Quantitative Finance
ISSN: 1469-7688 (Print) 1469-7696 (Online) Journal homepage: www.tandfonline.com/journals/rquf20
Modeling variance risk in financial markets using
power-laws: new evidence from the Garman-Klass
variance estimator
Masoumeh Fathi & Klaus Grobys
To cite this article: Masoumeh Fathi & Klaus Grobys (07 Aug 2025): Modeling variance risk in
financial markets using power-laws: new evidence from the Garman-Klass variance estimator,
Quantitative Finance, DOI: 10.1080/14697688.2025.2529485
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Quantitative Finance, 2025
https://doi.org/10.1080/14697688.2025.2529485
Modeling variance risk in financial markets using
power-laws: new evidence from the Garman-Klass
variance estimator
MASOUMEH FATHI†* and KLAUS GROBYS†‡∗∗
†Finance Research Group, School of Accounting and Finance, University of Vaasa, Wolffintie 34, Vaasa 65200, Finland
‡Innovation and Entrepreneurship (InnoLab), University of Vaasa, Wolffintie 34, Vaasa 65200, Finland
(Received 2 January 2025; accepted 26 June 2025 )
This study examines the range-based variance risk of five key financial asset markets—S&P 500,
gold, crude oil, the USD/GBP exchange rate, and Bitcoin—using the noise-efficient Garman-Klass
variance estimator. Our findings corroborate previous research by demonstrating that range-based
asset variances adhere to power-law behavior generating variance behavior that is effectively infi-
nite in practical terms. Furthermore, we provide novel evidence that the widely accepted log-normal
model is unequivocally rejected for all range-based asset variances, underscoring its inadequacy
in capturing the statistical properties of financial asset variances. A key contribution of this study
is the discovery that a power-law function with α ≈ 2.8 represents a universal law governing the
cross-sectional variances of otherwise unrelated asset markets. These findings have significant impli-
cations for risk management frameworks that rely on models developed within the mean-variance
space, as they highlight the limitations of traditional approaches in assessing and managing financial
risks.
Keywords: Garman-Klass estimator; Power-laws; Range-based variance; Second moment; Variance
of variance
JEL Classification: C18, G10, G12, G14
1. Introduction
The seminal study by Markowitz (1952) laid the ground-
work for portfolio analysis of financial assets, introducing
the mean-variance framework as the cornerstone for opti-
mizing portfolios of risky assets. Building on this founda-
tion, Treynor (1962), Sharpe (1964), Lintner (1965), and
Mossin (1966) developed the Capital Asset Pricing Model
(CAPM), which remains a foundational concept in asset pric-
ing literature. However, empirical failures of the CAPM have
inspired the development of various factor models, culminat-
ing in what Feng et al. (2020) term a ‘factor zoo.’ Among
these, the Fama-French factor models (Fama and French
1992, 1993, 2015, 2017, 2018, 2020) are widely recognized
∗Corresponding author. Email: masoumeh.fathi@uwasa.fi
∗∗Current address: Department of Monetary Economics and
International Finance, Christian-Albrechts University (CAU) of
Kiel, Wilhelm-Seelig Platz 1, 24118 Kiel, Germany
as benchmark models in empirical investigations aimed at
addressing CAPM’s limitations.
It is well understood that ‘true’ volatility is unobserv-
able, and the literature offers various proxies to estimate it.
These include the sample variance, conditional variance mod-
eled through GARCH or stochastic volatility frameworks,
and realized and range-based volatility derived from high-
frequency or intraday price data. Range-based estimators,
such as those proposed by Parkinson (1980) and Garman and
Klass (1980), have been shown to be particularly efficient
and robust, especially in the presence of market microstruc-
ture noise, and are widely used as nonparametric measures of
realized variance (Chou et al. 2010).
While these factor models offer an appealing theoretical
framework, their reliance on the mean-variance space presents
critical challenges if the variance is statistically undefined.
In his commentary on Mandelbrot’s (1963) ‘infinite variance
hypothesis,’ which profoundly disrupted traditional finance
theory, Fama (1963) cautioned that statistical tools assum-
ing finite variance—such as least-squares regression—could
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2 M. Fathi and K. Grobys
yield misleading results if variance is infinite. Subsequent
evidence suggests that many proposed asset pricing factors
may suffer from sample-specificity (Chan et al. 2000, Hirsh-
leifer 2001, Schwert 2003). Extending this discourse, Grobys
(2021) posited that replication failures in financial economics
could stem from infinite variances or infinite variance of
variances. Importantly, the qualitative implications of these
phenomena are effectively identical.
This study aims to address a fundamental question: does the
population mean of range-based variances in financial mar-
kets exist? Here, we focus on range-based realized variances,
which serve as ex-post estimators of volatility and approxi-
mate the latent conditional variance of returns. Unlike model-
based residual variance, which reflects unexplained variation
after regression and depends on model assumptions, range-
based variance is directly observable and model-free. If the
distribution of these range-based variances exhibits extremely
heavy tails, this implies that variance estimates themselves
are unstable and potentially without finite moments. Conse-
quently, the residual variance in linear regression models (e.g.
CAPM or Fama-French), which requires the theoretical vari-
ance of the inputs to be statistically well-defined, may itself
become undefined, thereby invalidating classical inference
tools such as standard errors, t-statistics, and R2. This makes
the investigation of tail behavior in, for example, range-based
variance distributions essential for assessing the reliability of
asset pricing models built on OLS regression.
To this end, we investigate range-based variances in five
major financial asset markets—S&P 500, gold, crude oil, the
USD/GBP exchange rate, and Bitcoin—using intraday data.
Range-based variances are computed using the Garman-Klass
range-based variance estimator. We also fit power-law models
to the data using maximum likelihood estimation (MLE) to
examine the statistical properties of extreme variance realiza-
tions, with a specific focus on the tail behavior of the distribu-
tion beyond the range of typical observations. To assess model
suitability, we conduct comprehensive goodness-of-fit tests,
comparing the power-law hypothesis against other plausi-
ble distributions, such as log-normal and exponential models.
Given the dependency structures in financial variance data
(e.g. volatility clustering), we orthogonalize the range-based
variance data through autoregressive modeling and reapply
goodness-of-fit tests to the resulting innovation processes.
Furthermore, we test for commonalities in the range-based
variances of unrelated asset markets by examining whether
power-law behavior is statistically invariant across these mar-
kets. Finally, we conduct an extensive set of robustness checks
to validate our findings, including sample-split tests, the use
of alternative volatility measures (e.g. realized variance), the
examination of additional asset classes (e.g. bonds), the appli-
cation of Bayesian inference to estimate tail exponents and
cutoffs, model comparisons using Akaike Information Crite-
rion (AIC) alongside the Kolmogorov–Smirnov test, rolling-
window estimations, and the fitting of tempered stable and
Generalized Beta of the Second Kind (GB2) distributions to
assess the stability and generality of the observed power-law
behavior.
This study makes several significant contributions to
the literature, addressing critical methodological, theoreti-
cal, and empirical gaps. These contributions advance our
understanding of range-based variances in financial markets
and their implications for asset pricing and risk management.
First, previous research, such as Grobys (2021), relied on
the Parkinson variance estimator (Parkinson 1980) to com-
pute range-based variances. While this approach has merits,
Molnár (2012) highlighted that range-based variance estima-
tors, such as those proposed by Garman and Klass (1980),
outperform alternatives when accounting for noise in high-
frequency data. Chou et al. (2010) also emphasized the
informational advantage of intraday price ranges over clos-
ing prices. By adopting the Garman-Klass variance estimator,
this study mitigates potential biases inherent in the use of
inflated variance estimators, as in the study of Grobys (2021).
Our methodological adjustment ensures greater precision in
capturing range-based variances, offering a robust founda-
tion for further analysis. This refinement is essential to derive
reliable insights, particularly when assessing heavy-tailed dis-
tributions like power laws, and aligns with calls from Molnár
(2012) for a more rigorous treatment of variance estimation
methods.
Second, realized volatility has long been theorized as
log-normal, with Andersen et al. (2001) providing seminal
empirical support for this assumption. However, evidence
challenging this perspective has been mounting. Renó and
Rizza (2003), for instance, demonstrated that volatility in
Italian futures markets aligns more closely with a Pareto dis-
tribution, necessitating the rejection of log-normality. Fathi
et al. (2024) further confirmed power law distributions as
superior in capturing realized variances in G10 currencies.
Our study expands on this work by rigorously comparing
the log-normal, exponential, and power-law models using
comprehensive goodness-of-fit tests. We provide robust evi-
dence that power laws offer the most plausible framework
for modeling daily range-based variances too. By reconcil-
ing theoretical assumptions with observed data, this study
extends Grobys’ (2021) findings, which focused exclusively
on power-law validity.
Third, Grobys (2021) used a relatively narrow sample, such
as range-based variances for gold from August 30, 2000, to
March 31, 2021, missing critical periods such as the gold price
bubbles of the 1980s. Data omissions of this nature can lead
to upward-biased estimates of power-law exponents, particu-
larly if extreme events are underrepresented. To address this
limitation, we extend the dataset for gold to cover February
7, 1980, to July 31, 2024, more than doubling the observation
count to 11 196. This comprehensive data expansion ensures
that periods of significant market stress are incorporated, mit-
igating biases and strengthening the reliability of our findings.
By filling this empirical gap, the study builds a robust foun-
dation for analyzing power-law behavior of range-based asset
market variances.
Fourth, studies have explored common power-law behavior
of range-based variances within related asset categories, such
as equity factors (Fathi et al. 2025) and foreign exchange mar-
kets (Grobys 2024). These works documented universal expo-
nents (e.g. α ≈ 2.6 for range-based variances of G10 curren-
cies) that imply risk homogeneity within specific asset classes.
However, no prior research has tested for such commonality
across unrelated markets. This study pioneers cross-sectional
risk analysis by examining whether the range-based variances
Modeling variance risk in financial markets using power-laws 3
of diverse financial markets, including equities, commodi-
ties, exchange rates, and cryptocurrencies are governed by
a common power law. The existence of such a commonal-
ity would suggest a shared underlying risk dynamic, likely
driven by similar systemic or behavioral forces, such as herd-
ing behavior. The identification of a cross-market universal
exponent would underscore the need for unified risk modeling
approaches and contributes a novel dimension to the literature
on power-law behavior.
Finally, robustness is a critical aspect of econometric anal-
ysis, especially when dealing with heavy-tailed distributions.
Unlike Grobys (2021), which relied primarily on raw data,
we employ autoregressive modeling to account for volatility
clustering, a well-documented phenomenon in financial mar-
kets. Analyzing the resulting innovation processes confirms
that power-law behavior persists even after accounting for
dynamic dependencies. Furthermore, our sample-split tests
validate the stability of power-law exponents across time
periods and subsamples. These methodological advancements
align with calls for more robust analytical frameworks in the
financial econometrics literature (Mandelbrot 1963, Grobys
2021, Fathi et al. 2024) and enhance the generalizability of
our findings.
Through these contributions, the study directly engages
with and builds upon prior literature, addressing gaps
in methodologies (Molnár 2012), theoretical assumptions
(Andersen et al. 2001, Renó and Rizza 2003), and empir-
ical scope (Grobys 2021, Fathi et al. 2024). By leverag-
ing improved variance estimators, expanding datasets, and
incorporating cross-sectional and dynamic robustness analy-
ses, the research advances our understanding of range-based
variances across financial markets. It also underscores the
limitations of traditional models, advocating for the integra-
tion of power-law frameworks to better capture the realities
of extreme market risks. These findings hold critical impli-
cations for both academic research and practical applica-
tions in risk management, portfolio optimization, and asset
pricing.
The results reveal that Garman-Klass range-based vari-
ances exhibit heavy tails across all five markets, strongly
supporting the power-law hypothesis. The MLE analysis of
tail exponents confirms that values consistently fall within
2 < α < 3, implying infinite variance. This corroborates ear-
lier findings (e.g. Grobys 2021, 2024, Fathi et al. 2024,
2025) and raises critical concerns about traditional econo-
metric techniques relying on finite-variance assumptions.
Goodness-of-fit tests firmly reject alternative distributions,
including log-normal and exponential models, highlighting
the systematic underestimation of tail risks by finite-moment
frameworks. Moreover, striking evidence supports the exis-
tence of a universal cross-sectional power-law exponent (α ≈
2.8) governing range-based variances across unrelated mar-
kets. Robustness checks using autoregressive modeling and
sample-split tests confirm these findings, underscoring their
consistency across time periods and market conditions.
This study provides critical insights into the statistical
properties of range-based variances across key financial mar-
kets, offering important implications for finance theory and
practice. First, the evidence of practically infinite variance
challenges traditional econometric methods, such as point
estimation and hypothesis testing, which assume finite vari-
ance, necessitating the development of alternative method-
ologies. Second, the rejection of the log-normal model in
favor of power-law distributions highlights the systematic
underestimation of tail risks by finite-moment frameworks,
emphasizing the need for power-law-based approaches in
assessing and managing extreme financial risks. Third, the
identification of a universal power-law exponent across unre-
lated markets implies limited diversification benefits, as com-
mon dynamics across asset variances may amplify systemic
risks, urging the adoption of unified risk modeling strategies.
Fourth, the robustness of power-law exponents across time
periods and subsamples, even after accounting for volatility
clustering, confirms the generality and consistency of these
findings, supporting their applicability to a wide range of
financial contexts. Collectively, these results challenge pre-
vailing assumptions underpinning modern portfolio theory
and traditional risk management frameworks, reinforcing the
necessity for alternative approaches that better capture the
realities of extreme financial risks. By addressing method-
ological, theoretical, and empirical gaps, this study paves
the way for more accurate risk modeling, portfolio optimiza-
tion, and the development of financial instruments resilient to
market extremes.
This study is organized as follows: Section 2 reviews the
relevant literature; Section 3 describes the data; Section 4 out-
lines the methodology; Section 5 presents the empirical results
and key findings; Section 6 discusses their implications; and
Section 7 concludes the study.
2. Literature review
The modeling of volatility in financial markets has evolved
into a rich and multifaceted area of research, resulting in
a variety of theoretical and empirical frameworks. These
models are designed to capture different aspects of volatil-
ity behavior, ranging from short-term clustering to long-
memory and heavy-tailed properties. This literature can be
grouped into six major research streams: GARCH-type mod-
els, stochastic volatility models, realized volatility models,
range-based volatility models, multifractal and scaling mod-
els, and power-law (namely, tail-index) approaches.
2.1. Generalized Autoregressive Conditional
Heteroskedasticity (GARCH) models
The Generalized Autoregressive Conditional Heteroskedas-
ticity (GARCH) model, introduced by Bollerslev (1986), and
its variants have become foundational in volatility model-
ing. These models describe volatility as a conditional pro-
cess that evolves based on past squared returns and past
variances. Key extensions include the Exponential GARCH
(EGARCH) model by Nelson (1991) and the Threshold
GARCH (TGARCH) model by Zakoian (1994), which allow
for asymmetry in volatility responses. Although highly flex-
ible, these models rely on parametric assumptions and are
typically calibrated on daily or lower-frequency data. They
perform well in capturing volatility clustering but struggle
with representing extreme tail behavior unless augmented
4 M. Fathi and K. Grobys
with fat-tailed distributions for example, Student’s t (Hansen
1994). A key limitation of GARCH-type models is that their
parameter estimates can change substantially when under-
lying distributional assumptions are relaxed—for example,
when the innovation process is altered from a normal to a Stu-
dent’s t distribution, or vice versa. In this regard, Mandelbrot
(2008) criticized the tendency of researchers to ‘fix’ GARCH
models when they failed to perform adequately, instead of
abandoning them altogether.
2.2. Stochastic volatility (SV) models
In contrast to GARCH models, stochastic volatility (SV)
models treat volatility as a latent stochastic process, indepen-
dent from return innovations (Taylor 1982, Shephard 2005).
These models provide greater flexibility in capturing volatil-
ity persistence and allow for richer dynamic structures. Recent
developments include score-driven SV models (Creal et al.
2013, Harvey and Palumbo 2019), which use the score of the
conditional likelihood to update latent volatility components.
SV models are often computationally intensive but can better
accommodate long-memory and structural breaks.
2.3. Realized volatility models
Realized volatility (RV) models utilize high-frequency intra-
day returns to construct ex-post volatility measures, offering
a data-driven alternative to conditionally specified models.
Andersen et al. (2001) played a pioneering role by formaliz-
ing the concept of realized volatility using the sum of squared
intraday returns. This approach avoids strong distributional
assumptions and allows researchers to observe and model
actual volatility over fixed intervals, for example, daily or
weekly periods. Subsequent studies have expanded upon this
framework to examine properties for example, the distribu-
tional characteristics of RV (Barndorff-Nielsen and Shephard
2002), the impact of microstructure noise and asynchronous
trading (Zhang et al. 2005), and the decomposition of volatil-
ity into continuous and jump components (Barndorff-Nielsen
and Shephard 2004). Realized volatility is now a central tool
in empirical finance, particularly in measuring and forecasting
short-term volatility with high precision.
2.4. Range-based volatility models
Range-based models use daily high, low, open, and close
prices to construct volatility estimators. The Parkinson (1980)
and Garman and Klass (1980) estimators are seminal exam-
ples. These methods are shown to be more efficient than
squared return-based measures, especially in the presence
of microstructure noise and thin trading. Molnár (2012) and
Chou et al. (2010) further highlight the robustness of range-
based estimators, making them suitable for low-frequency but
high-accuracy volatility measurement.
2.5. Multifractal and scaling models
Multifractal models attempt to capture the self-similar,
long-memory, and scaling properties observed in financial
volatility. The Markov-Switching Multifractal (MSM) model
developed by Calvet and Fisher (2004) models volatility as a
product of volatility components switching between different
regimes at multiple frequencies. Earlier work by Mandelbrot
et al. (1997) and Müller et al. (1997) emphasizes the hier-
archical and fractal nature of market volatility, linking it to
turbulence-like behavior in financial time series.
2.6. Power-law and tail-index approaches
A distinct stream focuses on the tail behavior of the volatil-
ity distribution. Rather than modeling conditional volatility,
these approaches examine whether realized volatility or vari-
ance follows a power-law distribution with infinite mean or
variance. Clauset et al. (2009) offer robust tools for tail-
index estimation and goodness-of-fit testing. Recent empir-
ical work (e.g. Grobys 2021, 2024) applies these methods
to range-based variance data, exploring the implications of
heavy-tailed volatility for the validity of classical econometric
models for example, OLS regression.
While all these models aim to capture features of financial
market volatility, they differ significantly in their assump-
tions and goals. GARCH and SV models are designed for
conditional volatility forecasting, relying on parametric or
semi-parametric methods. Realized and range-based models
use empirical data to estimate volatility more accurately but
often without strong assumptions about underlying distribu-
tions. Multifractal models and power-law approaches focus
on structural properties like scaling laws and tail behavior.
Notably, most traditional models do not explicitly address
extremely fat-tailed distributions or long-range dependence
in volatility, unless such features are specifically built into the
model. The power-law approach is unique in explicitly testing
for the existence (or lack) of moments, which has profound
implications for econometric modeling and asset pricing.
2.7. Alternative approaches to address heavy tails and long
memory in volatility
Notably, the literature on financial volatility offers a range
of alternative approaches that contribute to understanding
heavy-tailed behavior beyond traditional power-law model-
ing. For instance, Corsi (2009) introduced the Heterogeneous
Autoregressive (HAR) model, which captures long-memory
effects in realized volatility by aggregating volatility com-
ponents across different time horizons. This approach has
proven effective in modeling the persistent nature of volatility
without requiring high-frequency data. Similarly, Harvey and
Palumbo (2019) proposed a score-driven model for realized
volatility that dynamically adapts to changes in the underly-
ing data-generating process using a state-space framework.
These methods highlight the growing emphasis on flexi-
ble and robust volatility modeling frameworks. Furthermore,
Barndorff-Nielsen and Shephard (2002) introduced stochas-
tic volatility models driven by Lévy processes, allowing for
jumps and capturing the non-Gaussian nature of returns.
Similarly, Müller et al. (1997) proposed a time-deformation
model using multiplicative cascades, revealing scaling laws
and volatility components at different frequencies. Engle and
Modeling variance risk in financial markets using power-laws 5
Rangel (2008) addressed long-term volatility trends through
a spline-GARCH model that combines smooth structural
components with short-term GARCH effects. Calvet and
Fisher (2004) developed the MSM model, which generates
multifractal volatility patterns and long memory via regime-
switching cascades. Bollerslev and Todorov (2011) empha-
sized jump-tail risk by integrating extreme events into asset
price and volatility dynamics, accounting for the role of rare
but impactful shocks.
Collectively, these models offer alternative perspectives on
how volatility evolves over time under non-Gaussian con-
ditions, and they share common objectives: capturing fat-
tailed returns, long-range dependence, volatility clustering,
and asymmetric responses to market shocks. While traditional
GARCH-type models, including their extensions, are typi-
cally parametric and rely on specific distributional assump-
tions, several of the above frameworks (for example, MSM
and multiplicative cascades) adopt semi- or non-parametric
approaches, allowing for greater flexibility in capturing the
empirical irregularities of financial time series. Moreover,
models based on realized measures, for example, range-
based or high-frequency volatility, circumvent strong distri-
butional assumptions by leveraging more granular data. We
have now referenced these studies in the revised manuscript
to better position our research within this broader aca-
demic discourse and to clarify how our focus on power-
law behavior and extreme variance dynamics complements,
instead of contradicts, the insights offered by these alternative
models.
While the existing literature offers valuable insights into
the dynamics and persistence of volatility, our study diverges
by explicitly focusing on the tail behavior of the volatility
distribution—measured in terms of range-based variances—
itself. Rather than modeling the full distribution or volatility
path, this study investigates whether the extreme realiza-
tions of the data generating process—often dismissed as
noise—may actually be governed by a power-law distribu-
tion, suggesting that the tail, instead of the center, contains
the meaningful signal. This tail-centric perspective enables
us to characterize the statistical properties of extreme vari-
ance events, a topic that remains underexplored in much of
the existing volatility modeling literature.
3. Data
The data for this study were collected from Yahoo Finance.
We retrieved daily observations including open, high, low,
and closing prices on S&P 500, gold, crude oil, USD/GBP
exchange rate, and Bitcoin. Due to data availability, the data
sample for the S&P 500 spans from February 19, 1980, to
July 31, 2024, providing 11 207 observations. This exten-
sive period captures over four decades of market activity,
including significant events such as the 1987 market crash,
the dot-com bubble, the 2008 financial crisis, and the COVID-
19 pandemic. For gold, the dataset covers a similarly long
timeframe, from February 7, 1980, to July 31, 2024, result-
ing in 11 196 observations. This period reflects both periods
of stability and heightened volatility, such as 2008 financial
crisis. The crude oil dataset begins on January 2, 1991, and
runs until July 31, 2024, including 8556 observations. Specif-
ically, in our study we use data on West Texas Intermediate
(WTI) which is one of the main global benchmark prices for
crude oil. This period includes major fluctuations crude oil
prices due to events such as the Gulf War, the shale revolution,
and the COVID-19-related demand shock. The USD/GBP
exchange rate data is available from July 25, 2000, to July 31,
2024, yielding 6263 observations. This timeframe captures
significant currency market events, including the global finan-
cial crisis and Brexit. Finally, Bitcoin data is available from
September 17, 2014, to July 31, 2024, with 3603 observa-
tions. This relatively short period reflects the cryptocurrency’s
promising stage and its rapid growth as a speculative and
alternative asset, marked by extreme volatility.
To compute the Garman-Klass variance estimator, which
combines open and close prices along with daily highs and
lows, we follow the methodology proposed by Garman and
Klass (1980) and compute daily range-based variances as:
σˆ 2GK = 0.5
[
ln
(
Ht
Lt
)]2
− [2ln2 − 1]
[
ln
(
Ct
Ot
)]2
, (1)
where H represents the daily high price, L denotes the daily
low price, O is the daily opening price, and C refers the daily
closing price of each financial asset.
GARCH models typically estimate conditional variances
based on closing prices and thus may fail to fully capture
the intraday dynamics of price movements (e.g. Andersen et
al. 2003, 2004, Bubák et al. 2011). In contrast, range-based
variances—particularly those derived from range-based esti-
mators such as the Garman-Klass method—leverage intraday
data (high, low, open, and close prices) to provide a more
efficient and information-rich measure of daily volatility. As
shown by Garman and Klass (1980) and reinforced by Molnár
(2012), range-based estimators outperform return-based esti-
mators, especially in the presence of microstructure noise and
market frictions. Given our focus on modeling tail behavior
and extreme variance risk, range-based variances offer clear
advantages: they better capture the heavy-tailed nature of
financial markets and are less prone to distributional misspec-
ification. This allows for a more robust empirical foundation
when testing power-law behavior and modeling the variance
of variance—a key objective of our study.
Table 1 reports the descriptive statistics for the annual-
ized daily range-based variances of these assets. The statistics
show significant variations in the mean, standard deviation,
skewness, and kurtosis, indicating the existence of heavy tails
in the distribution of range-based variances, which could indi-
cate power-law behavior. The mean of Garman-Klass range-
based variance ranges from 0.95 for the USD/GBP exchange
rate to 36.18 for Bitcoin, indicating substantial variation in
the average level of volatility. Bitcoin’s high Garman-Klass
range-based variance mean value reflects its nature as a spec-
ulative and emergent asset class, characterized by extreme
price fluctuations. Similarly, crude oil exhibits a high mean
variance of 14.46. Conversely, traditional asset classes like
gold (2.79) and the S&P 500 (2.18) show relatively mod-
erate average variances, aligning with their roles as more
stable investment assets. The USD/GBP exchange rate, with a
6 M. Fathi and K. Grobys
mean of 0.95, suggests that currency markets involving major
economies tend to exhibit lower volatility. In addition, all
assets exhibit significant positive skewness and excess kur-
tosis, suggesting non-normality in their range-based variance
distributions. Crude oil exhibits the highest skewness and kur-
tosis (skewness: 88.83, kurtosis: 8095.44), while Bitcoin has
the lowest skewness and kurtosis (skewness: 9.50, kurtosis:
130.54) among the analyzed assets.
4. Methodology
4.1. Main analysis
4.1.1. Estimating power-law exponents via MLE and
comparing model fit. Following earlier studies (Grobys
2021, 2023, 2024, Fathi et al. 2024, 2025), we model the
Garman-Klass range-based variances of each asset in our
sample using a power-law function, expressed as:
p(x) = Cx−α , (2)
where C = (α − 1)xα−1MIN with α ∈ {R+|α > 1}, and x = σ 2j,t
denotes the respective range-based Garman-Klass variance of
financial asset j at time t. Note that x ∈ {R+|xMIN ≤ x < ∞},
where xMIN is the minimum threshold for variance values gov-
erned by the power-law distribution, and α represents the tail
exponent.
For range-based variances exceeding xMIN (X > xMIN ), the
conditional expected first moment is defined as:
E[X |X > xMIN ] =
∫ ∞
xMIN
xp(x)dx = (α − 1)
(α − 2)xMIN . (3)
The conditional expected second moment, or the variance of
range-based variances for X > xMIN , is expressed as:
E[X 2|X > xMIN ] =
∫ ∞
xMIN
x2p(x)dx = (α − 1)
(α − 3)x
2
MIN . (4)
Consequently, higher conditional expected moments of order
k follow the general form:
E[X k|X > xMIN ] = (α − 1)
(α − 1 − k)x
k
MIN . (5)
From Equations (3) and (4), it can be concluded that the
theoretical mean of range-based Garman-Klass variances is
defined only when α > 2, while the existence of the variance
for range-based Garman-Klass variances requires α > 3.
Next, as suggested by White et al. (2008) and Clauset et
al. (2009), we utilized MLE to estimate the power-law tail
exponents. The MLE estimator is given by:
αˆ = 1 + N
( N∑
i=1
ln
(
xi
xMIN
))−1
, (6)
where αˆ is the estimated tail exponent, N is the number
of observations exceeding xMIN , and other terms maintain
their previously defined meanings. The selection of xMIN is
determined by minimizing the distance between the empirical
data and the power-law model (Clauset et al. 2009). This is
achieved using the Kolmogorov–Smirnov (KS) distance, D,
which measures the maximum difference between the cumu-
lative distribution functions (CDFs) of the observed data and
the fitted model. The distance D is given by:
D = MAXx≥xMIN |S(x) − P(x)|, (7)
where S(x) represents the CDF of the empirical data for obser-
vations x ≥ xMIN , and P(x) is the CDF of the fitted power-
law model. The optimalxMIN is the value that minimizes
D. Additionally, as described by Clauset et al. (2009), the
standard deviation of the estimated power-law exponents is
expressed as:
σ = αˆ − 1√
N
+ O
(
1
N
)
. (8)
To verify the appropriateness of a power-law fit, we con-
ducted a goodness-of-fit test based on the KS distance, D,
following the methodology derived in Clauset et al. (2009)
which compares the empirical data to synthetic data gener-
ated from a power-law distribution using specific xMIN and
αˆ values. The null hypothesis of goodness-of-fit test is that
the empirical data and the synthetic data generated from a
power-law distribution using the estimated xMIN and αˆ val-
ues belong to the same distribution, suggesting the power-law
model is a plausible fit. While the alternative hypothesis sug-
gests that the empirical and synthetic data originate from
different distributions, implying that other models should be
considered.
Thus, we generated synthetic datasets using the estimated
parameters xMIN and αˆ in the prior analysis. For each synthetic
dataset, the distance (DS) was calculated and compared to the
distance (D) from the original dataset. The p-value for the
goodness-of-fit test is defined as the proportion of synthetic
datasets where DS > D, denoted as #(DS > D)/K, where K is
the total number of synthetic datasets. A significance level of
5% was applied, meaning the power-law model is not rejected
if #DS > D/K > 0.05. In addition, to ensure robust results,
1000 synthetic datasets were simulated for each financial asset
variance.
However, non-rejected power-law null models ascertained
by means of Clauset et al.’s (2009) goodness-of-fit test do not
necessarily mean that power-law distributions are the best fit
for the data, as alternative distributions could potentially offer
a better match. In addition, as mentioned earlier, prior stud-
ies such as Andersen et al. (2001), Andersen et al. (2001a,
2001b) have suggested that realized volatility often aligns
closely with a log-normal distribution. To address this, we
compare the power-law distributions with the log-normal dis-
tributions for each range-based asset variance to determine the
most suitable model for the data. As noted by Clauset et al.
(2009), comparing p-values of power-law models with those
of alternative distributions, such as log-normal or exponen-
tial, provides a robust basis for evaluating the appropriateness
of the power-law model. Thus, following the methodology of
Clauset et al. (2009), we use the same goodness-of-fit test
approach applied to the power-law distribution for assess-
ing the plausibility of other competing distributions (viz.,
Modeling variance risk in financial markets using power-laws 7
Table 1. Descriptive statistics.
Statistic S&P 500 Gold Crude oil USD/GBP BTC
Mean 2.18 2.79 14.46 0.95 36.18
Median 0.95 1.42 6.91 0.60 13.66
Maximum 161.31 200.71 15901.67 135.29 1809.48
Minimum 0.00 0.00 0.00 0.00 0.00
Std. Dev. 4.81 5.54 174.13 2.32 90.06
Skewness 14.32 11.90 88.83 35.59 9.50
Kurtosis 342.75 265.07 8095.44 1864.48 130.54
Observations 11207 11196 8556 6263 3603
Period (MM/DD/YYYY) 02/19/1980 02/07/1980 01/02/1991 07/25/2000 09/17/2014
07/31/2024 07/31/2024 07/31/2024 07/31/2024 07/31/2024
Note: This table presents the descriptive statistics of the annualized daily range-based variance for various assets,
including the S&P 500, gold, crude oil, USD/GBP exchange rate, and Bitcoin. The annualized daily range-based
variances for each asset calculated using the Garman-Klass methodology (Garman and Klass 1980). The statis-
tics reported in this table are sample statistics, which summarize empirical properties observed in the data. While
our analysis later demonstrates that the variance of variance is theoretically infinite due to heavy tails (α ∈ (2,3)),
these sample-based measures remain meaningful for descriptive purposes. They do not imply the existence of finite
theoretical moments.
log-normal, exponential). In this case, the null hypothesis
assumes that the alternative distribution is a plausible fit for
the data.
4.1.2. Assessing robust standard errors and testing for
a common power-law component. Motivated by recent
research documenting that range-based foreign exchange rate
variances are governed by a universal power-law exponent
(Grobys 2024), we explore this issue for the range-based
variances of our five asset markets. An important difference
between Grobys’ (2024) study and our research is that the
underlying asset markets in our study are otherwise unrelated,
whereas foreign exchange rates are not. Therefore, a common
power-law exponent governing the range-based variances of
otherwise unrelated asset markets would be a surprising find-
ing. To investigate this issue, we employ the test procedure
proposed in the study of Grobys (2024).
While the MLE proposed by Clauset et al. (2009) remains
asymptotically unbiased even under dependent data, the pres-
ence of autocorrelation—manifested, for example, through
volatility clustering—can lead to an underestimation of stan-
dard errors when standard i.i.d. assumptions are applied. This
would result in overly narrow confidence intervals and poten-
tially misleading inference. To address this issue, we imple-
ment a block bootstrap method, as recommended by Godfrey
(2009) and employed by Grobys (2024), which preserves tem-
poral dependencies within blocks. This adjustment produces
more realistic, wider standard errors and thereby ensures
that hypothesis testing related to tail exponents is appro-
priately conservative; that is, our block bootstrap approach
mitigates the issue of downward-biased standard errors due
to serial dependence. Hence, to address this issue, we first
estimate robust standard errors for each range-based asset
variance. This is achieved by employing the blocks bootstrap
methodology, as outlined in the study of Grobys (2024).
Specifically, a block bootstrap procedure is implemented
with the block length m, where E[m] = T1/3 following the
recommendation by Godfrey (2009) for variance estimation.
From a given data vector of range-based variances, x, blocks
of size m are randomly selected, where m follows a geomet-
ric distribution, m ∼ GEO(p) with an expected value E[m] =
(1 − p)/p. Thus, in the overall data sample, we use p =
0.0435, p = 0.0434, p = 0.0476, p = 0.0526 and p = 0.0625
for the range-based variances of the S&P 500, gold, crude oil,
USD/GBP exchange rate, and Bitcoin, respectively. Using this
bootstrap method, blocks drawn from the data vector x have
varying lengths. These randomly selected blocks m are then
stacked into a new vector xb as follows:
xbi =
⎡⎢⎢⎢⎣
m1
m2
m3
.
.
.
⎤⎥⎥⎥⎦ . (9)
Again, it is important to note that the lengths of the blocks
m1, m2, . . . vary. The process continues until the length of
the constructed vector xb exceeds T. Then, we cut off any
observations exceeding T , ensuring that the artificial data vec-
tor xb matches the length of the original data vector x. By
applying this blocks bootstrap, 1000 artificial data vectors[
x1 x2 · · · xB], are generated for each given original data
vector. Next, the point estimates αˆ for each bootstrap data
vector
[
x1 x2 · · · xB], are calculated and stored in a vec-
tor
[
αˆ1 αˆ2 · · · αˆB] following the method described by
Clauset et al. (2009). As a final step, the bootstrapped stan-
dard error σˆBOOT is calculated from
[
αˆ1 αˆ2 · · · αˆB] for
each given data vector of range-based asset variances as:
σˆBOOT =
√√√√ 1
B
B∑
b=1
(
αˆb −
(
1
B
B∑
b=1
αˆb
))2
. (10)
After obtaining the robust standard errors, we can con-
duct the joint test to examine whether the Garman-Klass
range-based variances of different assets share a common
component, represented by a common power-law exponent.
8 M. Fathi and K. Grobys
Identifying such a shared exponent would suggest a unified
risk of these assets. It is worth noting that our bootstrap-
ping method ensures COV(αˆi, αˆj) = 0∀i = j. Based on this,
the covariance matrix for estimated power law exponents is
defined as:
Σˆαˆ =
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝
σˆ 2
αˆboot ,S&P500 0 0 0 0
0 σˆ 2
αˆboot ,Gold 0 0 0
0 0 σˆ 2
αˆboot ,Crudeoil 0 0
0 0 0 σˆ 2
αˆboot ,USD/GBP 0
0 0 0 0 σˆ 2
αˆboot ,BTC
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
,
(11)
where, σˆ 2
αˆboot ,S&P500, σˆ
2
αˆboot ,Gold,σˆ
2
αˆboot ,Crudeoil, σˆ
2
αˆboot ,USD/GBP and
σˆ 2
αˆboot ,BTC represent the corresponding bootstrapped variances
(viz., σˆ 2BOOT ). Then, consistent with Grobys (2024), we define
the test statistic λˆ as follows:
λˆ = (αˆ − q1)′Σˆ−1αˆ (αˆ − q1), (12)
where the covariance matrix Σˆ αˆ is 5 × 5, αˆ represents a
5 × 1 vector of estimated power-law exponents, 1 is a 5 × 1
vector of ones, and q denoted the hypothesized common
power-law exponent. The estimated test statistic λˆ in Eq. (10)
follows a χ2(5) distribution under the null hypothesis, with
the corresponding critical value at the 5% significance level
being χ20.95(5) = 11.07. To evaluate whether the Garman-
Klass variances of our five asset markets share a common
power-law exponent, the test statistic λˆ is iteratively com-
puted across the economically significant range of q values,
spanning from 1.2 to 3.2 in increments of 0.1. This iterative
computation ensures that the analysis may capture the poten-
tial existence of a common power-law exponent governing
variance risks across the selected assets.
4.2. Additional analysis
4.2.1. Investigating the impact of autocorrelation on tail
index estimates in range-based variances. A potential con-
cern could be whether the observed power-law behavior in
the Garman-Klass range-based variance distributions of our
assets is genuine or a statistical effect resulting from autocor-
relation. It is well-recognized that time-varying volatility can
create patterns in the tails of return distributions that resem-
ble power-laws. This phenomenon could lead to inaccurate
conclusions, as standard statistical techniques for estimat-
ing power laws might incorrectly attribute thick tails to a
power-law structure when they are actually produced by a
combination of exponential-tail distributions rather than an
actual power law. This raises the important question as to
whether similar issues affect the estimation of power-law
behavior in the range-based variances of our assets?
To examine this, we fit autoregressive models of order p to
each asset’s Garman-Klass range-based variance. The models
are defined as:
σˆ 2j,t = β0,j + β1,jσˆ 2j,t−1 + β2,jσˆ 2j,t−2 + . . . + βp,jσˆ 2j,t−p + εj,t,
(13)
where σˆ 2j,t defines the Garman-Klass range-based variance
of asset j at time t, εj,t, represents the innovation pro-
cess, and p is the lag-order. The optimal lag order p is
determined by analyzing the partial autocorrelation func-
tion. This approach is similar to ARCH-type models, where
the conditional variance is modeled using the lagged values
of range-based variances directly. Next, we use the abso-
lute values of the estimated innovation processes of each
asset, |εˆS&P500|, |εˆGold |, |εˆCrudeoil|, |εˆUSD/GBP|, |εˆBTC|, to re-
evaluate the power-law exponents for each asset’s orthogo-
nalized range-based variance process. Then, we carried out
Clauset et al.’s (2009) goodness-of-fit test, following the
methodology outlined earlier for the absolute values of the
innovation processes for each asset variance. This procedure
helps to evaluate whether the observed power-law behav-
ior persists after accounting for autocorrelation, providing a
robust check on the validity of the power-law model for our
range-based variances.
4.2.2. Evaluating the reliability of power-law exponents:
evidence from sample-split tests. To further validate the
robustness of the observed power-law behavior in the inno-
vation processes of the range-based variances, we per-
form a sample-split test. We divide the residual series
for each asset, obtained from the autoregressive models
(viz., |εˆS&P500|, |εˆGold |, |εˆCrudeoil|, |εˆUSD/GBP|, |εˆBTC|), into
two non-overlapping subsamples of equal length. By esti-
mating the power-law exponents for the absolute values of
the residuals in each subsample separately, we aim to assess
the consistency of the power-law characteristics across dif-
ferent sample episodes. This approach allows us to examine
whether the power-law behavior persists independently within
each subsample, mitigating concerns about potential sample-
specificity. Additionally, we applied the goodness-of-fit tests
to the absolute values of the residuals in each subsample to
evaluate the validity of the power-law model in both cases.
4.2.3. Alternative measures of asset market variance: A
realized variance approach. While the main analysis in this
paper is based on the Garman-Klass range-based estimator,
concerns can be raised about the potential limitations of rely-
ing on a single volatility measure. Specifically, it has been
noted that different volatility estimators may capture different
aspects of return dynamics, and even the most efficient esti-
mators remain noisy approximations of true volatility (Shu
and Zhang 2006, Molnár 2012, Be˛dowska-Sójka and Kliber
2021). To address this concern and assess the robustness of
our findings, we extend our analysis by computing weekly
realized variances based on the sum of squared daily returns
over five-day intervals. This return-based volatility measure
offers a conventional benchmark and helps assess whether the
heavy-tailed behavior observed under the Garman-Klass esti-
mator persists under a different volatility estimation frame-
work. The daily return of asset market i at time t is computed
as follows:
RETi,t = 100 (Pi,t+1 − Pi,t)Pi,t , (14)
Modeling variance risk in financial markets using power-laws 9
where Pi,t denotes the corresponding daily closing price. Next,
we compute the weekly realized variance of each five assets
as the sum of the squared returns over five business days
according to the following Equation:
RVj,t = σ 2i,j =
∑
t∈j
(RETi,t)2, (15)
where j indicates the week and t ∈ j indicates the correspond-
ing trading days in the respective week. We then estimate
power-law exponents for the tail behavior of the weekly real-
ized variances of each asset as described in section 4.1.1.
Finally, to evaluate the appropriateness of the power-law
model relative to alternative specifications, we compare the
empirical data with synthetic data generated from power-law,
log-normal, and exponential distributions for each realized
asset variance to determine the most suitable model for the
weekly data as described in section 4.1.2.
4.2.4. Analysis of additional asset classes: evidence from
bonds. To test the generalizability of our findings across asset
classes, we extend the analysis to U.S. government bond
yields. Specifically, we include range-based bond yield vari-
ance data for the 2-year, 5-year, and 10-year Treasury yields,
computed using the Garman-Klass estimator on daily high,
low, open, and close prices. This addition allows us to explore
whether power-law behavior is also present in fixed income
markets, which are often considered less volatile than equi-
ties or cryptocurrencies. Following the methodology outlined
in Section 4.1.1, we estimate power-law exponents for each
bond series and conduct comparative goodness-of-fit tests
using synthetic data from power-law, log-normal, and expo-
nential distributions. This enables a robust evaluation of the
tail behavior in bond market volatility.
4.2.5. Alternative estimation of power-law exponents and
cutoffs: A Bayesian inference approach. While our main
results rely on the widely used Kolmogorov–Smirnov-based
method for estimating power-law parameters, one might be
concerned that the selection of the threshold xmin—a crucial
modeling choice—could influence the inferred tail behavior.
To address this potential sensitivity and enhance the credibil-
ity of our findings, we perform a robustness check using an
alternative, Bayesian estimation framework. This approach
follows the methodology introduced by Virkar and Clauset
(2014) and Grigaityte and Atwal (2019), which allows for
the simultaneous estimation of the power-law exponent αˆ
and the lower-bound threshold xmin. Unlike frequentist meth-
ods that rely on point estimates and deterministic threshold
selection, the Bayesian approach yields posterior distribu-
tions for both parameters. Using Markov Chain Monte Carlo
(MCMC) sampling, we generate estimates of αˆ and xmin.
This method not only enhances transparency around thresh-
old selection but also enables a more informative inference
by incorporating parameter uncertainty. To formally describe
the Bayesian approach, we consider a sample of continuous
observations x1, x2, . . . , xN , each satisfying xi ≥ xmin. The con-
tinuous power-law distribution is defined by the following
probability density function:
p(x|α, xmin) = α − 1
xmin
(
x
xmin
)−α
, for x ≥ xmin, α > 1. (16)
Given N observations above a threshold xmin, the likelihood
(L) of the data is:
L(α | x, xmin) =
N∏
i=1
α − 1
xmin
(
xi
xmin
)−α
. (17)
Taking the log, the log-likelihood becomes:
log L(α | x, xmin) = Nlog(α − 1) − Nlogxmin
− α
N∑
i=1
log
(
xi
xmin
)
. (18)
To complete the Bayesian model, we specify prior distribu-
tions where for the exponent α, we use a weakly informative
uniform prior:
α ∼ Uniform(1.01, 5). (19)
In addition, for the threshold xmin, we assume a discrete
uniform prior over a range of candidate values:
xmin ∈ {x(1), x(2), . . . , x(k)}, P(xmin) = 1k , (20)
where, P(xmin) denotes the prior probability assigned to each
candidate threshold, reflecting an assumption of equal plau-
sibility before observing the data. In this setting, the joint
posterior distribution over α and xmin is given by:
P(α, xmin|x) ∝ L(α|x, xmin) · P(α) · P(xmin), (21)
where P(α) is the prior distribution over the power-law expo-
nent α. Because xmin is discrete and α is continuous, we
marginalize over the possible values of xmin by computing:
(1) The posterior distribution P(α|x, xmin) for each candi-
date xmin.
(2) The marginal likelihood (also known as the model
evidence):
Z(xmin) = ∫L(α|x, xmin) · P(α)dα. (22)
(3) The posterior over xmin using Bayes’ rule:
P(xmin|x) = Z(xmin)∑K
k=1Z(xk)
. (23)
The overall posterior for α can be computed as a weighted
mixture of posteriors conditioned on each xmin, using the pos-
terior probabilities P(xmin, x) as weights. Finally, the power-
law exponent and threshold are being estimated by performing
MCMC sampling for each candidate xmin(k), compute their
marginal likelihoods, derive posterior probabilities for each
threshold, and calculate the weighted posterior mean.
10 M. Fathi and K. Grobys
4.2.6. An alternative model comparison metric: the
Akaike Information Criterion. In addition to the previously
reported goodness-of-fit tests based on Clauset et al. (2009),
we incorporated a formal model comparison framework to
strengthen the case for power-law behavior relative to alter-
native models. Specifically, we compare the power-law, log-
normal, and exponential distributions using the AIC. Unlike
the standalone goodness-of-fit tests, AIC provides a system-
atic approach that balances model fit and complexity through
the MLE. For each candidate distribution, model parameters
are estimated via MLE. The power-law parameters—the scal-
ing exponent α̂ and the lower bound xˆmin—are determined
following Clauset et al. (2009). The log-likelihood for each
model is computed based on the corresponding probability
density function and the AIC is calculated as:
AIC = 2k − 2log(Lˆ), (24)
where k represents the number of parameters (with k = 1 for
the power-law and exponential models, and k = 2 for the log-
normal model) and log(Lˆ), is the maximum log-likelihood of
the fitted model.
4.2.7. Rolling window estimations for retrieving time-
Varying power-law exponents. One could argue that finan-
cial volatility may be subject to structural changes over time,
potentially challenging the validity of our results from the
sample-split tests. In this context, implementing rolling win-
dow estimations offers a complementary approach to assess
the stability of the estimated power-law exponents. To address
this issue, we performed rolling window estimations as fol-
lows: for each series of range-based variances, we applied a
rolling window of 500 observations and estimated the MLE-
based power-law exponent as outlined in Section 4.1.1. The
window advances in steps of 50 observations, starting from
t = 500 until t = T.
4.2.8. Estimation of tempered stable distributions. A
finding of a shared power-law exponent governing all range-
based asset variances would represent a significant theoretical
insight. While it may be argued that certain financial assets
exhibit distinct distributional characteristics—particularly in
response to varying market conditions—such evidence would
motivate deeper investigation. Therefore, in this robustness
check, we assess whether the tempered stable distribution
can offer a more nuanced understanding of the common
dynamics underlying range-based variances. A tempered sta-
ble distribution generalizes classical stable distributions (e.g.
Lévy) by introducing exponential tempering to the heavy tails.
This modification allows for finite moments while preserving
power-law behavior over intermediate ranges. Formally, the
tail decays as
P(X > x) ∼ x−αe−λx, (25)
where α ∈ (0, 2) controls the tail heaviness and λ > 0 governs
the rate of tempering. This construction retains the scaling
properties of stable laws while ensuring statistical regularity
in the extreme tails. Tempered stable distributions are particu-
larly useful for modeling empirical phenomena that exhibit
approximate power-law scaling but experience faster-than-
power decay in the tails, as frequently observed in financial,
physical, and biological systems. In applications involving
multiple datasets that share a common power-law exponent
yet display distinct truncation or decay rates, the tempered
stable framework provides a coherent modeling strategy. By
fixing a common α across data series and allowing data series-
specific tempering parameters λi, one can jointly capture
universal scaling behavior and dataset-specific tail dynamics.
This approach is particularly suited for comparative studies or
hierarchical modeling contexts such as volatility dynamics. To
estimate the parameters via maximum likelihood, we numer-
ically invert the characteristic function, as the probability
density function lacks a closed-form expression. Parameters
are then estimated by maximizing the log-likelihood using
numerical optimization techniques (BFGS).
4.2.9. Comparison between power-law distributions and
the generalized beta of the second kind (GB2) family. One
might argue that the comparison of power-law distributions
with the exponential and log-normal distributions introduced
in Section 4.1.1 does not provide an informative benchmark
for evaluating the appropriateness of the power-law model.
As a consequence, we consider the Generalized Beta of the
Second Kind (GB2) family as a more flexible alternative, as
it accommodates both fat tails and finite second moments
across a wide range of parameterizations. The GB2 is a
highly flexible, four-parameter family of continuous proba-
bility distributions that can represent a wide variety of shapes,
including both heavy-tailed and light-tailed forms. It is fre-
quently employed in the analysis of skewed or heavy-tailed
data. The GB2 distribution is defined as:
f (x; a; b; p; q) = a(x/b)
ap−1
bB(p, q)[1 + (x/b)a]p+q , (26)
where B(p, q) denotes the Beta function, a > 0 is the tail
parameter controlling the tail heaviness, b > 0 is a scale
parameter, and p > 0, q > 0 determine the shape of the dis-
tribution, including skewness. In Table A.5 of the appendix,
we list several important subfamilies of the GB2 distribution,
highlighting its role as a ‘distributional umbrella’ that encom-
passes many well-known distributions. Asymptotically, the
GB2 exhibits power-law decay under certain conditions:
P(X > x) ∼ x−aq, (27)
as x → ∞, indicating that the product aq serves as an effec-
tive tail exponent, thereby linking the GB2 to power-law
modeling. For a valid comparison between the GB2 and
power-law models, both must be estimated using maximum
likelihood, and the likelihoods must be evaluated over the
same data range—specifically, the range above the power-
law cutoff xMIN . Accordingly, we restrict the GB2 model to
the same data ranges used for the power-law estimation, as
reported in table 3. We then compute the log-likelihood values
and the corresponding AIC for each model.
Modeling variance risk in financial markets using power-laws 11
5. Results
5.1. Main results
This section presents the findings from our analysis, applying
the methodological approach outlined in Section 4.1 to the
selected financial assets: S&P 500, gold, crude oil, USD/GBP
exchange rate, and Bitcoin.
The analysis begins by modeling the Garman-Klass range-
based variances using the power-law function described in
Section 4.1.1. Table 2 highlights the share of cumulative total
variance contributed by the top 1% and top 20% of obser-
vations for each asset, revealing a significant concentration.
A small proportion of extreme observations accounts for a
disproportionate share of total variance, consistent with the
heavy-tailed behavior expected under a power-law distribu-
tion. For instance, among the top 1% of observations, crude
oil exhibits the highest concentration at 27.23%, followed
by Bitcoin (20.03%), S&P 500 (16.10%), gold (15.05%),
and USD/GBP exchange rate (14.39%). Similarly, for the
top 20%, Bitcoin leads with 72.03%, followed by crude oil
(66.15%), S&P 500 (65.91%), gold (63.00%), and USD/GBP
exchange rate (55.45%). These results underscore the pres-
ence of extreme events driving range-based variances, partic-
ularly in crude oil and Bitcoin, aligning with the Pareto 80/20
rule.
Next, a power-law distribution is fitted to the annual-
ized daily range-based variances of each asset using MLE,
as detailed in Section 4.1.1 and proposed by White et al.
(2008) and Clauset et al. (2009). Table 3 reports the tail
exponents, ranging from αˆ = 2.62 for crude oil to αˆ = 2.94
for the S&P 500, indicating that variance of variance is
infinite for all assets. Such heavy tails invalidate founda-
tional assumptions underlying conventional statistical tools
like t-tests, which depend on finite variances. Standard met-
rics derived from ordinary least squares regression, such as
t-statistics, become unreliable due to their sample-specific
dependency. The evidence confirms that heavy-tailed distri-
butions render traditional econometric methods inadequate,
consistent with critiques by Grobys (2021) and Fathi et al.
(2025).
To further validate the plausibility of the power-law model,
goodness-of-fit tests (Clauset et al. 2009) are applied to assess
the plausibility of power-law and alternative distributions,
including log-normal and exponential models. Table 4 reports
p-values for these tests, demonstrating strong support for
the power-law model, as all p-values for the null hypothe-
sis of a power-law distribution exceed 5%. In contrast, the
log-normal model is rejected outright (p = 0.00) for all assets
except Bitcoin, where the power-law model (p = 0.66) out-
performs log-normal (p = 0.52). The exponential model is
conclusively ruled out, with p-values equating to zero for all
cases. These results collectively confirm the superior fit of the
power-law model to the data.
A joint test, described in Section 4.1.2, examines whether
the Garman-Klass range-based variances exhibit a shared
component characterized by a common power-law exponent.
To account for dependencies such as volatility clustering,
block bootstrap methods are employed, yielding robust stan-
dard errors. Table 5 illustrates the increased standard devia-
tions derived from bootstrapping. For instance, the standard
deviation for S&P 500 increases from 0.05 to 0.20, and for
gold, from 0.06 to 0.14. These adjustments underscore the
critical importance of incorporating dependency structures in
estimating tail exponents.
Subsequently, the joint test evaluates whether the assets
share a common power-law exponent. From table 6, the test
statistic λˆ, computed iteratively over a range of α-values
Table 2. Share of the top 1% and top 20%.
Metric S&P 500 Gold Crude oil USD/GBP BTC
Top 1% Share 16.10% 15.05% 27.23% 14.39% 20.03%
Top 20% Share 65.91% 63.00% 66.15% 55.45% 72.03%
Note: This table displays the share of the cumulative total represented by the top 1% and top
20% of the distribution for the annualized daily Garman-Klass range-based variance across
the S&P 500, gold, crude oil, the USD/GBP exchange rate, and Bitcoin.
Table 3. Estimation of power-law exponents for annualized daily range-based variance.
Metric S&P 500 Gold Crude oil USD/GBP BTC
αˆ 2.94 2.74 2.62 2.78 2.84
xˆMIN 4.59 6.62 23.97 1.09 187.71
D 0.02 0.02 0.02 0.01 0.04
NPL 11.93% 8.33% 10.30% 22.77% 3.41%
Std. Dev 0.05 0.06 0.05 0.04 0.17
Observations 11207 11196 8556 6263 3603
Period (MM/DD/YYYY) 02/19/1980 02/07/1980 01/02/1991 07/25/2000 09/17/2014
07/31/2024 07/31/2024 07/31/2024 07/31/2024 07/31/2024
Note: This table presents the estimation results for power-law exponents applied to the annualized daily range-based
variances of the S&P 500, gold, crude oil, the USD/GBP exchange rate, and Bitcoin. The parameter αˆ signifies
the estimated tail exponent, while σˆ denotes the estimated standard deviation. The lower threshold xˆMIN is identi-
fied through the optimized Kolmogorov-Smirnov distance (D), following the procedure described by Clauset et al.
(2009). The reported xˆMIN corresponds to the optimal distance D. Additionally, the fraction of observations obtained
by the power-law process, denoted as NPL is provided for each asset.
12 M. Fathi and K. Grobys
Table 4. Goodness-of-fit tests for the annualized daily
range-based variance.
Asset Power-law Log-normal Exponential
S&P 500 0.25 0.00 0.00
Gold 0.39 0.00 0.00
Crude oil 0.40 0.00 0.00
USD/GBP 0.46 0.00 0.00
BTC 0.66 0.52 0.00
Note: This table presents the results of the goodness-of-fit tests
conducted using the Kolmogorov-Smirnov (KS) method to assess
whether the empirical data and synthetic data generated from a
power-law distribution, specified by xmin and α, originate from
the same underlying distribution (column 2). Additionally, the
table includes the results of goodness-of-fit tests for log-normal
and exponential distributions in columns 3 and 4, respectively.
The null hypothesis for these tests states that the empirical data
and the synthetic data generated from the specified distributions
(log-normal and exponential) are consistent, suggesting that each
distribution provides an acceptable fit to the observed data.
from 1.2 to 3.2, fails to reject the null hypothesis of a
shared exponent within the economically important range of
2.5 < α < 3.1. This finding suggests a unified component
influencing variances. Notably, our results suggest that αˆ ≈
2.8 produces the highest p-value (p = 0.87), which indicates
that the second moment of the common power law govern-
ing the variance risk of unrelated financial asset markets is,
statistically, infinite.
5.2. Results from robustness checks
To ensure the observed power-law behavior is not an artifact
of statistical effects such as autocorrelation, robustness checks
are performed. As discussed in Section 4.2.1, autoregressive
models of order p are fitted to each asset’s Garman-Klass
range-based variance. Table 7 reports the point estimates,
confirming significant autocorrelation across all assets, with
partial autocorrelation functions indicating orders of p = 2,
6, 1, 4 for S&P 500, gold, crude oil, USD/GBP exchange rate,
and Bitcoin, respectively. Descriptive statistics for the result-
ing innovation processes, detailed in table 8, reveal extremely
heavy tails. For instance, kurtosis values range from 133.83
for Bitcoin to 8053.8 for crude oil, reinforcing the persistence
of extreme values even after autoregressive adjustments.
Table 9 presents power-law exponents derived from the
absolute innovation processes, closely matching those in table
3, further substantiating the presence of genuine power-law
behavior. The goodness-of-fit tests applied to the innova-
tion processes (table 10) corroborate these findings, failing
to reject the power-law null hypothesis for all assets except
gold, where the evidence is slightly weaker. Log-normal
and exponential distributions are once again rejected outright
(p = 0.00) across all assets.
Next, sample-split tests are conducted to validate the
robustness of power-law characteristics across subsamples.
Table 11 provides power-law exponent estimates for non-
overlapping subsamples, demonstrating consistent statistical
Table 5. Descriptive statistics for the power-law exponents estimated using block bootstrap methods.
Statistic αˆbootS&P 500 αˆ
boot
Gold αˆ
boot
Crudeoil αˆ
boot
USD/GBP αˆ
boot
BTC
Mean 2.95 2.74 2.61 2.81 2.33
Median 2.95 2.74 2.59 2.79 2.27
Maximum 3.57 3.37 3.48 3.75 3.44
Minimum 2.38 2.37 2.26 2.38 1.91
Std. Dev 0.20 0.14 0.17 0.18 0.25
Skewness 0.01 0.31 1.01 0.88 1.63
Kurtosis 2.80 3.42 4.94 4.51 5.93
Jarque-Bera (JB) 1.70 23.08 326.81 225.13 802.29
(p-value) JB 0.41 0.00 0.00 0.00 0.00
Observations 11207 11196 8557 6263 3603
Note: This table reports the block bootstrap methodology test to the annualized daily range-based variance for the S&P 500,
gold, crude oil, USD/GBP exchange rate, and Bitcoin. The block bootstrap procedure is implemented with the block length
m, where E[m] = T1/3 following the recommendation by Godfrey (2009) for variance estimation. From the given data vector
x, blocks of size m are randomly selected, where m follows a geometric distribution m ∼ GEO(p) with an expected value
E[m] = (1−p)p . Thus, in the overall data sample, we use p = 0.0435, p = 0.0434, p = 0.0476, p = 0.0526 and p = 0.0625
for the range-based variances of the S&P 500, gold, crude oil, USD/GBP exchange rate, and Bitcoin, respectively. Using
this bootstrap method, blocks drawn from the data vector x have varying lengths. The randomly selected blocks, m, are then
stacked into a new vector
xbi =
⎡⎢⎢⎣
m1
m2
m3
.
.
.
⎤⎥⎥⎦
where the lengths of the blocks m1, m2, . . . vary. The process continues until the length of the constructed vector xb exceeds
T. Then, we cut off any observations exceeding T , ensuring that the artificial data vector xb matches the length of the original
data vector x. By applying this blocks bootstrap, 1000 artificial data vectors
[
x1 x2 · · · xB], are generated for each
given original data vector. Next, the point estimates αˆ for each bootstrap data vector
[
x1 x2 · · · xB], are calculated and
stored in a vector
[
αˆ1 αˆ2 · · · αˆB] following the method described by Clauset et al. (2009). The block bootstrap method
is described in detail in Section 3.1.2.
Modeling variance risk in financial markets using power-laws 13
Table 6. Testing for a common power-law exponent governing from annualized daily range-based variance.
q λˆ p-value
1.2 390.68 0.00
1.3 342.52 0.00
1.4 297.53 0.00
1.5 255.73 0.00
1.6 217.11 0.00
1.7 181.68 0.00
1.8 149.42 0.00
1.9 120.34 0.00
2 94.45 0.00
2.1 71.74 0.00
2.2 52.21 0.00
2.3 35.87 0.00
2.4 22.70 0.00
2.5 12.72 0.03
2.6 5.91 0.31
2.7 2.29 0.81
2.8 1.85 0.87
2.9 4.60 0.47
3 10.52 0.06
3.1 19.63 0.00
3.2 31.92 0.00
Note: This table presents the results of a joint test designed to assess whether the range-based variances of S&P 500, gold, crude oil,
USD/GBP exchange rate, and Bitcoin share a common component. To investigate whether a common component underlies the power-law
behavior of these assets, the following test statistic is utilized:
λˆ = (αˆ − q1)′Σˆ−1αˆ (αˆ − q1),
where Σˆαˆ is the estimated 5 × 5 covariance matrix, αˆ is a 5 × 1 vector of estimated power-law exponents, 1 is a 5 × 1 vector of ones, and
q represents the hypothesized power-law exponent. The test statistic λˆ follows a χ2(5) distribution under the null hypothesis. The statistic
is iteratively calculated across the economically significant range of q = (1.2, 1.3, . . . , 3.1, 3.2). Bolded values in the results represents
statistical significance at the 5% level.
properties across both panels. Exceptions include the S&P
500 in Panel A and gold in Panel B, where the goodness-of-fit
test rejects the power-law hypothesis. Nevertheless, the log-
normal and exponential models remain consistently rejected
across all subsamples (p = 0.00). This consistency across
tests and subsamples strengthens the conclusion that gen-
uine power-law behavior governs the range-based variances
of these financial assets.
Building on the robustness analysis outlined in Section
4.2.3, we report the results derived from the weekly real-
ized variance estimator in this section. Table A.1 presents the
descriptive statistics for the weekly realized variances across
the five asset classes. Consistent with our earlier findings,
the realized variance distributions exhibit substantial skew-
ness and excess kurtosis—hallmarks of heavy-tailed behavior.
Weekly realized variances for crude oil and Bitcoin, in par-
ticular, display considerable variation and extreme upper-tail
values, further supporting the presence of volatility cluster-
ing and rare but impactful events. We then estimate power-
law exponents for the tail behavior of the weekly realized
variances using the MLE approach proposed by Clauset et
al. (2009). As reported in Table A.2 the estimated expo-
nents all lie within the critical range 2 < αˆ < 3, confirming
that the variance of variance remains theoretically unde-
fined for all asset markets. These results align closely with
those obtained using the Garman-Klass estimator, indicating
that our primary conclusions are not sensitive to the choice
of volatility measure. To further validate the plausibility of
the power-law model under this alternative volatility mea-
sure, we conduct KS goodness-of-fit tests for power-law,
log-normal, and exponential distributions (Table A.3). Based
on the results in Table A.3, the power-law model remains
statistically plausible for all assets (p-values > 0.05), except
gold, while the exponential model is decisively rejected
across all the assets. Additionally, although the log-normal
model is also accepted for all assets (p-values > 0.05),
the power-law model generally provides a superior fit, as
reflected in higher p-values—except in the case of gold.
In summary, the persistence of power-law behavior under
both range-based and return-based volatility estimators con-
firms the robustness of our core results. These findings
also underscore the broader insight that volatility risk in
financial markets may be governed by a common, heavy-
tailed structure—regardless of the specific estimation method
employed.
Additionally, as described in Section 4.2.4, we extended
the analysis to incorporate government bond markets in order
to examine whether the statistical properties of realized vari-
ances also apply to U.S. Treasury yields with 2-year, 5-year,
and 10-year maturities. This extension allows us to assess the
broader applicability of power-law behavior to fixed income
markets. The results of this extended analysis are presented
in tables 12–14. As shown in table 12, the descriptive statis-
tics for the range-based variances of the 2-year, 5-year, and
14 M. Fathi and K. Grobys
Table 7. Point estimates for autoregressive models of for annualized daily range-based variance.
βˆ0,j βˆ1,j βˆ2,j βˆ3,j βˆ4,j βˆ5,j βˆ6,j R2
σˆ 2S&P500,t 0.49
∗∗∗ 0.42∗∗∗ 0.10 0.08∗∗∗ 0.18∗∗∗ 0.42
(12.27) (44.89) (10.33) (7.81) (18.90)
σˆ 2Gold,t 0.74
∗∗∗ 0.36∗∗∗ 0.07∗∗ 0.03∗ 0.11∗∗∗ 0.08∗∗∗ 0.09∗∗∗ 0.30
(7.33) (8.71) (2.53) (1.69) (2.64) (2.95) (3.58)
σˆ 2Crudeoil,t 13.34
∗∗∗ 0.08∗∗∗ 0.01
(6.09) (3.37)
σˆ 2USD/GBP,t 0.48
∗∗∗ 0.16∗∗∗ 0.13∗∗ 0.09∗∗∗ 0.11∗∗∗ 0.10
(3.52) (3.11) (2.11) (3.29) (3.12)
σˆ 2BTC,t 20.35
∗∗∗ 0.44∗∗∗ 0.19
(10.73) (9.93)
∗
,
∗∗ and ∗∗∗ indicate statistical significance at the 10%, 5% and 1% levels, respectively.
Note: This table presents the estimated coefficients from autoregressive models of order P fitted to the annu-
alized daily range-based variance for the S&P 500, gold, crude oil, the USD/GBP exchange rate, and Bitcoin
(BTC). The models are specified as
σˆ 2j,t = β0,j + β1,jσˆ 2j,t−1 + β2,jσˆ 2j,t−2 + . . . + βp,jσˆ 2j,t−p + εj,t,
where εj,t is the innovation process, j refers to the respective asset. The optimal lag order P is determined
by the partial autocorrelation function. The reported coefficients include the intercept βˆ0,j and autoregressive
terms (β0,j, β1,j, β2,j, . . . , βp,j). The t-statistics are reported in parentheses.
Table 8. Descriptive statistics for the innovations process of annualized daily range-based
variance.
Statistic εˆS&P500 εˆGold εˆCrudeoil εˆUSD/GBP εˆBTC
Mean 0.00 0.00 0.00 0.00 0.00
Median − 0.40 − 0.63 − 7.14 − 0.24 − 14.29
Maximum 139.02 163.67 15818.40 132.17 1750.00
Minimum − 70.72 − 56.60 − 1048.3 − 21.49 − 676.04
Std. Dev 3.68 4.64 173.62 2.20 80.99
Skewness 11.83 10.18 88.46 38.32 8.33
Kurtosis 377.28 250.67 8053.8 2145.90 133.83
Observations 11203 11190 8555 6259 3602
Note: This table summarizes the descriptive statistics for the innovation process εi,t of the
annualized daily range-based variance for the S&P 500, gold, crude oil, the USD/GBP
exchange rate, and Bitcoin (BTC), as reported in Table 3. The innovation process of annual-
ized daily range-based variances for all five key assets is estimated by running autoregressive
models of order p:
σˆ 2j,t = β0,j + β1,jσˆ 2j,t−1 + β2,jσˆ 2j,t−2 + . . . + βp,jσˆ 2j,t−p + εj,t,
where εj,t is the innovation process, p is the lag-order, j refers to the respective asset. The
optimal lag order p is determined through an analysis of the partial autocorrelation function.
10-year Treasury yields exhibit high levels of skewness and
kurtosis, indicating heavy-tailed distributions similar to those
observed in the five major financial asset markets. The power-
law estimation results (table 13) show tail exponents of U.S.
Treasury yields with 2-year, 5-year, and 10-year maturities,
all falling within the critical interval 2 < αˆ < 3. This implies
that the second moment of range-based variances for these
bonds remains undefined, in line with our findings for major
financial asset markets. In addition, goodness-of-fit test results
in table 14 further confirm the plausibility of the power-
law model for bond market variances. The null hypothesis
of a power-law distribution is not rejected for any of the
three bond series (p-values: 0.94, 0.54, and 0.58, respec-
tively). In contrast, the log-normal and exponential models
are decisively rejected for all three bond series. These results
provide compelling evidence that power-law behavior is a
persistent and robust characteristic of range-based variances
across diverse financial markets—including government bond
markets. This suggests that heavy-tailed variance risk may be
a universal feature of modern financial systems, further rein-
forcing the need to reconsider traditional models based on
finite-variance assumptions.
Next, to complement our primary estimation strategy and
assess the robustness of the results, we apply a Bayesian
inference approach to estimate the power-law parameters
across all assets as described in section 4.2.5. Table 15
reports the Bayesian estimates of the power-law exponent αˆ,
the inferred lower-bound xˆmin. The results confirm that all
Modeling variance risk in financial markets using power-laws 15
Table 9. Estimating power-law exponents for the absolute amount of the innovation
processes of the annualized daily range-based variance.
Metric |εˆS&P500| |εˆGold | |εˆCrudeoil| |εˆUSD/GBP| |εˆBTC|
αˆ 2.54 2.46 2.41 2.40 2.30
xˆMIN 3.71 2.81 40.90 1.12 56.55
D 0.02 0.02 0.03 0.03 0.04
NPL 5.51% 14.59% 2.58% 6.65% 9.69%
Std. Dev 0.06 0.04 0.09 0.07 0.07
Observations 11203 11190 8555 6259 3602
Note: This table presents the estimation results for the power-law exponents applied to the
absolute amount of the residual series of annualized daily range-based variances for the
S&P 500, gold, crude oil, the exchange rate of the U.S. dollar and Bitcoin (BTC). The
innovation process for each asset is obtained by using the following autoregressive model:
σˆ 2j,t = β0,j + β1,jσˆ 2j,t−1 + β2,jσˆ 2j,t−2 + . . . + βp,jσˆ 2j,t−p + εj,t
where εj,t is the innovation process, p is the lag-order, and j refers to the respective asset. The
parameter αˆ represents the estimated tail exponent, while σˆ denotes the estimated standard
deviation. The lower threshold xˆMIN is determined using the optimized Kolmogorov-
Smirnov distance (D) approach, as outlined by Clauset et al. (2009). The reported xˆMIN
corresponds to the optimal distance D. Additionally, the fraction of observations governed
by the power-law process, denoted as NPL is provided for each asset.
Table 10. Goodness-of-fit tests for the innovation pro-
cesses of annualized daily range-based variance.
Power-law Log-normal Exponential
|εˆS&P500| 0.82 0.00 0.00
|εˆGold | 0.02 0.00 0.00
|εˆCrudeoil| 0.88 0.00 0.00
|εˆUSD/GBP| 0.22 0.00 0.00
|εˆBTC| 0.07 0.00 0.00
Note: This table presents the results of the goodness-of-
fit tests conducted using the Kolmogorov-Smirnov (KS)
method to assess whether the empirical data and synthetic
data generated from a power-law distribution, specified
by xmin and α, originate from the same underlying dis-
tribution (column 2). Additionally, the table includes the
results of goodness-of-fit tests for log-normal and exponen-
tial distributions in columns 3 and 4, respectively. The null
hypothesis for these tests states that the empirical data and
the synthetic data generated from the specified distributions
(log-normal and exponential) are consistent, suggesting that
each distribution provides an acceptable fit to the observed
data.
estimated exponents fall within the range typically associ-
ated with heavy-tailed behavior 2 < αˆ < 3, consistent with
our main findings based on Clauset et al.’s (2009) approach.
Importantly, the 95% credible intervals for αˆ across all assets
are relatively narrow, indicating precise and stable estimates
of the power-law exponent under the Bayesian framework. In
addition, figure 1 visualizes the posterior distribution of the
power-law exponent αˆ for each asset, illustrating the concen-
tration of posterior mass around the mean and the precision of
the Bayesian estimates. This robustness reinforces the cred-
ibility of our earlier conclusions regarding tail behavior in
financial range-based variances.
To complement the goodness-of-fit tests, we conducted a
formal model comparison using the AIC, as described in
Section 4.2.6. The AIC accounts for both model fit and
complexity, with lower values indicating a more parsimo-
nious and better-fitting model. Each distribution—power-law,
log-normal, and exponential—was fitted to the empirical data
using MLE. Table 16 reports the AIC values for each model
across the five asset classes. In all cases, the power-law dis-
tribution produces the lowest AIC, indicating a superior fit
to the extreme tails of the range-based variance distributions
when compared to the log-normal and exponential alterna-
tives. These findings provide consistent and robust evidence
that power-law models are better suited to capture the heavy-
tailed nature of range-based asset variance distributions. The
results reinforce the conclusions drawn from the goodness-of-
fit tests and underscore the relevance of power-law behavior
in modeling extreme financial events.
Furthermore, the results of the rolling-window estimations
outlined in Section 4.2.7 are presented in Table A.4 while
Figure A.1 visualizes the evolution of the time-varying power-
law exponents over the sample period. Notably, the S&P
500 dataset contains T = 11 207 observations of range-based
variance, allowing for the estimation of N = 205 power-law
exponents. In contrast, for Bitcoin, only N = 53 exponents
could be estimated due to the shorter sample. A comparison
between the full-sample estimates (table 3) and the rolling
window medians (Table A.4) reveals close alignment. For
example, Table A.4 reports median power-law exponents of
2.73 and 2.77 for the S&P 500 and Gold, respectively, while
the corresponding full-sample point estimates in table 3 are
2.94 and 2.74. Similar consistency is observed across other
range-based variance series. Finally, a visual inspection of
Figure A.1 suggests that the time-varying power-law expo-
nents derived from the rolling window estimation appear sta-
tionary, reinforcing the conclusion that the exponent estimates
are stable—that is, converging.
Next, we fit tempered stable distributions, as outlined in
Section 4.2.8, to the range-based variances of the S&P 500,
USD/GBP exchange rate, gold, crude oil, and Bitcoin, initially
fixing the power-law exponent αi = α = 2.8. The results
16 M. Fathi and K. Grobys
Table 11. Sample split analysis of power-law exponents for the innovation processes of annual-
ized daily range-based variance.
Panel A: Estimation results for the first subsample of power-law exponents and related metrics.
Metric S&P 500 Gold Crude oil USD/GBP BTC
αˆ 2.18 2.48 2.29 2.30 2.44
xˆMIN 0.65 2.82 33.07 1.24 26.81
(p-value)power−law 0.00 0.27 0.28 0.96 0.10
D 0.02 0.02 0.04 0.03 0.04
NPL 48.79 16.93 5.10 5.08 18.17
Std. Dev 0.02 0.05 0.09 0.10 0.08
(p-value)log−normal 0.00 0.00 0.00 0.00 0.00
(p-value)exponential 0.00 0.00 0.00 0.00 0.00
Observations 5601 5595 4277 3129 1801
Panel B: Estimation results for the second subsample of power-law exponents and related metrics.
Metric S&P 500 Gold Crude oil USD/GBP BTC
αˆ 2.76 2.31 2.49 2.47 2.21
xˆMIN 3.13 1.04 13.42 0.92 49.73
(p-value)power−law 0.85 0.00 0.39 0.12 0.40
D 0.02 0.02 0.04 0.04 0.03
NPL 7.10 43.66 5.35 9.90 14.71
Std. Dev 0.09 0.03 0.10 0.08 0.07
(p-value)log−normal 0.00 0.00 0.00 0.00 0.00
(p-value)exponential 0.00 0.00 0.00 0.00 0.00
Observations 5602 5595 4278 3130 1801
Note: This table presents the sample-split test of power-law exponents applied to the residual
series of annualized daily range-based variances, with results divided into two subsamples: Panel A
reports the first subsample, and Panel B reports the second subsample. The parameter αˆ represents
the estimated tail exponent, while σˆ denotes the estimated standard deviation. The lower threshold
xˆMIN is determined using the optimized Kolmogorov-Smirnov distance (D) approach, as outlined
by Clauset et al. (2009). The reported xˆMIN corresponds to the optimal distance D. Additionally,
the fraction of observations obtained by the power-law process, denoted as NPL is provided for
each asset.
Table 12. Descriptive statistics of range-based U.S. Treasury yield variances.
Statistic UST 2Y UST 5Y UST 10Y
Mean 51.41 35.47 21.70
Median 14.50 12.28 6.97
Maximum 3042.13 2975.99 5041.98
Minimum 0.00 0.00 0.00
Std. Dev. 157.46 114.40 120.87
Skewness 10.02 12.83 28.73
Kurtosis 133.63 233.29 1087.59
Observations 3010 3033 2818
Period (MM/DD/YYYY) 10/26/2012 10/26/2012 08/26/2013
07/31/2024 07/31/2024 07/31/2024
Note: This table presents the descriptive statistics of the annualized daily range-
based variances for U.S. Treasury yields with maturities of 2 years (UST 2Y),
5 years (UST 5Y), and 10 years (UST 10Y). The variances are calculated using
the Garman-Klass estimator, based on daily high, low, open, and close yields.
reveal that λi could be estimated only for the range-based vari-
ance for the USD/GBP exchange rate, while the estimation
failed to converge for the remaining series. To validate this
finding, we repeated the estimation without constraining α.
Again, both αi and λi could be reliably estimated only for
the range-based variance for the USD/GBP series. This pat-
tern is highly informative. It suggests that, for most assets, the
data do not exhibit statistically significant exponential trunca-
tion in the tails. Rather, the data appear to conform closely
to a pure power law over the observed range, rendering
the inclusion of a tempering parameter either superfluous or
numerically unstable. This interpretation aligns with earlier
empirical evidence of strong power-law decay in these series.
The consistent failure to estimate λi even when αi is uncon-
strained further supports the conclusion that the tail behavior
is well captured by a standard power law without the need for
exponential tempering.
Finally, restricting the data to x ≥ xMIN , we fit GB2 mod-
els to the range-based variances and computed the AIC
values, as outlined in Section 4.2.9. Table 17 presents the
Modeling variance risk in financial markets using power-laws 17
Table 13. Estimation of power-law exponents for range-based U.S. Treasury
yield variances.
Metric UST 2Y UST 5Y UST 10Y
αˆ 2.30 2.31 2.32
xˆMIN 87.42 56.62 27.45
D 0.02 0.03 0.02
NPL 12.82% 12.43% 14.05%
Std. Dev 0.07 0.07 0.07
Observations 3010 3033 2818
Period (MM/DD/YYYY) 02/19/1980 02/07/1980 01/02/1991
07/31/2024 07/31/2024 07/31/2024
Note: This table presents the estimation results for power-law exponents applied
to the annualized daily range-based variances of U.S. Treasury yields with 2-
year (UST 2Y), 5-year (UST 5Y), and 10-year (UST 10Y) maturities. The
parameter αˆ signifies the estimated tail exponent, while σˆ denotes the estimated
standard deviation. The lower threshold xˆMIN is identified through the opti-
mized Kolmogorov-Smirnov distance (D), following the procedure described
by Clauset et al. (2009). The reported xˆMIN corresponds to the optimal distance
D. Additionally, the fraction of observations obtained by the power-law process,
denoted as NPL is provided for each bond.
Figure 1. Posterior distribution of the power-law exponent αˆ, estimated using Bayesian inference. Note: This figure shows the posterior
distribution of the power-law exponent αˆ, estimated using Bayesian inference for values greater than the threshold xˆmin. The red dashed line
indicates the posterior mean of αˆ.
18 M. Fathi and K. Grobys
Table 14. Goodness-of-fit tests for range-based U.S.
Treasury yield variances.
Asset Power-law Log-normal Exponential
UST 2Y 0.94 0.04 0.00
UST 5Y 0.54 0.00 0.00
UST 10Y 0.58 0.00 0.00
Note: This table reports the results (viz., p-values) of the
goodness-of-fit tests for the annualized daily range-based
variances of U.S. Treasury yields with maturities of 2
years (UST 2Y), 5 years (UST 5Y), and 10 years (UST
10Y). The tests are conducted using the Kolmogorov-
Smirnov (KS) method to assess whether the empirical
data and the synthetic data—generated from a power-
law distribution specified by xmin and α —originate from
the same underlying distribution (column 2). Addition-
ally, the table includes the results of goodness-of-fit tests
for log-normal and exponential distributions in columns
3 and 4, respectively. The null hypothesis for these tests
states that the empirical data and the synthetic data gen-
erated from the specified distributions (log-normal and
exponential) are consistent, suggesting that each distri-
bution provides an acceptable fit to the observed data.
log-likelihoods and AIC values for both the power-law
models—using the parameterizations from table 3—and the
GB2 models restricted to the equivalent data ranges. Param-
eter estimates for the GB2 models are provided in Table A.6
in the appendix. The results in table 17 indicate that the AIC
values are consistently lower for the power-law models across
all range-based variance series, regardless of the underlying
asset class. This finding reinforces the results of our main
analysis, further supporting the conclusion that power-law
models are particularly well suited for capturing the dynamics
of range-based variances in financial assets.
6. Discussion
In this study we explored whether the Garman-Klass range-
based variances of five key financial assets—S&P 500,
gold, crude oil, USD/GBP exchange rate, and Bitcoin—
follow power-law distributions and examined the broader
implications of these findings for financial modeling and
risk analysis. By utilizing the Garman-Klass variance esti-
mator and conducting robust statistical tests, we provided
a comprehensive analysis of power-law behavior in range-
based variances of otherwise unrelated asset markets. Unlike
previous studies that primarily focused on absolute returns
or single-asset analyses, as highlighted by Lux and Alfarano
(2016), we employ the Garman-Klass range-based vari-
ances, which provide greater sensitivity to intraday price
dynamics and mitigate estimation bias. This methodolog-
ical improvement enables us to capture the complexity
of volatility structures across different asset classes more
effectively.
First, we confirmed the heavy-tailed nature of range-based
variances across all five assets, consistent with power-law
behavior. Data on range-based asset variances exhibited
Ta
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ex
ch
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itc
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n
(B
TC
).
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e
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es
th
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rm
ea
n
s
o
ft
he
ex
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ne
nt
αˆ
th
e
es
tim
at
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lo
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er
-
bo
un
d
th
re
sh
ol
d
xˆ m
in
,
th
e
st
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da
rd
de
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at
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ns
o
ft
he
po
ste
rio
r
di
str
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fo
rαˆ
,
an
d
th
e
95
%
cr
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ib
le
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te
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sf
or
αˆ
.
Th
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u
m
be
ro
fo
bs
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ed
.
Modeling variance risk in financial markets using power-laws 19
Table 16. AIC model comparison.
Asset Power-law∗ Log-normal Exponential
S&P 500 6346.79 36394.16 39929.06
Gold 5433.58 42811.29 45326.74
Crude oil 7592.74 58142.31 62010.57
USD/GBP 3044.78 10295.22 11925.14
BTC 1518.62 31259.29 33017.41
∗Bold values indicate the best-fitting model by AIC.
Note: This table presents the AIC values for power-law,
log-normal, and exponential distributions fitted to the annu-
alized range-based variances for each five assets, including
the S&P 500, gold, crude oil, USD/GBP exchange rate, and
Bitcoin (BTC). Each model is estimated using maximum
likelihood estimation methods. The AIC is calculated as
AIC = 2k − 2log(Lˆ), where k is the number of estimated
parameters and log(Lˆ) is the maximum log-likelihood of
the model. Lower AIC values indicate a better trade-off
between model fit and complexity.
Table 17. Comparison between power-law and GB2 functions.
Range-based variance
Log-likelihood
Power law
(AIC)
Log-likelihood
GB2 (AIC)
S&P 500 –1134.28 –3176.18
(2268.57) (6360.37)
Gold –953.78 –2714.71
(1907.56) (5437.42)
Crude oil –996.82 –3805.39
(1993.63) (7618.79)
USD/GBP –1397.88 –1528.31
(2795.76) (3064.61)
Bitcoin –114.42 –782.70
(228.85) (1573.40)
Note: We compare the GB2 distribution with our power-law
models by fitting the GB2 function to the data range x ≥ xmin for
each respective data vector, as reported in Table 3. The values
of the log-likelihood functions are computed, along with the cor-
responding AIC. This table reports the log-likelihood and AIC
values for both the power-law models—using the parameteriza-
tions documented in Table 3—and the GB2 models, which are
estimated over the same data ranges defined by the respective
power-law cutoffs.
significant skewness and kurtosis, with extreme values con-
centrated in the upper tails of the variance distributions.
For example, Bitcoin exhibited the highest mean value of
Garman-Klass range-based variance, reflecting its speculative
and volatile nature, while the USD/GBP exchange rate dis-
played relatively lower mean value of Garman-Klass range-
based variance.
Second, power-law fitting through MLE verified that tail
exponents for all assets fell within the range 2 < α < 3,
implying that the variance-of-variance is infinite. Interest-
ingly, these findings contrast with literature from the early
1990s, as discussed by Lux and Alfarano (2016), which
focused on the tail behavior of distributions and suggested that
financial asset returns exhibit power-law behavior with finite
variances. This result has profound implications, as it chal-
lenges traditional econometric assumptions that rely on finite
variances for hypothesis testing. The estimated power-law
exponent α has direct implications for economic risk. Lower
values of α indicate heavier tails in the return distribution,
implying a higher probability of extreme losses. This sug-
gests a more volatile and risk-prone market environment. For
instance, when α lies between 2 and 3, the theoretical mean
of the variance process is defined but higher-order moments
such as variance or skewness are undefined, posing chal-
lenges for traditional risk models. When α approaches 2 or
below, the theoretical mean of the variance process becomes
infinite, highlighting severe tail risk. Hence, the power-law
exponent serves as a quantifiable indicator of the intensity
of tail events, informing asset pricing, risk management, and
regulatory capital frameworks.
This finding also aligns with the previous literature (Grobys
2021, 2023, 2024, Fathi et al. 2024, 2025) which documented
tail exponents for realized variances or range-based variances
are within the range 2 < α < 3, indicating that the variance-
of-variances is infinite. In the presence of infinite variance-of-
variance, extreme observations can disproportionately influ-
ence the variance estimate, leading to results that may appear
significant in one sample but fail to replicate in others.
As highlighted by Fama (1963), if variances are undefined,
results derived from traditional statistical methods based on
the concept of correlation (i.e. t-statistics) become inherently
sample-dependent. According to Grobys (2021), unstable sec-
ond moments lead to t-statistics that are highly dependent
on the specific sample used. Thus, this study, in line with
Grobys (2021), argues that the replication failures observed in
financial research may stem from the infinite variance of vari-
ances, which leads to traditional statistical methods becoming
unreliable.
Third, our results derived from goodness-of-fit tests
strongly supported power-law distributions as plausible mod-
els for the range-based variances of these assets. These
findings align with Mandelbrot’s (1963) early insights that
financial asset returns may lack finite variances, and strength-
ens the notion that traditional statistical tools, such as OLS
and other moment-dependent methods, may be inadequate for
modeling financial data governed by power-law distributions.
Fourth, our analysis comprehensively compared power-
law distributions with alternative models (viz., log-normal
and exponential). The goodness-of-fit test strongly supported
power-law models as the most plausible representation of
the data for all assets, with alternative distributions being
consistently rejected. This finding strengthens the case for
power-law distributions as a fundamental characteristic of
financial asset variances. The rejection of log-normal and
exponential models aligns with Taleb’s (2020) view on the
necessity of adopting power-law frameworks for analyzing
financial data, particularly in the context of extreme events
and tail risks. Furthermore, these findings contrast with the
literature suggesting that realized asset volatility follows a
log-normal distribution (e.g. Andersen et al. 2001, 2001a,
2001b), but they align with studies proposing that a Pareto dis-
tribution better describes volatility in financial markets (Renò
and Rizza 2003, Grobys 2021, 2023, 2024, Fathi et al. 2024,
2025).
The presence of power-law behavior in financial returns
and volatility carries profound implications for risk measure-
ment frameworks such as Value-at-Risk (VaR) and stress test-
ing. Traditional models, including those based on log-normal
20 M. Fathi and K. Grobys
or Gaussian assumptions, underestimate the probability and
magnitude of extreme events due to their thin-tailed nature.
Empirical evidence from Bouchaud (2001) and Gencay et al.
(2003) demonstrates that financial markets often exhibit
heavy-tailed distributions, particularly in the tails of return
and variance distributions. This tail heaviness implies that
large losses occur far more frequently than predicted by nor-
mal models, which can severely compromise the reliability of
VaR estimates. Gencay and Selçuk (2004) show that applying
Extreme Value Theory (EVT) and the Generalized Pareto Dis-
tribution (GPD) provides a more robust framework for model-
ing the tails of financial distributions. These approaches allow
for better estimation of risk at high quantiles and facilitate
more realistic stress testing scenarios. Moreover, power-law
scaling properties are indicative of long memory and clustered
volatility, which further challenge the assumptions of inde-
pendence and stationarity in standard risk models. As such,
incorporating power-law features into risk modeling frame-
works is essential for accurately capturing the true nature of
financial market risk, particularly in the context of systemic
shocks.
Fifth, to assess whether the range-based variances of the
five assets share a common component, we implemented a
joint test hypothesizing a common power-law exponent. Rec-
ognizing the limitations of standard errors derived under inde-
pendence assumptions of Clauset et al. (2009), we employed
block bootstrap techniques following Grobys (2024) to com-
pute robust standard errors. In line with the study of Grobys
(2024), the bootstrap procedure showed that robust standard
errors were significantly larger than the ones derived in the
study of Clauset et al. (2009). Using the robust standard error,
we carried out conjoint tests, iteratively computed across a
range of hypothesized exponents (q). Our findings suggest
strong evidence for a shared power-law exponent governing
the range-based variances of these assets. Specifically, the
null hypothesis of a shared exponent could not be rejected
for the range 2.5 < q < 3.1, suggesting a unified compo-
nent influencing variance risks across the S&P 500, gold,
crude oil, USD/GBP exchange rate, and Bitcoin. This finding
aligns with theoretical expectations of power-law behavior
and underscores the interconnected nature of these assets
within a broader market framework.
Next, to address the potential impact of autocorrelation
on tail index estimates, we applied autoregressive models
of order p to the Garman-Klass range-based variances of
each asset. The results indicate significant autocorrelation
patterns in the processes generating range-based variances.
The descriptive statistics indicates that even after accounting
for higher-order autocorrelation, the innovation processes of
the annualized daily range-based variances for all the assets
display extremely heavy tails.
We then used the absolute values of the estimated inno-
vation processes for each asset to estimate the power-law
exponents. Notably, the estimated exponents closely align
with those observed for the original data, providing strong evi-
dence for the presence of genuine power-law behavior. Next,
we performed Clauset et al.’s (2009) goodness-of-fit test, for
the absolute values of the innovation processes of each asset.
This procedure helps to evaluate whether the observed power-
law behavior persists after accounting for autocorrelation,
providing a robust check on the validity of the power-law
estimates for our range-based variances. The goodness-of-fit
test indeed indicated that the power-law null hypothesis holds
for all innovation processes except gold, providing additional
confirmation of the consistent power-law characteristics in the
range-based variances. Additionally, the goodness-of-fit test
results, with zero p-values across all cases, reject log-normal
and exponential distributions, favoring the power-law distri-
bution instead. These results support recent findings of Fathi
et al. (2025) documenting that realized variances of Fama-
French factors are subject to power-law behavior even after
controlling for autocorrelations.
Finally, to validate the robustness of the power-law expo-
nents, we performed sample-split tests by dividing the resid-
ual series for each asset into two equal-length subsamples.
The consistency of tail exponents across both subsamples
demonstrated that power-law behavior persists independently
of sample-specific effects. Additionally, the goodness-of-fit
test supports the power-law model in both subsamples for
most assets, further enhancing the reliability of our findings.
A reader could be concerned about the robustness of tail
estimation due to limited data in the extreme upper tail. How-
ever, the tail sample sizes in our study consistently exceed
the empirical threshold suggested by Clauset et al. (2009),
who argue that reliable estimation of power-law parameters
typically requires a minimum of approximately 50 tail obser-
vations. For instance, in the case of Bitcoin, the number of
observations in the power-law tail constitutes at least 3.41%
of the total sample size of 3603—yielding more than 120
observations. Thus, the sample size in the tail comfortably
meets the minimum requirement for credible tail exponent
estimation using maximum likelihood techniques.
Another potential concern relates to whether our range-
based volatility estimates sufficiently address market micro
structure noise. In this regard, we note that our analysis
is based on daily OHLC data and employs the Garman–
Klass estimator, which has been identified as one of the
most efficient and least noisy among range-based volatility
measures (Molnár 2012). Importantly, microstructure noise
is particularly problematic in high-frequency data, whereas
range-based estimators constructed from daily prices are con-
siderably less susceptible to such distortions. Thus, while no
volatility estimator is entirely free from noise, the estimator
we use represents a methodologically sound and practically
robust choice that mitigates the typical concerns associated
with market microstructure effects.
Finally, the potential influence of structural market fea-
tures, such as liquidity and trading hours, on tail behavior
merits attention. However, our results suggest that markets
with widely varying characteristics—such as Bitcoin (traded
24/7 with relatively lower trading volume) and the S&P 500
(traded during regular hours with much higher volume)—
exhibit statistically indistinguishable power-law exponents
in range-based volatility. This finding is robust across asset
classes, including commodities and FX. Additionally, trad-
ing volume in the S&P 500 has shown a clear upward trend
over the sample period (see Figure A.2). As such, any attempt
to split the sample into low- and high-liquidity regimes
would effectively mirror our existing time-based splits (as in
table 11), which already reveal stable tail exponent estimates
Modeling variance risk in financial markets using power-laws 21
across subperiods. These observations suggest that structural
differences such as trading hours and volume have limited
impact on the tail behavior we observe.
7. Conclusion
This study presents novel evidence on the variance risk
of financial markets by modeling range-based variances
using the Garman-Klass variance estimator and analyzing
their properties under hypothesized power-law distributions.
Focusing on five key financial assets—S&P 500, gold, crude
oil, the USD/GBP exchange rate, and Bitcoin—our find-
ings reveal that range-based variances exhibit heavy tails,
aligning closely with power-law distributions. These findings
carry profound implications for financial modeling and risk
management.
First, the estimated tail exponents for range-based vari-
ances consistently fall within the range 2 < α < 3, indicating
characteristics of infinite variance-of-variance. As noted by
Mandelbrot (1967), such variance can be considered ‘so large
that it may in practice be assumed infinite.’ This challenges
the foundational assumption of finite variances in traditional
econometric models and highlights the limitations of statisti-
cal tools like t-tests, which are derived from the ordinary least
squares (OLS) regression framework. The evidence of sam-
ple specificity, as highlighted by Fama’s (1963) reflections
on Mandelbrot’s infinite variance hypothesis, underscores
the inadequacy of moment-dependent statistical methods for
handling the heavy-tailed distributions common in financial
markets.
Second, goodness-of-fit tests consistently reject alternative
distributions, including the widely used log-normal model,
in favor of power-law distributions, emphasizing the inade-
quacy of conventional models in capturing the tail risks that
define financial markets. Additionally, evidence derived from
blocks bootstrap underscores the importance of accounting for
dependency structures when estimating the standard devia-
tions of tail exponents. Our joint test for a common power-law
exponent reveals statistical invariance in power-law behavior
for the range (2.5 < q < 3.1) across all five assets, suggest-
ing a universal factor governing variance risk. Autoregressive
modeling of range-based variances further corroborates the
persistence of power-law behavior even after adjusting for
autocorrelation. Lastly, sample-split tests confirm the sta-
bility of power-law exponents over different time periods,
reinforcing the robustness of the findings.
The empirical finding that a common power-law expo-
nent governs realized variances across otherwise unre-
lated asset markets finds theoretical support in Gabaix
(2016). In his comprehensive review, Gabaix (2016) outlines
mechanisms—such as proportional random growth processes
and universality—that explain why power-law distributions
emerge across diverse economic systems. These mechanisms
suggest that similar tail behaviors can arise in structurally dif-
ferent markets due to shared statistical properties, rather than
asset-specific dynamics. Gabaix (2016) also emphasizes the
concept of universality, where systems governed by different
microfoundations may exhibit identical scaling laws at the
macro level. This perspective aligns with the observation in
our study that assets as diverse as crude oil, gold, and Bitcoin
exhibit similar tail exponents. Furthermore, his discussion of
aggregation effects and macroeconomic granularity provides
a theoretical foundation for interpreting power-law regular-
ities as emergent from complex but generalizable dynam-
ics. Hence, our finding is consistent with and theoretically
grounded in the framework developed by Gabaix (2016).
Furthermore, our results underscore the necessity of adopt-
ing alternative statistical frameworks for analyzing financial
data and managing extreme risks. Effectively infinite vari-
ances and the rejection of log-normality suggests that reliance
on traditional tools can significantly underestimate tail risks,
particularly in environments characterized by extreme mar-
ket events. The heavy-tailed nature of range-based variances
has substantial implications for portfolio optimization, asset
pricing, and risk management, necessitating frameworks that
effectively account for the inherent extremity in financial data.
By advancing a robust foundation for understanding variance
risks across diverse asset classes, this study contributes to a
more accurate and unified approach to financial modeling,
particularly under infinite variance phenomena.
Despite its significant contributions, this study has limi-
tations that merit consideration. For example, the analysis
is limited to five financial assets—S&P 500, gold, crude
oil, USD/GBP exchange rate, and Bitcoin—which constrains
the generalizability of the findings to other asset classes or
broader market conditions. Expanding the analysis to a wider
range of assets and markets would enhance the applicability
and breadth of the results.
8. Open Scholarship
This article has earned the Center for Open Science
badge for Open Data. The data are openly accessible
at (https://zenodo.org/records/15707465 (DOI 10.5281/zen-
odo.15707464).
Acknowledgments
The authors are grateful to the three anonymous referees for
their valuable comments.
Data availability statement
The dataset has been deposited in Zenodo and is pub-
licly available at: https://zenodo.org/records/15707465 (DOI
10.5281/zenodo.15707464).
Disclosure statement
No potential conflict of interest was reported by the author(s).
22 M. Fathi and K. Grobys
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Appendix
Figure A1. Time-varying estimated power-law exponents based on annualized daily range-based variances for various assets. Note: This
figure plots the estimated power-law exponents derived from rolling window estimations based on annualized daily range-based variances for
various assets, including the S&P 500, gold, crude oil, the USD/GBP exchange rate, and Bitcoin (BTC). The annualized daily range-based
variances for each asset are calculated using the Garman-Klass methodology. The power-law exponents are estimated using the maximum
likelihood estimation approach.
Figure A2. Trading volume of the S&P 500 over the sample period. Note: The figure displays the daily dollar trading volume over the
sample period from February 19, 1980 to July 31, 2024.
Modeling variance risk in financial markets using power-laws 25
Table A1. Descriptive statistics for weekly realized variance.
Statistic S&P 500 Gold Crude oil USD/GBP BTC
Mean 6.49 6.54 34.81 1.67 74.28
Median 2.99 3.35 15.36 1.09 35.46
Maximum 794.29 265.27 8330.27 78.36 4029.48
Minimum 0.02 0.02 0.14 0.01 0.00
Std. Dev. 22.28 12.33 220.41 3.01 189.70
Skewness 23.90 9.25 32.64 15.04 14.63
Kurtosis 758.30 142.91 1186.98 345.92 277.88
Observations 2242 2239 1711 1253 721
Period (MM/DD/YYYY) 02/19/1980 02/07/1980 01/02/1991 07/25/2000 09/17/2014
07/31/2024 07/31/2024 07/31/2024 07/31/2024 07/31/2024
Note: This table presents the descriptive statistics of the weekly realized variance for various assets, including the
S&P 500, gold, crude oil, USD/GBP exchange rate, and Bitcoin (BTC). The weekly realized variances for each asset
calculated using the sum of the squared returns over five business days.
Table A2. Estimation of power-law exponents for weekly realized variance.
Metric S&P 500 Gold Crude oil USD/GBP BTC
αˆ 2.51 2.24 2.43 2.84 2.93
xˆMIN 10.51 4.13 23.42 1.96 162.39
D 0.02 0.04 0.03 0.03 0.05
NPL 12.53% 43.19% 33.26% 24.18% 10.40%
Std. Dev 0.09 0.04 0.06 0.11 0.22
Observations 2242 2239 1711 1253 721
Period (MM/DD/YYYY) 02/19/1980 02/07/1980 01/02/1991 07/25/2000 09/17/2014
07/31/2024 07/31/2024 07/31/2024 07/31/2024 07/31/2024
Note: This table presents the estimation results for power-law exponents applied to the weekly realized variances
of the S&P 500, gold, crude oil, the USD/GBP exchange rate, and Bitcoin. The weekly realized variances for each
asset calculated using the sum of the squared returns over five business days. The parameter αˆ signifies the estimated
tail exponent, while σˆ denotes the estimated standard deviation. The lower threshold xˆMIN is identified through
the optimized Kolmogorov-Smirnov distance (D), following the procedure described by Clauset et al. (2009). The
reported xˆMIN corresponds to the optimal distance D. Additionally, the fraction of observations obtained by the
power-law process, denoted as NPL is provided for each asset.
Table A3. Goodness-of-fit tests for the weekly realized variance.
Asset market Power-law Log-normal Exponential
S&P 500 0.99 0.58 0.00
Gold 0.04 0.65 0.00
Crude oil 0.34 0.11 0.00
USD/GBP 0.67 0.31 0.00
BTC 0.73 0.31 0.00
Note: This table presents the results (viz., p-values) of the goodness-
of-fit tests for weekly realized variance, conducted using the
Kolmogorov-Smirnov (KS) method to assess whether the empirical
data and the synthetic data—generated from a power-law distribu-
tion specified by xmin and α —originate from the same underlying
distribution (column 2). Additionally, the table includes the results
of goodness-of-fit tests for log-normal and exponential distributions
in columns 3 and 4, respectively. The null hypothesis for these tests
states that the empirical data and the synthetic data generated from
the specified distributions (log-normal and exponential) are consis-
tent, suggesting that each distribution provides an acceptable fit to
the observed data. The weekly realized variances for each asset mar-
ket are calculated using the sum of the squared returns over five
business days.
26 M. Fathi and K. Grobys
Table A4. Descriptive statistics of estimated power-law exponents derived from rolling
window estimations.
Statistic S&P 500 Gold Crude Oil USD/GBP BTC
Minimum 1.83 1.91 1.79 1.76 1.98
5% Qnt. 1.99 2.17 1.92 2.28 2.04
10% Qnt. 2.10 2.31 2.22 2.54 2.05
Median 2.73 2.77 3.11 3.24 2.54
90% Qnt. 3.88 3.34 4.04 7.01 2.84
95% Qnt. 4.59 3.55 4.27 8.41 2.91
Maximum 7.61 4.63 5.33 11.48 3.38
Mean 2.96 2.79 3.09 3.78 2.45
Std. Dev 0.92 0.44 0.68 1.82 0.31
Excess Kurtosis 7.48 2.67 0.32 4.19 − 0.06
Skewness 2.26 1.15 0.39 2.17 0.35
N 205 204 152 106 53
Note: This table reports descriptive statistics for estimated power-law exponents derived
from rolling window estimations based on annualized daily range-based variances for var-
ious assets, including the S&P 500, gold, crude oil, the USD/GBP exchange rate, and
Bitcoin (BTC). The annualized daily range-based variances for each asset are calculated
using the Garman-Klass methodology. The power-law exponents are estimated using the
maximum likelihood estimation approach. Quintiles (Qnt) are reported from the 5th to the
95th percentile.
Table A5. Important subfamilies of the GB2 distribution.
Subfamily Condition
Generalized Gamma q → ∞
Beta Prime a = 1
F-distribution a = 1, b = 1, p = d12 , q = d22
Singh–Maddala p = 1
Dagum distribution q = 1
Pareto type IV p = 1, b = 1
Note: The Generalized Beta of the Second Kind (GB2) is a four-parameter family of continuous probability distributions that encompasses
a wide variety of shapes. The GB2 distribution is defined as:
f (x; a; b; p; q) = a(x/b)
ap−1
bB(p, q)[1 + (x/b)a]p+q ,
where B(p, q) is the Beta function, a > 0 is the tail parameter controlling the tail heaviness, b > 0 is a scale parameter, p > 0, q > 0 are
shape parameters that also govern skewness. This table reports important subfamilies of the GB2 distribution.
Table A6. Estimated parameters of the GB2 distribution.
Range-based variance a b p q
S&P 500 125.30 4.44 67.73 0.016
Gold 114.95 6.36 116.26 0.015
Crude oil 73.95 22.47 113.15 0.022
USD/GBP 108.31 1.05 105.28 0.016
BTC 223.86 243.08 0.0255 0.0094
Note: The Generalized Beta of the Second Kind (GB2) is a four-parameter family of continuous
probability distributions that encompasses a wide variety of shapes. The GB2 distribution is defined
as:
f (x; a; b; p; q) = a(x/b)
ap−1
bB(p, q)[1 + (x/b)a]p+q ,
where B(p, q) is the Beta function, a > 0 is the tail parameter controlling the tail heaviness, b > 0 is
a scale parameter, p > 0, q > 0 are shape parameters that influence the distribution’s form, including
its skewness. This table reports the parameter estimates for GB2 distributions fitted to the data ranges
defined by the power-law cutoffs listed in table 3.