arXiv subjects

R. A. Serota

Publications and source records attributed to R. A. Serota.

At least 19 recordsLinked to original sources

arXiv2026

Fitting Accumulated Stock Returns with Tempered Skew t-Distribution

We analyze distributions of historic S&P500 multi-day returns, for the number of days of accumulation from 20 to 120. With the increase of the number of days of accumulation, we observe clear tempering of power-law tails toward a seemingly finite value. To explain this phenomenon, we employ a model that produces a "capped Inverse Gamma" stationary (steady-state) distribution for stochastic volatility which, in turn, produces a "tempered Student-t" distribution for returns. We then employ Jones-Faddy-like symmetry breaking mechanism that produces a "tempered Skew-t" distribution. This distribution provides rather good fits to the distributions of accumulated multi-day S&P500 returns, which exhibit symmetry breaking between gains and losses -- as reflected by positive mean and negative skew. Tempered Skew-t fits are also consistent with near perfect linear dependence on the number of days of accumulation of the mean values and, even more so, of the variances (mean squared realized volatility) of the distributions.

q-fin.ST↗

arXiv2026

Broken Symmetry, Conservation Law, and Scaling in Accumulated Stock Returns -- a Modified Jones-Faddy Skew t-Distribution Perspective

We analyze historic S&P500 multi-day returns: from daily returns to those accumulated over up to ten days. Despite symmetry breaking between gains and losses in the distribution of returns, resulting in its positive mean and negative skew, realized variance (volatility squared) exhibits remarkably good linear dependence on the number of days of accumulation. Mean of the distribution also shows near perfect linear dependence as well. We analyze this phenomenon both analytically and numerically using a modified Jones-Faddy skew t-distribution.

q-fin.ST↗

arXiv2026

Broken Symmetry of Stock Returns -- a Modified Jones-Faddy Skew t-Distribution

We argue that negative skew and positive mean of the distribution of stock returns are largely due to the broken symmetry of stochastic volatility governing gains and losses. Starting with stochastic differential equations for stock returns and for stochastic volatility we argue that the distribution of stock returns can be effectively split in two -- for gains and losses -- assuming difference in parameters of their respective stochastic volatilities. A modified Jones-Faddy skew t-distribution utilized here allows to reflect this in a single organic distribution which tends to meaningfully capture this asymmetry. We illustrate its application on distribution of daily S&P500 returns, including analysis of its tails.

q-fin.ST↗

arXiv2025

Asymmetry in Distributions of Accumulated Gains and Losses in Stock Returns

We study decades-long historic distributions of accumulated S\&P500 returns, from daily returns to those over several weeks. The time series of the returns emphasize major upheavals in the markets -- Black Monday, Tech Bubble, Financial Crisis and Covid Pandemic -- which are reflected in the tail ends of the distributions. De-trending the overall gain, we concentrate on comparing distributions of gains and losses. Specifically, we compare the tails of the distributions, which are believed to exhibit power-law behavior and possibly contain outliers. Towards this end we find confidence intervals of the linear fits of the tails of the complementary cumulative distribution functions on a log-log scale, as well as conduct a statistical U-test in order to detect outliers. We also study probability density functions of the full distributions of the returns with the emphasis on their asymmetry. The key empirical observations are that the mean of de-trended distributions increases near-linearly with the number of days of accumulation while the overall skew is negative -- consistent with the heavier tails of losses -- and depends little on the number of days of accumulation. At the same time the variance of the distributions exhibits near-perfect linear dependence on the number of days of accumulation, that is it remains constant if scaled to the latter. Finally, we discuss the theoretical framework for understanding accumulated returns. Our main conclusion is that the current state of theory, which predicts symmetric or near-symmetric distributions of returns cannot explain the aggregate of empirical results.

q-fin.ST↗

arXiv2023

Exploring Distributions of House Prices and House Price Indices

We use house prices (HP) and house price indices (HPI) as a proxy to income distribution. Specifically, we analyze sale prices in the 1970-2010 window of over 116,000 single-family homes in Hamilton County, Ohio, including Cincinnati metro area of about 2.2 million people. We also analyze HPI, published by Federal Housing Finance Agency (FHFA), for nearly 18,000 US ZIP codes that cover a period of over 40 years starting in 1980's. If HP can be viewed as a first derivative of income, HPI can be viewed as its second derivative. We use generalized beta (GB) family of functions to fit distributions of HP and HPI since GB naturally arises from the models of economic exchange described by stochastic differential equations. Our main finding is that HP and multi-year HPI exhibit a negative Dragon King (nDK) behavior, wherein power-law distribution tail gives way to an abrupt decay to a finite upper limit value, which is similar to our recent findings for realized volatility of S\&P500 index in the US stock market. This type of tail behavior is best fitted by a modified GB (mGB) distribution. Tails of single-year HPI appear to show more consistency with power-law behavior, which is better described by a GB Prime (GB2) distribution. We supplement full distribution fits by mGB and GB2 with direct linear fits (LF) of the tails. Our numerical procedure relies on evaluation of confidence intervals (CI) of the fits, as well as of p-values that give the likelihood that data come from the fitted distributions.

econ.EM↗

arXiv2023

Are there Dragon Kings in the Stock Market?

We undertake a systematic study of historic market volatility spanning roughly five preceding decades. We focus specifically on the time series of realized volatility (RV) of the S&P500 index and its distribution function. As expected, the largest values of RV coincide with the largest economic upheavals of the period: Savings and Loan Crisis, Tech Bubble, Financial Crisis and Covid Pandemic. We address the question of whether these values belong to one of the three categories: Black Swans (BS), that is they lie on scale-free, power-law tails of the distribution; Dragon Kings (DK), defined as statistically significant upward deviations from BS; or Negative Dragons Kings (nDK), defined as statistically significant downward deviations from BS. In analyzing the tails of the distribution with RV > 40, we observe the appearance of "potential" DK which eventually terminate in an abrupt plunge to nDK. This phenomenon becomes more pronounced with the increase of the number of days over which the average RV is calculated -- here from daily, n=1, to "monthly," n=21. We fit the entire distribution with a modified Generalized Beta (mGB) distribution function, which terminates at a finite value of the variable but exhibits a long power-law stretch prior to that, as well as Generalized Beta Prime (GB2) distribution function, which has a power-law tail. We also fit the tails directly with a straight line on a log-log scale. In order to ascertain BS, DK or nDK behavior, all fits include their confidence intervals and p-values are evaluated for the data points to check if they can come from the respective distributions.

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arXiv2022

Rethinking Generalized Beta Family of Distributions

We approach the Generalized Beta (GB) family of distributions using a mean-reverting stochastic differential equation (SDE) for a power of the variable, whose steady-state (stationary) probability density function (PDF) is a modified GB (mGB) distribution. The SDE approach allows for a lucid explanation of Generalized Beta Prime (GB2) and Generalized Beta (GB1) limits of GB distribution and, further down, of Generalized Inverse Gamma (GIGa) and Generalized Gamma (GGa) limits, as well as describe the transition between the latter two. We provide an alternative form to the "traditional" GB PDF to underscore that a great deal of usefulness of GB distribution lies in its allowing a long-range power-law behavior to be ultimately terminated at a finite value. We derive the cumulative distribution function (CDF) of the "traditional" GB, which belongs to the family generated by the regularized beta function and is crucial for analysis of the tails of the distribution. We analyze fifty years of historical data on realized market volatility, specifically for S\&P500, as a case study of the use of GB/mGB distributions and show that its behavior is consistent with that of negative Dragon Kings.

q-fin.ST↗

arXiv2020

Distributions of Historic Market Data -- Relaxation and Correlations

We investigate relaxation and correlations in a class of mean-reverting models for stochastic variances. We derive closed-form expressions for the correlation functions and leverage for a general form of the stochastic term. We also discuss correlation functions and leverage for three specific models -- multiplicative, Heston (Cox-Ingersoll-Ross) and combined multiplicative-Heston -- whose steady-state probability density functions are Gamma, Inverse Gamma and Beta Prime respectively, the latter two exhibiting "fat" tails. For the Heston model, we apply the eigenvalue analysis of the Fokker-Planck equation to derive the correlation function -- in agreement with the general analysis -- and to identify a series of time scales, which are observable in relaxation of cumulants on approach to the steady state. We test our findings on a very large set of historic financial markets data.

q-fin.ST↗

arXiv2019

Modeling Response Time Distributions with Generalized Beta Prime

We use Generalized Beta Prime distribution, also known as GB2, for fitting response time distributions. This distribution, characterized by one scale and three shape parameters, is incredibly flexible in that it can mimic behavior of many other distributions. GB2 exhibits power-law behavior at both front and tail ends and is a steady-state distribution of a simple stochastic differential equation. We apply GB2 in contrast studies between two distinct groups -- in this case children with dyslexia and a control group -- and show that it provides superior fitting. We compare aggregate response time distributions of the two groups for scale and shape differences (including several scale-independent measures of variability, such as Hoover index), which may in turn reflect on cognitive dynamics differences. In this approach, response time distribution of an individual can be considered as a random variate of that individual's group distribution.

q-bio.NC↗

arXiv2019

Generalized Beta Prime Distribution: Stochastic Model of Economic Exchange and Properties of Inequality Indices

We argue that a stochastic model of economic exchange, whose steady-state distribution is a Generalized Beta Prime (also known as GB2), and some unique properties of the latter, are the reason for GB2's success in describing wealth/income distributions. We use housing sale prices as a proxy to wealth/income distribution to numerically illustrate this point. We also explore parametric limits of the distribution to do so analytically. We discuss parametric properties of the inequality indices -- Gini, Hoover, Theil T and Theil L -- vis-a-vis those of GB2 and introduce a new inequality index, which serves a similar purpose. We argue that Hoover and Theil L are more appropriate measures for distributions with power-law dependencies, especially fat tails, such as GB2.

econ.EM↗

arXiv2019

Implied and Realized Volatility: A Study of Distributions and the Distribution of Difference

We study distributions of realized variance (squared realized volatility) and squared implied volatility, as represented by VIX and VXO indices. We find that Generalized Beta distribution provide the best fits. These fits are much more accurate for realized variance than for squared VIX and VXO -- possibly another indicator that the latter have deficiencies in predicting the former. We also show that there are noticeable differences between the distributions of the 1970-2017 realized variance and its 1990-2017 portion, for which VIX and VXO became available. This may be indicative of a feedback effect that implied volatility has on realized volatility. We also discuss the distribution of the difference between squared implied volatility and realized variance and show that, at the basic level, it is consistent with Pearson's correlations obtained from linear regression.

q-fin.ST↗

arXiv2018

Combined Mutiplicative-Heston Model for Stochastic Volatility

We consider a model of stochastic volatility which combines features of the multiplicative model for large volatilities and of the Heston model for small volatilities. The steady-state distribution in this model is a Beta Prime and is characterized by the power-law behavior at both large and small volatilities. We discuss the reasoning behind using this model as well as consequences for our recent analyses of distributions of stock returns and realized volatility.

q-fin.MF↗

arXiv2018

Distributions of Historic Market Data -- Implied and Realized Volatility

We undertake a systematic comparison between implied volatility, as represented by VIX (new methodology) and VXO (old methodology), and realized volatility. We compare visually and statistically distributions of realized and implied variance (volatility squared) and study the distribution of their ratio. We find that the ratio is best fitted by heavy-tailed -- lognormal and fat-tailed (power-law) -- distributions, depending on whether preceding or concurrent month of realized variance is used. We do not find substantial difference in accuracy between VIX and VXO. Additionally, we study the variance of theoretical realized variance for Heston and multiplicative models of stochastic volatility and compare those with realized variance obtained from historic market data.

q-fin.MF↗

arXiv2017

Distributions of Historic Market Data - Stock Returns

We show that the moments of the distribution of historic stock returns are in excellent agreement with the Heston model and not with the multiplicative model, which predicts power-law tails of volatility and stock returns. We also show that the mean realized variance of returns is a linear function of the number of days over which the returns are calculated. The slope is determined by the mean value of the variance (squared volatility) in the mean-reverting stochastic volatility models, such as Heston and multiplicative, independent of stochasticity. The distribution function of stock returns, which rescales with the increase of the number of days of return, is obtained from the steady-state variance distribution function using the product distribution with the normal distribution.

q-fin.MF↗

arXiv2017

On absence of steady state in the Bouchaud-Mézard network model

In the limit of infinite number of nodes (agents), the Itô-reduced Bouchaud-Mézard network model of economic exchange has a time-independent mean and a steady-state inverse gamma distribution. We show that for a finite number of nodes the mean is actually distributed as a time-dependent lognormal and inverse gamma is quasi-stationary, with the time-dependent scale parameter.

cond-mat.stat-mech↗

arXiv2016

Correlation and Relaxation Times for a Stochastic Process with a Fat-Tailed Steady-State Distribution

We study a stochastic process defined by the interaction strength for the return to the mean and a stochastic term proportional to the magnitude of the variable. Its steady-state distribution is the Inverse Gamma distribution, whose power-law tail exponent is determined by the ratio of the interaction strength to stochasticity. Its time-dependence is characterized by a set of discrete times describing relaxation of respective cumulants to their steady-state values. We show that as the progressively lower cumulants diverge with the increase of stochasticity, so do their relaxation times. We analytically evaluate the correlation function and show that it is determined by the longest of these times, namely the inverse interaction strength, which is also the relaxation time of the mean. We also investigate relaxation of the entire distribution to the steady state and the distribution of relaxations times, which we argue to be Inverse Gaussian.

cond-mat.stat-mech↗

arXiv2015

Modeling response time with power-law distributions

Understanding the properties of response time distributions is a long-standing problem in cognitive science. We provide a tutorial overview of several contemporary models that assume power law scaling is a plausible description of the skewed heavy tails that are typically expressed in response time distributions. We discuss several properties and markers of these distribution functions that have implications for cognitive and neurophysiological organization supporting a given cognitive activity. We illustrate how a power law assumption suggests that collecting larger samples, and combining individual subjects' data into a single set for a distribution-function analysis allows for a better comparison of a group of interest to a control group. We demonstrate our techniques in contrasts of response time measurements of children with and without dyslexia.

q-bio.NC↗

arXiv2013

Spectral and Parametric Averaging for Integrable Systems

We analyze two theoretical approaches to ensemble averaging for integrable systems in quantum chaos - spectral averaging and parametric averaging. For spectral averaging, we introduce a new procedure - rescaled spectral averaging. Unlike traditional spectral averaging, it can describe the correlation function of spectral staircase and produce persistent oscillations of the interval level number variance. Parametric averaging, while not as accurate as rescaled spectral averaging for the correlation function of spectral staircase and interval level number variance, can also produce persistent oscillations of the global level number variance and better describes saturation level rigidity as a function of the running energy. Overall, it is the most reliable method for a wide range of statistics.

quant-ph↗