Dynamics, behaviours, and anomaly persistence in cryptocurrencies and equities surrounding COVID-19
DDynamics, behaviours, and anomaly persistence incryptocurrencies and equities surrounding COVID-19
Nick James
School of Mathematics and Statistics, University of Sydney, NSW, Australia
Abstract
This paper uses new and recently introduced methodologies to study the similarityin the dynamics and behaviours of cryptocurrencies and equities surroundingthe COVID-19 pandemic. We study two collections; 45 cryptocurrencies and72 equities, both independently and in conjunction. First, we examine theevolution of cryptocurrency and equity market dynamics, with a particular focuson their change during the COVID-19 pandemic. We demonstrate markedlymore similar dynamics during times of crisis. Next, we apply recently introducedmethods to contrast trajectories, erratic behaviours, and extreme values amongthe two multivariate time series. Finally, we introduce a new framework fordetermining the persistence of market anomalies over time. Surprisingly, wefind that although cryptocurrencies exhibit stronger collective dynamics andcorrelation in all market conditions, equities behave more similarly in theirtrajectories and extremes, and show greater persistence in anomalies over time.
Keywords:
Market dynamics, Cryptocurrency, Time series analysis, Nonlineardynamics, COVID-19
1. Introduction
Over the last several years there has been growing interest in the cryptocur-rency market. The sector has experienced impressive growth in asset inflows andits level of sophistication. More recently, the COVID-19 pandemic has causedimmense social and economic impacts, including changes in the behaviour offinancial markets. The goal of this paper is to analyze the evolution of cryptocur-rencies and equities over time, and in particular, assess whether the increase ininterest from sophisticated investors has led to more uniformity in the dynamicsand behaviours of the two asset classes. We use the COVID-19 pandemic as amotivating example to ascertain whether this similarity changes during marketcrises.The study of financial market correlation structure has been a topic of greatinterest to the nonlinear dynamics community over the past several decades
Email address: [email protected] (Nick James)
Preprint submitted to Elsevier February 19, 2021 a r X i v : . [ q -f i n . S T ] F e b
1, 2, 3]. Evolutionary market dynamics have been studied through a wide varietyof techniques such as clustering [4] and principal components analysis (PCA)[2, 5, 6, 7]. Until the past decade, the primary asset classes of interest to theresearch community were equities [7], fixed income [8], and foreign exchange [9].More recently, select research has focused on the study of trajectory modelling[10], extreme behaviours and structural breaks [11, 12] and these methods havebeen applied to a variety of asset classes.There is a current wave of interest from econophysics researchers in thedevelopment and application of methods for understanding cryptocurrencydynamics. Areas attracting interest from researchers include studies of Bitcoin’sbehaviour [13, 14, 15, 16, 17], fractal patterns, [18, 19, 20, 21], cross-correlationand scaling effects [22, 23, 24, 25, 26]. Many of these studies are concerned withthe time-varying nature of such dynamics, or the behaviour of cryptocurrenciesduring various market regimes. Quite naturally, the impact of COVID-19 oncryptocurrency behaviours has been widely studied [27, 28, 29, 30]The evolution of COVID-19 and its impact on financial markets has attractedbroad interest from various research communities. COVID-19’s spread andcontainment measures have been studied by the epidemiology community [31,32, 33, 34, 35, 36, 37, 38, 39], while clinically-inclined research has detailednew treatments for various COVID-19 strains [40, 41, 42, 43, 44, 45, 46]. Thepandemic’s varied impact on financial markets has also been studied [47, 48, 49],with many papers exploring financial contagion and market stability [50, 51, 52].In the nonlinear dynamics community, COVID-19 research has used new andexisting techniques to study the temporal evolution of cases and deaths [53, 54, 55,56], with a substantial emphasis on SIR models [57, 58, 59, 60, 61, 62, 63], powerlaw models [64, 65, 66, 67] and the use of networks [68]. More recently there hasbeen work that explores the impact of COVID-19 cases on the performance ofcountry financial markets [10].The goal of this paper is to explore the similarity in the dynamics andbehaviours of cryptocurrencies and equities over the past two years. In doing so,we make several contributions. First, we complement current methods with theintroduction of a new measure between eigenspectra to study the similarity in twotime series’ evolutionary dynamics. Next, we apply recently developed techniquesto study the trajectories, extremes and erratic behaviour of cryptocurrenciesand equities, and analyze their similarity. Finally, we introduce a pithy methodto study the persistence of financial anomalies over time.This paper is structured as follows. Section 2 describes the data used inthis paper. Section 3 studies the time-varying dynamics of cryptocurrencies andequities, and contrasts their behaviour during different market states. Sections4, 5 and 6 study trajectories, erratic behaviour and extremes respectively. InSection 7 we contrast the consistency in anomalies among our two collections.In Section 8, we conclude. 2 . Data
In the proceeding analysis, the two primary objects of study are cryptocur-rency and equity multivariate time series between 03-12-2018 to 08-12-2020. Weanalyze the 45 largest cryptocurrencies by market capitalisation (excluding thosepreviously identified as anomalous) [11] and 72 global equities whose marketcapitalisation is greater than US$100 billion. We report and contrast on thedynamics, behaviours, and anomaly persistence between cryptocurrencies andequities. In select sub-sections, we refer to the period 03-12-2018 to 28-02-2020 as Pre-COVID, 02-03-2020 to 29-05-2020 as Peak COVID, and 01-06-2020to 08-12-2020 as Post-COVID. Cryptocurrency and equity data are sourcedfrom https://coinmarketcap.com/ and Bloomberg respectively. A full list ofcryptocurrencies and equities studied in this paper is available in Appendix B.
3. Market dynamics
In this section we follow the framework introduced in [1] to study the temporalevolution of correlation structure in cryptocurrencies and equities, and contrastthese collections’ evolutionary dynamics. Our analysis in this section differs from[1] in several ways, however. First, rather than applying this framework to a singlecollection of securities from different asset classes, we apply the time-evolvingmodel to two separate asset classes (cryptocurrencies and equities) and comparethe respective time-varying dynamics. Second, we use a shorter time windowto study correlation structures, allowing correlations to change more quickly tovarying market conditions. This allows us to study the impact of COVID-19on both collections. Third, we introduce a new dynamics deviation measurebetween surfaces to determine the similarity in two time-varying eigenspectraacross different time periods. Finally, we use daily data rather than weekly data.Let c i ( t ) and e j ( t ) be the multivariate time series of cryptocurrency andequity daily closing prices, for t = 1 , ..., T , i = 1 , ..., N , and j = 1 , ..., K . Wegenerate two multivariate time series of log returns, R ci ( t ) and R ej ( t ) , wherecryptocurrency and equity log returns are computed as follows R ci ( t ) = log (cid:18) c i ( t ) c i ( t − (cid:19) , (1) R ej ( t ) = log (cid:18) e j ( t ) e j ( t − (cid:19) . (2)We define standardized cryptocurrency returns as ˆ R ci ( t ) = [ R ci ( t ) −(cid:104) R ci (cid:105) ] /σ ( R ci ) ,where σ ( R ci ) = (cid:112) (cid:104) ( R ci ) (cid:105) − (cid:104) R ci (cid:105) represents the standard deviation of cryptocur-rency time series R ci and (cid:104) . (cid:105) denotes an average over time. Standardized equityreturns are computed similarly and we denote this time series ˆ R ej ( t ) . Havingnormalized the returns, we may construct empirical correlation matrices3 c = 1 T ˆ R c ˆ R cT , (3) Ω e = 1 T ˆ R e ˆ R eT , (4)for cryptocurrency and equity time series. Elements of both correlationmatrices ω c ( i, j ) and ω e ( i, j ) lie ∈ [ − , . We study the evolution of thesecorrelation matrices, using a rolling window of T = 60 days. One must bejudicious in the choice of the smoothing parameter T , as correlation coefficientscan be excessively smooth or noisy if T is too large or too small respectively.Our choice of 60 days corresponds approximately to the length of the COVID-19market crash. This allows us to capture the entirety of the COVID-19 marketshock, without including unrelated data outside the COVID-related crash inour calculation. Next, we study the dynamics of the cryptocurrency and equitymarket security collections by applying principal components analysis (PCA)to the two time-varying correlation matrices. For each correlation matrix, wewish to estimate the linear maps Φ c and Φ e that transform our standardizedcryptocurrency returns ˆ R c and equity returns ˆ R e into uncorrelated variables Z c and Z e respectively. That is, Z c = Φ c ˆ R c , (5) Z e = Φ e ˆ R e . (6)where the rows of Z c and Z e represent PCs of the matrices R c and R e . Therows of Φ c and Φ e , which contain PC coefficients, are ordered such that thefirst rows are along the axes of most variation in the data, with subsequent PCs,subject to the optimization constraint that they are all mutually orthogonal, eachaccounting for maximal variance along their respective axes. The correlationmatrices, which are symmetric and diagonalizable matrices can be written inthe form Ω c = 1 T Λ c D c Λ Tc , (7) Ω e = 1 T Λ e D e Λ Te , (8)where D c , D e are diagonal matrices with eigenvalues λ ci , λ ej , and Λ c , Λ e areorthogonal matrices with the associated eigenvectors from the cryptocurrencyand equity correlation matrices respectively. The PCs are estimated using thediagonalizations above.Finally, we contrast the proportion of variance produced by a sub-collectionof eigenvalues within each of the two collections. The total variance of thecryptocurrency returns ˆ R c and equity returns ˆ R e for the N and K assetsrespectively, is equal to the sum of all eigenvalues λ c + ... + λ cN and λ e + .... + λ eK .This is equivalently the trace of the two diagonal matrices of eigenvalues tr ( D c ) = N and tr ( D e ) = K . To compute the proportion of total variance explained4 a) Cryptocurrency (b) Equity Figure 1: Time-varying eigenspectrum for cryptocurrencies and equities. by the m th PC in ˆ R c and ˆ R e is therefore ˜ λ cm = λ cm /N and ˜ λ em = λ em /K . Formore details on this construction readers should visit [1, 69, 70]. In some cases,the correlation matrix under examination may not be full rank. This is not aconcern for the proceeding analysis, where we focus on the behavior of the first10 eigenvectors. Since PCs are mutually orthogonal, and by definition linearlyindependent, this would not cause any issues in the conclusions resulting fromour methodology.Prior work has demonstrated that the eigenvector corresponding to thelargest eigenvalue represents the significance of ‘the market’ within the collection[1]. Bearing this in mind, there are several noteworthy insights revealed inFigure 1 regarding the similarity in cryptocurrency and equity dynamics. First,both Figure 1a and Figure 1b exhibit a broadly similar shape; the majorityof explanatory variance is provided by the first several eigenvectors, with theremaining proportion of total variance falling away quickly over the entire timeperiod. Next, the explanatory variance provided by the first cryptocurrencyeigenvalue ˜ λ c seen in figure 1a is consistently higher than the correspondingequity market eigenvalue ˜ λ e in figure 1b. This demonstrates that the collectiveforce of the market is more pronounced in cryptocurrencies than equities duringour window of analysis. The second observation one may make from Figure 1is the significant variability in ˜ λ e when compared with ˜ λ c . This may highlightthat within equity markets, there is more variability related to the market’simpact on collective behaviour than that of cryptocurrencies. The time-varyingexplanatory variance of the first PC is displayed in Figure 2, where ˜ λ c > ˜ λ e foralmost the entire window of analysis. Figure 2 indicates that the difference inthe first eigenvalue, | ˜ λ c − ˜ λ e | , is smallest during the Peak COVID period. In this section we study the similarity in the dynamics of cryptocurrenciesand equities during three discrete time periods which are characterized bydifferent systematic (market) risk profiles. Our goal is to determine whether5 igure 2: Rolling explained variance ratio for cryptocurrencies λ c /N and equities λ e /K . the similarity in cryptocurrency and equity market dynamics changes in varyingmarket conditions. The three periods are defined• Pre-COVID: 03-01-2018 to 28-02-2020,• Peak COVID: 02-03-2020 to 29-05-2020,• Post-COVID: 01-06-2020 to 08-12-2020,with corresponding lengths | T PRE | , | T PEAK | and | T POST | . For the two sequencesof time-varying correlation matrices Ω ct and Ω et , we study the similarity in theexplanatory variance of the first 10 eigenvalues. ˜ λ c ,t , ..., ˜ λ c ,t and ˜ λ e ,t , ..., ˜ λ e ,t .We define the difference in these spectral surfaces dynamics deviation andcompute as follows DD PRE = 1 | T PRE | (cid:88) t =60 10 (cid:88) i =1 | ˜ λ ci,t − ˜ λ ei,t | (9)DD PEAK = 1 | T PEAK | (cid:88) t =312 10 (cid:88) i =1 | ˜ λ ci,t − ˜ λ ei,t | (10)DD POST = 1 | T POST | (cid:88) t =376 10 (cid:88) i =1 | ˜ λ ci,t − ˜ λ ei,t | . (11)The measure is normalized by the length of each time period, allowing us to com-pare dynamics during periods of varying lengths. As the majority of explanatory6 a) Pre Covid (b) Peak Covid(c) Post Covid Figure 3: Time-varying eigenspectrum Pre Covid, Peak Covid, and Post Covid. a) Cryptocurrency (b) Equity Figure 4: Kernel density estimates of w c ( i, j ) and w e ( i, j ) from Pre Covid, Peak Covid andPost Covid periods. variance is provided by the first 10 eigenvalues in both the cryptocurrency andequity collections, we ignore the negligible difference in total variance explainedby the remaining elements of the eigenspectrum.Dynamics deviation scoresPeriod ScoreDD PRE
PEAK
POST
Table 1: Dynamics deviation from 3 periods of analysis
Figure 3 shows the cryptocurrency and equity eigenspectra during the threewindows of analysis. Of the three analysis windows, the two eigenspectra appearto be most similar during the Peak COVID period, which is displayed in Figure3b. This is confirmed in Table 1, which shows dynamics deviation scores forthe three windows of analysis. The results highlight a significant increase in thesimilarity of the two collections’ collective behaviour during the Peak COVIDperiod, with a score of 0.160. The Pre-COVID and Post-COVID scores are 0.369and 0.298 respectively, highlighting less similarity in the dynamics of equitiesand cryptocurrencies outside periods of market crisis. This is primarily due tothe equity eigenspectrum’s first eigenvalue exhibiting lower explanatory variancein the Pre-COVID and Post-COVID periods. This is shown in Figure 3a andFigure 3c respectively.Next, we contrast the cryptocurrency and equity correlation coefficientsduring the same three windows of analysis seen in Figure 4. This analysis issimilar to that conducted in [24], where the probability density functions ofoff-diagonal correlation matrix elements are studied for six base cryptocurrenciesand the USD. [24]’s analysis also focuses on the distribution of correlation matrixeigenvalues, and the time-varying behavior of the correlation matrix’s largest8igenvalues (seen in Figure 7 of [24]). However, the work presented in thissection focuses more prominently on changes in the distribution of correlationmatrix elements seen in different markets across cryptocurrencies and equities.Figures 4a and 4b display kernel density estimates of cryptocurrency and equitycorrelation matrix elements during the Pre-COVID, Peak COVID and Post-COVID periods. There are two important insights. First, both cryptocurrencyand equity markets highlight a sharp increase in collective correlations duringthe Peak COVID period. Both cryptocurrencies and equities displayed markedlylower correlation coefficients in the Pre-COVID and Post-COVID periods. Inall three periods, cryptocurrency correlation coefficients were more stronglypositive than that of equities. These findings are consistent with the results inthe Section 3.1, where dynamics deviations were lowest during the Peak COVIDmarket period; suggesting that during market crises cryptocurrency and equitybehaviours are most similar.
4. Trajectory modelling
In this section, we study the trajectory dynamics [71] of equity and cryp-tocurrency closing prices for the entirety of our time period, a single period of T = 508 days. To compare trajectories of securities with markedly differentprices, we normalize the cryptocurrency time series c i ( t ) and equity time series e j ( t ) . Analyzing a candidate individual cryptocurrency provides a function c i ∈ R T . We let (cid:107) c i (cid:107) = (cid:80) Tt =1 | c i ( t ) | be the L norm of the function, and definea normalized cryptocurrency price trajectory by T ci = c i (cid:107) c i (cid:107) . Similarly, we define (cid:107) e j (cid:107) = (cid:80) Tt =1 | e j ( t ) | , and the corresponding normalized equity trajectory as T ej = e j (cid:107) e j (cid:107) . Distances between such vectors highlight the relative change in cryp-tocurrency and equity securities during the time period. To study such changes,we define two trajectory matrices , D T Cij = (cid:107) T ci − T cj (cid:107) and D T Eij = (cid:107) T ei − T ej (cid:107) ,and perform hierarchical clustering .First, we compare norms of the two trajectory matrices to better understandsimilarity within each collection. Both of these matrices are symmetric, realand have trace 0. As the two collections are of different sizes, we normalize thenorm computations by the number of elements in each trajectory matrix. Thenormalized cryptocurrency trajectory matrix norm (cid:107) D T C (cid:107) ∗ = N − (cid:113)(cid:80) i,j | d tcij | and the normalized equity trajectory matrix norm (cid:107) D T E (cid:107) ∗ = K − (cid:113)(cid:80) i,j | d teij | .The normalized cryptocurrency trajectory matrix norm (cid:107) D T C (cid:107) ∗ = 0 . andthe normalized equity trajectory matrix norm (cid:107) D T E (cid:107) ∗ = 0 . , demonstratingmore similarity in normalized trajectories among equities than cryptocurrencies.There are several possible explanations for this finding. First, there could besome bias in our sample of equities, having chosen the 72 largest equities inthe world by market capitalization. It is likely that the volatility in their pricebehaviour will be less than that of smaller equities. On the other hand, thisfinding may quite reasonably reflect the volatile nature of cryptocurrency marketsentiment. Although correlations among the cryptocurrency constituents are9 igure 5: Hierarchical clustering on D TC . higher than that of equity constituents, the consistently strong influence of ‘themarket’ may make trajectories highly responsive to sharp changes in sentiment.Figures 5 and 6 display the cryptocurrency and equity trajectory dendrogramsrespectively. Each dendrogram highlights several noteable insights regardingtrajectory clusters. Figure 5 shows two major clusters, a predominant clusterwith high self-similarity, a smaller more amorphous cluster and a single outlier inRevain. The predominant cluster contains most cryptocurrencies analyzed withless volatile price trajectories. The smaller cluster is composed of cryptocurren-cies having exhibited major volatility in their price trajectory, such as Chainlink,which experienced a price increase of almost 42 times during our window of anal-ysis. This dendrogram has markedly different structure to the equity trajectorydendrogram, shown in Figure 6. The equity trajectory dendrogram exhibitsmore substantial self-similarity, with three clearly defined clusters; one predomi-nant cluster and two smaller, well-defined clusters exhibiting high self-similarity.The first minority cluster contains stocks such as; Microsoft, Amazon, Alibaba,Tencent, Facebook, Apple and TSMC - all of which are technology companies.The second small cluster contains stocks such as: Chevron, Exxon, BP, Shell,Wells Fargo and HSBC - primarily financial services and energy companies. Bothof these sectors tend to perform well in buoyant equity markets, and badly indeclining markets. The largest, most predominant third cluster contains theremaining equities. The growth in passive and factor based investing may haveincreased the similarity in these equities’ behaviours, as investors increasinglyseek to buy stocks within a sector or ‘theme’ of the market.10 igure 6: Hierarchical clustering on D TE .
5. Erratic behaviour modelling
In this section we study the similarity of structural breaks in our two mul-tivariate time series of log returns, R ci ( t ) and R ej ( t ) defined earlier. For eachsecurity in the respective time series, we apply the two-phase change point detec-tion algorithm described by [72] to generate a set of structural breaks for eachlog return time series. Each change point represents a point in time where thealgorithm determines the statistical properties of the time series to have changed.We apply the Kolmogorov-Smirnov test, detecting general distributional changesin the underlying time series. The change point detection method could insteadfocus on changes in specific distributional moments such as mean or variance,however. We obtain two collections of finite sets ξ c , ..., ξ cN , and ξ e , ..., ξ eK forcryptocurrency and equity time series respectively, where all sets are a subset of { , ..., T } .Next, we compute distances between the cryptocurrency structural break sets ξ ci and equity structural break sets ξ ej . There is significant literature highlightingissues in using metrics such as the Hausdorff distance, due to its sensitivity tooutliers, [73, 74] and so we use a recently introduced semi-metric modification[74] between candidate sets within our two collections of structural breaks ξ ci and ξ ej . Normalized distances between sets of cryptocurrencies are computed: D ( ξ ci , ξ cj ) = 12 (cid:32) (cid:80) b ∈ ξ cj d ( b, ξ ci ) | ξ cj | + (cid:80) a ∈ ξ ci d ( a, ξ cj ) | ξ ci | (cid:33) , (12)11 igure 7: Hierarchical clustering on D BC . where d ( b, ξ ci ) is the minimal distance from b ∈ ξ cj to the set ξ ci . Distancesbetween sets of equities are computed similarly: D ( ξ ei , ξ ej ) = 12 (cid:32) (cid:80) b ∈ ξ ej d ( b, ξ ei ) | ξ ej | + (cid:80) a ∈ ξ ei d ( a, ξ ej ) | ξ ei | (cid:33) , (13)where d ( b, ξ ei ) is the minimal distance from b ∈ ξ ej to the set ξ ei . This semi-metric is the L norm average of all minimal distances between any two sets.As all time series are of equal length, it is not necessary to normalize by thelength of the time series. Finally, we form two breaks matrices between sets ofcryptocurrency structural breaks, D BCij = D ( ξ ci , ξ cj ) and equity structural breaks, D BEij = D ( ξ ei , ξ ej ) . To better understand collective similarity in structural breaks,we perform hierarchical clustering on our two breaks matrices.Like Section 4, we compare norms of the two distance matrices to betterunderstand breaks similarity within each collection. As the two collectionsare of different sizes, again, we normalize the two norm computations by thesize of the distance matrix. The normalized cryptocurrency breaks matrixnorm (cid:107) D BC (cid:107) ∗ = N − (cid:113)(cid:80) i,j | d bcij | and the normalized equity breaks matrixnorm (cid:107) D BE (cid:107) ∗ = K − (cid:113)(cid:80) i,j | d beij | . The normalized cryptocurrency breaksmatrix norm (cid:107) D BC (cid:107) ∗ = 23 . and the normalized equity breaks matrix norm (cid:107) D BE (cid:107) ∗ = 23 . , highlighting approximately equivalent similarity in structural12 igure 8: Hierarchical clustering on D BE . breaks among equities and cryptocurrencies. This result is consistent with earlierfindings [11], which suggest that although one collection of time series may exhibitmore volatility and possibly warrant a higher number of structural breaks, theunivariate nature of the change point detection algorithm [72] detects structuralbreaks relative to the properties of the particular time series. Therefore, althoughcryptocurrency time series may exhibit more volatility, the behaviour in theircollective structural breaks is not necessarily more or less similar than that ofthe equity collection.Next, we compare the cluster structures of the two collections. The cryptocur-rency breaks dendrogram, seen in Figure 7 consists of one primary cluster withthree, diffuse sub-clusters. The primary cluster has one predominant sub-clusterof concentrated similarity, with the three remaining, smaller clusters having amore indeterminate form. By contrast, the equity breaks dendrogram in Figure8 has a more easily interpreted cluster structure. There are four small clusters,each of which contains two or three equities, and a predominant cluster which iscomprised of the remaining equities. The four small clusters appear to clusterbased on sector, where cluster one consists of Facebook and Comcast (technol-ogy), cluster two consists of BP and Shell (energy), cluster three consists ofMerck and Amgen (biotechnology/pharmaceuticals), and cluster four consists ofTencent, Apple and Microsoft (technology). These results suggest that, with theexception of select equities within specific sectors that exhibit similar structuralbreak patterns, most equities have similar structural breaks behaviour.13 . Extreme behaviour modelling In this section, we study anomalies with respect to total returns and extremebehaviors within our collections of cryptocurrencies and equities. First, weoutline the procedure to measure distance between extreme values of candidatetime series. We let µ be a probability distribution that stores the extreme valuesof a cryptocurrency time series c i ( t ) or equity time series e j ( t ) . We assumethat µ is a continuous probability measure of the form µ = f ( x ) dx , where dx isLebesgue measure, and f ( x ) is a probability density function that is non-negativeeverywhere and integrates to . We study the % and % points of density,respectively, by the equations (cid:90) l −∞ f ( x ) dx = 0 . (14) (cid:90) ∞ u f ( x ) dx = 0 . (15)The range x ≤ l gives the most extreme 10% of the distribution on the left sideof the distribution, while the range x ≥ u gives the 10% right most extremevalues. The restricted function is defined g ( x ) = f ( x ) { x ≤ l }∪{ x ≥ u } = f ( x ) , x ≤ l , l < x < uf ( x ) , x ≥ u. (16)Next, we construct an associated measure ν = g ( x ) dx , with dx as Lebesguemeasure. We generate N associated probability measures µ c , ..., µ cN , and returnsmeasures ν c , ..., ν cN for our cryptocurrency time series. Similarly we generate K probability measures µ e , ..., µ eK , and return measure ν e , ..., ν eK for our equitytime series. As all restricted measures are of total size 0.2, we use the Wassersteinmetric to compute distances between these truncated distributions. Finally, weform a matrix between the distributional extremities of all time series. Let D ECij = d w ( ν ci , ν cj ) be the matrix between the cryptocurrency extreme returndistributions, and D EEij = d w ( ν ei , ν ej ) be the matrix between equity extremereturn distributions.We define total returns for cryptocurrencies z ci = (cid:80) Tt =1 R ci ( t ) and equities z ej = (cid:80) Tt =1 R ej ( t ) . We now compute returns matrices for cryptocurrency returns, D RCij = | z ci − z cj | and equity returns D REij = | z ei − z ej | . To identify anomalies withrespect to returns and extreme values, we transform the four distance matricesinto affinity matrices. That is, a candidate affinity matrix A , is defined as A ij = 1 − D ij max { D } , (17)where A is symmetric, A ii = 1 , ≤ A ij ≤ , ∀ i, j. We now define an affinityreturns matrix A R and an affinity extremes matrix A E both of which are of14 a) A R (b) A E Figure 9: Affinity matrices returns and extremes. dimension 117 x 117, which include all 45 cryptocurrencies and 72 equitiesanalyzed in this paper.We study Figure 9 and compare collective similarities in returns and extremesamong our two collections. Figure 9a shows the affinity returns matrix A R . Itis clear that both equities and cryptocurrencies exhibit strong self-similarity,with the equity collection exhibiting slightly more similar returns than thecryptocurrency collection. Although not as strong as the intra-asset similarity,there is still reasonable similarity in the returns profile between that of equitiesand cryptocurrencies. Figure 9b displays a clear difference in the collectivebehaviours. Similarly to returns, equities exhibit more self-similarity than that ofcryptocurrencies. However unlike returns, there is markedly less similarity whencomparing the extremes of cryptocurrencies and equities. These findings areconsistent with the results presented in Sections 4 and 5, where cryptocurrencieswere shown to be more varied in their trajectories and less consistent in theirstructural breaks. This finding supports the high degree of dependence identifiedbetween extreme and erratic behaviour in the cryptocurrency market.
7. Anomaly persistence
In this section, we study the evolution of ranks among cryptocurrencies and eq-uities with respect to risk-adjusted returns. Rather than using correlation, we ap-ply the concept of ranks, as financial analysts often apply ‘ranking’ systems whenidentifying anomalous securities (equities, bonds, currencies, etc.). We definerolling risk-adjusted returns κ ci ( t ) = (cid:80) tm = t − R ci ( m ) σ ci ( t ) and κ ej ( t ) = (cid:80) tm = t − R ej ( m ) σ ej ( t ) where σ ci ( t ) and σ ej ( t ) are the standard deviations of rolling cryptocurrency andequity log returns for t ∈ { , ..., T } . We use a rolling window of 61 days and15 igure 10: Elements κ c ( s, t ) and κ e ( s, t ) . (a) K c ( s, t ) (b) K e ( s, t ) Figure 11: Anomaly persistence matrices K c ( s, t ) and K e ( s, t ) . s, t ∈ { , ..., T } , we define K c ( s, t ) and K e ( s, t ) as matrices measuringthe Kendall rank correlation coefficient between any two risk-adjusted returnrank vectors s and t for all possible points in time. A higher score indicates moresimilarity in the securities exhibiting high and low risk-adjusted returns at anytwo points in time. Elements of both matrices k c ( s, t ) and k e ( s, t ) lie ∈ [ − , .First, we study the norms of the two anomaly persistence matrices (cid:107) K c (cid:107) = (cid:113)(cid:80) s,t | k cst | and (cid:107) K e (cid:107) = (cid:113)(cid:80) s,t | k est | . The cryptocurrency anomaly persis-tence norm, (cid:107) K c (cid:107) = 109.69 and the equity anomaly persistence norm (cid:107) K e (cid:107) = 122.62. The higher score for the equity collection suggests that there ismore consistency in the stocks that are anomalous on a risk-adjusted returnbasis over time. Figure 10 which plots two distributions of the elements k c ( s, t ) and k e ( s, t ) , demonstrates a higher average correlation for equities than that ofcryptocurrencies. Further interesting structure over time is revealed in Figure11. Both Figures 11a and 11b have high correlation scores around the diagonal,which is indicative of short-term dependence in anomalous behaviours withinboth collections. However, Figure 11b clearly displays a higher level of similarityover time - indicating more persistence in anomalous behaviours. This is furthersupported in our hierarchical clustering analysis, where the dendrograms for K c ( s, t ) and K e ( s, t ) are displayed in Figure 12. There are two primary takeawaysfrom this analysis. First, K e ( s, t ) is demonstrably more positive than K c ( s, t ) for the vast majority of the dendrogram. This indicates that there is greatersimilarity in risk-adjusted return ranks, over all comparative measurements intime, for our collection of equities. Second, the K e ( s, t ) dendrogram determines atotal of 9 clusters, while K c ( s, t ) has a total of 11 clusters. This finding indicatesmore stability in equity rank correlation scores over the entirety of our analysiswindow. 17 a) K c ( s, t ) (b) K e ( s, t ) Figure 12: Hierarchical clustering on anomaly persistence matrices K c ( s, t ) and K e ( s, t ) . . Conclusion Our work in Section 3 demonstrates that collective dynamics in the cryp-tocurrency market are significantly stronger than that of the equity market. Theexplanatory variance provided by the largest eigenvector is consistently larger,and more stable among cryptocurrencies than equities. Partitioning our analysisinto three discrete windows highlights that collective dynamics are most similarbetween our two collections during the Peak COVID period, demonstrating thatequities and cryptocurrencies behave most similarly during market crises. This isfurther supported in our correlation matrix analysis, where both cryptocurrenciesand equities experience a sharp increase in their correlations during the peak ofCOVID-19. In periods surrounding the crisis (Pre-COVID and Post-COVID),cryptocurrency correlation coefficients are more strongly positive than that ofequities. The findings in this section are also consistent with the work presentedin [75]. In [75] the authors demonstrate that although cryptocurrency dynamicsare decoupled with other asset classes during 2019, during select market eventsin 2020 such as the COVID-19 pandemic, the dynamics of cryptocurrencies andother asset classes behave much more similarly.The work presented in this manuscript certainly has its limitations. The workof [76] demonstrates that after explicitly accounting for time zone differencesbetween indices, (Dow Jones and DAX), there is a significant increase in thesimilarity of the dynamics of such collections. When initially analyzing bothcollections in conjunction, the authors show there are two dominant eigenvaluesrepresenting the dynamics of each collection. After accounting for differencesbetween the collections by translating the returns of the DAX, one dominanteigenvalue exhibits itself - highlighting a marked increase in the similarity of thedynamics among the total collection. This has not been explicitly considered inthis work, and could alter the results and subsequent interpretation. Furtherresearch comparing the dynamics of cryptocurrencies and equities using tech-niques from [76], and studying the resulting change in the dynamics deviationscores could be of interest to the econophysics community.Section 4 examines the similarity in normalized price trajectories amongboth collections. Distance matrix norms indicate that equities exhibit moresimilarity among their trajectories than cryptocurrencies. This may be due tothe significant price volatility exhibited by cryptocurrencies over the past twoyears, making their trajectories (generally, but not universally) more dissimilar.Hierarchical clustering on both time series displays marked differences in clusterstructures. Equity trajectories display more self-similarity than cryptocurrencies.We suspect that the latent phenomenon may be the growth of passive andfactor-based investing over the last several years.Section 5 compares the similarity in erratic behaviour among equities andcryptocurrencies. Distance matrix norms display comparable similarity in theerratic behaviours of cryptocurrencies and equities. Although cryptocurrenciesmay be more volatile, the univariate nature of our changepoint detection algo-rithm is unable to determine structural breaks with respect to the rest of thecollection. Using other change point detection methodologies to detect structural19reaks [77] may result in different findings.Our results in Section 6 are consistent with those in Section 4. Figure 9 showsmore homogeneity among equity extremes in comparison to cryptocurrencies.Analyzing distance matrix norms and affinity matrices highlights a substantialdifference in self-similarity. When contrasting the similarity in all 117 time seriesfor total returns and extreme values, the distinction in extreme value similarityis most evident.Finally in Section 7, equities are shown to exhibit more persistent anomaliesthan cryptocurrencies. We apply hierarchical clustering to our proposed anomalypersistence matrix. Hierarchical clustering determines the existence of 9 and 11clusters respectively in the K c ( s, t ) and K e ( s, t ) . A lower number of clusterssignals greater consistency in anomaly ranks over time. This is further supportedanalyzing the elements of our matrix, which indicate a higher correlation inanomaly rankings over time in the equity time series.This work bridges several disparate areas of research: nonlinear dynamics,econophysics, COVID-19 and cryptocurrency market dynamics. There are severalinteresting avenues for future research. First, our analysis could be applied tomore asset classes beyond cryptocurrencies and equities. Second, other techniquescould be introduced to study phenomena such as: market dynamics, trajectories,extreme and erratic behaviour, and anomaly persistence. Finally, this analysiscould be run on different time windows and on a more timely basis. The chaoticand non-deterministic nature of financial markets necessitates timely researchon topics of interest. Acknowledgements
I would like to thank Peter Radchenko and Max Menzies for helpful discus-sions. 20 ppendix A. Mathematical objects glossaryMathematical objects table: Section 3
Object Description N K c i ( t ) Cryptocurrency price time series e j ( t ) Equity price time series R ci ( t ) Cryptocurrency returns time series R ej ( t ) Equity returns time series ˆ R ci ( t ) Standardized cryptocurrency returns time series ˆ R ej ( t ) Standardized equity returns time series Ω c Cryptocurrency correlation matrix Ω e Equity correlation matrix ω c ( i, j ) Element ( i, j ) in cryptocurrency correlation matrix ω e ( i, j ) Element ( i, j ) in equity correlation matrix Φ c Cryptocurrency PC coefficient matrix Φ e Equity PC coefficient matrix Z c Cryptocurrency PC matrix Z e Equity PC matrix D c Cryptocurrency diagonal matrix D e Equity diagonal matrix λ c Cryptocurrency eigenvalues λ e Equity eigenvalues Λ c Cryptocurrency orthogonal eigenvector matrix Λ e Equity orthogonal eigenvector matrix ˜ λ c Cryptocurrency eigenvalue explanatory variance ˜ λ e Equity eigenvalue explanatory variance | T PRE | Length of Pre-COVID period | T PEAK | Length of Peak COVID period | T POST | Length of Post-COVID periodDD
PRE
Pre-COVID dynamics deviationDD
PEAK
Peak COVID dynamics deviationDD
POST
Post-COVID dynamics deviation
Table A.2: Mathematical objects and definitions athematical objects table: Sections 4, 5, 6, 7 Object Description T ci Cryptocurrency normalized price trajectory T ej Equity normalized price trajectory D T C
Cryptocurrency trajectory matrix D T E
Equity trajectory matrix (cid:107) D T C (cid:107) ∗ Normalized cryptocurrency trajectory matrix norm (cid:107) D T E (cid:107) ∗ Normalized equity trajectory matrix norm ξ c , ..., ξ cN Cryptocurrency structural break sets ξ e , ..., ξ eK Equity structural break sets D BC Cryptocurrency breaks matrix D BE Equity breaks matrix (cid:107) D BC (cid:107) ∗ Normalized cryptocurrency breaks matrix norm (cid:107) D BE (cid:107) ∗ Normalized equity breaks matrix norm D EC Cryptocurrency extremes matrix D EE Equity extremes matrix z ci ( t ) Cryptocurrency total returns time series z ej ( t ) Equity total returns time series D RC Cryptocurrency returns matrix D RE Equity returns matrix A R Affinity returns matrix (Cryptocurrencies and equities) A E Affinity extremes matrix (Cryptocurrencies and equities) κ c ( t ) Cryptocurrency risk-adjusted return vector at time tκ e ( t ) Equity risk-adjusted return vector at time tσ ci ( t ) Cryptocurrency realized volatility at time tσ ej ( t ) Equity realized volatility at time tK c ( s, t ) Cryptocurrency anomaly persistence matrix K e ( s, t ) Equity anomaly persistence matrix k c ( s, t ) Element ( s, t ) in cryptocurrency anomaly persistence matrix k e ( s, t ) Element ( s, t ) in equity anomaly persistence matrix (cid:107) K c (cid:107) Cryptocurrency anomaly persistence matrix norm (cid:107) K e (cid:107) Equity anomaly persistence matrix norm
Table A.3: Mathematical objects and definitions ppendix B. Securities analyzedCryptocurrency tickers and names Ticker Coin Name Ticker Coin NameBTC Bitcoin THETA THETAETH Ethereum MKR MakerXRP XRP SNX SynthetixLINK Chainlink OMG OMG NetworkBCH Bitcoin Cash DOGE DogecoinADA Cardano ONT OntologyBNB Binance Coin DCR DecredXLM Stellar BAT Basic AttentionBSV Bitcoin SV NEXO NexoEOS EOS ZRX 0xXMR Monero REN RenTRX Tron QTUM QtumXEM NEM ICX ICONXTZ Tezos LRC LoopringNEO NEO Token KNC Kyber NetworkVET VeChain REP AugurREV Revain Classic LSK LiskDASH Dash BTG Bitcoin GoldWAVES Waves SC SiacoinHT Huobi Token QNT QUANTMIOTA IOTA MAID MaidSafeCoinZEC ZCash NANO NanoETC Ethereum Classic
Table B.4: Cryptocurrency tickers and names quity tickers and names Ticker Equity Name Ticker Equity NameNYSE: C Citigroup EPA: OR L’OrealNYSE: MRK Merck NYSE: UNH UnitedHealth GroupNYSE: KO Coca-Cola EPA: FP TotalNASDAQ: AMGN Amgen SHA: 601988 Bank of ChinaNYSE: T AT&T SHA: 601288 A.B. ChinaLON: BATS British American Tobacco LON: HSBA HSBCNYSE: JPM JP Morgan Chase NYSE: VZ VerizonTYO: 7203 JP Toyota Motor NYSEARCA: SPY SPDR S&P 500ASX: CBA CBA NYSE: SLB SchlumbergerSHA: 601939 China Construction Bank EPA: SAN SanofiNASDAQ: CSCO Cisco NYSE: IBM IBMNYSE: MDT Medtronic NYSE: PG Procter & GambleLON: BP BP NASDAQ: FB FacebookNYSE: BRK Berkshire Hathaway SHA: 601398 ICBCSWX: NOVN Novartis SHA: 600028 SinopecETR: SIE Siemens NASDAQ: MSFT MicrosoftNYSE: WMT Walmart NYSE: WFC Wells FargoNYSE: DIS Walt Disney SWX: RO Roche HoldingsNYSE: JNJ Johnson and Johnson NASDAQ: PEP PepsiCoNASDAQ: INTC Intel NYSE: PFE PfizerNYSE: GE General Electric NYSE: XOM Exxon MobilNASDAQ: AAPL Apple NYSE: BMY Bristol-Myers SquibbNYSE: GS Goldman Sachs NASDAQ: CMCSA ComcastKRX: 005930 Samsung Electronics NYSE: HD Home DepotHKG: 0941 China Mobile NYSE: MA MastercardNASDAQ: AMZN Amazon LON: ULVR UnilverNYSE: ORCL Oracle SWX: NESN NestleNYSE: MMM 3M NYSE: V VisaAMS: RDSA Royal Dutch Shell NYSE: PM Philip MorrisBME: ITX INDITEX NYSE: ABBV AbbVie IncNYSE: MO Altria Group HKG: 9988 AlibabaNYSE: CVX Chevron NASDAQ: KHC Kraft HeinzTSE: RY Royal Bank of Canada NASDAQ: GOOGL AlphabetHKG: 0700 Tencent EBR: ABI Anheuser Busch Inbev NVTPE: 2330 TSMC NYSE: BAC Bank of AmericaETR: SAP SAP SHA: 601857 PetroChina
Table B.5: Equity tickers and names
Appendix C. Change point detection algorithm
In this section, we provide an outline of change point detection algorithms, anddescribe the specific algorithm that we implement. Many statistical modellingproblems require the identification of change points in sequential data. By24efinition, these are points in time at which the statistical properties of a timeseries change. The general setup for this problem is the following: a sequenceof observations x , x , ..., x n are drawn from random variables X , X , ..., X n and undergo an unknown number of changes in distribution at points τ , ..., τ m .One assumes observations are independent and identically distributed betweenchange points, that is, between each change points a random sampling of thedistribution is occurring. Following Ross [72], we notate this as follows: X i ∼ F if i ≤ τ F if τ < i ≤ τ F if τ < i ≤ τ ,. . . While this requirement of independence may appear restrictive, dependence cangenerally be accounted for by modelling the underlying dynamics or drift process,then applying a change point algorithm to the model residuals or one-step-aheadprediction errors, as described by Gustafsson [78]. The change point modelsapplied in this paper follow Ross [72].
Appendix C.1. Batch change detection (Phase I)
This phase of change point detection is retrospective. We are given a fixedlength sequence of observations x , . . . , x n from random variables X , . . . , X n .For simplicity, assume at most one change point exists. If a change point existsat time k , observations have a distribution of F prior to the change point, anda distribution of F proceeding the change point, where F (cid:54) = F . That is, onemust test between the following two hypotheses for each k : H : X i ∼ F , i = 1 , ..., nH : X i ∼ (cid:40) F i = 1 , , ..., kF , i = k + 1 , k + 2 , ..., n and end with the choice of the most suitable k .One proceeds with a two-sample hypothesis test, where the choice of testis dependent on the assumptions about the underlying distributions. To avoiddistributional assumptions, non-parametric tests can be used. Then one ap-propriately chooses a two-sample test statistic D k,n and a threshold h k,n . If D k,n > h k,n then the null hypothesis is rejected and we provisionally assume thata change point has occurred after x k . These test statistics D k,n are normalisedto have mean and variance and evaluated at all values < k < n , and thelargest value is assumed to be coincident with the existence of our sole changepoint. That is, the test statistic is then D n = max k =2 ,...,n − D k,n = max k =2 ,...,n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ D k,n − µ ˜ D k,n σ ˜ D k,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ D k,n were our unnormalised statistics.The null hypothesis of no change is then rejected if D n > h n for someappropriately chosen threshold h n . In this circumstance, we conclude that a(unique) change point has occurred and its location is the value of k whichmaximises D k,n . That is, ˆ τ = argmax k D k,n . This threshold h n is chosen to bound the Type 1 error rate as is standard instatistical hypothesis testing. First, one specifies an acceptable level α for theproportion of false positives, that is, the probability of falsely declaring that achange has occurred if in fact no change has occurred. Then, h n should be chosenas the upper α quantile of the distribution of D n under the null hypothesis. Forthe details of computation of this distribution, see [72]. Computation can oftenbe made easier by taking appropriate choice and storage of the D k,n . Appendix C.2. Sequential change detection (Phase II)
In this case, the sequence ( x t ) t ≥ does not have a fixed length. New ob-servations are received over time, and multiple change points may be present.Assuming no change point exists so far, this approach treats x , ..., x t as a fixedlength sequence and computes D t as in phase I. A change is then flagged if D t > h t for some appropriately chosen threshold. If no change is detected, thenext observation x t +1 is brought into the sequence. If a change is detected, theprocess restarts from the following observation in the sequence. The proceduretherefore consists of a repeated sequence of hypothesis tests.In this sequential setting, h t is chosen so that the probability of incurringa Type 1 error is constant over time, so that under the null hypothesis of nochange, the following holds: P ( D > h ) = α,P ( D t > h t | D t − ≤ h t − , ..., D ≤ h ) = α, t > . 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