 36.728 (0.0002) | 8.3691 (0.0038) |

Closing Returns—COVID-19 | −13.4422 (0.0100) | −13.7433 (0.0001) | 0.3249 (0.0100) | 73.918 (0.0000) | 5.8386 (0.0157) |

GJR-GARCH-SGED COVID-19 Period | ||||||||

mu | ar1 | ma1 | $\omega $ | ${\alpha}_{1}$ | ${\beta}_{1}$ | ${\gamma}_{1}$ | skew | shape |

0.0002 (0.0010) | 0.8163 (0.1413) | −0.7490 (0.1661) | 0.0000 (0.0000) | 0.0000 (0.0126) | 0.8222 (0.0312) | 0.2437 (0.0820) | 0.8801 (0.1162) | 1.6361 (0.3015) |

EGARCH-STD before-COVID-19 Period | ||||||||

mu | ar1 | ma1 | $\omega $ | ${\alpha}_{1}$ | ${\beta}_{1}$ | ${\gamma}_{1}$ | shape | |

−0.0013 (0.0000) | −0.1308 (0.0000) | 0.3022 (0.0000) | −0.1310 (0.0000) | −0.1792 (0.0000) | 0.9854 (0.0001) | −0.1452 (0.0000) | 18.7972 (0.0030) |

Estimation | Volatility | Prices | ||
---|---|---|---|---|

Method | COVID-19 | Before COVID-19 | COVID-19 | Before COVID-19 |

ML | 5.3661 | 5.4902 | 5.5037 | 5.5050 |

Jeffreys | 5.5052 | 5.5053 | 5.5037 | 5.5050 |

Laplace | 5.5053 | 5.5053 | 5.5037 | 5.5050 |

SG | 5.4207 | 5.4971 | 5.5037 | 5.5050 |

minimax | 5.4439 | 5.4994 | 5.5037 | 5.5050 |

Shrink | 5.5053 | 5.5053 | 5.5037 | 5.5050 |

Parameter | Volatility | Prices | ||
---|---|---|---|---|

q | COVID-19 | Before COVID-19 | COVID-19 | Before COVID-19 |

1 | 5.3661 | 5.4902 | 5.5037 | 5.5050 |

1.2 | 3.2792 | 3.3314 | 3.3365 | 3.3370 |

1.4 | 2.1998 | 2.2212 | 2.2232 | 2.2234 |

1.6 | 1.5958 | 1.6045 | 1.6052 | 1.6053 |

1.8 | 1.2309 | 1.2344 | 1.2346 | 1.2347 |

2 | 0.9944 | 0.9958 | 0.9959 | 0.9959 |

**Table 6.**Estimates of the approximate entropy for estimated volatility, closing prices, and closing returns of PSX-100.

$\mathit{r}$ | $\mathit{l}$ | Returns | Volatility | Prices | |||
---|---|---|---|---|---|---|---|

COVID-19 | Before COVID-19 | COVID-19 | Before COVID-19 | COVID-19 | Before COVID-19 | ||

0.05 | 1 | 2.0077 | 1.9918 | 0.5665 | 1.051 | 0.8225 | 0.876 |

2 | 0.6347 | 0.5192 | 0.475 | 0.7457 | 0.5948 | 0.6834 | |

3 | 0.0958 | 0.047 | 0.3587 | 0.3605 | 0.4043 | 0.4813 | |

0.1 | 1 | 1.9011 | 2.0217 | 0.4579 | 0.9009 | 0.5764 | 0.5923 |

2 | 0.9052 | 0.7616 | 0.3918 | 0.7500 | 0.482 | 0.565 | |

3 | 0.2774 | 0.1014 | 0.3073 | 0.5177 | 0.3998 | 0.4872 | |

0.15 | 1 | 1.7757 | 1.9621 | 0.4091 | 0.7828 | 0.4231 | 0.4143 |

2 | 0.995 | 0.9847 | 0.3325 | 0.6667 | 0.3639 | 0.4258 | |

3 | 0.4658 | 0.222 | 0.2761 | 0.5267 | 0.3085 | 0.3828 | |

0.2 | 1 | 1.6367 | 1.8465 | 0.3535 | 0.6795 | 0.3358 | 0.3074 |

2 | 1.0500 | 1.1772 | 0.3105 | 0.6127 | 0.3141 | 0.3251 | |

3 | 0.6265 | 0.3374 | 0.2585 | 0.5123 | 0.2757 | 0.3164 |

**Table 7.**Estimates of the approximate entropy for estimated volatility, closing prices, and closing returns of PSX-100.

r | l | Returns | Volatility | Prices | |||
---|---|---|---|---|---|---|---|

COVID-19 | Before COVID-19 | COVID-19 | Before COVID-19 | COVID-19 | Before COVID-19 | ||

0.05 | 1 | 2.4511 | 2.6077 | 0.3638 | 1.0016 | 0.7022 | 0.8336 |

2 | 2.3931 | 2.4849 | 0.3316 | 1.0452 | 0.613 | 0.8029 | |

3 | 2.9444 | 1.7047 | 0.3039 | 0.8627 | 0.5625 | 0.8052 | |

0.1 | 1 | 2.0316 | 2.4002 | 0.2712 | 0.7824 | 0.4747 | 0.5476 |

2 | 1.875 | 2.4022 | 0.2397 | 0.8176 | 0.4061 | 0.5454 | |

3 | 1.9879 | 2.3514 | 0.2128 | 0.7807 | 0.3757 | 0.5301 | |

0.15 | 1 | 1.7655 | 2.165 | 0.2215 | 0.6359 | 0.3431 | 0.385 |

2 | 1.5924 | 2.1313 | 0.19 | 0.6476 | 0.2913 | 0.3878 | |

3 | 1.678 | 2.7515 | 0.1632 | 0.6107 | 0.2723 | 0.3644 | |

0.2 | 1 | 1.5554 | 1.9543 | 0.1939 | 0.5431 | 0.271 | 0.2969 |

2 | 1.434 | 2.0363 | 0.1655 | 0.5518 | 0.2335 | 0.2982 | |

3 | 1.5666 | 2.3026 | 0.143 | 0.5125 | 0.2184 | 0.2895 |

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**MDPI and ACS Style**

Sheraz, M.; Nasir, I.
Information-Theoretic Measures and Modeling Stock Market Volatility: A Comparative Approach. *Risks* **2021**, *9*, 89.
https://doi.org/10.3390/risks9050089

**AMA Style**

Sheraz M, Nasir I.
Information-Theoretic Measures and Modeling Stock Market Volatility: A Comparative Approach. *Risks*. 2021; 9(5):89.
https://doi.org/10.3390/risks9050089

**Chicago/Turabian Style**

Sheraz, Muhammad, and Imran Nasir.
2021. "Information-Theoretic Measures and Modeling Stock Market Volatility: A Comparative Approach" *Risks* 9, no. 5: 89.
https://doi.org/10.3390/risks9050089