The Great Moderation Flattens Fat Tails: Disappearing Leptokurtosis

Recently, Fagiolo et al. (2008) find fat tails of economic growth rates after adjusting outliers, autocorrelation and heteroskedasticity. This paper employs US quarterly real output growth, showing that this finding of fat tails may reflect the Great Moderation. That is, leptokurtosis disappears after GARCH adjustment once we incorporate the break in the variance equation.

An Empirical Investigation of Value-at-Risk in Long and Short Trading Positions

This paper uses the Value-at-Risk approach to define the risk in both long and short trading positions. The investigation is done on some major market indices(Japanese, UK, German and US). The performance of models that takes into account skewness and fat-tails are compared to symmetric models in relation to both the specific model for estimating the variance, and the distribution of the variance estimate used as input in the VaR estimation. The results indicate that more flexible models not necessarily perform better in predicting the VaR forecast; the reason for this is most probably the complexity of these models. A general result is that different methods for estimating the variance are needed for different confidence levels of the VaR, and for the different indices. Also, different models are to be used for the left respectively the right tail of the distribution.

Unusual eigenvalue spectrum and relaxation in the Levy-Ornstein-Uhlenbeck process

We consider the rates of relaxation of a particle in a harmonic well, subject to Levy noise characterized by its Levy index mu. Using the propagator for this Levy-Ornstein-Uhlenbeck process (LOUP), we show that the eigenvalue spectrum of the associated Fokker-Planck operator has the form (n + m mu)nu where nu is the force constant characterizing the well, and n, m is an element of N. If mu is irrational, the eigenvalues are all nondegenerate, but rational mu can lead to degeneracy. The maximum degeneracy is shown to be 2. The left eigenfunctions of the fractional Fokker-Planck operator are very simple while the right eigenfunctions may be obtained from the lowest eigenfunction by a combination of two different step-up operators. Further, we find that the acceptable eigenfunctions should have the asymptotic behavior vertical bar x vertical bar(-n1-n2 mu) as vertical bar x vertical bar -> infinity, with n(1) and n(2) being positive integers, though this condition alone is not enough to identify them uniquely. We also assert that the rates of relaxation of LOUP are determined by the eigenvalues of the associated fractional Fokker-Planck operator and do not depend on the initial state if the moments of the initial distribution are all finite. If the initial distribution has fat tails, for which the higher moments diverge, one can have nonspectral relaxation, as pointed out by Toenjes et al. Phys. Rev. Lett. 110, 150602 (2013)].

Scale-wise evolution of rainfall probability density functions fingerprints the rainfall generation mechanism

© 2010 by the American Geophysical Union.The cross-scale probabilistic structure of rainfall intensity records collected over time scales ranging from hours to decades at sites dominated by both convective and frontal systems is investigated. Across these sites, intermittency build-up from slow to fast time-scales is analyzed in terms of heavy tailed and asymmetric signatures in the scale-wise evolution of rainfall probability density functions (pdfs). The analysis demonstrates that rainfall records dominated by convective storms develop heavier-Tailed power law pdfs toward finer scales when compared with their frontal systems counterpart. Also, a concomitant marked asymmetry build-up emerges at such finer time scales. A scale-dependent probabilistic description of such fat tails and asymmetry appearance is proposed based on a modified q-Gaussian model, able to describe the cross-scale rainfall pdfs in terms of the nonextensivity parameter q, a lacunarity (intermittency) correction and a tail asymmetry coefficient, linked to the rainfall generation mechanism.

Do benchmark African equity indices exhibit the stylized facts?

This paper investigates if benchmark African equity indices exhibit the stylized facts reported for financial time-series returns. The returns distributions of the Africa All-Share, Large, Medium and Small Company Indices were found to be leptokurtotic, had fat-tails, over time experienced volatility clustering and exhibited long memory in volatility. Both the All-Share and Large Company Indices were found to exhibit leverage effects. In contrast, positive shocks had a greater impact on future volatility for the Small Company Index which implies a reverse leverage effect. This finding could reflect a bull/bubble market for small capitalisation stocks in Africa.

An Eigenfunction Approach for Volatility Modeling.

In this paper, we introduce a new approach for volatility modeling in discrete and continuous time. We follow the stochastic volatility literature by assuming that the variance is a function of a state variable. However, instead of assuming that the loading function is ad hoc (e.g., exponential or affine), we assume that it is a linear combination of the eigenfunctions of the conditional expectation (resp. infinitesimal generator) operator associated to the state variable in discrete (resp. continuous) time. Special examples are the popular log-normal and square-root models where the eigenfunctions are the Hermite and Laguerre polynomials respectively. The eigenfunction approach has at least six advantages: i) it is general since any square integrable function may be written as a linear combination of the eigenfunctions; ii) the orthogonality of the eigenfunctions leads to the traditional interpretations of the linear principal components analysis; iii) the implied dynamics of the variance and squared return processes are ARMA and, hence, simple for forecasting and inference purposes; (iv) more importantly, this generates fat tails for the variance and returns processes; v) in contrast to popular models, the variance of the variance is a flexible function of the variance; vi) these models are closed under temporal aggregation.

How the full opening of the capital account to highly liquid financial markets led Latin America to two and a half cycles of ‘mania, panic and crash’

Latin America has recently experienced three cycles of capital inflows, the first two ending in major financial crises. The first took place between 1973 and the 1982 ‘debt-crisis’. The second took place between the 1989 ‘Brady bonds’ agreement (and the beginning of the economic reforms and financial liberalisation that followed) and the Argentinian 2001/2002 crisis, and ended up with four major crises (as well as the 1997 one in East Asia) — Mexico (1994), Brazil (1999), and two in Argentina (1995 and 2001/2). Finally, the third inflow-cycle began in 2003 as soon as international financial markets felt reassured by the surprisingly neo-liberal orientation of President Lula’s government; this cycle intensified in 2004 with the beginning of a (purely speculative) commodity price-boom, and actually strengthened after a brief interlude following the 2008 global financial crash — and at the time of writing (mid-2011) this cycle is still unfolding, although already showing considerable signs of distress. The main aim of this paper is to analyse the financial crises resulting from this second cycle (both in LA and in East Asia) from the perspective of Keynesian/ Minskyian/ Kindlebergian financial economics. I will attempt to show that no matter how diversely these newly financially liberalised Developing Countries tried to deal with the absorption problem created by the subsequent surges of inflow (and they did follow different routes), they invariably ended up in a major crisis. As a result (and despite the insistence of mainstream analysis), these financial crises took place mostly due to factors that were intrinsic (or inherent) to the workings of over-liquid and under-regulated financial markets — and as such, they were both fully deserved and fairly predictable. Furthermore, these crises point not just to major market failures, but to a systemic market failure: evidence suggests that these crises were the spontaneous outcome of actions by utility-maximising agents, freely operating in friendly (‘light-touch’) regulated, over-liquid financial markets. That is, these crises are clear examples that financial markets can be driven by buyers who take little notice of underlying values — i.e., by investors who have incentives to interpret information in a biased fashion in a systematic way. Thus, ‘fat tails’ also occurred because under these circumstances there is a high likelihood of self-made disastrous events. In other words, markets are not always right — indeed, in the case of financial markets they can be seriously wrong as a whole. Also, as the recent collapse of ‘MF Global’ indicates, the capacity of ‘utility-maximising’ agents operating in (excessively) ‘friendly-regulated’ and over-liquid financial market to learn from previous mistakes seems rather limited.

How the full opening of the capital account to highly liquid financial markets led Latin America to two and a half cycles of ‘mania, panic and crash’

Latin America has recently experienced three cycles of capital inflows, the first two ending in major financial crises. The first took place between 1973 and the 1982 ‘debt-crisis’. The second took place between the 1989 ‘Brady bonds’ agreement (and the beginning of the economic reforms and financial liberalisation that followed) and the Argentinian 2001/2002 crisis, and ended up with four major crises (as well as the 1997 one in East Asia) — Mexico (1994), Brazil (1999), and two in Argentina (1995 and 2001/2). Finally, the third inflow-cycle began in 2003 as soon as international financial markets felt reassured by the surprisingly neo-liberal orientation of President Lula’s government; this cycle intensified in 2004 with the beginning of a (purely speculative) commodity price-boom, and actually strengthened after a brief interlude following the 2008 global financial crash — and at the time of writing (mid-2011) this cycle is still unfolding, although already showing considerable signs of distress. The main aim of this paper is to analyse the financial crises resulting from this second cycle (both in LA and in East Asia) from the perspective of Keynesian/ Minskyian/ Kindlebergian financial economics. I will attempt to show that no matter how diversely these newly financially liberalised Developing Countries tried to deal with the absorption problem created by the subsequent surges of inflow (and they did follow different routes), they invariably ended up in a major crisis. As a result (and despite the insistence of mainstream analysis), these financial crises took place mostly due to factors that were intrinsic (or inherent) to the workings of over-liquid and under-regulated financial markets — and as such, they were both fully deserved and fairly predictable. Furthermore, these crises point not just to major market failures, but to a systemic market failure: evidence suggests that these crises were the spontaneous outcome of actions by utility-maximising agents, freely operating in friendly (light-touched) regulated, over-liquid financial markets. That is, these crises are clear examples that financial markets can be driven by buyers who take little notice of underlying values — investors have incentives to interpret information in a biased fashion in a systematic way. ‘Fat tails’ also occurred because under these circumstances there is a high likelihood of self-made disastrous events. In other words, markets are not always right — indeed, in the case of financial markets they can be seriously wrong as a whole. Also, as the recent collapse of ‘MF Global’ indicates, the capacity of ‘utility-maximising’ agents operating in unregulated and over-liquid financial market to learn from previous mistakes seems rather limited.

Verificação e análise dos fatos estilizados no mercado de ações brasileiro

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Estudo e análise estatística no mercado de ações brasileiro

The objective of this paper is to verify and analyze the existence in Brazil of stylized facts observed in financial time series: volatility clustering, probability distributions with fat tails, the presence of long run memory in absolute return time series, absence of linear return autocorrelation, gain/loss asymmetry, aggregative gaussianity, slow absolute return autocorrelation decay, trading volume/volatility correlation and leverage effect. We analyzed intraday prices for 10 stocks traded at the BM&FBovespa, responsible for 52.1% of the Ibovespa portfolio on Sept. 01, 2009. The data analysis confirms the stylized facts, whose behavior is consistent with what is observed in international markets.

Modelización de series temporales complejas: de las redes de telecomunicación a los mercados financieros

Este trabajo aborda el problema de modelizar sistemas din´amicos reales a partir del estudio de sus series temporales, usando una formulaci´on est´andar que pretende ser una abstracci´on universal de los sistemas din´amicos, independientemente de su naturaleza determinista, estoc´astica o h´ıbrida. Se parte de modelizaciones separadas de sistemas deterministas por un lado y estoc´asticos por otro, para converger finalmente en un modelo h´ıbrido que permite estudiar sistemas gen´ericos mixtos, esto es, que presentan una combinaci´on de comportamiento determinista y aleatorio. Este modelo consta de dos componentes, uno determinista consistente en una ecuaci´on en diferencias, obtenida a partir de un estudio de autocorrelaci´on, y otro estoc´astico que modeliza el error cometido por el primero. El componente estoc´astico es un generador universal de distribuciones de probabilidad, basado en un proceso compuesto de variables aleatorias, uniformemente distribuidas en un intervalo variable en el tiempo. Este generador universal es deducido en la tesis a partir de una nueva teor´ıa sobre la oferta y la demanda de un recurso gen´erico. El modelo resultante puede formularse conceptualmente como una entidad con tres elementos fundamentales: un motor generador de din´amica determinista, una fuente interna de ruido generadora de incertidumbre y una exposici´on al entorno que representa las interacciones del sistema real con el mundo exterior. En las aplicaciones estos tres elementos se ajustan en base al hist´orico de las series temporales del sistema din´amico. Una vez ajustados sus componentes, el modelo se comporta de una forma adaptativa tomando como inputs los nuevos valores de las series temporales del sistema y calculando predicciones sobre su comportamiento futuro. Cada predicci´on se presenta como un intervalo dentro del cual cualquier valor es equipro- bable, teniendo probabilidad nula cualquier valor externo al intervalo. De esta forma el modelo computa el comportamiento futuro y su nivel de incertidumbre en base al estado actual del sistema. Se ha aplicado el modelo en esta tesis a sistemas muy diferentes mostrando ser muy flexible para afrontar el estudio de campos de naturaleza dispar. El intercambio de tr´afico telef´onico entre operadores de telefon´ıa, la evoluci´on de mercados financieros y el flujo de informaci´on entre servidores de Internet son estudiados en profundidad en la tesis. Todos estos sistemas son modelizados de forma exitosa con un mismo lenguaje, a pesar de tratarse de sistemas f´ısicos totalmente distintos. El estudio de las redes de telefon´ıa muestra que los patrones de tr´afico telef´onico presentan una fuerte pseudo-periodicidad semanal contaminada con una gran cantidad de ruido, sobre todo en el caso de llamadas internacionales. El estudio de los mercados financieros muestra por su parte que la naturaleza fundamental de ´estos es aleatoria con un rango de comportamiento relativamente acotado. Una parte de la tesis se dedica a explicar algunas de las manifestaciones emp´ıricas m´as importantes en los mercados financieros como son los “fat tails”, “power laws” y “volatility clustering”. Por ´ultimo se demuestra que la comunicaci´on entre servidores de Internet tiene, al igual que los mercados financieros, una componente subyacente totalmente estoc´astica pero de comportamiento bastante “d´ocil”, siendo esta docilidad m´as acusada a medida que aumenta la distancia entre servidores. Dos aspectos son destacables en el modelo, su adaptabilidad y su universalidad. El primero es debido a que, una vez ajustados los par´ametros generales, el modelo se “alimenta” de los valores observables del sistema y es capaz de calcular con ellos comportamientos futuros. A pesar de tener unos par´ametros fijos, la variabilidad en los observables que sirven de input al modelo llevan a una gran riqueza de ouputs posibles. El segundo aspecto se debe a la formulaci´on gen´erica del modelo h´ıbrido y a que sus par´ametros se ajustan en base a manifestaciones externas del sistema en estudio, y no en base a sus caracter´ısticas f´ısicas. Estos factores hacen que el modelo pueda utilizarse en gran variedad de campos. Por ´ultimo, la tesis propone en su parte final otros campos donde se han obtenido ´exitos preliminares muy prometedores como son la modelizaci´on del riesgo financiero, los algoritmos de routing en redes de telecomunicaci´on y el cambio clim´atico. Abstract This work faces the problem of modeling dynamical systems based on the study of its time series, by using a standard language that aims to be an universal abstraction of dynamical systems, irrespective of their deterministic, stochastic or hybrid nature. Deterministic and stochastic models are developed separately to be merged subsequently into a hybrid model, which allows the study of generic systems, that is to say, those having both deterministic and random behavior. This model is a combination of two different components. One of them is deterministic and consisting in an equation in differences derived from an auto-correlation study and the other is stochastic and models the errors made by the deterministic one. The stochastic component is an universal generator of probability distributions based on a process consisting in random variables distributed uniformly within an interval varying in time. This universal generator is derived in the thesis from a new theory of offer and demand for a generic resource. The resulting model can be visualized as an entity with three fundamental elements: an engine generating deterministic dynamics, an internal source of noise generating uncertainty and an exposure to the environment which depicts the interactions between the real system and the external world. In the applications these three elements are adjusted to the history of the time series from the dynamical system. Once its components have been adjusted, the model behaves in an adaptive way by using the new time series values from the system as inputs and calculating predictions about its future behavior. Every prediction is provided as an interval, where any inner value is equally probable while all outer ones have null probability. So, the model computes the future behavior and its level of uncertainty based on the current state of the system. The model is applied to quite different systems in this thesis, showing to be very flexible when facing the study of fields with diverse nature. The exchange of traffic between telephony operators, the evolution of financial markets and the flow of information between servers on the Internet are deeply studied in this thesis. All these systems are successfully modeled by using the same “language”, in spite the fact that they are systems physically radically different. The study of telephony networks shows that the traffic patterns are strongly weekly pseudo-periodic but mixed with a great amount of noise, specially in the case of international calls. It is proved that the underlying nature of financial markets is random with a moderate range of variability. A part of this thesis is devoted to explain some of the most important empirical observations in financial markets, such as “fat tails”, “power laws” and “volatility clustering”. Finally it is proved that the communication between two servers on the Internet has, as in the case of financial markets, an underlaying random dynamics but with a narrow range of variability, being this lack of variability more marked as the distance between servers is increased. Two aspects of the model stand out as being the most important: its adaptability and its universality. The first one is due to the fact that once the general parameters have been adjusted , the model is “fed” on the observable manifestations of the system in order to calculate its future behavior. Despite the fact that the model has fixed parameters the variability in the observable manifestations of the system, which are used as inputs of the model, lead to a great variability in the possible outputs. The second aspect is due to the general “language” used in the formulation of the hybrid model and to the fact that its parameters are adjusted based on external manifestations of the system under study instead of its physical characteristics. These factors made the model suitable to be used in great variety of fields. Lastly, this thesis proposes other fields in which preliminary and promising results have been obtained, such as the modeling of financial risk, the development of routing algorithms for telecommunication networks and the assessment of climate change.

Unifractality and multifractality in the Italian stock market

Tests for random walk behaviour in the Italian stock market are presented, based on an investigation of the fractal properties of the log return series for the Mibtel index. The random walk hypothesis is evaluated against alternatives accommodating either unifractality or multifractality. Critical values for the test statistics are generated using Monte Carlo simulations of random Gaussian innovations. Evidence is reported of multifractality, and the departure from random walk behaviour is statistically significant on standard criteria. The observed pattern is attributed primarily to fat tails in the return probability distribution, associated with volatility clustering in returns measured over various time scales. © 2009 Elsevier Inc. All rights reserved.

On the Lp-Norm Regression Models for Estimating Value-at-Risk

Analysis of risk measures associated with price series data movements and its predictions are of strategic importance in the financial markets as well as to policy makers in particular for short- and longterm planning for setting up economic growth targets. For example, oilprice risk-management focuses primarily on when and how an organization can best prevent the costly exposure to price risk. Value-at-Risk (VaR) is the commonly practised instrument to measure risk and is evaluated by analysing the negative/positive tail of the probability distributions of the returns (profit or loss). In modelling applications, least-squares estimation (LSE)-based linear regression models are often employed for modeling and analyzing correlated data. These linear models are optimal and perform relatively well under conditions such as errors following normal or approximately normal distributions, being free of large size outliers and satisfying the Gauss-Markov assumptions. However, often in practical situations, the LSE-based linear regression models fail to provide optimal results, for instance, in non-Gaussian situations especially when the errors follow distributions with fat tails and error terms possess a finite variance. This is the situation in case of risk analysis which involves analyzing tail distributions. Thus, applications of the LSE-based regression models may be questioned for appropriateness and may have limited applicability. We have carried out the risk analysis of Iranian crude oil price data based on the Lp-norm regression models and have noted that the LSE-based models do not always perform the best. We discuss results from the L1, L2 and L∞-norm based linear regression models. ACM Computing Classification System (1998): B.1.2, F.1.3, F.2.3, G.3, J.2.

Three essays on asset pricing

In this dissertation, I investigate three related topics on asset pricing: the consumption-based asset pricing under long-run risks and fat tails, the pricing of VIX (CBOE Volatility Index) options and the market price of risk embedded in stock returns and stock options. These three topics are fully explored in Chapter II through IV. Chapter V summarizes the main conclusions. In Chapter II, I explore the effects of fat tails on the equilibrium implications of the long run risks model of asset pricing by introducing innovations with dampened power law to consumption and dividends growth processes. I estimate the structural parameters of the proposed model by maximum likelihood. I find that the stochastic volatility model with fat tails can, without resorting to high risk aversion, generate implied risk premium, expected risk free rate and their volatilities comparable to the magnitudes observed in data. In Chapter III, I examine the pricing performance of VIX option models. The contention that simpler-is-better is supported by the empirical evidence using actual VIX option market data. I find that no model has small pricing errors over the entire range of strike prices and times to expiration. In general, Whaley’s Black-like option model produces the best overall results, supporting the simpler-is-better contention. However, the Whaley model does under/overprice out-of-the-money call/put VIX options, which is contrary to the behavior of stock index option pricing models. In Chapter IV, I explore risk pricing through a model of time-changed Lvy processes based on the joint evidence from individual stock options and underlying stocks. I specify a pricing kernel that prices idiosyncratic and systematic risks. This approach to examining risk premia on stocks deviates from existing studies. The empirical results show that the market pays positive premia for idiosyncratic and market jump-diffusion risk, and idiosyncratic volatility risk. However, there is no consensus on the premium for market volatility risk. It can be positive or negative. The positive premium on idiosyncratic risk runs contrary to the implications of traditional capital asset pricing theory.

Three Essays on Asset Pricing

In this dissertation, I investigate three related topics on asset pricing: the consumption-based asset pricing under long-run risks and fat tails, the pricing of VIX (CBOE Volatility Index) options and the market price of risk embedded in stock returns and stock options. These three topics are fully explored in Chapter II through IV. Chapter V summarizes the main conclusions. In Chapter II, I explore the effects of fat tails on the equilibrium implications of the long run risks model of asset pricing by introducing innovations with dampened power law to consumption and dividends growth processes. I estimate the structural parameters of the proposed model by maximum likelihood. I find that the stochastic volatility model with fat tails can, without resorting to high risk aversion, generate implied risk premium, expected risk free rate and their volatilities comparable to the magnitudes observed in data. In Chapter III, I examine the pricing performance of VIX option models. The contention that simpler-is-better is supported by the empirical evidence using actual VIX option market data. I find that no model has small pricing errors over the entire range of strike prices and times to expiration. In general, Whaley’s Black-like option model produces the best overall results, supporting the simpler-is-better contention. However, the Whaley model does under/overprice out-of-the-money call/put VIX options, which is contrary to the behavior of stock index option pricing models. In Chapter IV, I explore risk pricing through a model of time-changed Lévy processes based on the joint evidence from individual stock options and underlying stocks. I specify a pricing kernel that prices idiosyncratic and systematic risks. This approach to examining risk premia on stocks deviates from existing studies. The empirical results show that the market pays positive premia for idiosyncratic and market jump-diffusion risk, and idiosyncratic volatility risk. However, there is no consensus on the premium for market volatility risk. It can be positive or negative. The positive premium on idiosyncratic risk runs contrary to the implications of traditional capital asset pricing theory.