Efficiency of the financial markets during the COVID-19 crisis: time-varying parameters of fractional stable dynamics
EEfficiency of the financial marketsduring the COVID-19 crisis:time-varying parameters of fractional stable dynamics
Ayoub Ammy-Driss a , Matthieu Garcin b, ∗ July 22, 2020
Abstract
This paper investigates the impact of COVID-19 on financial markets. It focuses on theevolution of the market efficiency, using two efficiency indicators: the Hurst exponent andthe memory parameter of a fractional L´evy-stable motion. The second approach combines, inthe same model of dynamic, an alpha-stable distribution and a dependence structure betweenprice returns. We provide a dynamic estimation method for the two efficiency indicators. Thismethod introduces a free parameter, the discount factor, which we select so as to get thebest alpha-stable density forecasts for observed price returns. The application to stock indicesduring the COVID-19 crisis shows a strong loss of efficiency for US indices. On the opposite,Asian and Australian indices seem less affected and the inefficiency of these markets duringthe COVID-19 crisis is even questionable.
Keywords – alpha-stable distribution, dynamic estimation, efficient market hypothesis, financialcrisis, Hurst exponent
The COVID-19 pandemic has strongly affected many persons, either for medical reasons or for theeconomic aftermath of the various prophylactic measures decided by governments, in particularlockdowns. The evolution of financial markets during the pandemic provides an illustration of theeconomic impact of these measures. According to several empirical studies, financial markets haveindeed been strongly disturbed during this period [3, 5, 34, 62]. The question of the reaction offinancial markets to a crisis is not specific to the COVID-19 pandemic. For example, we can cite astudy of the impact of crisis in the 80s and the 90s on a model of dynamic for a stock market [18].In general, these studies focus on variations of several statistics, such as jump intensity, impliedvolatility, parameters of factor models, divergence of price return densities, etc. To the best ofour knowledge, no paper focuses on measuring the impact of the COVID-19 pandemic on marketefficiency.Market efficiency is the ability of market prices to reflect all the available information, so that noarbitrage is possible. In other words, if markets are efficient, price returns are not correlated with ∗ Corresponding author: [email protected]. a ESILV, 92916 Paris La D´efense, France. b L´eonard de Vinci Pˆole Universitaire, Research center, 92916 Paris La D´efense, France. a r X i v : . [ q -f i n . S T ] J u l ach other and investors are not able to statistically determine what is more profitable betweenselling and buying a financial asset. Even if this dogma is sometimes questionable in calm periods,we wonder if it can resist to a crisis. We also wonder if we can observe regional disparities,hypothetically related to the magnitude of the outbreak in these regions, and how fast financialmarkets recover.The Hurst exponent is a widespread indicator of market efficiency. The long-range feature associ-ated to a Hurst exponent above 1 / τ is τ H σ , where H is theHurst exponent and σ the variance of increments of duration 1. The fBm assumes that incrementsfollow a Gaussian distribution. The fractal property of the fBm is then obtained by introducing apositive (respectively negative) correlation among increments if the Hurst exponent is above (resp.below) 1 /
2. In the case where the Hurst exponent is 1 /
2, the fBm is simply a standard Brownianmotion (Bm), that is with independent increments. In the fBm framework, this value of the Hurstexponent is thus the only one consistent with the efficient market hypothesis (EMH).However, the approach using the fBm assumes Gaussian price returns. It is not very realistic,since the presence of fat tails in the distribution of price returns is well documented [48, 60, 29].As an alternative to the Gaussian distribution, the alpha-stable distribution is appealing becauseit includes the Gaussian distribution as a particular case and entails fat tails, whose amplitude isdirectly related to α , a parameter of this distribution. When combining alpha-stable distributionsand dependence between the successive price returns, we can get the fractional L´evy-stable motion(fLsm), which is thus a non-Gaussian extension of the fBm [70, 73, 71]. In this framework, theHurst exponent is to be decomposed in m + 1 /α , where m is a memory parameter. If m = 0,adjacent price returns are independent and the market is efficient. We thus use this memoryparameter as an alternative efficiency indicator.The estimation of the fLsm partly relies on the estimation of alpha-stable distributions. Manyestimation methods exist for this kind of distribution [2, 12]. In particular, we will focus onMcCulloch’s method, which is based on empirical quantiles [55]. In this paper, we are in additioninterested in the dynamic estimation of this distribution, because we want to depict the chronologyof the crisis, day after day. Several articles deal with the estimation of time-varying non-parametricdensities [38, 34]. The case of parametric densities is simpler as it consists in estimating time-varying parameters. In both the non-parametric and parametric cases, the estimation at a givendate takes into account the estimation at the previous date updated by the new observation. Thebalance between the previous estimation and the new observation is tuned by a discount factor.Several rules are possible for the selection of this free parameter. We focus on the minimization ofa criterion related to arguments coming from the field of validation of density forecast [34, 22].In the empirical part of the paper, we study the evolution of the two indicators of market efficiency, H and m for several stock indices, with a significance analysis. We have discovered that the Hurstexponent H detects less often a market inefficiency than does the memory parameter m of anfLsm. We foster the use of this parameter as efficiency indicator instead of the Hurst exponent.It indeed improves the standard Hurst approach insofar as it filters the kurtosis of price returns,which biases the Hurst indicator.Besides the analysis of the impact of the COVID-19 on market efficiency, the innovative aspects ofthis paper include a selection rule for the discount factor of a dynamic parametric distribution, anestimation method of dynamic Hurst exponents, and the introduction of the memory parameterof an fLsm as efficiency indicator. 2he rest of the paper is organized as follows. Section 2 introduces the estimation of dynamicalpha-stable distributions, along with the selection rule of the discount factor. Section 3 providessome elements on market efficiency and details how indicators are built. Section 4 empiricallystudies the impact of COVID-19 on market efficiency. Section 5 concludes. The alpha-stable distribution is a generalization of the Gaussian distribution, appreciated for en-tailing fat tails. For this reason, it has been widely invoked in signal processing [69, 32, 33], withapplications for example in medicine [68, 71] or in finance [48, 60, 29].We present below the estimation of a time-varying alpha-stable distribution, which we will applylater in this paper to financial time series. For this purpose, we present first the static estimationas well as the various representations of alpha-stable distributions. Regarding the dynamic distri-bution, a free parameter is to be selected. It is the discount factor. We propose a rule of selectionin the last subsection.
Four parameters are used to depict a random variable following a stable distribution: X ∼ S α ( γ, β, µ ). The parameter α ∈ (0 ,
2] is the one we will mostly be interested in. It determines thethickness of the tails. The parameter β ∈ [ − ,
1] is a skewness parameter. If α = 2 and β = 0, weretrieve the Gaussian distribution. The last two parameters stand for the location ( µ ∈ R ) and thescale ( γ >
0) of the distribution. We do not have any analytic expression for the probability densityof X , but we can characterize the stable distribution by the mean of its characteristic function: t (cid:55)→ E [exp ( itX )] = (cid:26) exp (cid:0) iµt − γ α | t | α (cid:2) − iβ sign( t ) tan πα (cid:3)(cid:1) if α (cid:54) = 1exp (cid:0) iµt − γ | t | (cid:2) iβ sign( t ) π log | t | (cid:3)(cid:1) if α = 1 . We could use the Fourier transform to get the pdf from the characteristic function [68], but theabove parameterization is not totally satisfactory insofar as the pdf is not continuous in the pa-rameters, in particular when α = 1 [59, 2]. Indeed, when β >
0, the density is shifted right when α < α >
1, with a shift toward + ∞ (respectively −∞ ) when α tends toward 1by below (resp. above) [59]. For applications to data and interpretation of the coefficients, thisparameterization is thus to be avoided. For this reason, Nolan has proposed to use Zolotarev’s(M) parameterization [76], which is also often called the S parameterization. The characteristicfunction corresponding to X ∼ S α ( γ, β, µ ) is [59, 2]: t (cid:55)→ E [exp ( itX )] = (cid:26) exp (cid:0) iµ t − γ α | t | α (cid:2) iβ sign( t ) tan πα (cid:0) γ − α | t | − α − (cid:1)(cid:3)(cid:1) if α (cid:54) = 1exp (cid:0) iµ t − γ | t | (cid:2) iβ sign( t ) π (log | t | + log γ ) (cid:3)(cid:1) if α = 1 . (1)This alternative parameterization is not far from the S α one. The only difference is about thelocation parameter, which, in this new setting, corrects the shift exposed above for values of α close to 1: µ = (cid:26) µ + βγ tan πα if α (cid:54) = 1 µ + β π γ log γ if α = 1 . (2)A Fourier transform makes it possible to get the pdf of a standard variable S α (1 , β,
0) [59]: p ( x ; α, β ) = 1 π (cid:90) + ∞ cos ( h ( x, t ; α, β )) e − t α dt, h ( x, t ; α, β ) = (cid:26) xt + β tan πα ( t − t α ) if α (cid:54) = 1 xt + β π t log t if α = 1 . We also obtain the pdf of a variable X ∼ S α ( γ, β, µ ) with γ (cid:54) = 1 and µ (cid:54) = 0, by the meanof a translation, a scaling, and the substitution s = γt , starting from the characteristic functionprovided in equation (1): f ( x ; α, γ, β, µ ) = π (cid:82) + ∞ cos (cid:16) h (cid:16) x − µ γ , γt ; α, β (cid:17)(cid:17) e − ( γt ) α dt = γπ (cid:82) + ∞ cos (cid:16) h (cid:16) x − µ γ , s ; α, β (cid:17)(cid:17) e − s α ds = γ p (cid:16) x − µ γ ; α, β (cid:17) . The formula of p contains an integral on an unbounded interval. For this reason, other formulationshave been proposed [2, 50, 42, 12]. However, in the application to financial series, we find valuesof α far from 0, so that the truncated integral converges rapidly. We also get the correspondingcdf by numerically integrating f ( x ; α, γ, β, µ ). Many estimation methods of stable distributions exist [2, 12]. Some of them focus on the sole α parameter, using for instance a regression of extreme quantiles along with the extreme valuetheory [37]. Other methods make it possible to estimate all the parameters. In this class of methods,we can cite the estimation using L-moments, provided that α > α > β = 0, and µ = 0 [26]. This method is asymptoti-cally biased. McCulloch proposed an extended version of the method, in which he corrected theasymptotic bias [55]. This version is also less restrictive with respect to the parameters, insofar asit only requires to have α ≥ . v α = Q (0 . − Q (0 . Q (0 . − Q (0 . v β = Q (0 .
95) + Q (0 . − Q (0 . Q (0 . − Q (0 . , where Q ( p ) is the theoretical quantile of probability p for a S α ( γ, β, µ ) variable [55, 2, 12]. Neither v α nor v β depend on γ and µ , so that α and β are functions of v α and v β : α = φ ( v α , v β ) and β = φ ( v α , v β ). In practice, φ and φ are provided by tables [55]. Replacing the theoreticalquantiles Q ( p ) by empirical quantiles (cid:98) Q ( p ), we get estimators (cid:98) v α for v α and (cid:98) v β for v β , as well asthe following estimators for α and β : (cid:26) (cid:98) α = φ ( (cid:98) v α , (cid:98) v β ) (cid:98) β = φ ( (cid:98) v α , (cid:98) v β ) . γ and µ relies on two other intermediate quantities which only depend on thealready estimated α and β . For simplicity, we introduce the variable ζ defined by: ζ = (cid:26) µ + βγ tan πα if α (cid:54) = 1 µ if α = 1 . (3)The intermediate quantities are: v γ = Q (0 . − Q (0 . γ and v ζ = ζ − Q (0 . γ . They are such that v γ = φ ( α, β ) and v ζ = φ ( α, β ). Their estimators (cid:98) v γ and (cid:98) v ζ are obtained byreplacing the quantiles by empirical ones. We thus have the following estimators for γ and ζ : (cid:40) (cid:98) γ = (cid:98) Q (0 . − (cid:98) Q (0 . φ ( (cid:98) α, (cid:98) β ) (cid:98) ζ = (cid:98) γφ ( (cid:98) α, (cid:98) β ) + (cid:98) Q (0 . . The deduction of the estimator of µ from (cid:98) ζ is straightforward using equation (3), as well as theversion µ of the location parameter in the parameterization S using equation (2). In particular (cid:98) µ = (cid:98) ζ as soon as α (cid:54) = 1. The time-varying adaptation of McCulloch’s estimation technique amounts to estimating time-varying quantiles. Indeed, if we are able to infer dynamic quantiles Q t ( p ), the first two McCullochstatistics are defined in a dynamic fashion: (cid:40) v α,t = Q t (0 . − Q t (0 . Q t (0 . − Q t (0 . v β,t = Q t (0 . Q t (0 . − Q t (0 . Q t (0 . − Q t (0 . , so that we get time-varying α and β parameters. Time-varying quantiles make it also possibleto define the last two McCulloch statistics and finally to fully estimate a dynamic alpha-stableprobability distribution.The subject of estimating dynamic quantiles is largely handled by the econometric literature. Thefavoured appraoch is based on quantile autoregression [43, 20, 35], like in the application to value-at-risk known as CAViaR [24]. A drawback of quantile regression is that different quantiles maycross: the monotonicity of quantiles is not necessarily preserved. The dynamic additive quantile,while keeping the autoregressive approach, deals with this limitation [36].However, all the quantile autoregressions are based on a model of dynamic and we prefer tointroduce a method in which we do not specify the evolution of quantiles with respect to previouslyestimated ones. The inspiration of such a model-free approach comes from the non-parametricstatistics literature, in which we can estimate for example time-varying moments or even time-varying probability densities [38]. In this approach, close to an exponentially weighting movingaverage (EWMA), for each new observation, the weight of past observations is discounted at aconstant rate, whereas the new observation is provided with the highest weight. The rationaleis the following: the more recent the observation, the more likely its future occurrence. In the5WMA approach, the average A t , estimated at time t , incorporates the average estimated at time t − t , X t , in the following manner: A t = ωA t − + (1 − ω ) X t , (4)where ω ∈ (0 ,
1] is the discount factor. The higher ω , the more stable the series A t . We can alsointerpret equation (4) in a probabilistic perspective, that is by considering that weights are proba-bilities. Indeed, A t − is a weighted average of observations till time t −
1, and it is a mathematicalexpectation under a specific probability if the sum of the weights is equal to 1. If we suppose itis the case for the statistic A t − , then the probability associated in A t to each past observation ismultiplied by ω , and is thus globally equal to ω , whereas the new observation has the probabilitycomplementary to 1, that is 1 − ω . We thus have an underlying probability distribution at eachtime t , which associates the probability p ωt,i to the i -th observation, for 1 ≤ i ≤ t : p ωt,i = 1 − ω − ω t ω t − i . For ω (cid:54) = 1 and big values of t , ω t is close to zero and we have the approximation p ωt,i ≈ (1 − ω ) ω t − i .From these updating probabilities, it is possible to estimate time-varying kernel densities [38] butalso time-varying quantiles. These dynamic quantiles may rely on the dynamic kernel densitiesbut we prefer to work here with simple empirical quantiles, in order to avoid the complexity ofestimating dynamic kernel densities [74, 38]. For extreme quantiles, the algorithmic complexity ofour method is less than linear with respect to the number of observations considered at each date.We begin with a first estimation at date t of the quantile of probability p , (cid:98) Q ωt ( p ). To estimatethe quantile, we build the matrix M ωt containing the past observed price returns till t , { X i } i ≤ t ,sorted in descending order, along with their corresponding historical EWMA-like probability: M ωt = X π t ( t ) p ωt ,π t ( t ) ... ... X π t ( i ) p ωt ,π t ( i ) ... ... X π t (1) p ωt ,π t (1) , where X π t ( i ) is the i -th order statistic among the t first observations and is obtained with the helpof a permutation π t : min i ≤ t { X i } = X π t (1) ≤ X π t (2) ≤ ... ≤ X π t ( t ) = max i ≤ t { X i } . Following thedefinition of the generalized quantile, we get the following estimator for the quantile of probability p : (cid:98) Q ωt ( p ) = X π t (cid:18) min (cid:26) τ ∈{ ,...,t } (cid:12)(cid:12)(cid:12)(cid:12)(cid:80) τi =1 p ωt ,πt i ) ≥ p (cid:27)(cid:19) , (5)which is simply the lowest price return such that the cumulated probability associated to lowerreturns reaches p .Iteratively, we can update this quantile in the following manner. We suppose we are given theprobability distribution till time t − t . We firstapply a probability decay to M ωt − by a simple matrix product, so that we get a new matrix (cid:102) M ωt − containing sorted past observations till time t − t : (cid:102) M ωt − = M ωt − (cid:18) ω (cid:19) . t in the matrix (cid:102) M ωt − thanks to a binary search inits first column. In the inserted line, we write the corresponding probability 1 − ω in the secondcolumn. If we write I t the position of the new observation, we get the new probability distributionmatrix M ωt by: M ωt = X π t ( t ) p ωt,π t ( t ) ... ... X π t ( I t +1) p ωt,π t ( I t +1) X π t ( I t ) p ωt,π t ( I t ) X π t ( I t − p ωt,π t ( I t − ... ... X π t (1) p ωt,π t (1) = X π t − ( t − ωp ωt − ,π t − ( t − ... ... X π t − ( I t ) ωp ωt − ,π t − ( I t ) X t − ωX π t − ( I t − ωp ωt − ,π t − ( I t − ... ... X π t − (1) ωp ωt − ,π t − (1) . We can then calculate the empirical quantile in a manner similar to equation (5): (cid:98) Q ωt ( p ) = X π t (cid:16) min (cid:110) τ ∈{ ,...,t } (cid:12)(cid:12)(cid:12)(cid:80) τi =1 p ωt,πt ( i ) ≥ p (cid:111)(cid:17) . In order to diminish the algorithmic complexity of this method, if we are looking for a quantileabove the probability 0 .
5, we prefer the following definition of the quantile, which is mathematicallyconsistent with the one provided above: (cid:98) Q ωt ( p ) = X π t (cid:16) min (cid:110) τ ∈{ ,...,t } (cid:12)(cid:12)(cid:12)(cid:80) ti = τ +1 p ωt,πt ( i ) ≤ − p (cid:111)(cid:17) . Finally, using McCulloch’s method, these time-varying estimations of quantiles make it possible toestimate the dynamic parameters of the stable distribution, that we note (cid:98) α ωt , (cid:98) β ωt , (cid:98) γ ωt , and (cid:98) µ ωt . The above dynamic estimation of quantiles and of the parameters of stable distributions relies ona free parameter, the discount factor ω , which depicts how fast the dynamic distribution evolves.If ω is close to 1, the distribution is almost constant. If ω is lower, the evolution of the distributionis faster and the description of the last observations will be more accurate. Nevertheless, thisaccuracy may be excessive and the evolution of the distribution may be non-significant. Indeed, inthe extreme situation where ω is close to zero, the distribution will be very narrow and centeredon the last observation, with a big divergence between two successive distributions. In order tofind a good balance between accuracy and robustness, we decide to select the ω maximizing theability of the density (cid:98) f ωt estimated at date t to forecast the density of X t +1 , the price return attime t + 1.Several definitions of what a good density forecast is are possible. Indeed, the true density at time t + 1 is never observed, so that we can only rely on one observation drawn in this density. Inthe non-parametric literature about time-varying densities, we find for instance a selection rule forthe free parameter of the densities based on the maximization of a likelihood criterion [38]. Wethink that this criterion does not take properly into account the possibility of the occurrence ofextreme events. The alternative solution we follow is based on an adaptation of a method comingfrom the literature of density forecast evaluation [34]. Indeed, even if (cid:98) f ωt varies with t , so thatwe are provided with only one observation in this distribution, a simple transformation of eachprice return defines a distribution which remains the same through time. This transformation is7he probability integral transform (PIT), usually introduced in the perspective of density forecastevaluation [22]: Z ωt = (cid:98) F ωt − ( X t ) , (6)where (cid:98) F ωt − is the cdf corresponding to (cid:98) f ωt − . In this literature, two conditions are required for thePITs: the Z ωt must follow a uniform distribution in [0 ,
1] and they must be independent from eachother.The translation of these rules to the selection of free parameters in density estimation leads to twoproperties regarding ω [34]: (cid:46) uniformity of the PITs Z ωt , ..., Z ωT : ω is to be selected so as to minimize the divergencebetween their empirical distribution and a uniform distribution, (cid:46) independence of the PITs: ω is to be selected so as to minimize the discrepancy, that is,for each subinterval of [ t , T ] of size greater than a threshold ν , the divergence betweenthe empirical distribution of the corresponding PITs and a uniform distribution is to beminimized.In the perspective of the selection of a free parameter of a time-varying distribution, we can definethe above divergence as a Kolmogorov-Smirnov statistic with an adaptation to compare directlydivergences of distributions estimated on samples of different sizes [34]. Moreover, the minimalsize ν of the subintervals considered is intended to be a threshold above which the asymptoticframework required by the Kolmogorov-Smirnov statistic is satisfied. We consider ν = 22 days, sothat we expect the PITs to be uniform for scales larger than one month. As a consequence, thecriterion to be minimized is: d ν ( Z ωt +1 , ..., Z ωT ) = max t +1 ≤ s
2, the fBm is a Bm. If
H > / H < / H − / / − H ) of a Bm; increments are thus positively(resp. negatively) correlated.The absolute-moment method uses another definition of the fBm, which is consistent with theintegral form provided above. Indeed, an fBm is also the only zero-mean Gaussian process, withzero at the origin, such that, for s, t ≥ E { B H ( t ) B H ( s ) } = σ | t | H + | s | H − | t − s | H ) . From this covariance, we get the self-similarity property: E {| B H ( t ) − B H ( s ) | k } = σ | t − s | kH , for k >
0. This property states that the k -order absolute moment of the increments of duration | t − s | is proportional to | t − s | kH . Comparing two scales thus makes it possible to estimate H . Ifwe focus on an order k = 2 and on the two smallest scales, we indeed get the following estimator,for a time series of log-prices X , X , ..., X t [28]: (cid:98) H = 12 log (cid:32) ( t − (cid:80) t − i =1 ( X i +2 − X i ) ( t − (cid:80) t − i =1 ( X i +1 − X i ) (cid:33) , which converges almost surely toward H [8, 7].We could use this estimator of H as an indicator of market efficiency. But, we are mostly interestedhere in evaluating the evolution of market efficiency through time. We need therefore a time-varyingversion of this estimator. This question is not new and the solution put forward in the literature10s often based on the estimation in sliding windows [16, 10], possibly with a smoothing of the rawseries of Hurst exponents as a post-processing [28]. But we prefer a method more consistent withthe smoothing applied above for estimating the parameters of a distribution, that is in the mannerof an EWMA, insofar as it overweights more recent observations and is thus a more relevant pictureof the current state of the market. The closest method in the literature is a time-varying GHE usingthe EWMA [57]. The main difference with the method we propose is that the GHE is always basedon a linear regression of log-absolute moments of increments on several log time scales. Contraryto the GHE, we focus on only two scales, so that we get a simpler closed-form estimator: (cid:98) H ωt = 12 log (cid:32) ( t − (cid:80) t − i =1 ω t − − i ( X i +2 − X i ) ( t − (cid:80) t − i =1 ω t − − i ( X i +1 − X i ) (cid:33) . Beyond this simple formula, a more efficient implementation method is possible. Given the one-stepstatistic M ,t = (cid:80) t − i =1 ω t − − i ( X i +1 − X i ) and the two-step statistic M ,t = (cid:80) t − i =1 ω t − − i ( X i +2 − X i ) ,it consists in the following recurrence: M ,t +1 = ωM ,t + ( X t +1 − X t ) M ,t +1 = ωM ,t + ( X t +1 − X t − ) (cid:98) H ωt +1 = log (cid:16) tM ,t +1 ( t − M ,t +1 (cid:17) . The question of the optimal choice of the discount factor ω in the estimator (cid:98) H ωt is also to beaddressed. We could imagine to adapt the method exposed in section 2.4 to the case of an fBm.But it sounds more relevant to use the same discount factor for all the statistics of our work. Wewill thus simply use the discount factor chosen for estimating the time-varying stable distribution. The fLsm is a generalization of the fBm in which increments follow an alpha-stable distribution,which admits the Gaussian distribution as a particular case. The fLsm is defined as the fractionalintegral or fractional derivative of a symmetric L´evy-stable motion [70, 73, 71]: B H,α ( t ) = σC H,α (cid:90) ∞−∞ (cid:16) ( t − s ) H − /α + − ( − s ) H − /α + (cid:17) dL α ( s ) , where L α ( t ) is a symmetric α -stable process and C H,α = (cid:18)(cid:90) R (cid:12)(cid:12)(cid:12) ( t − s ) H − /α + − ( − s ) H − /α + (cid:12)(cid:12)(cid:12) α ds (cid:19) /α . In other words, increments follow a symmetric stable law and, if and only if H − /α >
0, non-overlapping increments are positively dependent, that is with a positive codifference or a positivecovariation [49]. The parameter H controls the scaling behaviour of the process, in the samemanner as in the fBm, and the parameter α controls the thickness of the tails. The lower α , thefatter the tails. If α = 2, increments are Gaussian and the fLsm is an fBm. When comparing anfBm and an fLsm, the fractal feature of the latter is not obtained only by adjusting the dependenceof the increments but, in addition, by tuning the kurtosis of the underlying law. Therefore, we canwrite the Hurst exponent as the combination of a tail component, 1 /α , and a memory parameter, m : H = 1 α + m.
11n this framework, the Hurst exponent is not the most relevant indicator of market efficiency andone should instead use m . As we have proposed time-varying estimators both for α and for H , wewrite the following time-varying estimator for m : (cid:98) m ωt = (cid:98) H ωt − (cid:98) α ωt , where (cid:98) α ωt is the estimator of the parameter of the stable law defined in section 2.3. Efficientmarkets then correspond to (cid:98) m ωt close to zero. We apply the above method to ten stock indices of various regions: USA (S&P 500, S&P 100),Europe (EURO STOXX 50, Euronext 100, DAX, CAC 40), Asia (Nikkei, KOSPI, SSE 180), andAustralia (S&P/ASX 200). We have used data from Yahoo finance in the time interval betweenthe 1st May 2015 and the 29th June 2020. The first date at which we estimate stable densitiesand the parameters of an fBm and an fLsm, that is t , is the 1st November 2019. The period ofstudy includes the financial crisis sparked by the COVID-19 pandemic.We first determine for each stock index the optimal discount factor ω as in equation (8). The resultsare displayed in Table 1. We observe that the optimal discount factors are close to 0.95, whateverthe index considered. For the rest of the empirical study, we consider a common discount factor ω m , so that we can make fair comparisons between stock indices. We choose ω m = 0 . ω (cid:63) S&P 500 0.946S&P 100 0.939EURO STOXX 50 0.949Euronext 100 0.954DAX 0.949CAC 40 0.952Nikkei 225 0.951KOSPI 0.952SSE 180 0.946S&P/ASX 200 0.956Mean value 0.949Median value 0.950Table 1: Optimal discount factor ω (cid:63) for several stock indices for stabledensities between November 2019 and June 2020.Using ω m , we are able to determine the dynamic pdf of daily price returns. We display in Figure 1these densities for four stock indices corresponding to regions with a different timing in the growthof the outbreak: S&P 500, the French CAC 40, the Chinese SSE 180, and the S&P/ASX 200indices. For each of these indices, we plot the pdf before the crisis, in November 2019, at the peakof the crisis, and a the end of our sample, late June 2020. The peak of the crisis is not the samefor all the indices. We define the peak date as the one leading to the maximal value for | m | . This12ate is in March for the four indices. Exact dates are provided in Table 3. For the four indices, thepdf at the peak shows fat tails and asymmetry. Late June, the pdf still has these features, exceptfor SSE 180 index, for which the pdf is very similar to the one before the crisis, indicating a veryfast recovery in China. The case of CAC 40 at the peak is also of interest, because the advent offat tails and asymmetry is so abrupt that it does not crush the body of the pdf, contrary to otherindices. Figure 1: Estimated dynamic stable density of daily price returns forS&P 500 (top left), CAC 40 (top right), S&P/ASX 200 (bottom left),and SSE 180 (bottom right) indices. The dotted line is the densityat the 5th November 2019, the black continuous line at the peak ofthe crisis, the grey line at the 29th June 2020.In this paper, we are mostly interested in determining whether the financial markets are efficientduring a financial crisis. For this purpose, we have introduced two indicators. The first one is thewidespread Hurst exponent, estimated here in a dynamic fashion as exposed in Section 3.1. Butthe Hurst exponent H provided above is an indicator of dependence between price returns only ifthese price returns are Gaussian. In a more general framework, if we consider the possibility offat tails by the mean of an alpha-stable distribution, we define another efficiency indicator by thememory parameter m of an fLsm, as exposed in Section 3.2. In the first approach, the market isefficient for H = 1 /
2. In the second approach, which is more accurate because it takes into accountthe kurtosis of price returns thanks to the α parameter, the market is efficient for m = 0. The nullhypothesis H is the efficiency of each market. In order to know for which threshold of efficiencyindicator we can reject H with a given confidence p , we perform a simulation. We consider that13he right price model corresponding to H is a geometric Bm. We thus simulate a time seriesfollowing this model. The length of the simulated series used for estimating the first parametersin t is the same as in our financial dataset. We then simulate 4,000 other dates. For each of thesedates, we estimate H , m , and α dynamically, using the discount factor ω m . We consider that thebounds of the confidence interval with confidence level p , for the estimated H , m , and α , are theempirical quantiles of the corresponding parameters, estimated on the simulations, for probabilities(1 − p ) / p ) / /
2, so that we cannot reject H during this period. During thecrisis, we observe very low Hurst exponents for the S&P 500 index which make it possible to reject H with a confidence of more than 99%. But for the three other indices, the drawdown of theHurst exponent is much less significant, in particular for the SSE 180 index. According to thisapproach, the market becomes clearly inefficient if we consider the S&P 500 index, whereas wecannot ascertain the inefficiency of the three other indices. It is also worth noting that after a peakdownward, the Hurst exponent reaches abnormally high values for the French and the Australianindices. It suggests a persistence of the crisis: a short mean-reverting phenomenon followed bypositively correlated price returns.Figure 2: Daily evolution through time of the estimated Hurst ex-ponent for S&P 500 (top left), CAC 40 (top right), S&P/ASX 200(bottom left), and SSE 180 (bottom right) indices. The grey lines arethe bounds of simulated confidence intervals with a confidence levelof 95%, 99%, and 99 . m of an14Lsm, the conclusions regarding the impact of COVID-19 on the market efficiency are not thesame. In Figure 3, we observe first that m is in general negative, even before the crisis. Itindicates a dominating mean-reversion phenomenon in the stock markets. But it is in fact oftennot significantly different from 0. When the crisis occurs, m goes downward and H is rejectedwith a confidence higher than 99%, whatever the stock index. The most significant drawdown of m is again for the S&P 500 index. The duration of the significant inefficiency varies among theindices. The longer period is for the S&P 500 index. The S&P/ASX 200 has a very short periodof inefficiency. Figure 3: Daily evolution through time of the estimated m parameterof an fLsm for S&P 500 (top left), CAC 40 (top right), S&P/ASX 200(bottom left), and SSE 180 (bottom right) indices. The grey lines arethe bounds of simulated confidence intervals with a confidence levelof 95%, 99%, and 99 . α , which depicts thesize of the tails of the distribution of price returns. When α = 2, the price returns follow a Gaussiandistribution. The lower α , the fatter the tails. For the American, French, and Australian indices,we observe in Figure 4 a negative impact of the crisis on α . It means that extreme events tendto occur more frequently and with a larger magnitude. This stylized fact is confirmed by anotherapproach relying on non-parametric densities [34]. The evolution of α in the Chinese market isnot similar to the three other indices and the values reached are less significantly different from2. A progressive recovery toward high values of α is visible for the CAC 40 and the S&P/ASX200 indices. For the S&P 500 index, we also observe an abrupt increase of α after the peak of thecrisis, but it is of limited amplitude and α remains at a fairly low value.We display in Tables 2 and 3 the range of values reached during the period by the two efficiency15igure 4: Daily evolution through time of the estimated α parameterof an fLsm for S&P 500 (top left), CAC 40 (top right), S&P/ASX 200(bottom left), and SSE 180 (bottom right) indices. The grey lines arethe bounds of simulated confidence intervals with a confidence levelof 95%, 99%, and 99 . H and m , for the ten stock indices considered. These tables confirm that the greatestimpact of COVID-19 on market efficiency occurred for US indices, whatever the efficiency indicator.We also note that inefficiency always leads to negative values of m . Indeed, the observed upperbounds in the period are never significantly different from 0. On the contrary, for the Hurstexponent, we find downward peaks below 1 / /
2, so that it isunclear whether inefficiency leads to high or low values for H . In fact, the presence of fatter tailsduring the crisis biases this efficiency indicator. So, focusing on the m indicator, we find fairlysynchronized peaks for indices of the same region: on 10th March 2020 in the USA, on 16th March2020 in Europe. Other regions seem less affected: the maximal | m | is indeed lower in Asia andAustralia. However, the biased H indicator suggests a similar loss of efficiency in Europe andin Asia. This underpins again the relevance of the refinement of the efficiency indicator to takeinto account the kurtosis. The two efficiency indicators also lead to opposite conclusions whenconsidering the situation at the end of the sample: according to H , markets are efficient againeverywhere, whereas they are significantly inefficient in the USA and in Japan according to m .16tock index Min H Date of the min Max H Date of the max H on T S&P 500 0.122 2020-03-04 0.565 2020-02-25 0.452S&P 100 0.098 2020-03-04 0.565 2020-02-25 0.427EURO STOXX 50 0.288 2020-02-20 0.665 2020-03-12 0.488Euronext 100 0.269 2020-02-20 0.680 2020-03-12 0.491DAX 0.296 2020-02-20 0.678 2020-03-12 0.502CAC 40 0.250 2020-02-20 0.676 2020-03-12 0.497Nikkei 225 0.237 2020-01-08 0.677 2020-03-24 0.458KOSPI 0.343 2020-01-23 0.757 2020-03-25 0.506SSE 180 0.307 2020-03-05 0.576 2019-12-18 0.440S&P/ASX 200 0.231 2020-02-20 0.686 2020-03-09 0.482Table 2: Range of values reached by the efficiency indicator H be-tween November 2019 and June 2020 for ten stock indices. T is the29th June 2020. An efficient market corresponds to H close to 1 / m Date of the min Max m Date of the max m on T S&P 500 -1.128 2020-03-10 0.065 2020-02-25 -0.355S&P 100 -1.065 2020-03-10 0.065 2020-02-25 -0.464EURO STOXX 50 -0.474 2020-03-18 0.043 2020-02-25 -0.188Euronext 100 -0.603 2020-03-16 0.055 2020-05-15 -0.190DAX -0.576 2020-03-16 0.152 2020-03-09 -0.250CAC 40 -0.690 2020-03-16 0.043 2020-02-25 -0.222Nikkei 225 -0.417 2020-06-29 0.139 2020-03-19 -0.417KOSPI -0.355 2020-06-24 0.148 2020-03-24 -0.290SSE 180 -0.460 2020-03-10 0.048 2019-11-06 -0.247S&P/ASX 200 -0.504 2020-03-26 0.097 2020-02-25 -0.225Table 3: Range of values reached by the efficiency indicator m be-tween November 2019 and June 2020 for ten stock indices. T is the29th June 2020. An efficient market corresponds to m close to 0.17 Conclusion
We have shown to which extent the stock markets become inefficient during the COVID-19 crisis.The efficiency is clearly rejected in the case of the S&P index. On the contrary, the Hurst exponentdoes not make it possible to conclude about a loss of efficiency for the CAC 40, SSE 180, andS&P/ASX 200 indices. However, we have highlighted a limitation of this indicator: it does nottake into account the dynamic kurtosis of price returns. Therefore, if we use the more appropriatememory parameter of an fLsm, we observe the occurrence of an inefficiency period almost at thebeginning of the crisis, even though it is less noticeable for the Chinese and the Australian indices.We have also introduced in this paper the tools used for this analysis, namely the estimation of adynamic stable distribution along with the estimation of the dynamic Hurst exponent and memoryparameter of an fLsm. An important free parameter in this approach is the discount factor whichis related to the speed at which the weight of past information decreases. We have used a selectionrule based on the minimization of a criterion depicting the uniformity and the independence of thePITs, consistently with the literature on the validation of density forecasts.
Acknowledgement
The authors deeply thank Akin Arslan, Thomas Barrat, and Sarah Bouabdallah for their valuablehelp in the implementation of some of the methods described in this paper.
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