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Are there Dragon Kings in the Stock Market?
Realized volatility $RV$ is the square root of realized variance, which is defined as follows
where
is the average realized variance over $n$ days and
are the daily returns with $S_{i}$ being the reference (closing) price on day $i$. This is an annualized value, where $252$ represents the number of trading days in a year. In particular, $n=1$ represent daily returns and $n=21$, being a typical number of trading days in a month, is useful for evaluating monthly RV. We point out however that in our calculation $n$ is simply a number of consecutive trading days that can fall on different weeks and months. Specifically, we performed our analysis for $n=1, 2, 3, 5, 7, 9, 13, 17, 21$. Here we present results for $n=1, 7, 21$ which already succinctly illustrate the changes in the RV distribution with $n$.
Since it is based on actual trades, realized volatility (RV) is the ultimate measure of market volatility, although the latter is more often associated with the implied volatility, most commonly measured by the VIX index cboevix,cboevixhistoric -- the so called market "fear index" -- that tries to predict RV of the S&P500 index for the following month. Its model-independent evaluation demeterfi1999guide is based on options contracts, which are meant to predict future stock prices fluctuations whitepaper2003cboe. The question of how well VIX predicts future realized volatility has been of great interest to researchers christensen1998relation, vodenska2013understanding, kownatzki2016howgood, russon2017nonlinear. Recent results dashti2019implied,dashti2021realized show that VIX is only marginally better than past RV in predicting future RV. In particular, it underestimates future low volatility and, most importantly, future high volatility. In fact, while both RV and VIX exhibit scale-free power-law tails, the distribution of the ratio of RV to VIX also has a power-law tail with a relatively small power exponent dashti2019implied,dashti2021realized, meaning that VIX is incapable of predicting large surges in volatility.
It should be emphasized that RV is agnostic with respect to gains or losses in stock returns. Nonetheless, it has been habitual that large gains and losses occur at around the same time. Here we wish to address the question of whether the largest values of RV fall on the power-law tail of the RV distribution. As is well known, the largest upheavals in the stock market happened on, and close to, the Black Monday, which was a precursor to the Savings and Loan crisis, the Tech Bubble, the Financial Crisis and the COVID Pandemic. Plotted on a log-log scale, power-law tails of a distribution show as a straight line. If the largest RV fall on the straight line they can be classified as Black Swans (BS). If, however, they show statistically significant deviations upward or downward from this straight line, they can be classified as Dragon Kings (DK) sornette2009,sornette2012dragon or negative Dragon Kings (nDK) respectively pisarenko2012robust.
The main result of this paper is that the largest values of RV are in fact nDK. We find that daily returns are the closest to the BS behavior. However, with the increase of $n$ we observe the development of "potential" DK with statistically significant deviations upward from the straight line. This trend terminates with the data points returning to the straight line and then abruptly plunging into nDK territory.
To gain further insight into this phenomenon, we start in Sec. (ref) with the time series of RV from 1970 to 2021, including expanded views of the aforementioned periods of market upheavals. In Sec. (ref) we give analytical expressions of the two distribution functions used to fit the entire RV distribution: modified Generalized Beta (mGB), which is discussed in great detail in a companion paper liu2023rethinking, and Generalized Beta Prime (GB2), which is essentially a limiting case of mGB and is chosen because it has power-law tails. mGB is chosen because it exhibits long stretch of power-law dependence before dropping off and terminating at a finite value of the variable, thus mimicking the nDK behavior of RV liu2023rethinking. Additionally, both mGB and GB2 emerge as steady-state distributions of a stochastic differential equation for stochastic volatility liu2023rethinking. In Sec. (ref) we describe fits of RV with mGB and GB2 and give a detailed description of the tails, specifically in regards to possible DK/nDK. Towards this end we also use a linear fit (LF) of the tails. For all three fits, we provide confidence intervals janczura2012black and, more importantly, the results of a U-test pisarenko2012robust, which evaluates a $p$-value for the null hypothesis that a data point comes from a fitting distribution pisarenko2012robust. Sec. (ref) is a discussion of results obtained in Sec. (ref).
Fig. (ref) shows the time series of RV for $n=1$, based on daily returns, $n=7$, and $n=21$, where $n$ is the number of days over which daily RV is averaged in ((ref)) and ((ref)). Only values with $RV>17$ are shown and black dots mark values $RV>10^{1.75} \approx 56$. It is clear that the time series progression with the increase of $n$ is towards a more pronounced amplification of singularly important events, such as periods corresponding to Black Monday, Financial Crisis and COVID pandemic -- even though the maximum values of RV understandably decrease in the same progression as averaging is taken over a larger number of days $n$. While such progression naturally leads to a question of whether these events might belong to the DK category, we shall see in what follows that they are actually nDK.
Figs. (ref) and (ref) give snapshots of the time series in Fig. (ref) around the largest volatility events: Fig. (ref) based on daily returns, $n=1$, and Fig. (ref) for $n=7$ and $n=21$ respectively. Based on Figs. (ref) -- (ref), Black Monday was clearly the most singular volatility event, while Financial Crisis and COVID Pandemic were distinguished by more prolonged periods of sustained extraordinarily large RV.
A companion paper liu2023rethinking discusses in great detail the modified Generalized Beta distribution function (mGB) used here to fit the distributions of RV. A generalization of the traditional mcdonald1995generalization GB can be written, in a slightly modified form relative to that of liu2023rethinking, as follows:
where $\beta _1$ and $\beta _2$ are scale parameters and $\alpha$, $p$ and $q$ are shape parameters, all positive, $B(p,q)$ is the beta function and $x \leq \beta _1$. Although it has a concise and transparent form, it does not come out as a solution of a stochastic differential equation (SDE) hertzler2003classical, which is desirable for the purpose of modeling behavior of quantities, such as stochastic volatility, important for understanding of RV dashti2021combined.
The probability density function (PDF) of mGB, which comes out as a solution of an SDE (with minor caveats explained in liu2023rethinking) and which is used here to model the RV distribution, can be written as
The cumulative distribution function (CDF) and complimentary CDF (CCDF) of mGB are given respectively by
and
where the first term in ((ref)) and ((ref)) represent, respectively, CDF and CCDF of GB (whose PDF is given by((ref))), while $I(y;p,q)=B(y;p,q)/B(p,q)$ and $B(y;p,q)$ are, respectively, the regularized and incomplete beta functions nist2022digital.
In what follows, we will be specifically interested in the $\beta_2\ll\beta_1$ circumstance since for $\beta_2 \ll x \ll \beta_1$ GB and mGB exhibit a power-law dependence,
In the limit of $\beta_1 \rightarrow \infty$, mGB and GB become, respectively, mGB2 and GB2 (the latter also known as Generalized Beta Prime) and are given by liu2023rethinking
and
Unlike mGB and GB, for whom the power-law dependences in ((ref)) eventually terminate at $\beta_1$, mGB2 and GB2 will sustain these power-law dependences indefinitely.
Below, we will use ((ref)) to fit CCDF of distributions of RV. As explained in liu2023rethinking, mGB2 and GB2 are equivalent since $q$ and $p$ are independently defined at this level GB family of distributions and $q$ can be shifted by unity in the definition of mGB2/GB2. Consequently, we choose a more familiar CCDF of GB2
to fit CCDF of the RV data. Insofar as the main difference between mGB and GB is concerned, it is their behavior near $\beta_1$ in the present context liu2023rethinking. Namely,
and
that is $1-F_{mGB}$ drops off to zero ($F_{mGB}$ saturates to unity) faster than $1-F_{GB}$ due to the factor $\left(\frac{\beta _2}{\beta _1}\right)^\alpha$. This feature accounts for a better fit via mGB versus GB, which may be due to the fact that mGB emerges from a physically motivated stochastic model liu2023rethinking.
We fit CCDF of the full RV distribution -- for the entire time span discussed in Sec. (ref) -- using mGB ((ref)) and GB2 ((ref)). The fits are shown on the log-log scale in Figs. (ref) -- (ref), together with the linear fit (LF) of the tails with $RV>40$. LF excludes the end points, as prescribed in pisarenko2012robust, that visually may be nDK candidates. (In order to mimic LF we also excluded those points in GB2 fits, which has minimal effect on GB2 fits, including the slope and KS statistic). To make the progression of the fits as a function of $n$ clearer, we included results for $n=5$ and $n=17$, in addition to $n=1,7,21$ that we used in Sec. (ref). Confidence intervals (CI) were evaluated per janczura2012black, via inversion of the binomial distribution. $p$-values were evaluated in the framework of the U-test, which is discussed in pisarenko2012robust and is based on order statistics:
where $x_{k,n}$ is the $k$'s member of numbers between $1$ and $n$ ordered by increasing magnitude (RV values in this case), and $F(x_{k,n})$ is the assumed CDF (mGB, GN2 and LF here).
For each $n$, from top to bottom, Figs. (ref) -- (ref) are organized as follows:
In the CI plots, upward pointing triangles indicate $p$-values consistent with DK, while downward pointing triangles indicate $p$-values consistent with nDK pisarenko2012robust. Fig. (ref) shows LF for $n=7,17,21$, where the last 10% of the range of values were excluded, that is values greater than $0.9\max\{RV\}$, as opposed to excluding points visually as in Figs. (ref) -- (ref). Fig. (ref) shows LF and GB2 slopes and Kolmogorv-Smirnov (KS) statistic for GB2 and mGB as a function of $n$; the horizontal line with the table value of KS statistic for our sample size massey1985kolmogorov is shown for guidance only since mGB and GB2 here are distributions with estimated parameters.
While the standard search for Dragon Kings involves performing a linear fit of the tails of the distribution pisarenko2012robust,janczura2012black, here we tried to broaden our analysis by also fitting the entire distribution using mGB ((ref)) and GB2 ((ref)) -- the two members of the Generalized Beta family of distributions liu2023rethinking, mcdonald1995generalization. As explained in the paragraph that follows ((ref)), the central feature of mGB is that, after exhibiting a long power-law dependence, it eventually terminates at a finite value of the variable. GB2, on the other hand, has a power-law tail that extends mGB's power-law dependence to infinity.
The key to understanding the results of fits in Sec. (ref) is the analysis of the structure of RV used by the markets -- a square root of realized variance ((ref)). At its core is the average of the consecutive daily realized variances ((ref)). Distribution of daily realized variance can be modeled using a duo of stochastic differential equations -- for stock returns and stochastic volatility -- which produces distributions of daily variance such as mGB liu2023rethinking and GB2 dashti2021combined. Via a simple change of variable, daily RV would then follow the same distributions but with renormalized parameters.
Even assuming the knowledge of the distribution of daily realized variance, finding the distribution of the averages constitutes a daunting task. To begin with, using convolution to evaluate the distribution of a sum of just two such complex distributions as mGB and GB2 is already not amenable to analytical evaluation. To complicate things further, the consecutive daily RV cannot be treated as independent identically distributed variable (i.i.d.) due to the correlations that persists up to roughly 5 - 7 days dashti2021realized.
With the above in mind, we first address Figs. (ref) -- (ref). According to Figs. (ref) and (ref), daily RV appears to be the closest of being commensurate with the Black Swan behavior as both LF and GB2 approximate the tail of the distribution better than mGB and LF does not point to existence of either DK, $p<0.05$, or nDK, $p>0.95$. The $n=1$ behavior undergoes a dramatic change with the increase of $n$, as seen in Figs. (ref) -- (ref), where we observe that, first, the "potential" DK, $p<0.05$, develop at the earlier portions of the tails, only to terminate in nDK at the tail ends.
Generally speaking, the existence of the large number of "potential" DK in the tail of the distribution indicates that the distribution is not describing the tail adequately. This becomes pronounced for large $n$ for all three fits -- LF, GB2 and mGB -- although less so for LF, which also does not exhibit "potential" DK for small $n$. However, if we adopt a different procedure for LF, whereby instead of visually excluding nDK candidates at the tail end we exclude the values whose RV is greater than $0.9$ of the maximum RV, we observe in Fig. (ref) that it has little effect on LF for small $n$ but all but eliminates "potential" DK.
For large $n$ we also observe that mGB approximates the tail end better than GB2 -- consistent with smaller KS values in Fig. (ref) and smaller number of nDK. However, neither approximates the preceding portion of the tail well as indicated by the "potential" DK. This has to do with the fact that neither of the distributions appear as a solution of a first-principle model describing average RV. Finally, in the first plot in Fig. (ref), we observe that after roughly 5 - 7 days the slope of the GB2 tail saturates, consistent with the correlation range of daily RV dashti2021realized. The slope of LF, on the other hand, increases with $n$. However neither is consistence with a naive assumption of the distribution having the same slope as that of the daily RV.
In conclusion, we showed that for daily returns distribution of realized volatility likely has a power-law tail, consistent with the Black Swan behavior. Multi-day realized volatility develops strong negative Dragon King signature as the number of days involved in averaging of daily realized variances increases. The breadth and strength of the S&P index analyzed here may be a contributing factor in suppressing the runaway power-law behavior. A natural extension of this work will be analysis of gains and losses of stock returns as well as of other large data sets that call into question possible power-law tails, such as incomes.