# Efficiency of the financial markets
during the COVID-19 crisis:
time-varying parameters of fractional stable dynamics

Ayoub Ammy-Driss${}^{\text{a}}$, Matthieu Garcin${}^{\text{b,}}$ Corresponding
author<EMAIL_ADDRESS>
${}^{\text{a}}$ ESILV, 92916 Paris La Défense, France.
${}^{\text{b}}$ Léonard de Vinci Pôle Universitaire, Research center, 92916
Paris La Défense, France.

###### Abstract

This paper investigates the impact of COVID-19 on financial markets. It
focuses on the evolution of the market efficiency, using two efficiency
indicators: the Hurst exponent and the memory parameter of a fractional Lévy-
stable motion. The second approach combines, in the same model of dynamic, an
alpha-stable distribution and a dependence structure between price returns. We
provide a dynamic estimation method for the two efficiency indicators. This
method introduces a free parameter, the discount factor, which we select so as
to get the best alpha-stable density forecasts for observed price returns. The
application to stock indices during 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 during the COVID-19 crisis
is even questionable.

Keywords – alpha-stable distribution, dynamic estimation, efficient market
hypothesis, financial crisis, Hurst exponent

##  1 Introduction

The COVID-19 pandemic has strongly affected many persons, either for medical
reasons or for the economic aftermath of the various prophylactic measures
decided by governments, in particular lockdowns. The evolution of financial
markets during the pandemic provides an illustration of the economic impact of
these measures. According to several empirical studies, financial markets have
indeed been strongly disturbed during this period [4, 6, 37, 65]. The question
of the reaction of financial markets to a crisis is not specific to the
COVID-19 pandemic. For example, we can cite a study of the impact of crisis in
the 80s and the 90s on a model of dynamic for a stock market [20]. In general,
these studies focus on variations of several statistics, such as jump
intensity, implied volatility, parameters of factor models, divergence of
price return densities, etc. To the best of our knowledge, no paper focuses on
measuring the impact of the COVID-19 pandemic on market efficiency.

Market efficiency is the ability of market prices to reflect all the available
information, so that no arbitrage is possible. In other words, if markets are
efficient, price returns are not correlated with each other and investors are
not able to statistically determine what is more profitable between selling
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 financial markets recover.

The Hurst exponent is a widespread indicator of market efficiency. The long-
range feature associated to a Hurst exponent above $1/2$ is indeed
traditionally interpreted as indicating predictability of a time series. It is
used in finance [53, 15, 58, 8] but also in many other fields such as
meteorology, for models of temperature [21]. It is often related to a model of
dynamic with a specific fractal property, namely the fractional Brownian
motion (fBm), introduced by Mandelbrot and van Ness [56]. The fractal property
states that the variance of increments of duration $\tau$ is
$\tau^{2H}\sigma^{2}$, where $H$ is the Hurst exponent and $\sigma^{2}$ the
variance of increments of duration 1. The fBm assumes that increments follow a
Gaussian distribution. The fractal property of the fBm is then obtained by
introducing a positive (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 Brownian motion (Bm), that is
with independent increments. In the fBm framework, this value of the Hurst
exponent 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 [50, 63, 34]. As an alternative to the Gaussian
distribution, the alpha-stable distribution is appealing because it includes
the Gaussian distribution as a particular case and entails fat tails, whose
amplitude is directly related to $\alpha$, a parameter of this distribution.
When combining alpha-stable distributions and dependence between the
successive price returns, we can get the fractional Lévy-stable motion (fLsm),
which is thus a non-Gaussian extension of the fBm [73, 74, 75]. In this
framework, the Hurst exponent is to be decomposed in $m+1/\alpha$, where $m$
is a memory parameter. If $m=0$, adjacent price returns are independent and
the market is efficient. We thus use this memory parameter as an alternative
efficiency indicator.

The estimation of the fLsm partly relies on the estimation of alpha-stable
distributions. Many estimation methods exist for this kind of distribution [2,
14]. In particular, we will focus on McCulloch’s method, which is based on
empirical quantiles [57]. In this paper, we are in addition interested in the
dynamic estimation of this distribution, because we want to depict the
chronology of the crisis, day after day. Several articles deal with the
estimation of time-varying non-parametric densities [40, 37]. 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 given date takes into account the estimation at the previous date updated by
the new observation. The balance 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 of a criterion
related to arguments coming from the field of validation of density forecast
[37, 25].

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 Hurst exponent $H$ detects
less often a market inefficiency than does the memory parameter $m$ of an
fLsm. 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 of this paper include a selection rule for the discount
factor of a dynamic parametric distribution, an estimation method of dynamic
Hurst exponents, and the introduction of the memory parameter of an fLsm as
efficiency indicator.

The rest of the paper is organized as follows. Section 2 introduces the
estimation of dynamic alpha-stable distributions, along with the selection
rule of the discount factor. Section 3 provides some elements on market
efficiency and details how indicators are built. Section 4 empirically studies
the impact of COVID-19 on market efficiency. Section 5 concludes.

##  2 Alpha-stable distributions

The alpha-stable distribution is a generalization of the Gaussian
distribution, appreciated for entailing fat tails. For this reason, it has
been widely invoked in signal processing [72, 35, 36], with applications for
example in medicine [71, 75] or in finance [50, 63, 34].

We present below the estimation of a time-varying alpha-stable distribution,
which we will apply later in this paper to financial time series. For this
purpose, we present first the static estimation as well as the various
representations of alpha-stable distributions. Regarding the dynamic
distribution, a free parameter is to be selected. It is the discount factor.
We propose a rule of selection in the last subsection.

###  2.1 Representation

Four parameters are used to depict a random variable following a stable
distribution: $X\sim S_{\alpha}(\gamma,\beta,\mu)$. The parameter
$\alpha\in(0,2]$ is the one we will mostly be interested in. It determines the
thickness of the tails. The parameter $\beta\in[-1,1]$ is a skewness
parameter. If $\alpha=2$ and $\beta=0$, we retrieve the Gaussian distribution.
The last two parameters stand for the location ($\mu\in\mathbb{R}$) and the
scale ($\gamma>0$) of the distribution. We do not have any analytic expression
for the probability density of $X$, but we can characterize the stable
distribution by the mean of its characteristic function:

$t\mapsto\mathbb{E}\left[\exp\left(itX\right)\right]=\left\\{\begin{array}[]{ll}\exp\left(i\mu
t-\gamma^{\alpha}|t|^{\alpha}\left[1-i\beta\text{sign}(t)\tan\frac{\pi\alpha}{2}\right]\right)&\text{if
}\alpha\neq 1\\\ \exp\left(i\mu
t-\gamma|t|\left[1+i\beta\text{sign}(t)\frac{2}{\pi}\log|t|\right]\right)&\text{if
}\alpha=1.\end{array}\right.$

We could use the Fourier transform to get the pdf from the characteristic
function [71], but the above parameterization is not totally satisfactory
insofar as the pdf is not continuous in the parameters, in particular when
$\alpha=1$ [62, 2]. Indeed, when $\beta>0$, the density is shifted right when
$\alpha<1$ and left when $\alpha>1$, with a shift toward $+\infty$
(respectively $-\infty$) when $\alpha$ tends toward 1 by below (resp. above)
[62]. For applications to data and interpretation of the coefficients, this
parameterization is thus to be avoided. For this reason, Nolan has proposed to
use Zolotarev’s (M) parameterization [79], which is also often called the
$S^{0}$ parameterization. The characteristic function corresponding to $X\sim
S^{0}_{\alpha}(\gamma,\beta,\mu_{0})$ is [62, 2]:

$t\mapsto\mathbb{E}\left[\exp\left(itX\right)\right]=\left\\{\begin{array}[]{ll}\exp\left(i\mu_{0}t-\gamma^{\alpha}|t|^{\alpha}\left[1+i\beta\text{sign}(t)\tan\frac{\pi\alpha}{2}\left(\gamma^{1-\alpha}|t|^{1-\alpha}-1\right)\right]\right)&\text{if
}\alpha\neq 1\\\
\exp\left(i\mu_{0}t-\gamma|t|\left[1+i\beta\text{sign}(t)\frac{2}{\pi}\left(\log|t|+\log\gamma\right)\right]\right)&\text{if
}\alpha=1.\end{array}\right.$ (1)

This alternative parameterization is not far from the $S_{\alpha}$ one. The
only difference is about the location parameter, which, in this new setting,
corrects the shift exposed above for values of $\alpha$ close to 1:

$\mu_{0}=\left\\{\begin{array}[]{ll}\mu+\beta\gamma\tan\frac{\pi\alpha}{2}&\text{if
}\alpha\neq 1\\\ \mu+\beta\frac{2}{\pi}\gamma\log\gamma&\text{if
}\alpha=1.\end{array}\right.$ (2)

A Fourier transform makes it possible to get the pdf of a standard variable
$S^{0}_{\alpha}(1,\beta,0)$ [62]:

$p(x;\alpha,\beta)=\frac{1}{\pi}\int_{0}^{+\infty}{\cos\left(h(x,t;\alpha,\beta)\right)e^{-t^{\alpha}}dt},$

with

$h(x,t;\alpha,\beta)=\left\\{\begin{array}[]{ll}xt+\beta\tan\frac{\pi\alpha}{2}\left(t-t^{\alpha}\right)&\text{if
}\alpha\neq 1\\\ xt+\beta\frac{2}{\pi}t\log t&\text{if
}\alpha=1.\end{array}\right.$

We also obtain the pdf of a variable $X\sim
S^{0}_{\alpha}(\gamma,\beta,\mu_{0})$ with $\gamma\neq 1$ and $\mu_{0}\neq 0$,
by the mean of a translation, a scaling, and the substitution $s=\gamma t$,
starting from the characteristic function provided in equation (1):

$\begin{array}[]{ccl}f(x;\alpha,\gamma,\beta,\mu_{0})&=&\frac{1}{\pi}\int_{0}^{+\infty}{\cos\left(h\left(\frac{x-\mu_{0}}{\gamma},\gamma
t;\alpha,\beta\right)\right)e^{-(\gamma t)^{\alpha}}dt}\\\
&=&\frac{1}{\gamma\pi}\int_{0}^{+\infty}{\cos\left(h\left(\frac{x-\mu_{0}}{\gamma},s;\alpha,\beta\right)\right)e^{-s^{\alpha}}ds}\\\
&=&\frac{1}{\gamma}p\left(\frac{x-\mu_{0}}{\gamma};\alpha,\beta\right).\end{array}$

The formula of $p$ contains an integral on an unbounded interval. For this
reason, other formulations have been proposed [2, 52, 44, 14]. However, in the
application to financial series, we find values of $\alpha$ far from 0, so
that the truncated integral converges rapidly. We also get the corresponding
cdf by numerically integrating $f(x;\alpha,\gamma,\beta,\mu_{0})$.

###  2.2 Estimation

Many estimation methods of stable distributions exist [2, 14]. Some of them
focus on the sole $\alpha$ parameter, using for instance a regression of
extreme quantiles along with the extreme value theory [22]. 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 $\alpha>1$ [2, 41],
empirical quantiles [57], the empirical characteristic function [69, 64, 47,
46, 75], and the maximum likelihood method [26]. This last method relies on
the knowledge of the pdf, which may be approximated as exposed in the previous
subsection, using Nolan’s work [62].

We focus on the method using empirical quantiles, which has been introduced
first by Fama and Roll, with the following assumptions: $\alpha>1$, $\beta=0$,
and $\mu=0$ [29]. This method is asymptotically biased. McCulloch proposed an
extended version of the method, in which he corrected the asymptotic bias
[57]. This version is also less restrictive with respect to the parameters,
insofar as it only requires to have $\alpha\geq 0.6$.

The McCulloch’s method consists in linking the four parameters to five
empirical quantiles of probability levels $5\%$, $25\%$, $50\%$, $75\%$, and
$95\%$. To this end, we first have to define two intermediate quantities:

$v_{\alpha}=\frac{Q(0.95)-Q(0.05)}{Q(0.75)-Q(0.25)}$

and

$v_{\beta}=\frac{Q(0.95)+Q(0.05)-2Q(0.50)}{Q(0.95)-Q(0.05)},$

where $Q(p)$ is the theoretical quantile of probability $p$ for a
$S_{\alpha}(\gamma,\beta,\mu)$ variable [57, 2, 14]. Neither $v_{\alpha}$ nor
$v_{\beta}$ depend on $\gamma$ and $\mu$, so that $\alpha$ and $\beta$ are
functions of $v_{\alpha}$ and $v_{\beta}$:
$\alpha=\phi_{1}(v_{\alpha},v_{\beta})$ and
$\beta=\phi_{2}(v_{\alpha},v_{\beta})$. In practice, $\phi_{1}$ and $\phi_{2}$
are provided by tables [57]. Replacing the theoretical quantiles $Q(p)$ by
empirical quantiles $\widehat{Q}(p)$, we get estimators $\widehat{v}_{\alpha}$
for $v_{\alpha}$ and $\widehat{v}_{\beta}$ for $v_{\beta}$, as well as the
following estimators for $\alpha$ and $\beta$:

$\left\\{\begin{array}[]{l}\widehat{\alpha}=\phi_{1}(\widehat{v}_{\alpha},\widehat{v}_{\beta})\\\
\widehat{\beta}=\phi_{2}(\widehat{v}_{\alpha},\widehat{v}_{\beta}).\end{array}\right.$

The estimation of $\gamma$ and $\mu$ relies on two other intermediate
quantities which only depend on the already estimated $\alpha$ and $\beta$.
For simplicity, we introduce the variable $\zeta$ defined by:

$\zeta=\left\\{\begin{array}[]{ll}\mu+\beta\gamma\tan\frac{\pi\alpha}{2}&\text{if
}\alpha\neq 1\\\ \mu&\text{if }\alpha=1.\end{array}\right.$ (3)

The intermediate quantities are:

$v_{\gamma}=\frac{Q(0.75)-Q(0.25)}{\gamma}$

and

$v_{\zeta}=\frac{\zeta-Q(0.50)}{\gamma}.$

They are such that $v_{\gamma}=\phi_{3}(\alpha,\beta)$ and
$v_{\zeta}=\phi_{4}(\alpha,\beta)$. Their estimators $\widehat{v}_{\gamma}$
and $\widehat{v}_{\zeta}$ are obtained by replacing the quantiles by empirical
ones. We thus have the following estimators for $\gamma$ and $\zeta$:

$\left\\{\begin{array}[]{l}\widehat{\gamma}=\frac{\widehat{Q}(0.75)-\widehat{Q}(0.25)}{\phi_{3}(\widehat{\alpha},\widehat{\beta})}\\\
\widehat{\zeta}=\widehat{\gamma}\phi_{4}(\widehat{\alpha},\widehat{\beta})+\widehat{Q}(0.50).\end{array}\right.$

The deduction of the estimator of $\mu$ from $\widehat{\zeta}$ is
straightforward using equation (3), as well as the version $\mu_{0}$ of the
location parameter in the parameterization $S^{0}$ using equation (2). In
particular $\widehat{\mu}_{0}=\widehat{\zeta}$ as soon as $\alpha\neq 1$.

###  2.3 Dynamic estimation

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 McCulloch statistics are defined in a
dynamic fashion:

$\left\\{\begin{array}[]{ccl}v_{\alpha,t}&=&\frac{Q_{t}(0.95)-Q_{t}(0.05)}{Q_{t}(0.75)-Q_{t}(0.25)}\\\
v_{\beta,t}&=&\frac{Q_{t}(0.95)+Q_{t}(0.05)-2Q_{t}(0.50)}{Q_{t}(0.95)-Q_{t}(0.05)},\end{array}\right.$

so that we get time-varying $\alpha$ and $\beta$ parameters. Time-varying
quantiles make it also possible to define the last two McCulloch statistics
and finally to fully estimate a dynamic alpha-stable probability distribution.

The subject of estimating dynamic quantiles is largely handled by the
econometric literature. The favoured appraoch is based on quantile
autoregression [45, 23, 38], like in the application to value-at-risk known as
CAViaR [27]. A drawback of quantile regression is that different quantiles may
cross: the monotonicity of quantiles is not necessarily preserved. The dynamic
additive quantile, while keeping the autoregressive approach, deals with this
limitation [39].

All the quantile autoregressions are based on a model of dynamic and we prefer
to introduce a method in which we do not specify the evolution of quantiles
with respect to previously estimated ones. The inspiration of such a model-
free approach comes from the non-parametric statistics literature, in which we
can estimate for example time-varying moments or even time-varying probability
densities [40]. In this perspective, we simply consider that the price returns
$X_{t}$ are alpha-stable random variables, each following its own alpha-stable
distribution $F_{t}$, defined by parameters $\alpha_{t}$, $\beta_{t}$,
$\omega_{t}$, and $\mu_{t}$. In each distribution $F_{t}$, we only observe one
variable $X_{t}$. This is not enough for estimating $F_{t}$. For this reason,
we add another assumption on the dynamic of this time-varying distribution. We
indeed consider that it evolves smoothly, so that price returns close in time
will follow a close distribution.

Thanks to this assumption, we can use several price returns at times close to
$t$ in order to estimate $F_{t}$ or its related quantiles $Q_{t}(p)$. We could
estimate these quantiles $Q_{t}(p)$ by their empirical versions. For example,
if one considers 100 observations, the empirical quantile of probability $p$
is the $\lceil 100p\rceil$-th smallest of the 100 observations. In other
words, each of the 100 observations is associated with a probability $1/100$.
However, price returns closer in time induce a lower bias in the estimation of
$Q_{t}(p)$, so that the probability associated with these recent returns in
the cdf related to $Q_{t}$ should be higher. Thus, we use an exponentially-
weighted quantile estimator (EWQ) and the corresponding discrete probability
$\widehat{P}_{t}^{EWQ}$, such that $\widehat{P}_{t}^{EWQ}(x)$ is the
probability for the price return at time $t$ to be equal to $x$ [60]. To this
end, we estimate the EWQ-style discrete probability function by assigning to
each observation $X_{i}$, for $i\in\llbracket 1,t\rrbracket$, a probability
$p^{\omega}_{t,i}$ depending on a discount factor $\omega\in(0,1)$, so that
$\widehat{P}_{t}^{EWQ}(X_{i})=p^{\omega}_{t,i}$. The probability
$p^{\omega}_{t,i}$ follows a standard form of exponential weight:

$p^{\omega}_{t,i}=\frac{1-\omega}{1-\omega^{t}}\omega^{t-i}.$

This expression ensures that the probabilities sum to 1, as required for
probabilities:

$\sum_{i=1}^{t}{p^{\omega}_{t,i}}=1.$

For $\omega\neq 1$ and big values of $t$, $\omega^{t}$ is close to zero and we
have the approximation $p^{\omega}_{t,i}\approx(1-\omega)\omega^{t-i}$. In
addition to the discrete probability $\widehat{P}_{t}^{EWQ}$, we can define
the cdf $\widehat{F}_{t}^{EWQ}$ associated with the EWQ:

$\widehat{F}_{t}^{EWQ}(x)=\sum_{i=1}^{t}{\widehat{P}_{t}^{EWQ}(X_{i})\mathds{1}_{X_{i}\leq
x}}=\sum_{i=1}^{t}{p^{\omega}_{t,i}\mathds{1}_{X_{i}\leq x}}.$ (4)

We can also write recursively this cdf [60]:

$\widehat{F}_{t}^{EWQ}(x)=\omega\widehat{F}_{t-1}^{EWQ}(x)+(1-\omega)\mathds{1}_{X_{t}\leq
x}.$

More precisely, for each new observation $X_{t}$, the EWQ-style cdf associated
with each past observations is discounted at the constant rate $\omega$,
whereas the new observation is provided with the highest probability,
$1-\omega$. The rationale is the following: the more recent the observation,
the more likely its future occurrence.

These discrete probabilities make it possible to calculate easily time-varying
quantiles, as generalized inverse functions of $\widehat{F}_{t}^{EWQ}$.
Compared to quantiles determined from dynamic kernel densities, which suffer
from a high algorithmic complexity [78, 40], the EWQ approach seems
computationally more performing. Indeed, for extreme quantiles, the
algorithmic complexity of our method is less than linear with respect to the
number of observations considered at each date.

We now describe precisely the procedure for calculating the EWQs. We begin
with a first estimation, at date $t_{0}$, of the quantile of probability $p$,
$\widehat{Q}^{\omega}_{t_{0}}(p)$, using the EWQ approach. To estimate the
quantiles, we build the matrix $M^{\omega}_{t_{0}}$ containing the past
observed price returns till $t_{0}$, $\\{X_{i}\\}_{i\leq t_{0}}$, sorted in
descending order, along with their corresponding historical EWQ-style
probability:

$M^{\omega}_{t_{0}}=\left(\begin{array}[]{cc}X_{\pi_{t_{0}}(t_{0})}&p^{\omega}_{t_{0},\pi_{t_{0}}(t_{0})}\\\
\vdots&\vdots\\\ X_{\pi_{t_{0}}(i)}&p^{\omega}_{t_{0},\pi_{t_{0}}(i)}\\\
\vdots&\vdots\\\
X_{\pi_{t_{0}}(1)}&p^{\omega}_{t_{0},\pi_{t_{0}}(1)}\end{array}\right),$

where $X_{\pi_{t_{0}}(i)}$ is the $i$-th order statistic among the $t_{0}$
first observations and is obtained with the help of a permutation
$\pi_{t_{0}}$: $\underset{i\leq t_{0}}{\min}\\{X_{i}\\}=X_{\pi_{t_{0}}(1)}\leq
X_{\pi_{t_{0}}(2)}\leq...\leq X_{\pi_{t_{0}}(t_{0})}=\underset{i\leq
t_{0}}{\max}\\{X_{i}\\}$. Thanks to the definition of the generalized quantile
and to equation (4), we get the following estimator for the quantile of
probability $p$:

$\widehat{Q}^{\omega}_{t_{0}}(p)=X_{\pi_{t_{0}}\left(\min\left\\{\tau\in\\{1,...,t_{0}\\}\left|\sum_{i=1}^{\tau}{p^{\omega}_{t_{0},\pi_{t_{0}}(i)}}\geq
p\right.\right\\}\right)},$ (5)

which is simply the lowest price return $X$ such that the cumulated
probability associated to lower returns, $\widehat{F}_{t_{0}}^{EWQ}(X)$,
reaches $p$.

Iteratively, we can update this quantile in the following manner. We suppose
we are given the probability distribution till time $t-1$ and that we want to
estimate a quantile at time $t$. We first apply a probability decay to
$M^{\omega}_{t-1}$ by a simple matrix product, so that we get a new matrix
$\widetilde{M}^{\omega}_{t-1}$ containing sorted past observations till time
$t-1$ along with their new probability at time $t$:

$\widetilde{M}^{\omega}_{t-1}=M^{\omega}_{t-1}\left(\begin{array}[]{cc}1&0\\\
0&\omega\end{array}\right).$

Then, we insert the new observation of time $t$ in the matrix
$\widetilde{M}^{\omega}_{t-1}$ thanks to a binary search in its first column.
In the inserted line, we write the corresponding probability $1-\omega$ in the
second column. If we write $\mathcal{I}_{t}$ the position of the new
observation, we get the new probability distribution matrix $M^{\omega}_{t}$
by:

$M^{\omega}_{t}=\left(\begin{array}[]{cc}X_{\pi_{t}(t)}&p^{\omega}_{t,\pi_{t}(t)}\\\
\vdots&\vdots\\\
X_{\pi_{t}\left(\mathcal{I}_{t}+1\right)}&p^{\omega}_{t,\pi_{t}\left(\mathcal{I}_{t}+1\right)}\\\
X_{\pi_{t}\left(\mathcal{I}_{t}\right)}&p^{\omega}_{t,\pi_{t}\left(\mathcal{I}_{t}\right)}\\\
X_{\pi_{t}\left(\mathcal{I}_{t}-1\right)}&p^{\omega}_{t,\pi_{t}\left(\mathcal{I}_{t}-1\right)}\\\
\vdots&\vdots\\\
X_{\pi_{t}(1)}&p^{\omega}_{t,\pi_{t}(1)}\end{array}\right)=\left(\begin{array}[]{cc}X_{\pi_{t-1}(t-1)}&\omega
p^{\omega}_{t-1,\pi_{t-1}(t-1)}\\\ \vdots&\vdots\\\
X_{\pi_{t-1}\left(\mathcal{I}_{t}\right)}&\omega
p^{\omega}_{t-1,\pi_{t-1}\left(\mathcal{I}_{t}\right)}\\\ X_{t}&1-\omega\\\
X_{\pi_{t-1}\left(\mathcal{I}_{t}-1\right)}&\omega
p^{\omega}_{t-1,\pi_{t-1}\left(\mathcal{I}_{t}-1\right)}\\\ \vdots&\vdots\\\
X_{\pi_{t-1}(1)}&\omega p^{\omega}_{t-1,\pi_{t-1}(1)}\end{array}\right).$

We can then calculate the empirical quantile in a manner similar to equation
(5):

$\widehat{Q}^{\omega}_{t}(p)=X_{\pi_{t}\left(\min\left\\{\tau\in\\{1,...,t\\}\left|\sum_{i=1}^{\tau}{p^{\omega}_{t,\pi_{t}(i)}}\geq
p\right.\right\\}\right)}.$

In order to diminish the algorithmic complexity of this method, if we are
looking for a quantile above the probability $0.5$, we prefer the following
definition of the quantile, which is mathematically consistent with the one
provided above:

$\widehat{Q}^{\omega}_{t}(p)=X_{\pi_{t}\left(\min\left\\{\tau\in\\{1,...,t\\}\left|\sum_{i=\tau+1}^{t}{p^{\omega}_{t,\pi_{t}(i)}}\leq
1-p\right.\right\\}\right)}.$

Finally, using McCulloch’s method, these time-varying estimations of quantiles
make it possible to estimate the dynamic parameters of the stable
distribution, that we note $\widehat{\alpha}^{\omega}_{t}$,
$\widehat{\beta}^{\omega}_{t}$, $\widehat{\gamma}^{\omega}_{t}$, and
$\widehat{\mu}^{\omega}_{t}$. We stress the fact that the EWQ method is only
intended to estimate non-parametric quantiles in order to infer time-varying
parameters. In particular, in what follows, the time-varying cdf on which we
focus is not the non-parametric EWQ-style cdf but the parametric alpha-stable
cdf with the time-varying parameters estimated above.

###  2.4 Selection of the discount factor

The above dynamic estimation of quantiles and of the parameters of stable
distributions relies on a free parameter, the discount factor $\omega$, which
depicts how fast the dynamic distribution evolves. If $\omega$ is close to 1,
the distribution is almost constant. If $\omega$ is lower, the evolution of
the distribution is faster and the description of the last observations will
be more accurate. Nevertheless, this accuracy may be excessive and the
evolution of the distribution may be non-significant. Indeed, in the extreme
situation where $\omega$ is close to zero, the distribution will be very
narrow and centered on the last observation, with a big divergence between two
successive distributions. In order to find a good balance between accuracy and
robustness, we decide to select the $\omega$ maximizing the ability of the
density $\widehat{f}^{\omega}_{t}$ estimated at date $t$ to forecast the
density of $X_{t+1}$, the price return at time $t+1$.

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. In the non-parametric literature about
time-varying densities, we find for instance a selection rule for the free
parameter of the densities based on the maximization of a likelihood criterion
[40]. We think that this criterion does not take properly into account the
possibility of the occurrence of extreme events. The alternative solution we
follow is based on an adaptation of a method coming from the literature of
density forecast evaluation [37]. Indeed, even if $\widehat{f}^{\omega}_{t}$
varies with $t$, so that we are provided with only one observation in this
distribution, a simple transformation of each price return defines a
distribution which remains the same through time. This transformation is the
probability integral transform (PIT), usually introduced in the perspective of
density forecast evaluation [25]:

$Z^{\omega}_{t}=\widehat{F}^{\omega}_{t-1}(X_{t}),$ (6)

where $\widehat{F}^{\omega}_{t-1}$ is the cdf corresponding to
$\widehat{f}^{\omega}_{t-1}$. In this literature, two conditions are required
for the PITs: the $Z^{\omega}_{t}$ must follow a uniform distribution in
$[0,1]$ and they must be independent from each other.

The translation of these rules to the selection of free parameters in density
estimation leads to two properties regarding $\omega$ [37]:

  * $\triangleright$

uniformity of the PITs $Z^{\omega}_{t_{0}},...,Z^{\omega}_{T}$: $\omega$ is to
be selected so as to minimize the divergence between their empirical
distribution and a uniform distribution,

  * $\triangleright$

independence of the PITs: $\omega$ is to be selected so as to minimize the
discrepancy, that is, for each subinterval of $[t_{0},T]$ of size greater than
a threshold $\nu$, the divergence between the empirical distribution of the
corresponding PITs and a uniform distribution is to be minimized.

In the perspective of the selection of a free parameter of a time-varying
distribution, we can define the above divergence as a Kolmogorov-Smirnov
statistic with an adaptation to compare directly divergences of distributions
estimated on samples of different sizes [37]. Moreover, the minimal size $\nu$
of the subintervals considered is intended to be a threshold above which the
asymptotic framework required by the Kolmogorov-Smirnov statistic is
satisfied. We consider $\nu=22$ days, so that we expect the PITs to be uniform
for scales larger than one month. As a consequence, the criterion to be
minimized is:

$d_{\nu}(Z^{\omega}_{t_{0}+1},...,Z^{\omega}_{T})=\underset{t_{0}+1\leq
s<s+\nu-1\leq t\leq T}{\max}\left(\sqrt{t-s+1}\times
k(Z^{\omega}_{s},...,Z^{\omega}_{t})\right),$

in which $k$ is the standard Kolmogorov-Smirnov statistic with respect to the
uniform distribution:

$k(Z^{\omega}_{s},...,Z^{\omega}_{t})=\underset{s\leq u\leq
t}{\max}\left|\frac{u-s}{t-s}-Z^{\omega}_{\rho(u;s,t)}\right|,$ (7)

where $v\in\llbracket a,b\rrbracket\mapsto\rho(v;a,b)$ is a permutation of
$\llbracket a,b\rrbracket$ defining the new order
$Z^{\omega}_{\rho(a;a,b)}\leq Z^{\omega}_{\rho(a+1;a,b)}\leq...\leq
Z^{\omega}_{\rho(b-1;a,b)}\leq Z^{\omega}_{\rho(b;a,b)}$ [37]. In equation
(7), $(u-s)/(t-s)$ is the empirical cdf of the PITs, whereas the sorted PIT
$Z^{\omega}_{\rho(u;s,t)}$ is the theoretical uniform cdf. Finally, the
optimal discount factor is defined as the solution of the following equation:

$\omega^{\star}=\underset{0<\omega<1}{\operatorname{argmin}}\text{
}d_{\nu}(Z^{\omega}_{t_{0}+1},...,Z^{\omega}_{T}).$ (8)

We can easily apply the above method to the case of an alpha-stable
distribution. We only have to pay attention to the definition of the PIT in
equation (6). Indeed, it relies on the cdf, which does not follow a simple
expression for alpha-stable distributions. Nevertheless, as exposed in
Subsection 2.1, numerical methods make it possible to calculate both the pdf
and the cdf of an alpha-stable distribution. Therefore, the estimated cdf
$x\mapsto\widehat{F}^{\omega}_{t-1}(x)$ in equation (6) is the numerically
evaluated cdf of an alpha-stable distribution with estimated parameters:
$x\mapsto
F(x;\widehat{\alpha}^{\omega}_{t-1},\widehat{\gamma}^{\omega}_{t-1},\widehat{\beta}^{\omega}_{t-1},(\widehat{\mu}_{0})^{\omega}_{t-1})$.

##  3 Market efficiency

The market efficiency is a usual assumption in finance, which is in particular
invoked when pricing derivatives. The EMH states that asset price series
follow a random walk [28]. Investors and market makers update their
expectations and their quotes at each instant, using the available
information, so that it is not possible to beat the market. The EMH is
convenient, because working with independent price returns makes the financial
mathematics easier. But practitioners know that the EMH is not very realistic.
The asset management industry in general aims at performing statistical
arbitrages, whatever the transaction time scale, whether it is less than a
second or more than a month. Maybe the asset manager will not win at each
time, but in average he should win. Standard models relying on the EMH, such
as the widespread geometric Brownian motion, are not consistent with the
existence of statistical arbitrages.

We can also stress other unrealistic components of the EMH, such as the
availability of the same information for every investor and market maker, as
well as the rationality of all the agents, or the fact that we can substitute
two assets provided that they are equally risky. Several alternatives to the
EMH aim at describing more realistically the financial markets. It is the case
of the fractal market hypothesis [68] and of the adaptive market hypothesis
(AMH) [55]. The AMH is a good compromise between the standard EMH and the
reality of statistical arbitrage. It states that a model can predict the
market in average. But such a model can provide investors with performing
forecasts during a limited time only, because other investors and market
makers will progressively adapt their own models and decisions to this
performing model. The AMH thus leads to a long-term efficiency of the market
and allows statistical arbitrages for small time scales only.

Besides, it is worth noting that the price series of some assets are not far
from the EMH. It is the case for major stock indices. But empirical studies
show that these indices have fluctuating efficiency and may encounter a loss
of efficiency during financial crisis [3, 54]. Since the decisions of market
makers and investors may not be the same if markets are efficient or if they
aren’t, it is important for them to determine from time series of prices
whether the markets are efficient or not. We can cite many statistical
indicators of market efficiency [70]. Some of them look for a predictability
of price returns [18], using for example the amplitude of the parameters of a
time-varying AR model [43, 61]. Other indicators measure a deviation from a
random walk, using for example variance ratios [17] or a combination of
several statistics such as fractal dimension and entropy [49, 48]. But the
most widespread indicator of market efficiency seems to be the Hurst exponent
[53, 15, 58, 31].

The Hurst exponent is an indicator of long-range memory. Mandelbrot and van
Ness introduced the fBm as a model consistent with a given Hurst exponent and
extending the standard Bm by making the increments dependent on each other
[56]. For this reason, the fBm is a popular model in finance, provided that
practitioners do not want to comply with the EMH. In the fBm, the Hurst
exponent $H$ is also linked to the fractal property of the series, insofar as
the variance of the increments of duration $\tau$ is $\tau^{2H}$ times the
variance of increments of duration 1. Some estimators of the Hurst exponent
use this fractal property instead of the long-range memory. However, given the
fractal property estimated on the dataset, the fBm is not the only possible
model. Other models may indeed have the same estimated Hurst exponent as the
fBm but, relying on another specification, they may lead to other conclusions
regarding the dependence of the increments and thus regarding the efficiency
of the markets [1]. This fact has been documented for instance for foreign
exchange rates, for which the stationarity of the time series biases the
estimation of the Hurst exponent in the perspective of an fBm [32, 33].
Indicators of market efficiency should not bypass the diversity of models
featuring a fractal property and consistent with the estimated Hurst exponent.
Extensions of the fBm include some specificities for the Hurst exponent: it
may vary deterministically through time, as in the multifractional Brownian
motion [66, 10, 19, 31], it may be a random process, as in the multifractional
process with random exponent [5, 13, 30, 34], or it may even be asymmetric
[16, 76].

In what follows, after a presentation of the Hurst exponent in the perspective
of the fBm, we will focus on a model in which increments are not necessarily
Gaussian. In this framework, the fLsm is a natural extension of the fBm, in
which increments follow an alpha-stable distribution [74, 77, 75, 34]. The
fractal property of this process takes into account both the dependence
between increments and the tail parameter of the alpha-stable distribution.
This model thus makes it possible to disentangle the kurtosis of price returns
and the efficiency of the market. It provides us with richer information than
the sole Hurst exponent. This refinement is not superfluous. We will indeed
see in the empirical section that the conclusions regarding the efficiency of
stock indices are not the same if we consider the Hurst exponent or the fLsm-
based indicator of market efficiency.

###  3.1 Hurst exponents

The Hurst exponent was originally introduced by Harold Edwin Hurst as an
indicator of long-range memory that could be obtained thanks to the rescaled
range (R/S) analysis [42]. Later, alternative estimation methods appeared,
such as the detrended fluctuation analysis [67], or the absolute-moment method
[10, 9, 11, 31], which is related to the notion of generalized Hurst exponent
(GHE) [7, 24]. The absolute-moment method is a method mainly used by the
community of statisticians of stochastic processes, because, contrary to the
R/S analysis, it is strongly related to a stochastic process, namely the fBm.

The fBm is a generalization of the standard Bm. Increments of the fBm are
Gaussian, since the fBm is a fractional integral or a fractional derivative of
a Bm, $W_{t}$ [56]:

$B_{H}(t)=\frac{\sigma}{\Gamma\left(H+\frac{1}{2}\right)}\int_{-\infty}^{\infty}{\left((t-s)_{+}^{H-1/2}-(-s)_{+}^{H-1/2}\right)dW(s)}.$

The two parameters of the fBm are the Hurst exponent $H$, and the volatility
parameter $\sigma$. If $H=1/2$, the fBm is a Bm. If $H>1/2$ (respectively
$H<1/2$), the fBm is the fractional integral (resp. fractional derivative) of
order $H-1/2$ (resp. $1/2-H$) of a Bm; increments are thus positively (resp.
negatively) correlated.

The absolute-moment method uses another definition of the fBm, which is
consistent with the integral form provided above. Indeed, an fBm is also the
only zero-mean Gaussian process, with zero at the origin, such that, for
$s,t\geq 0$:

$\mathbb{E}\\{B_{H}(t)B_{H}(s)\\}=\frac{\sigma^{2}}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}).$

From this covariance, we get the self-similarity property:

$\mathbb{E}\\{|B_{H}(t)-B_{H}(s)|^{k}\\}=\sigma^{2}|t-s|^{kH},$

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 $H$. If we 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 $X_{1},X_{2},...,X_{t}$ [31]:

$\widehat{H}=\frac{1}{2}\log_{2}\left(\frac{(t-1)\sum_{i=1}^{t-2}{(X_{i+2}-X_{i})^{2}}}{(t-2)\sum_{i=1}^{t-1}{(X_{i+1}-X_{i})^{2}}}\right),$

which converges almost surely toward $H$ [10, 9].

We could use this estimator of $H$ as an indicator of market efficiency. But,
we are mostly interested here in evaluating the evolution of market efficiency
through time. We need therefore a time-varying version of this estimator. This
question is not new and the solution put forward in the literature is often
based on the estimation in sliding windows [19, 12], possibly with a smoothing
of the raw series of Hurst exponents as a post-processing [31]. But we prefer
a method more consistent with the smoothing applied above for estimating the
parameters of a distribution, that is with an exponential weighting, insofar
as it overweights more recent observations and is thus a more relevant picture
of the current state of the market. The closest method in the literature is a
time-varying GHE using the exponentially-weighted moving average [59]. The
main difference with the method we propose is that the GHE is always based on
a linear regression of log-absolute moments of increments on several log time
scales. Contrary to the GHE, we focus on only two scales, so that we get a
simpler closed-form estimator:

$\widehat{H}^{\omega}_{t}=\frac{1}{2}\log_{2}\left(\frac{(t-1)\sum_{i=1}^{t-2}{\omega^{t-2-i}(X_{i+2}-X_{i})^{2}}}{(t-2)\sum_{i=1}^{t-1}{\omega^{t-1-i}(X_{i+1}-X_{i})^{2}}}\right).$

Beyond this simple formula, a more efficient implementation method is
possible. Given the one-step statistic
$M_{1,t}=\sum_{i=1}^{t-1}{\omega^{t-1-i}(X_{i+1}-X_{i})^{2}}$ and the two-step
statistic $M_{2,t}=\sum_{i=1}^{t-2}{\omega^{t-2-i}(X_{i+2}-X_{i})^{2}}$, it
consists in the following recurrence:

$\left\\{\begin{array}[]{ccl}M_{1,t+1}&=&\omega M_{1,t}+(X_{t+1}-X_{t})^{2}\\\
M_{2,t+1}&=&\omega M_{2,t}+(X_{t+1}-X_{t-1})^{2}\\\
\widehat{H}^{\omega}_{t+1}&=&\frac{1}{2}\log_{2}\left(\frac{tM_{2,t+1}}{(t-1)M_{1,t+1}}\right).\end{array}\right.$

The question of the optimal choice of the discount factor $\omega$ in the
estimator $\widehat{H}^{\omega}_{t}$ is also to be addressed. 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. We will thus simply use the discount factor chosen for estimating
the time-varying stable distribution.

###  3.2 Fractional Lévy-stable motion

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 fractional integral or fractional derivative
of a symmetric Lévy-stable motion [73, 74, 75]:

$B_{H,\alpha}(t)=\frac{\sigma}{C_{H,\alpha}}\int_{-\infty}^{\infty}{\left((t-s)_{+}^{H-1/\alpha}-(-s)_{+}^{H-1/\alpha}\right)dL_{\alpha}(s)},$

where $L_{\alpha}(t)$ is a symmetric $\alpha$-stable process and

$C_{H,\alpha}=\left(\int_{\mathbb{R}}{\left|(t-s)_{+}^{H-1/\alpha}-(-s)_{+}^{H-1/\alpha}\right|^{\alpha}ds}\right)^{1/\alpha}.$

In other words, increments follow a symmetric stable law and, if and only if
$H-1/\alpha>0$, non-overlapping increments are positively dependent, that is
with a positive codifference or a positive covariation [51]. The parameter $H$
controls the scaling behaviour of the process, in the same manner as in the
fBm, and the parameter $\alpha$ controls the thickness of the tails. The lower
$\alpha$, the fatter the tails. If $\alpha=2$, increments are Gaussian and the
fLsm is an fBm. When comparing an fBm and an fLsm, the fractal feature of the
latter is not obtained only by adjusting the dependence of the increments but,
in addition, by tuning the kurtosis of the underlying law. Therefore, we can
write the Hurst exponent as the combination of a tail component, $1/\alpha$,
and a memory parameter, $m$:

$H=\frac{1}{\alpha}+m.$

In this framework, the Hurst exponent is not the most relevant indicator of
market efficiency and one should instead use $m$. As we have proposed time-
varying estimators both for $\alpha$ and for $H$, we write the following time-
varying estimator for $m$:

$\widehat{m}^{\omega}_{t}=\widehat{H}^{\omega}_{t}-\frac{1}{\widehat{\alpha}^{\omega}_{t}},$

where $\widehat{\alpha}^{\omega}_{t}$ is the estimator of the parameter of the
stable law defined in section 2.3. Efficient markets then correspond to
$\widehat{m}^{\omega}_{t}$ close to zero.

##  4 Empirical study

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), and Australia (S&P/ASX 200). We have used data from
Yahoo finance in the time interval between the 1st May 2015 and the 29th June
2020. The first date at which we estimate stable densities and the parameters
of an fBm and an fLsm, that is $t_{0}$, is the 1st November 2019. The period
of study includes the financial crisis sparked by the COVID-19 pandemic.

We first determine for each stock index the optimal discount factor $\omega$
as in equation (8). The results are displayed in Table 1. We observe that the
optimal discount factors are close to 0.95, whatever the index considered. For
the rest of the empirical study, we consider a common discount factor
$\omega^{m}$, so that we can make fair comparisons between stock indices. We
choose $\omega^{m}=0.956$, which is the highest optimal discount factor
measured for the various stock indices. This conservative choice limits the
risk of providing a new observation with a too high weight in the dynamic
estimators. A lower value could lead to spurious conclusions regarding the
market efficiency for some stock indices.

Stock index | $\omega^{\star}$
---|---
S&P 500 | 0.946
S&P 100 | 0.939
EURO STOXX 50 | 0.949
Euronext 100 | 0.954
DAX | 0.949
CAC 40 | 0.952
Nikkei 225 | 0.951
KOSPI | 0.952
SSE 180 | 0.946
S&P/ASX 200 | 0.956
Mean value | 0.949
Median value | 0.950

Table 1: Optimal discount factor $\omega^{\star}$ for several stock indices
for stable densities between November 2019 and June 2020.

Using $\omega^{m}$, we are able to determine the dynamic pdf of daily price
returns. We display in Figure 1 these densities for four stock indices
corresponding to regions with a different timing in the growth of the
outbreak: S&P 500, the French CAC 40, the Chinese SSE 180, and the S&P/ASX 200
indices. For each of these indices, we plot the pdf before the crisis, in
November 2019, at the peak of the crisis, and a the end of our sample, late
June 2020. The peak of the crisis is not the same for all the indices. We
define the peak date as the one leading to the maximal value for $|m|$. This
date is in March for the four indices. Exact dates are provided in Table 3.
For the four indices, the pdf at the peak shows fat tails and asymmetry. Late
June, the pdf still has these features, except for SSE 180 index, for which
the pdf is very similar to the one before the crisis, indicating a very fast
recovery in China. The case of CAC 40 at the peak is also of interest, because
the advent of fat tails and asymmetry is so abrupt that it does not crush the
body of the pdf, contrary to other indices.

Figure 1: Estimated dynamic stable density of daily price returns for S&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 density at the 5th November 2019, the
black continuous line at the peak of the crisis, the grey line at the 29th
June 2020.

In this paper, we are mostly interested in determining whether the financial
markets are efficient during a financial crisis. For this purpose, we have
introduced two indicators. The first one is the widespread Hurst exponent,
estimated here in a dynamic fashion as exposed in Section 3.1. But the Hurst
exponent $H$ provided above is an indicator of dependence between price
returns only if these price returns are Gaussian. In a more general framework,
if we consider the possibility of fat tails by the mean of an alpha-stable
distribution, we define another efficiency indicator by the memory parameter
$m$ of an fLsm, as exposed in Section 3.2. In the first approach, the market
is efficient for $H=1/2$. In the second approach, which is more accurate
because it takes into account the kurtosis of price returns thanks to the
$\alpha$ parameter, the market is efficient for $m=0$. The null hypothesis
$H_{0}$ is the efficiency of each market. In order to know for which threshold
of efficiency indicator we can reject $H_{0}$ with a given confidence $p$, we
perform a simulation. We consider that the right price model corresponding to
$H_{0}$ is a geometric Bm. We thus simulate a time series following this
model. The length of the simulated series used for estimating the first
parameters in $t_{0}$ is the same as in our financial dataset. We then
simulate 4,000 other dates. For each of these dates, we estimate $H$, $m$, and
$\alpha$ dynamically, using the discount factor $\omega^{m}$. We consider that
the bounds of the confidence interval with confidence level $p$, for the
estimated $H$, $m$, and $\alpha$, are the empirical quantiles of the
corresponding parameters, estimated on the simulations, for probabilities
$(1-p)/2$ and $(1+p)/2$.

We display the time-varying Hurst exponent in Figure 2 for the four focal
stock indices of our study. Before the crisis, which begins in February or
March with respect to the region, the Hurst exponent is not significantly
different from $1/2$, so that we cannot reject $H_{0}$ during this period.
During the crisis, we observe very low Hurst exponents for the S&P 500 index
which make it possible to reject $H_{0}$ with a confidence of more than
$99\%$. But for the three other indices, the drawdown of the Hurst exponent is
much less significant, in particular for the SSE 180 index. According to this
approach, the market becomes clearly inefficient if we consider the S&P 500
index, whereas we cannot ascertain the inefficiency of the three other
indices. It is also worth noting that after a peak downward, the Hurst
exponent reaches abnormally high values for the French and the Australian
indices. It suggests a persistence of the crisis: a short mean-reverting
phenomenon followed by positively correlated price returns.

Figure 2: Daily evolution through time of the estimated Hurst exponent 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 are the bounds of simulated confidence
intervals with a confidence level of $95\%$, $99\%$, and $99.5\%$.

If we consider another indicator of market efficiency, namely the memory
parameter $m$ of an fLsm, the conclusions regarding the impact of COVID-19 on
the market efficiency are not the same. In Figure 3, we observe first that $m$
is in general negative, even before the crisis. It indicates a dominating
mean-reversion phenomenon in the stock markets. But it is in fact often not
significantly different from 0. When the crisis occurs, $m$ goes downward and
$H_{0}$ is rejected with 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 the indices. The
longer period is for the S&P 500 index. The S&P/ASX 200 has a very short
period of inefficiency.

Figure 3: Daily evolution through time of the estimated $m$ parameter of 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 are the bounds of simulated
confidence intervals with a confidence level of $95\%$, $99\%$, and $99.5\%$.

It is also interesting to track another important parameter of the fLsm,
namely $\alpha$, which depicts the size of the tails of the distribution of
price returns. When $\alpha=2$, the price returns follow a Gaussian
distribution. The lower $\alpha$, the fatter the tails. For the American,
French, and Australian indices, we observe in Figure 4 a negative impact of
the crisis on $\alpha$. It means that extreme events tend to occur more
frequently and with a larger magnitude. This stylized fact is confirmed by
another approach relying on non-parametric densities [37]. The evolution of
$\alpha$ in the Chinese market is not similar to the three other indices and
the values reached are less significantly different from 2. A progressive
recovery toward high values of $\alpha$ is visible for the CAC 40 and the
S&P/ASX 200 indices. For the S&P 500 index, we also observe an abrupt increase
of $\alpha$ after the peak of the crisis, but it is of limited amplitude and
$\alpha$ remains at a fairly low value.

Figure 4: Daily evolution through time of the estimated $\alpha$ parameter of
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 are the bounds of simulated
confidence intervals with a confidence level of $95\%$, $99\%$, and $99.5\%$.

We display in Tables 2 and 3 the range of values reached during the period by
the two efficiency indicators, $H$ and $m$, for the ten stock indices
considered. These tables confirm that the greatest impact 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 upper bounds in the period are never significantly different from
0. On the contrary, for the Hurst exponent, we find downward peaks below $1/2$
as well as upward peaks above $1/2$, so that it is unclear whether
inefficiency leads to high or low values for $H$. In fact, the presence of
fatter tails during the crisis biases this efficiency indicator. So, focusing
on the $m$ indicator, we find fairly synchronized peaks for indices of the
same region: on 10th March 2020 in the USA, on 16th March 2020 in Europe.
Other regions seem less affected: the maximal $|m|$ is indeed lower in Asia
and Australia. However, the biased $H$ indicator suggests a similar loss of
efficiency in Europe and in Asia. This underpins again the relevance of the
refinement of the efficiency indicator to take into account the kurtosis. The
two efficiency indicators also lead to opposite conclusions when considering
the situation at the end of the sample: according to $H$, markets are
efficient again everywhere, whereas they are significantly inefficient in the
USA and in Japan according to $m$.

Stock 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 | 0.452
S&P 100 | 0.098 | 2020-03-04 | 0.565 | 2020-02-25 | 0.427
EURO STOXX 50 | 0.288 | 2020-02-20 | 0.665 | 2020-03-12 | 0.488
Euronext 100 | 0.269 | 2020-02-20 | 0.680 | 2020-03-12 | 0.491
DAX | 0.296 | 2020-02-20 | 0.678 | 2020-03-12 | 0.502
CAC 40 | 0.250 | 2020-02-20 | 0.676 | 2020-03-12 | 0.497
Nikkei 225 | 0.237 | 2020-01-08 | 0.677 | 2020-03-24 | 0.458
KOSPI | 0.343 | 2020-01-23 | 0.757 | 2020-03-25 | 0.506
SSE 180 | 0.307 | 2020-03-05 | 0.576 | 2019-12-18 | 0.440
S&P/ASX 200 | 0.231 | 2020-02-20 | 0.686 | 2020-03-09 | 0.482

Table 2: Range of values reached by the efficiency indicator $H$ between
November 2019 and June 2020 for ten stock indices. $T$ is the 29th June 2020.
An efficient market corresponds to $H$ close to $1/2$.

Stock index | Min $m$ | 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 | -0.355
S&P 100 | -1.065 | 2020-03-10 | 0.065 | 2020-02-25 | -0.464
EURO STOXX 50 | -0.474 | 2020-03-18 | 0.043 | 2020-02-25 | -0.188
Euronext 100 | -0.603 | 2020-03-16 | 0.055 | 2020-05-15 | -0.190
DAX | -0.576 | 2020-03-16 | 0.152 | 2020-03-09 | -0.250
CAC 40 | -0.690 | 2020-03-16 | 0.043 | 2020-02-25 | -0.222
Nikkei 225 | -0.417 | 2020-06-29 | 0.139 | 2020-03-19 | -0.417
KOSPI | -0.355 | 2020-06-24 | 0.148 | 2020-03-24 | -0.290
SSE 180 | -0.460 | 2020-03-10 | 0.048 | 2019-11-06 | -0.247
S&P/ASX 200 | -0.504 | 2020-03-26 | 0.097 | 2020-02-25 | -0.225

Table 3: Range of values reached by the efficiency indicator $m$ between
November 2019 and June 2020 for ten stock indices. $T$ is the 29th June 2020.
An efficient market corresponds to $m$ close to $0$.

##  5 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 exponent does not make it possible to
conclude about a loss of efficiency for the CAC 40, SSE 180, and S&P/ASX 200
indices. However, we have highlighted a limitation of this indicator: it does
not take into account the dynamic kurtosis of price returns. Therefore, if we
use the more appropriate memory parameter of an fLsm, we observe the
occurrence of an inefficiency period almost at the beginning 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 a dynamic stable distribution along with the estimation of
the dynamic Hurst exponent and memory parameter of an fLsm. An important free
parameter in this approach is the discount factor which is related to the
speed at which the weight of past information decreases. We have used a
selection rule based on the minimization of a criterion depicting the
uniformity and the independence of the PITs, consistently with the literature
on the validation of density forecasts.

## Acknowledgement

The authors deeply thank Akin Arslan, Thomas Barrat, and Sarah Bouabdallah for
their valuable help in the implementation of some of the methods described in
this paper.

## References

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