A study of the international stock market crash and recovery during COVID-19 pandemic using a modified chaos game representation

This paper deals with a novel approach to visualize and compare financial markets across the globe using chaos game representation of iterated function systems. We modified a widely used fractal method to study genome sequences and applied it to study the effect of COVID-19 on global financial markets. We investigate the financial market reaction and volatility to the current pandemic by comparing its behavior before and after the onset of COVID-19. Our method clearly demonstrates the imminent bearish and a surprise bullish pattern of the financial markets across the world.


Introduction
Financial markets all over the world are currently witnessing sharp volatility due to the ongoing COVID-19 pandemic. This is something the world has not seen earlier. Recessions in the last 75 years can usually be categorized as one of the three factorsfinancial bubbles, oil shocks, or policy mistakes. According to Deloitte U.S.A., the rapid economic deterioration of economies and stock markets amid the COVID-19 threat represents a new category: a global societal shock (Renjen, 2020). We saw a sharp decline in the stock market from 19th February 2020 through 23rd March 2020.. This crash was, however, recovered significantly quicker than expected, creating a 'V' shaped recovery.
The economic impact of pandemics has been studied earlier, such as how HIV/AIDS impacted the economy discussed by Haacker (2004), while Santaeulalia-Llopis (2008) focused on the impact of the HIV/AIDS pandemic on development. Yach, Stuckler, and Brownell (2006) discussed the costs of the global growth of obesity and diabetes. All of them conclude that these pandemics bring economic disruption, loss of employment, and loss of foreign direct investment which resulted in a global recession in the case of the COVID-19 pandemic (Goodell, 2020). A black swan event that is not related to the pandemic but resulted in similar behavior in the financial market was the financial crisis of 2008-09, as discussed by Grout and Zalewska (2016). Berkmen et al. (2012) studied how an economic crisis affects different countries. The study concluded that the countries having more leveraged domestic financial systems, stronger credit growth, and have short-term debt suffer the most economy-wise. The recovery of the market has also been tried using mathematical tests (Yanglin et.al.2020).
The current pandemic crisis has forced financial researchers to study its effects in a short time (Nicola et. al. 2020;Zhang, Hu and Ji, 2020;Zaremba et.al. 2020, Ali et.al. 2020. In this paper, we aim to investigate the impact of the novel coronavirus disease (COVID-19) on the financial markets worldwide using the chaos game method, which uses the concept of fractals. Fractals have widely been used for the study of financial markets (Lux, 1998; Kristoufek, 2013; Bianchi and Frezza, 2017; Alves, 2019). We use the widely used concepts of chaos game representation (CGR) of D.N.A. sequences (Jeffrey, 1990;Almeida et.al., 2001;Randhawa et.al., 2020, Pratibha et. al. 2020) and modify the CGR method to accommodate the financial data.
In the next section, the data and the methodology used in this study are described in detail, followed by presentation and discussions of the results obtained from this study in section 3.
The major outcome of the study is highlighted in the conclusions section 4.

Data and Methods
Here we describe a novel and simple approach to quantify the similarity/dissimilarity between two data sequences using a modified CGR. CGR is an iterative mapping technique to convert a time series of a given length into a single image based on the movement of a point controlled by the amplitudes in the time series. As the time series of different financial market indices differ in length, a direct comparison of the stock prices based solely on the CGR image is difficult. To make the CGR image length independent, we first convert it into a Percentage CGR plot (PC-plot) by plotting, at each pixel level, the percentage of the korder frequencies in a data sequence, which, in our case, is the 1-minute percentage variations in the index funds around the globe. Visually, the PC-plots and traditional CGR images are identical. Since we aim to study the effect of COVID-19 on the financial markets, it is important to compare the stock prices before the COVID-19 with those after the pandemic.
To quantify the similarity/dissimilarity between the two time periods, we introduce two new concepts, a subtraction percentage CGR plot (SP-plot) and the k-order proximity index (Pr).
An SP-plot is obtained by subtracting the percentage points of respective k-orders in each sequence. The SP-plot consists of positive and negative values indicating the differences of k-orders percentage distribution between two series. The sum of the positive differences will always be equal to the sum of negative differences. This sum is named as k-order proximity index (Pr), which represents the degree of similarity between the financial market variations during two time periods. Obviously, the value of this proximity index will increase with the degree of dissimilarity between the two species. The value of this index also changes with the value of 'k' because the distribution of a specific k-length combination of variations in the market will change as 'k' changes.

CGR -
We consider a unit square U and name corners Ci (i=1,2,3,4) as A, B, C, and D, respectively, which corresponds to the value of X(k) (Figure 1a). The initial point P(0) is the midpoint of the square. Now the second point P(1) is the midpoint between P(0) and CX(1) and so on. In General, P(k) is plotted as the midpoint between P(k-1) and CX(k) (Jeffrey, 1990). This is called a chaos game. If the points in the series are truly random, then this game will ultimately fill the square else will produce a fractal( figure 2).
An example for movement of points in CGR is shown with the first eight members of the data sequence DACCBADC in Figure 1a. After plotting the financial data sequence X in unit square U, the unit square is divided into 2N x 2N sub squares; each sub-square represents a unique sub-sequence of length k (k-order) which are called the address of the sub-squares (Figure 1b). An example of addresses of the sub-squares for different order followed in the sequence is given in Figure 1b. The corresponding address in which a particular point falls is noted. Many points may end up in the same address sub-square, which determines the density of the points at a given address.  2.3 SP-plots and k-order proximity Index (Pr) -Subtraction plot between series 1 (s1) and series 2 (s2) is plotted as

PC-plots
From the subtraction plot S, the sum of all the positive numbers (also the sum of modulus of negative numbers) is a measure of similarity or dissimilarity (Pr) between two genetic sequences. Or, , ℎ < 0 (4)

Data Set
We Using the modified chaos game representation, we compare the stock markets using their PCplots, SP-plots, and proximity indices (Pr). We also studied the most frequent addresses in each of the data sets, indicating the repeated patterns and the forbidden patterns in the time series.

Results and Discussions
The modified chaos game representation, along with proximity index, was programmed in MATLAB and applied to the data series described in section 2.4. PC-plots, SP-plots, and the proximity indices Pr are obtained for the five bi-monthly periods J, K, L, M, and N as defined above. Some of the representative plots are shown here. Figure 3 shows the 1minute variations in the HangSeng index (upper panel) along with its PC-plots (lower panel) over five bi-monthly periods. The first three PC-plots are for the period before the COVID-19, and the last two plots are for the aftermath period. The vertices of these squares in these plots are defined in Figure 1     We consider the period L, just before the pandemic as the reference period. We computed the proximity indices of L-J, L-K, L-M. and L-N for each of the six markets ( Figure 6). The Pr is a very good indicator of the level of market variations. Figure 6 clearly demonstrates the high volatility of various international markets after the COVID-19 pandemic declaration. It also quantifies the magnitude of the relative variation of any market with respect to a calm period. In Figure 6, we observe that the French market had the highest volatility, both during the bearish period M and the bullish period N. In fact it was much more volatile during the recovery phase (L4 -N4 in Figure 6)  (Table 1). These addresses indicate the trend of the market during a particular period for which the PC-plot is drawn.  A combination 'CBBC' indicates a low rise, low fall, lowfall, and a low rise in successive order; whereas a 'DDAD' means a high rise, high rise, high fall, and high rise in the market.
As we see in Table 1, the period M in all the markets see a more occurrence of A's (bears) as compared to D's, whereas during the period N the situation reverses. We observe more D's (bulls) than A's.
Thus from the modified chaos game representations of the international markets, we clearly see the effect of the global societal shock during the pandemic (Renjen, 2020). We also observe an equally strong recovery (sometimes even stronger) in the market which can be quantified in terms of the proximity index.