Econometric and stochastic analysis of stock price before and during COVID-19 in India

Abstract

Unexpected and sudden spread of the novel Coronavirus disease (COVID-19) has opened up many scopes for researchers in the fields of biotechnology, health care, educational sectors, agriculture, manufacturing, service sectors, marketing, finance, etc. Hence, the researchers are concerned to study, analyze and predict the impact of infection of COVID-19. The COVID-19 pandemic has affected many fields, particularly the stock markets in the financial sector. In this paper, we have proposed an econometric approach and stochastic approach to analyze the stochastic nature of stock price before and during a COVID-19-specific pandemic period. For our study, we considered the BSE SENSEX INDEX closing pricing data from the Bombay Stock Exchange for the period before and during COVID-19. We have applied the statistical tools, namely descriptive statistics for testing the normal distribution of data, unit root test for testing the stationarity, and GARCH and stochastic model for measuring the risk, also investigated drift and volatility (or diffusion) coefficients of the stock price SDE by using R Environment software and formulated the 95% confidence level bound with the help of 500 times simulations. Finally, the results have been obtained from these methods and simulations are discussed.

Introduction

In 1826–1827, Scottish Botanist Robert Brown noticed the erratic mobility of pollen particles suspended in water. The French Mathematician L. Bachelier first described the fluctuation in the stock price mathematically and obtained certain results in 1900 and extended by Albert Einstein in 1905 (Evans, 2012). These were described by an equation called stochastic differential equations (SDEs).

Nowadays, SDEs are used in many fields like physics, astronomy, mechanics, economics, mathematical finance, biology, epidemiology, political analysis, environmental science and a wide range of other scientific and engineering domains (Wilmott et al., 1995; Mun, 2004; Benth, 2003; Beichelt, 2006; Arnold, 1974; Brémaud, 2012; Oksendal, 2013; Baxter et al., 1996; Hull, 2003). Many researchers have worked on econometric and stochastic models on the stocks (Lingaraja et al., 2014; Fama , 1995; Dmouj, 2006; Iyer et al., 2013; Ugbebor et al., 2001; Ogwuche et al., 2014; Boccia & Sarnacchiaro, 2020; Sorin, 1997).

Particularly, Nicolau (2008) modeled the financial time series by second-order SDEs, Adeosun et al. (2015) stochastically analyzed the behavior of Nigerian Stock Exchange (NSE), Ofomata et al. (2017) formulated a stochastic model of a few chosen stocks in the NSE and was compared with the other stock for better investments, Bank and Kramkov (2013) provided adequate conditions for the presence and distinctiveness of SDE solutions which emerge in the price impact model, Tian and Zhang (2020) described the pricing of European options using stochastic volatility jump diffusion approaches with transaction cost, and Lima & Melgaço (2021) analyzed the behavior of stock prices using the Black–Scholes equation and nonlinear SDE. Moreover, recently Cheng et al. (2022) have examined the COVID-19 pandemic effects on the global stock market’s volatility connectedness network and Zhang et al. (2022) studied the COVID-19 pandemic impacts on the stock market volatility among the countries with advanced technologies.

In 1986, for the 30 standard companies, S &P BSE SENSEX was compiled and calculated using the methodology “Market Capitalization-Weight” taking 1978–79 as the base year. “Free-float Market Capitalization Methodology” approach to construct the S &P BSE SENSEX has been used since 2003. This methodology is also used by MSCI, FTSE, STOXX and Dow Jones ((BSEXXSlahUndXXHistory, 2021)).

An analysis of a day-to-day-based statistical data of the stock price will help the investors in their decision-making on that stocks. More specifically, we have analyzed the stock prices before and during the COVID-19-specific period in India between March 01, 2019, and February 22, 2021. Since the first report of COVID-19 infection in India occurred on January 30, 2020 (PIB, 2020; Tomar & Gupta, 2020), we collected Bombay Stock Exchange closing price data from March 01, 2019, to February 28, 2020, as the Study Period- I for before COVID-19 and from March 01, 2020, to February 22, 2021, as the Study Period-II for during COVID-19 from the official website of BSE India BSEXXSlahUndXXIndices (2020-2021) and we have demonstrated the descriptive statistics, normal distribution test, unit toot test and GARCH model by using E-Views (version 7.0) for the econometric approach and we fit the stock price stochastic differential equation through finding drift and volatility constant for stochastic analysis by using the R Environment (version 3.6.2). The prediction of a stock price is the biggest challenge for a stock investor for investment (Adeosun et al., 2015; Dmouj, 2006; Ofomata et al., 2017). In this paper, we have analyzed the Bombay Stock Exchange stock prices during the specified period using both econometrical and stochastical approaches.

Scope of the study

This research offers a detailed analysis of econometric and stochastic differential equation of the index of Indian stock market (SENSEX) for before and during the COVID-19. As it has been mentioned earlier, this research is to find the possible risks, returns and opportunities in the Bombay Stock Exchange, India during the study periods. Besides, using descriptive statistics and the unit root test, this study examines the normality and stationarity for the applicability of the sample data. Additionally, the risk and the volatility for the study periods are investigated, using GARCH (1,1) and stochastic differential equation model. Hence, this research tries to fill up this research gap.

Sample and period of the study

The Bombay Stock Exchange is one of Asia’s most prestigious and historic stock exchanges. The BSE SENSEX is one of the best indicators, devised and introduced by BSE. Hence, the researcher selected BSE Sensex and collecting required daily BSE Sensex index data from https://www.bseindia.com, the official website of the Bombay Stock Exchange. The present study covers Study Period -I and Study Period -II.

Hypotheses

The proposed hypotheses are as follows:

• $H_{01}$: there is no normality of BSE Sensex index before and during COVID-19.

• $H_{02}$: there is no stationarity of BSE Sensex index before and during COVID-19.

• $H_{03}$: there is no volatility (risk) of BSE Sensex index before COVID-19.

• $H_{04}$: there is no volatility (risk) of BSE Sensex index during COVID-19.

Basics of probability theory

Probability theory helps us study the mathematical system which involves randomness. For example, we need probability theory to predict the outcomes of flipping a coin, throwing a dice, playing a game, weather report of rain or dryness, the closing price of a stock in the stock market, etc. To describe the stock price mathematically, we need the concepts of probability theory, stochastic process and Brownian motion. We present the basics of probability theory in this section and then stochastic processes and Brownian motion in the next section.

Let us consider a set $\mathrm{\Omega }$ that includes all the potential results of a mathematical system, every subset of $\mathrm{\Omega }$ is not an observable or intriguing event. We classify these observable or intriguing events as a family $\mathcal{F}$ of subsets of $\mathrm{\Omega }$. Here we have taken the basic definitions from (Evans, 2012; Mao, 2007).

Definition 1

A $\sigma$-algebra is a collection $\mathcal{F}$ of subsets of $\mathrm{\Omega }$ with

1. The empty set $\mathrm{\varnothing },\mathrm{\Omega }$ are in $\mathcal{F}$.

2. A in $\mathcal{F}$ implies $A^C=\mathrm{\Omega }-A$ in $\mathcal{F}$.

3. ${\left\{A_k\right\}}_{k\ge 1}$ is the subset of $\mathcal{F}$ implies $\bigcup _{k=1}^{\infty }{A}_k$ in $\mathcal{F}$.

The members of $\mathcal{F}$ are referred to as $\mathcal{F}$-measurable sets and $\left(\mathrm{\Omega },,,\mathcal{F}\right)$ is referred to as a measurable space.

Definition 2

Let $\mathcal{F}$ be a $\sigma$-algebra of $\mathrm{\Omega }$ and let P is a function from $\mathcal{F}$ to [0, 1]. P is said to be a probability measure if

1. $P(\mathrm{\varnothing })$ is equal to zero and $P(\mathrm{\Omega })$ is equal to one.

2. If $A_1,A_2,...$ are in $\mathcal{F}$, then $P\left(\bigcup _{k=1}^{\infty },A_k\right)$ is less than or equal to ${\sum }_{k=1}^{\infty }P\left(A_k\right)$.

3. If $A_1,A_2,...$ are disjoint sets in $\mathcal{F}$, then $P\left(\bigcup _{k=1}^{\infty },A_k\right)$ is equal to ${\sum }_{k=1}^{\infty }P\left(A_k\right)$.

Definition 3

Let $\mathrm{\Omega }$ be any set. Consider ($\mathrm{\Omega }$, $\mathcal{F}$) be a measurable space and P is a probability measure defined on $\mathcal{F}$; then, the triple $(\mathrm{\Omega },\mathcal{F},P)$ is called as a probability space.

Definition 4

The Borel $\sigma$-algebra, denoted by $\mathcal{B}$, is the $\sigma$-algebra with minimum cardinality containing all open subsets of ${\mathbb{R}}^n$.

Definition 5

Consider a probability space $(\mathrm{\Omega },\mathcal{F},P)$. A function $\mathbf{X}$ from $\mathrm{\Omega }$ to ${\mathbb{R}}^n$ is a n-dimensional random variable if $\forall B\in \mathcal{B},{\mathbf{X}}^{-1}(B)\in \mathcal{F}$, $\mathbf{X}$ is said to be $\mathcal{F}$-measurable. Moreover, E(X)= ${\int }_{\mathrm{\Omega }}\mathbf{X}dP$ and V(X)=${\int }_{\mathrm{\Omega }}{\Vert \mathbf{X}-E(\mathbf{X})\Vert }^{2}dP$.

Stochastic process and Brownian motion

Stochastic process

A stochastic process is a set of random variables specified on a universal probability space ($\mathrm{\Omega }$, $\mathcal{F}$, P), indexed by some set $\mathbf{S}$.

Non-anticipating

If for all $s\ge 0$, X(s) is $\mathcal{F}(s)$-measurable, then a real-valued stochastic process $X(·)$ is described as non-anticipating.

Filtration

In a probability space ($\mathrm{\Omega }$, $\mathcal{F}$, P), a collection ${\left\{{\mathcal{F}}_s\right\}}_{s\ge 0}$ of growing sub $\sigma -$ algebras of $\mathcal{F}$ is called a filtration (Mao, 2007).

1-D Brownian motion

Consider the probability space $(\mathrm{\Omega },\mathcal{F},P)$ with ${\left\{{\mathcal{F}}_s\right\}}_{s\ge 0}$ as a filtration. 1D Brownian motion is a real-valued continuous $\left\{{\mathcal{F}}_s\right\}-$ adapted process ${\left\{B_s\right\}}_{s\ge 0}$ with

1. $B_0$ is zero almost surely.

2. for $0\le s_1<s_2<\infty$, the increment $B_{s_2}-B_{s_1}$ is $\mathcal{N}(0,s_2-s_1)$.

3. for $0\le s_1<s_2<\infty$, the increment $B_{s_2}-B_{s_1}$ is independent of ${\mathcal{F}}_{s_1}$.

Stochastic integrals and SDEs

Stochastic integrals

Due to the Brownian sample path $B.(\omega )$’s nowhere differentiability for almost $\omega$ in $\mathrm{\Omega }$, we are unable to define the integral ordinarily. However, Brownian motion’s stochastic character enables us to derive the integral for a broad class of stochastic processes. It$\widehat{o}$ stochastic integral is the name given to this integral that K. It$\widehat{o}$ initially described in 1949.

Definition 6

Consider a complete probability space $(\mathrm{\Omega },\mathcal{F},P)$ with ${\left\{{\mathcal{F}}_s\right\}}_{s\ge 0}$ as a filtration meeting the usual requirements. Also consider $B={\left\{B_s\right\}}_{s\ge 0}$ be an 1D Brownian motion specified on the probability space adapted to the filtration. Let $h={\{h(s)\}}_{a\le s\le b}$ be a real-valued stochastic process. If $\exists$ a partition $a=s_0<s_1<\cdots <s_k=b$ of [a, b], random variables ${\eta }_i,0\le i\le k-1$ such that ${\eta }_i$ is bounded, ${\mathcal{F}}_{s_i}$ measurable and

$$\begin{array}{c}\hfill h(s)={\eta }_0{I}_{\left[s_0,,,s_1\right]}(s)+\sum _{i=1}^{k-1}{\eta }_i{I}_{\left(s_i,,,s_{i+1}\right]}(s)\end{array}$$

Then h is referred to be a simple (or step) process.

Definition 7

Consider a step process H in ${\mathcal{L}}^2(0,S)$. Then

$$\begin{array}{c}\hfill {\int }_0^{S}HdB:=\sum _{i=0}^{k-1}{H}_i(B(s_{i+1})-B(s_i))\end{array}$$

is called as the It$\widehat{o}$’s integral of H on (0, S).

Stochastic differential equations

Consider a complete probability space $(\mathrm{\Omega },\mathcal{F},P)$ with ${\left\{{\mathcal{F}}_s\right\}}_{s\ge 0}$ as a filtration meeting the usual requirements. Let $B(s)={\left(B_1,(s),,,\cdots ,,,B_m,(s)\right)}^T,s\ge 0$ be an m-D Brownian motion specified on the space. Let $0\le s_0<S<\infty$ and $X_0$ be a $R^d$ -valued, ${\mathcal{F}}_{s_0}$ measurable random variable with $E{\left[X_0\right]}^2<\infty$. Consider $\mu :R^d\times \left[s_0,,,S\right]\to R^d$ and $\sigma :R^d\times \left[s_0,,,S\right]\to R^{d\times m}$ be both Borel measurable functions (Mao, 2007).

The d-D SDE of It$\widehat{o}$ type is given by

$$\begin{array}{c}\hfill dX(s)=\mu (X(s),s)ds+\sigma (X(s),s)dB(s)\text{on}{s}_0\le s\le S\end{array}$$ 1

with $X\left(s_0\right)=X_0$.

That is,

$$\begin{array}{c}\hfill X(s)=X_0+\underset{s_0}{\overset{s}{\int }}\mu (X(t),t)dt+\underset{s_0}{\overset{s}{\int }}\sigma (X(t),t)dB(t)\text{on}{s}_0\le s\le S\end{array}$$ 2

In the above integral, the first integral is ordinary integral, and the second is an Ito integral.

It$\widehat{o}$’s formula

Let X$\left(.\right)$ satisfy the differential $dX=\mu ds+\sigma dB$ where $\mu \in {\mathbb{L}}^1\left(0,,,S\right)$ and $\sigma \in {\mathbb{L}}^2\left(0,,,S\right)$. Assume that $v:\mathbb{R}\times \left[0,,,S\right]\to \mathbb{R}$ is continuous with $\frac{\partial v}{\partial s}$, $\frac{\partial v}{\partial x}$, $\frac{{\partial }^{2}v}{\partial x^2}$ exist and are continuous. Let $Y\left(s\right):=v\left(X,\left(s\right),,,s\right)$. Then Y has the stochastic differential

$$\begin{array}{cc}\hfill dY=& \frac{\partial v}{\partial s}ds+\frac{\partial v}{\partial x}dX+\frac{1}{2}\frac{{\partial }^{2}v}{\partial x^2}{\sigma }^{2}ds\hfill \\ \hfill =& \left(\frac{\partial v}{\partial s},+,\frac{\partial v}{\partial x},\mu ,+,\frac{1}{2},\frac{{\partial }^{2}v}{\partial x^2},{\sigma }^2\right)ds+\frac{\partial v}{\partial x}\sigma dB\hfill \end{array}$$ 3

This is known as the It$\widehat{o}$’s chain rule or It$\widehat{o}$’s formula (Evans, 2012).

Stochastic differential equation of the stock prices

Assume that S(t) represents the price of stock at time t. Then the relative change of price is given by the SDE (Evans, 2012)

$$\begin{array}{c}\hfill \frac{\mathit{dS}}{S}=\mu dt+\sigma dB\end{array}$$ 4

where the drift and the volatility of the stock are denoted by the constants $\mu >0$ and $\sigma$, respectively.

Equation (4) can be written as

$$\begin{array}{cc}& dS=\mu Sdt+\sigma SdB\hfill \\ \hfill & \text{Apply It}{\widehat{o}}^{\prime }s\text{formula}\hfill \\ \hfill & d(log(S))=\frac{\mathit{dS}}{S}-\frac{1}{2}\frac{{\sigma }^2{S}^{2}dt}{S^2}=\left(\mu ,-,\frac{{\sigma }^2}{2}\right)dt+\sigma dB\hfill \\ \hfill & \text{On stochastic integration,}S(t)=s_0{e}^{\sigma B(t)+\left(\mu ,-,\frac{{\sigma }^2}{2}\right)t}\hfill \end{array}$$

In the above equation, the stock price at a future time t is expressed in terms of the stock price at a current moment $s_0$, drift constant $\mu >0$ and the volatility of the stock $\sigma$.

Econometric analysis for the Study Period-I

The original closing prices of the BSE SENSEX index during the Study Period-I are represented in the graph shown in Fig. 1.

Fig. 1.

BSE SENSEX INDEX closing prices during the Study Period-I. Source: The data collected from https://www.bseindia.com and computed by using R Environment software

The outcomes of descriptive statistics in the index of BSE Sensex for the Study Periods I and II are presented in Fig. 2. Figure 2 unequivocally demonstrates that both periods mean returns are positive. The mean return value for the Study Period-I is 0.00246 and for the Study Period-II is 0.001088. It is to be noted that, out of the two sample periods, the Study Period-II earned the maximum standard deviation (0.020852) and the Study Period-I earned the minimum standard deviation (0.009274). It is inferred from the analysis that high risk is found during the Study Period-II. In accordance with the analysis of skewness, both the periods of skewness fall between $-3$ and $+3$. The level of kurtosis is high for both the periods. The sample periods earned a value of 3 or a high-level peak tail, which indicate Leptokurtic. It is confirmed that the two sample periods under the BSE Sensex index follow normal distribution. Hence, the hypothesis, $H_{01}$, there is no normality of BSE Sensex index data before and during COVID-19, is rejected.

Fig. 2.

Results of descriptive statistics in the Index of BSE Sensex for the sample periods: before COVID-19 (Study Period-I) and during COVID-19 (Study Period-II). Source: The data collected from the Bombay Stock Exchange (https://www.bseindia.com) and analyzed by using E-Views 7

Table 1 illustrates the cumulative distribution of the Study Period-I data which shows that 50.82% of data lies between 0 and 0.02 with the maximum counting 124, cumulative counting 239 out of 244 and cumulative is 97.95%.

Table 1.

Cumulative distribution table for the Study Period-I

Value	Frequency	%	Cumulative frequency	Cumulative %
$[-0.04,-0.02)$	4	1.64	4	1.64
$[-0.02,0)$	111	45.49	115	47.13
[0, 0.02)	124	50.82	239	97.95
[0.02, 0.04)	4	1.64	243	99.59
[0.04, 0.06)	1	0.41	244	100
Total	244	100	244	100

Source: The data collected from the Bombay Stock Exchange (https://www.bseindia.com) and calculated by E-Views 7

Table 2 presents the empirical findings of augmented Dickey–Fuller (ADF) test for daily Index returns (closing) of BSE Sensex during the Study Period-I. The test critical values at significant levels at 1 pct, 5 pct and 10 pct are $-3.457173$, $-2.87324$ and $-2.57308$, respectively. It is found that the value of t-statistic of ADF is $-14.01185$. Table 2 presents that the t-statistic value of ADF test for sample index is lesser than that of the test critical values at 1 pct, 5 pct and 10 pct significance levels. Further, the R-squared value under ADF test is 0.449 and it is not only significant but also nearly 0.50. It is noted that the Durbin–Watson test is used to test the stationarity of BSE Sensex index for Study Period-I and the value is 1.94 nearly two. The results of test statistics (Durbin–Watson and R-squared) further confirmed the fact that the index attained the stationarity. This demonstrates that the daily returns data of BSE Sensex index is stationarity in the Study Period-I.

Table 2.

Unit root test for the Study Period-I

Null hypothesis: log normal of Study Period-I has a unit root
Exogenous constant
Length of Lag: 0
(Automatic-based on SIC, maxlag is 14)
			t-Statistic	Prob.*
Augmented Dickey–Fuller
(ADF) test statistic			$-14.01185$	0
Values of test critical	1 pct level		$-3.457173$
	5 pct level		$-2.87324$
	10 pct level		$-2.57308$

Variable	Coefficient	SE	t-Statistic	Prob.
Bef_Covid_Log(-1)	$-0.929987$	0.066371	$-14.01185$	0
C	0.000176	0.000595	0.296205	0.7673
$R^2$	0.448931	Mean dep. var		$-0.000196$
Adjusted $R^2$	0.446644	SD dep. var		0.012459
SER	0.009268	Akaike info criter.		$-6.516309$
RSS	0.020701	Schwarz criter.		$-6.487559$
Log likelihood	793.7315	Hannan–Quinn
		criter.		$-6.504729$
F-statistic	196.3319	Durbin–Watson stat		1.937727
Prob(F-statistic)	0

Source: The data collected from the Bombay Stock Exchange (https://www.bseindia.com) and analyzed by E-Views 7, * - 1% level of significance

GARCH analysis is given in Table 3 for the Study Period-I. Based on the analysis of GARCH model, the $\alpha +\beta$ value of Study Period-I is 0.776905 which is not very closer to one. This demonstrates that the daily returns data of BSE Sensex index not highly volatile during the Study Period-I. Hence, we accept the hypothesis $H_{03}$: there is no volatility (risk) of BSE Sensex index before COVID-19.

Table 3.

GARCH analysis during the Study Period-I

Variable	Coefficient variance equation	SE	z-Statistic	Prob.
C	2.42E−05	1.17E−05	2.065196	0.0389
RESID(-1)${}^2$ ($\alpha )$	0.292638	0.072779	4.020901	0.0001
GARCH(-1) ($\beta$)	0.484267	0.141251	3.428414	0.0006
$R^2$	$-0.000708$	Mean dep. var		0.000246
Adjusted $R^2$	0.003393	SD dep. var		0.009274
SER	0.009258	Akaike info criter.		$-6.552844$
RSS	0.020916	Schwarz criter.		$-6.509846$
Log likelihood	802.4469	Hannan–Quinn
		Criter.		$-6.535527$
Durbin–Watson stat	1.796459

Source: The data collected from the Bombay Stock Exchange (https://www.bseindia.com) and analyzed by E-Views 7

Econometric analysis for the Study Period-II

The original closing price of the BSE stock index during Study Period-II is represented in the graph shown in Fig. 3.

Fig. 3.

BSE SENSEX INDEX closing prices in the Study Period-II. Source: the data collected from https://www.bseindia.com and computed by using R Environment software

The descriptive statistics for lognormal of the closing price is described in Fig. 2.

Table 4 illustrates the cumulative distribution for the Study Period-II which shows that 59.43% of data lies between 0 and 0.05 with a maximum counting of 145 and a cumulative counting of 241 out of 244 and a cumulative is 98.77%.

Table 4.

Cumulative distribution table for the Study Period-II

Value	Frequency	%	Cumulative frequency	Cumulative %
$[-0.15,-0.1)$	1	0.41	1	0.41
$[-0.1,-0.05)$	5	2.05	6	2.46
$[-0.05,0)$	90	36.89	96	39.34
[0, 0.05)	145	59.43	241	98.77
[0.05, 0.1)	3	1.23	244	100
Total	244	100	244	100

Source: The data collected from the Bombay Stock Exchange (https://www.bseindia.com) and calculated by using E-Views 7

Table 5 presents the empirical findings of augmented Dickey–Fuller (ADF) test for daily Index returns (closing) of BSE Sensex during the Study Period-II. The test critical values at significant levels at 1, 5 and 10% are $-3.45729$, $-2.87329$ and $-2.57311$, respectively. It is found that the value of t-statistic of ADF is $-10.9071$. Table 5 shows that the t-statistic value of ADF test for sample index is lesser than that of the test critical values at 1 pct, 5 pct and 10 pct significant levels. Further, the R-squared value under ADF test is 0.559 and it is not only significant but also very nearly 0.50. It should be noted that the Durbin–Watson test, which is used for assessing the stationarity of the BSE Sensex index during the Study Period-II, yields a value of 2.0029 which is very nearly 2. The results of test statistics (Durbin–Watson and R-squared) further confirmed the fact that the index attained the stationarity. This demonstrates that the daily returns data of BSE Sensex index is stationarity in the Study Period-II. Furthermore, the daily returns data of BSE Sensex index are stationarity in both the study periods. Hence, the hypothesis $(H_{02})$, namely there is no stationarity of BSE Sensex index before and during COVID-19, is rejected.

Table 5.

Unit root test for the Study Period-II

Null hypothesis: log normal of Study Period-II has a unit root
Exogenous constant
Length of lag: 1
(Automatic-based on SIC, maxlag is 14)
			t-Statistic	Prob.*
Augmented Dickey–Fuller
(ADF) test statistic			$-10.9071$	0
Values of test critical:	1 pct level		$-3.45729$
	5 pct level		$-2.87329$
	10 pct level		$-2.57311$

Variable	Coefficient	SE	t-Statistic	Prob.
During_Covid_Log(-1)	$-1.05862$	0.097057	$-10.9071$	0
C	0.001131	0.001344	0.841401	0.401
$R^2$	0.559962	Mean dep. var		$-7.11$E−05
Adjusted $R^2$	0.55628	S.D. dep. var		0.031266
SER	0.020827	Akaike info criter.		$-4.89279$
RSS	0.103672	Schwarz criter.		$-4.84954$
Log likelihood	595.0275	Hannan–Quinn
		criter.		$-4.87537$
F-Statistic	152.0675	Durbin–Watson stat		2.002904
Prob (F-statistic)	0

Source: The data collected from the Bombay Stock Exchange (https://www.bseindia.com) and analyzed by E-Views 7, * - 1% level of significance

GARCH analysis is given in Table 6 for the Study Period-II. Based upon the analysis of GARCH model, the $\alpha +\beta$ value of Study Period-II is 0.955691 which is near to one. This demonstrates that the daily returns data of BSE Sensex index is highly volatile during the Study Period-II. Hence, we reject the hypothesis $N{H}_{04}$: There is no volatility (risk) of the BSE Sensex index during COVID-19.

Table 6.

GARCH analysis during the Study Period-II

Variable	Coefficient variance equation	SE	z-Statistic	Prob.
C	7.82E−06	2.84E−06	2.754794	0.0059
RESID(-1)${}^2$ ($\alpha$)	0.098765	0.025641	3.851895	0.0001
GARCH(-1) ($\beta$)	0.856926	0.033627	25.48347	0
$R^2$	$-0.00274$	Mean dep. var		0.001088
Adjusted $R^2$	0.001375	S.D. dep. var		0.020852
SER	0.020838	Akaike info criter.		$-5.56922$
RSS	0.10595	Schwarz criter.		$-5.52623$
Log likelihood	682.4452	Hannan–Quinn
		Criter.		$-$5.55191
Durbin–Watson stat	2.226772

Source: The data collected from the Bombay Stock Exchange (https://www.bseindia.com) and analyzed by E-Views 7

Stochastic analysis

We have used the R Environment of statistical software to estimate the drift $\mu$ and volatility parameters $\sigma$ of the stock price stochastic differential equation by Kessler method. There are other methods available for parameter estimation such as Euler method, Ozaki method and Shoji–Ozaki method. An Ito–Taylor expansion of higher order was suggested by Kessler in 1997 as a method for determining the drift and volatility coefficients in a conditional Gaussian density. Consider the following SDE

$$\begin{array}{c}\hfill dX(s)=f(X(s),\theta )ds+g(X(s),\theta )dB(s)\text{on}{s}_0\le s\le S\end{array}$$ 5

with $X\left(s_0\right)=X_0$.

The transition density which is given by Kessler method follows a normal distribution. That is, $X_{s+\mathrm{\Delta }s}\mid X_s$ is equal to x which is a normal distribution $\mathcal{N}(E_x,V_x)$ where

$$\begin{array}{cc}\hfill E_x=& x+f(s,x)\mathrm{\Delta }s+(f(s,x){\partial }_{x}f(s,x)+\frac{1}{2}{g}^2(s,x){\partial }_{\mathit{xx}}g(s,x))\frac{{(\mathrm{\Delta }s)}^2}{2}\hfill \\ \hfill V_x=& x^2+(2f(s,x)x+g^2(s,x)\mathrm{\Delta }s)+(2f(s,x)({\partial }_{x}f(s,x)x+f(s,x)+g(s,x){\partial }_{x}g(s,x))\hfill \\ \hfill & +g^2(s,x)({\partial }_{\mathit{xx}}f(s,x)x+2{\partial }_{x}f(s,x)+{\partial }_x{g}^2(s,x)+g(s,x){\partial }_{\mathit{xx}}g(s,x)))\frac{{(\mathrm{\Delta }s)}^2}{2}-E_x^2\hfill \end{array}$$

From the above equations, we can estimate the drift and volatility parameters for our study periods with the help of R Environment software using “Sim. DiffProc” package (Guidoum & Boukhetala, 2016).

Table 7 gives the values of the estimated drift and volatility parameters $\mu$ and $\sigma$ of the stock price stochastic differential equation by Kessler method for both the Study Period-I and Study Period-II.

Table 7.

Estimation of drift and volatility parameters

Parameters	Study period-I	Study period-II
$\mu$	0.0002875	0.0013012
$\sigma$	0.0093718	0.0205254

Source: The data collected from https://www.bseindia.com and computed by using R Environment software

$95\%$ confidence interval for the estimation of parameters is given in Table 8. We have formulated a stochastic differential equation model for both the study periods with estimated parameters given in Table 7 and simulated 500 times solution of stock price stochastic differential equation.

Table 8.

95% Confidence interval for estimation of parameters

Parameters	Study period-I		Study period-II
	2.50%	97.50%	2.50%	97.50%
$\mu$	$-0.000888343$	0.001463527	$-0.001274397$	0.003876878
$\sigma$	0.008571477	0.010172281	0.018718109	0.022332843

Source: The data collected from https://www.bseindia.com and computed by using R environment software

Figure 4 illustrates the 500 times stochastic solution simulations of the stock price stochastic differential equation for the closing price during the Study Period-I. Also, Fig. 4 presents the 95% confidence level bound and mean path of the simulations.

Fig. 4.

Stochastic simulations of BSE SENSEX INDEX closing prices during the Study Period-I Source: The data collected from https://www.bseindia.com and computed by using R Environment software

Figure 5 demonstrates the 500 times stochastic solution simulations of the stock price stochastic differential equation for the closing price during the Study Period-II. Also, Fig. 5 presents the 95% confidence level bound and mean path of the simulations.

Fig. 5.

Stochastic simulations of BSE SENSEX INDEX closing prices during the Study Period-II Source: The data collected from the Bombay Stock Exchange (https://www.bseindia.com) and computed by using R Environment software

Results and discussion

Descriptive statistics: We have found that the skewness values range from $-3$ and $+3$. The level of kurtosis is positive for S &P BSE Sensex during both the study periods. This implies that data of both the study periods follow a normal distribution. Therefore, the hypothesis, $H_{01}$, There is no normality of BSE Sensex index data before and during COVID-19, is rejected.

Unit root: It is to be found that at 1, 5 and 10% significant levels, the statistical value (ADF) is lower than the test critical values during both study periods. This indicates that the data have stationarity. Hence, the hypothesis $(H_{02})$, namely there is no stationarity of BSE Sensex index before and during COVID-19, is rejected.

GARCH: Based on the analysis of the GARCH (1, 1) model, the value $\alpha +\beta$ for the sample index is very close to one during the Study Period-II. This demonstrates the volatility in the market index returns is higher during the Study Period-II. Hence, we accept the hypothesis $H_{03}$: There is no volatility (risk) of BSE Sensex index for before COVID-19 and we reject the hypothesis $H_{04}$: There is no volatility (risk) of BSE Sensex index during COVID-19.

The results of econometric analysis show that the Study Period-II is a critical period for the investors to invest due to more fluctuation occur in BSE Sensex index in Study Period-II than Study Period-I.

SDE: The BSE Sensex index closing price during both the study periods are fitted by the stock price stochastic differential equation model. We observed that the BSE Sensex Index closing price during the Study Period-I lies between the bounds of 95% confidence level, whereas the Study Period-II is out of 95% confidence level bound. This shows that there are more fluctuations occured during the Study Period-II than Study Period-I. Hence, we accept the hypothesis $H_{03}$: There is no volatility (risk) of BSE Sensex index before COVID-19 and we reject the hypothesis $H_{04}$: there is no volatility (risk) of BSE Sensex index during COVID-19. Consequently, the econometric and the stochastic analysis confirmed the same results.

Furthermore, Fig. 6 presents the outcomes of graphical expression, for BSE Sensex index data before the Study Period-I and Study Period-II. It should be observed that the line of BSE Sensex index log return price during the Study Period-II is highly fluctuated (volatile) than the Study Period-I. Therefore, there is no interrelationship between the index returns for both the periods. Further, it is found that the stock market SENSEX index of BSE RED line is heavily affected during the Study Period-II. It is evident that the sample index return price, during the COVID-19 is highly volatile (risk). Therefore, the hypothesis $H_{04}$, there is no volatility (risk) of BSE Sensex index during COVID-19, is rejected.

Fig. 6.

Log normal of BSE Sensex index closing prices for the Study Period-I (before COVID-19) and Study Period-II (during COVID-19). Source: The data collected from the Bombay Stock Exchange (https://www.bseindia.com) and figured by E-Views 7

Conclusion

Our results show that during Study Period-II there are more fluctuations occured than the Study Period-I. Hence, COVID-19 period would be suitable for investors knowing the risk factors but the speculators can make use of this period for better returns. The findings of the present study, relating to econometric and stochastic analysis of stock price before and during COVID-19, supported the results of Cheng et al. Cheng et al. (2022) and Zhang et al. Zhang et al. (2022). Also, it shows that a COVID-19-like shock would result in a sharp and significant decrease in BSE Index Returns, and might constitute an existential danger to all the sectors due to the probability of extremely severe stock market recessions. The outcomes of this research will help policymakers to draft and make innovation policies.

Limitations of the study

This study is focuses only on the BSE Sensex index in India. It is mainly based on the secondary data, and hence, it may have certain drawbacks that are inextricably linked to the secondary data. For the purpose of the study, the sample study periods are restricted between March 2019 and February 2021, i.e., before and during COVID pandemic. Finally, all the limitations are associated with mathematical and econometric tools like mean, standard deviation, skewness, kurtosis, GARCH and stochastic simulation applied to this research.

Scope for future research

The aim of this research is to assess the impact of the Indian stock market before and during the COVID-19. Similar objectives could be initiated concerning other Asian countries’ stock market, currency market (FOREX), Sectoral Indices, etc.

Data availability Statement

The datasets generated and analyzed during the current study are openly available in the official website of BSEINDIA repository, https://www.bseindia.com/Indices/IndexArchiveData.html. Furthermore, the following links provide the data for Study Period-I and Study Period-II separately. For the Study Period-I: https://api.bseindia.com/BseIndiaAPI/api/ProduceCSVForDate/w?strIndex=SENSEX &dtFromDate=01/03/2019 &dtToDate=28/02/2020 For the Study Period-II:https://api.bseindia.com/BseIndiaAPI/api/ProduceCSVForDate/w?strIndex=SENSEX &dtFromDate=02/03/2020 &dtToDate=26/02/2021.