Renyi entropy and mutual information measurement of market expectations and investor fear during the COVID-19 pandemic

 The COVID-19 pandemic has truly influenced world economies. In such manner, it is normal that data level and dividing among value, computerized money, and energy markets has been modified because of the pandemic flare-up. In particular, the subsequent curved danger among business sectors is ventured to ascend during the unusual condition of world economy. The reason for the current investigation is twofold. To start with, by utilizing Renyi entropy, we dissect the multiscale entropy work in the return time series of Bitcoin, S&P500, WTI, Brent, Gas, Gold, Silver, and financial backer dread list addressed by VIX. Second, by assessing common data, we break down the data dividing among these business sectors. The examinations are led previously and during the COVID-19 pandemic. The exact outcomes from Renyi entropy demonstrate that for all market files, irregularity and confusion are more moved in less plausible occasions. The exact outcomes from common data showed that the data dividing network among business sectors has changed during the COVID-19 pandemic. According to an administrative viewpoint, we infer that during the pandemic (I) portfolios made out of Bitcoin and Silver, Bitcoin and WTI, Bitcoin and Gold, Bitcoin and Brent, or Bitcoin and S&P500 could be unsafe, (ii) expansion openings exist by putting resources into portfolios made out of Gas and Silver, Gold and Silver, Gold and Gas, Brent and Silver, Brent and Gold, or Bitcoin and Gas, and that (iii) the VIX showed the most minimal degree of data problem at all scales previously and during the pandemic. Accordingly, it appears to be that the pandemic has not impacted the assumptions for financial backers. Our outcomes give a knowledge of the reaction of stocks, cryptographic forms of money, energy, valuable metal business sectors, to assumptions for financial backers in the repercussions of the COVID-19 pandemic as far as data requesting and sharing. 

Watchwords: COVID-19 Pandemic, Bitcoin, S&P500, WTI, Brent, Gas, Gold, Silver, VIX, Renyi entropy, Mutual data 

1. Introduction 

There is no question that the COVID-19 pandemic has altogether and contrarily influenced economies around the world. In such manner, while the pandemic episode isn't finished at this point, a couple of studies has been distributed to research its impact on worldwide business sectors. For example, it was tracked down that financial exchanges quickly respond to COVID-19 pandemic and this response varies over the long haul contingent upon the phase of flare-up [1], COVID-19 episode biggerly affects the US international danger and monetary vulnerability than on the US securities exchange [2], day by day development altogether affirmed cases and in complete instances of death brought about by COVID-19 essentially and adversely influence stock returns of Chinese organizations [3], unpredictability generously increments in worldwide business sectors because of the pandemic disturbance [4], presence of fractal virus impact on the securities exchanges [5], proof of enormous impact on the cross-relationship of multifractal property between unrefined petroleum and agrarian future business sectors [6], and that digital forms of money show bigger flimsiness and abnormality contrasted with securities exchange during the pandemic [7]. 

This paper expects to examine the multiscale data request, in one hand, and the data sharing, then again, in stocks, digital currency, energy, and valuable metal business sectors previously and during the COVID-19 pandemic. For this reason, we inspect the conduct of Renyi entropy [8] and common data [9] in Bitcoin, S&P500, WTI, Brent, Gas, Gold, Silver, and VIX. In reality, examining data problem at various scales permits uncovering the degree of data misfortune at uncommon and regular occasions happening in the sign. In addition, investigating common data helps discovering nonlinear conditions between two unique factors as far as data sharing. For sure, common data is a symmetric measurement used to check the shared reliance between two signs according to the viewpoint of data hypothesis. 

In this work, we depend on the idea of entropy as it permits a nonlinear investigation of the sign under examination dependent on data request and partaking to all the more likely comprehend the impact of the COVID-19 on world significant business sectors: value (Bitcoin and S&P500), energy (WTI, Brent, and Gas), and valuable metals (Gold and Silver). Likewise, we analyze the Chicago Board Options Exchange (CBOE) Volatility Index (VIX) used to check dread and assumptions for financial backers. Thusly, contrasted with the not many works on the impact of the COVID-19 on world economies [1], [2], [3], [4], [5], [6], [7], the commitments of our examination follow. To start with, we uncover the impact of the pandemic on the continuous occasions in a set made out of world significant business sectors. Second, we shed light on the impact of the pandemic on data dividing among world significant business sectors. Third, we investigate the impact of the pandemic the market on financial backer's assumptions, a significant issue missing in existing examinations. This would help seeing how the pandemic formed dread and assumptions in financial backers around the world. Fourth, the examination investigations are applied with and between market across time (before as opposed to during pandemic). Fifth, we depend on the idea of multiscale entropy and common data since they have not been considered in researching the reaction of economies to the pandemic, as far as we could possibly know. Being the significant subject in data hypothesis, entropy has been broadly applied in a lot of fields in econophysics [10], [11], [12], [13], [14], [15], [16], [17], [18], guaging stock value anticipating [19], securities exchange procuring [20], portfolio streamlining [21], block chain assessment [22], and market grouping [23]. 

The remainder of the examination follows. Segment 2 presents Renyi entropy and common data. Segment 3 presents information and observational outcomes. At long last, Section 4 finishes up. 

2. Methods 

In ata hypothesis, the Rényi entropy [8] is a speculation of Shannon entropy. Undoubtedly, as a speculation of Shannon entropy, the Rényi entropy displays an engaging property meant to assess the sign intricacy by fusing an extra boundary used to catch uncommon and successive occasions independently. For example, let X be a limited worth of the irregular variable, and P be a limited discrete likelihood circulation which is assume P = {p 1, p 2,… , p n} and ∑ipi=1. Then, at that point, the Rényi entropy (RE) of request q, where q≥ 0 and q ≠ 0, is given by: 



where q is the request for the entropy. At the point when q < 1, uncommon occasions are special. In opposite, when q > 1, continuous occasions are favored. At long last, when q → 1, Rq unites to the Shannon entropy. In such manner, Rényi entropy is valuable to evaluate the variety, vulnerability or haphazardness of a given framework [24]. In this examination, the Gaussian piece work is taken on to assess the probabilities in Eq. (1) [25]. 

The common data (MI) between two arbitrary factors X and Y is characterized in bits as follows: 



where p(x) = Prob{X = x} and p(y) = Prob{Y = y} are separately the peripheral likelihood thickness work (PDF) of X and Y. The p(x,y) addresses the joint PDF of X and Y. The higher is the common data esteem, the more grounded is the reliance between the factors X and Y. On the off chance that the shared data esteem is zero, the two factors are free. In this work, the multivariate Gaussian capacity thickness assessor is utilized to estimated the joint probabilities in Eq. (2) [9]. 

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3. Data and results 

We assembled day by day shutting costs of Bitcoin, S&P500, West Texas Instrument (WTI), Brent, Gas, Gold, Silver, and Chicago Board Options Exchange (CBOE) Volatility Index (VIX) for the period from 1 August 2019 to 26 May 2020. Since the COVID-19 episode was proclaimed by the world wellbeing association as pandemic on 30th January 2020, the pre-pandemic period traverses 1 August 2019 to 31 December 2019 and the pandemic period from 2 January 2020 to 26 May 2020. In the wake of eliminating missing information, the quantity of perceptions in the pre-pandemic time frame is 102 and the quantity of perceptions during the pandemic time frame is 96. The information was gotten from Yahoo finance. All measurable investigations are applied to return series, where the return is processed as the principal logarithmic contrasts of costs. 

The plot of Renyi entropy (RE) as an element of scale q for Bitcoin, S&P500, WTI, Brent, Gas, Gold, Silver, and VIX previously and during COVID-19 pandemic are displayed in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8 individually. It is noticed for all business sectors, with the exception of Gas, that at all scales, the degree of RE before the COVID-19 pandemic is higher than that during the pandemic. These discoveries propose that haphazardness in high likelihood occasions has diminished during the COVID-19 pandemic in Bitcoin, S&P500, WTI, Brent, Gold, Silver, and VIX. In opposite, it has expanded during COVID-19 pandemic in Gas market. Furthermore, the experimental outcomes show that there is a solid expansion in RE at scales 2 and 3 followed by a reduction for all excess scales for all business sectors, with the exception of Gas market. Subsequently, for Bitcoin, S&P500, WTI, Brent, Gold, Silver, and VIX, haphazardness and turmoil are more amassed in less likelihood occasions. For Gas market, the RE before COVID-19 pandemic is higher than that during the pandemic at low scopes (q = 2,3,4,5). Actually, RE before COVID-19 pandemic is lower than that during the pandemic at high scales (q = 6 to 20). Accordingly, for Gas market, arbitrariness and turmoil are more gathered in less likelihood occasions, yet at longer range (q = 2,3,4,5) contrasted with the other market (q = 2,3). At last, it merits seeing that for the VIX, the degree of data problem in successive occasions has diminished during the pandemic. This outcome recommend that the financial backers showed lower level of dread and in the long run more elevated level of future assumptions about most successive occasions.

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