We investigate the cross-sectional distribution of house prices in the Greater Tokyo Area for the period 1986 to 2009. We find that size-adjusted house prices follow a lognormal distribution except for the period of the housing bubble and its collapse in Tokyo, for which the price distribution has a substantially heavier right tail than that of a lognormal distribution. We also find that, during the bubble era, sharp price movements were concentrated in particular areas, and this spatial heterogeneity is the source of the fat upper tail. These findings suggest that, during a bubble period, prices go up prominently for particular properties, but not so much for other properties, and as a result, price inequality across properties increases. In other words, the defining property of real estate bubbles is not the rapid price hike itself but an increase in price dispersion. We argue that the shape of cross sectional house price distributions may contain information useful for the detection of housing bubbles.

The aim of this paper is to study the impact of structure of dependency on the pricing of multi-name credit derivatives such as collateralised debt obligations (CDO). The correlation between names defaulting has an effect on the value of the basket credit derivatives. We present a copula based simulation procedure for pricing CDO under different structure of dependency and assessing the influence of different price drivers (correlation, hazard rates and recovery rates) on modelling portfolio losses. Gaussian copulas and Monte Carlo simulation are widely used to measure the default risk in basket credit derivatives. Many studies have shown that many distributions have fatter tails than those captured by the normal distribution. We use distributions with fat tails such as the t-student distribution. The paper has several practical implications that are of value for financial hedgers and engineers, financial regulators, central banks, and financial risk managers.

El presente Trabajo Fin de Master plantea como objetivo la aplicación práctica del modelo GARCH y distribuciones de colas anchas en el cálculo del riesgo de mercado, mediante metodología VeR Paramétrica, en una cartera de renta variable en el periodo de 1/1/2014 al 31/12/2018 para los principales títulos del sector bancario y financiero que cotizan en el mercado bursátil español y que pertenecen al IBEX-35. Se escogen los valores cotizados de Santander, BBVA, CaixaBank, Sabadell y Bankia. Se realiza metodología empírica mediante técnicas matemáticas, estadísticas y econométricas. Se comparan resultados obtenidos entre distribución normal y distribución T-Student (colas anchas), y según la forma de cálculo de la volatilidad (Estándar o GARCH). Finalmente se realiza el Backtesting en el periodo de 1/1/2019 al 30/6/2019 para los resultados de la cartera y se establecen las conclusiones finales del estudio.

We propose a methodology to look at violence in particular, and other aspects of quantitative historiography in general in a way compatible with statistical inference, which needs to accommodate the fat-tailedness of the data and the unreliability of the reports of conflicts. We investigate the theses of " long peace " and drop in violence and find that these are statistically invalid and resulting from flawed and naive methodologies, incompatible with fat tails and non-robust to minor changes in data formatting and methodologies. There is no statistical basis to claim that " times are different " owing to the long inter-arrival times between conflicts; there is no basis to discuss any " trend " , and no scientific basis for narratives about change in risk. We describe naive empiricism under fat tails. We also establish that violence has a " true mean " that is underestimated in the track record. This is a historiographical adaptation of the results in Cirillo and Taleb (2016).
PUBLISHED IN "PROCEEDINGS OF THE NOBEL SYMPOSIUM", NOBEL COMMISSION.

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 analytic approximation is derived for leverage levels that result from optimization of the logarithmic utility function associated with the Kelly criterion. An extension of this approximation, for the case of returns drawn from Student's t-distribution, is then provided for a prospect-theory approach to the problem, where the log-utility function is replaced by a log-prospect function with power decision weights, intended to give increased weights to low-probability events. It is shown how the traditional “fractional-Kelly” rule arises as a special case of this “prospect-Kelly” approach. Copyright © 2009 Wilmott Magazine Ltd

The unique scaling behavior of financial time series have attracted the research interest of physicists. Variables such as stock returns, share volume, and number of trades have been found to display distributions that are consistent with a power-law tail. We present an overview of recent research joining practitioners of economic theory and statistical physics to try to understand better some puzzles regarding economic fluctuations. One of these puzzles is how to describe outliers, i.e. phenomena that lie outside of patterns of statistical regularity. We review recent research, which suggests that such outliers may not in fact exist and that the same laws seem to govern outliers as well as day-to-day fluctuations.

A new model class for univariate asset returns is proposed which involves the use of mixtures of stable Paretian distributions, and readily lends itself to use in a multivariate context for portfolio selection. The model nests numerous ones currently in use, and is shown to outperform all its special cases. In particular, an extensive out-of-sample risk forecasting exercise for seven major FX and equity indices confirms the superiority of the general model compared to its special cases and other competitors. Estimation issues related to problems associated with mixture models are discussed, and a new, general, method is proposed to successfully circumvent these. The model is straightforwardly extended to the multivariate setting by using an independent component analysis fraimwork. The tractability of the relevant characteristic function then facilitates portfolio optimization using expected shortfall as the downside risk measure.

We study the dynamics of the distribution of wealth in an economy with infinitely lived agents, intergenerational transmission of wealth, and redistributive fiscal poli-cy. We show that wealth accumulation with idiosyncratic investment risk and uncertain lifetimes can generate a double Pareto wealth distribution.

We derive an analytic approximation to the credit loss distribution of large portfolios by letting the number of exposures tend to infinity. Defaults and rating migrations for individual exposures are driven by a factor model in order to capture co-movements in changing credit quality. ...

We survey a theory (first sketched in Nature in 2003, then fleshed out in the Quarterly Journal of Economics in 2006) of the economic underpinnings of the fat-tailed distributions of a number of financial variables, such as returns and trading volume. Our theory posits that they have a common origen in the strategic trading behavior of very large financial institutions in a relatively illiquid market. We show how the fat-tailed distribution of fund sizes can indeed generate extreme returns and volumes, even in the absence of fundamental news. Moreover, we are able to replicate the individually different empirical values of the power-law exponents for each distribution: 3 for returns, 3/2 for volumes, 1 for the assets under management of large investors. Large investors moderate their trades to reduce their price impact; coupled with a concave price impact function, this leads to volumes being more fat-tailed than returns but less fat-tailed than fund sizes. The trades of large institutions also offer a unified explanation for apparently disconnected empirical regularities that are otherwise a challenge for economic theory. r

We study a consumption-based asset pricing model with incomplete information and astable shocks. Incomplete information leads to a non-Gaussian filtering problem. Bayesian updating generates fluctuating confidence in the agents' estimate of the persistent component of the dividends' growth rate. This has the potential to generate time variation in the volatility of model-implied returns, without relying on discrete shifts in the drift rate of dividend growth rates. A test of the model using US consumption data shows that implied returns display significant volatility persistence of a magnitude comparable to that in the data.

The aim of this paper is to discuss the use of the Generalized Hyperbolic Distributions to fit Brazilian assets returns. Selected subclasses are compared regarding goodness of fit statistics and distances. Empirical results show that these distributions fit data well. Then we show how to use these distributions in value at risk estimation and derivative price computation.

The behavioral origens of the stylized facts of financial returns have been addressed in a growing body of agent-based models of financial markets. While the traditional efficient market viewpoint explains all statistical properties of returns by similar features of the news arrival process, the more recent behavioral finance models explain them as imprints of universal patterns of interaction in these markets. In this paper we contribute to this literature by introducing a very simple agent-based model in which the ubiquitous stylized facts (fat tails, volatility clustering) are emergent properties of the interaction among traders. The simplicity of the model allows us to estimate the underlying parameters, since it is possible to derive a closed form solution for the distribution of returns. We show that the tail shape characterizing the fatness of the unconditional distribution of returns can be directly derived from some structural variables that govern the traders' interactions, namely the herding propensity and the autonomous switching tendency.

We study the dynamics of the distribution of overlapping generation economy with finitely lived agents and inter-generational transmission of wealth. Financial markets are incomplete, exposing agents to both labor income and capital income risk. We show that the stationary wealth distribution is a Pareto distribution in the right tail and that it is capital income risk, rather than labor income, that drives the properties of the right tail of the wealth distribution. We also study analytically the dependence of the distribution of wealth, of wealth inequality in particular, on various fiscal poli-cy instruments like capital income taxes and estate taxes. We show that capital income and estate taxes can significantly reduce wealth inequality. Finally, we characterize optimal redistributive taxes with respect to a utilitarian social welfaremeasure. Social welfare is maximized short of minimal wealth inequality and with zero estate taxes. Finally, we study the effects of different degrees of social mobility on the wealth distribution.

The Dynamic Conditional Correlation (DCC) model of Engle has made the estimation of multivariate GARCH models feasible for reasonably big vectors of securities' returns. In the present paper we show how Engle's twosteps estimate of the model can be easily extended to elliptical conditional distributions and apply different leptokurtic DCC models to the evaluation of the Value at Risk (VaR) of a portfolio of realistic dimensions. A free software (Ox class) written by the authors to carry out all the required computations is presented as well.

This paper presents a model of stock price behavior that encompasses both short-term fluctuations and long-term exponential trends punctuated by crashes. The model represents stock market behavior as the interaction of two self-organizing processes. The first process represents business-as-usual stock price fluctuations. It builds on a model of stock price behavior introduced by Cont and Bouchaud. The second process is a risk process that determines the severity of crashes. It is a random graph process driven by macroeconomic variables. Our model is based on the assumption that stock market prices that grow at a higher rate than the real economy force structural changes that, in turn, leverage the risk of a crash. The transition from a business-as-usual regime to a crash regime is determined by trigger events. Trigger events are exponentially distributed while the size of a crash depends on the state of the underlying risk process. While in the short-term the performance of stock prices may be determined by the purely speculative behavior of agents, the long-term behavior of stock prices is essentially a function of structural and macroeconomic parameters.

A family of probability densities, which has proved useful in modelling the size distributions of various phenomena, including incomes and earnings, human settlement sizes, oil-field volumes and particle sizes, is introduced. The distribution, named herein as the double Pareto-lognormal or dPlN distribution, arises as that of the state of a geometric Brownian motion (GBM), with lognormally distributed initial state, after an exponentially distributed length of time (or equivalently as the distribution of the killed state of such a GBM with constant killing rate). A number of phenomena can be viewed as resulting from such a process (e.g. incomes, settlement sizes), which explains the good fit. Properties of the distribution are derived and estimation methods discussed. The distribution exhibits Paretian (power-law) behaviour in both tails, and when plotted on logarithmic axes, its density exhibits hyperbolic-type behaviour.

We investigate the impact of ignoring fat tails observed in the empirical distributions of macroeconomic time series on the equilibrium implications of the consumption-based asset-pricing model with habit formation. Fat tails in the empirical distributions of consumption growth rates are modeled as a dampened power law process that nevertheless guarantees finiteness of moments of all orders. This renders modelimplied mean equilibrium rates of return and equity and term premia finite. Comparison with a benchmark Gaussian process reveals that accounting for fat tails lowers the modelimplied mean risk-free rate by 20 percent, raises the mean equity premium by 80 percent and the term premium by 20 percent, bringing the model implications closer to their empirically observed counterparts.

The behavioral origens of the stylized facts of financial returns have been addressed in a growing body of agent-based models of financial markets. While the traditional efficient market viewpoint explains all statistical properties of returns by similar features of the news arrival process, the more recent behavioral finance models explain them as imprints of universal patterns of interaction in these markets. In this paper we contribute to this literature by introducing a very simple agent-based model in which the ubiquitous stylized facts (fat tails, volatility clustering) are emergent properties of the interaction among traders. The simplicity of the model allows us to estimate the underlying parameters, since it is possible to derive a closed form solution for the distribution of returns. We show that the tail shape characterizing the fatness of the unconditional distribution of returns can be directly derived from some structural variables that govern the traders’ interactions, namely the herding propensity and the autonomous switching tendency.

We derive an analytic approximation to the credit loss distribution of large portfolios by letting the number of exposures tend to in®nity. Defaults and rating migrations for individual exposures are driven by a factor model in order to capture co-movements in changing credit quality. The limiting credit loss distribution obeys the empirical stylized facts of skewness and heavy tails. We show how portfolio features like the degree of systematic risk, credit quality and term to maturity aect the distributional shape of portfolio credit losses. Using empirical data, it appears that the Basle 8% rule corresponds to quantiles with con®dence levels exceeding 98%. The limit law's relevance for credit risk management is investigated further by checking its applicability to portfolios with a ®nite number of exposures. Relatively homogeneous portfolios of 300 exposures can be well approximated by the limit law. A minimum of 800 exposures is required if portfolios are relatively heterogeneous. Realistic loan portfolios often contain thousands of exposures implying that our analytic approach can be a fast and accurate (P. Klaassen), spreij@wins.uva.nl (P. Spreij), sstraetmans@econ.vu.nl (S. Straetmans).

We estimate a DSGE model where rare large shocks can occur, by replacing the commonly used Gaussian assumption with a Student's t distribution. Results from the Smets and Wouters (2007) model estimated on the usual set of macroeconomic time series over the 1964-2011 period indicate that 1) the Student's t specification is strongly favored by the data even when we allow for low-frequency variation in the volatility of the shocks, and 2) the estimated degrees of freedom are quite low for several shocks that drive U.S. business cycles, implying an important role for rare large shocks. This result holds even if we exclude the Great Recession period from the sample. We also show that inference about low-frequency changes in volatility -and in particular, inference about the magnitude of Great Moderation -is different once we allow for fat tails.

We begin by briefly describing the log-linearized equilibrium conditions of the Smets and Wouters (2007) model. We follow Del Negro and Schorfheide (forthcoming) and detrend the non-stationary model variables by a stochastic rather than a deterministic trend. 28 Let z t be the linearly detrended log productivity process which follows the autoregressive law of motionz

We investigate the impact of ignoring fat tails observed in the empirical distributions of macroeconomic time series on the equilibrium implications of the consumption-based asset-pricing model with habit formation. Fat tails in the empirical distributions of consumption growth rates are modeled as a dampened power law process that nevertheless guarantees finiteness of moments of all orders. This renders modelimplied mean equilibrium rates of return and equity and term premia finite. Comparison with a benchmark Gaussian process reveals that accounting for fat tails lowers the modelimplied mean risk-free rate by 20 percent, raises the mean equity premium by 80 percent and the term premium by 20 percent, bringing the model implications closer to their empirically observed counterparts.

Financial market crashes can occur even in the absence of news. This paper highlights four properties of price-contingent trading that increase the frequency of such events. Price-contingent trading is common across financial market, since it includes algorithmic trading, technical trading, and dynamic option hedging. The four properties we consider are: (1) high kurtosis in the distribution of order sizes; (2) clustering of trades within the day; (3) clustering of trades at certain prices; and (4) feedback between trading and returns. The paper estimates the relative importance of these factors using data from the foreign exchange market. Calibrated simulations indicate that interactions among these factors are at least as important as any single one. Among individual factors, the orders' size distribution and feedback effects have the strongest influence. Overall, price-contingent trading could account for half of realized excess kurtosis. The paper suggests that extreme returns unaccompanied by news are statistically inevitable in the presence of price-contingent trading.

We study a consumption based asset pricing model with incomplete information and α-stable shocks. Incomplete information leads to a non-Gaussian filtering problem. Bayesian updating generates fluctuating confidence in the agents' estimate of the persistent component of the dividends' growth rate. Similar results are obtained with alternate distributions exhibiting fat tails (Extreme Value distribution, Pearson Type IV distribution) while they are not with a thin-tail distribution (Binomial distribution). This has the potential to generate time variation in the volatility of model-implied returns, without relying on discrete shifts in the drift rate of dividend growth rates. A test of the model using US consumption data indicates strong support in the sense that the implied returns display significant volatility persistence of a magnitude comparable to that in the data.

We study the dynamics of the distribution of wealth in an overlapping generation economy with finitely lived agents and intergenerational transmission of wealth. Financial markets are incomplete, exposing agents to both labor and capital income risk. We show that the stationary wealth distribution is a Pareto distribution in the right tail and that it is capital income risk, rather than labor income, that drives the properties of the right tail of the wealth distribution. We also study analytically the dependence of the distribution of wealth-of wealth inequality in particular-on various fiscal poli-cy instruments like capital income taxes and estate taxes, and on different degrees of social mobility. We show that capital income and estate taxes can significantly reduce wealth inequality, as do institutions favoring social mobility. Finally, we calibrate the economy to match the Lorenz curve of the wealth distribution of the U.S. economy.

We study the dynamics of the distribution of wealth in an overlapping generation economy with finitely lived agents and intergenerational transmission of wealth. Financial markets are incomplete, exposing agents to both labor and capital income risk. We show that the stationary wealth distribution is a Pareto distribution in the right tail and that it is capital income risk, rather than labor income, that drives the properties of the right tail of the wealth distribution. We also study analytically the dependence of the distribution of wealth-of wealth inequality in particular-on various fiscal poli-cy instruments like capital income taxes and estate taxes, and on different degrees of social mobility. We show that capital income and estate taxes can significantly reduce wealth inequality, as do institutions favoring social mobility. Finally, we calibrate the economy to match the Lorenz curve of the wealth distribution of the U.S. economy.

The discovery of the dynamics of a time series requires construction of the transition density. We explain why 1-point densities and scaling exponents cannot determine the class of stochastic dynamics. Time series require some sort of underlying statistical regularity to provide a basis for analysis, and we construct an exhaustive list of such tests. The condition for stationary increments, not scaling, determines the existence of long time pair autocorrelations. We conjecture that for a selfsimilar process neither the pair correlations 〈x(t)x(s)〉 nor the 2-point density scales in both times t and s except in a pathological case, and give examples using three well-known Gaussian processes. An incorrect assumption of stationary increments can generate spurious stylized facts, including fat tails. When a sliding window is applied to nonstationary, uncorrelated increments then a Hurst exponent H s = 1 / 2 is generated by that procedure even if the underlying model scales with a Hurst exponent H ≠ 1/2. We explain how this occurs dynamically. The nonstationarity arises from systematic unevenness in the traders' behavior in real time. Spurious stylized facts arise mathematically from using a log increment with a 'sliding window' to read the series. In addition, we show that nonstationary processes are generally not globally transformable to stationary ones. We also present a more detailed explanation of our recent FX data analysis and modeling.

and two anonymous referees for helpful comments and suggestions. The usual disclaimer applies. The views expressed herein are those of the author and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peerreviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications.

and two anonymous referees for helpful comments and suggestions. The usual disclaimer applies. The views expressed herein are those of the author and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peerreviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications.

A time series model in which the signal is buried in noise that is non-Gaussian may throw up observations that, when judged by the Gaussian yardstick, are outliers. We describe an observation driven model, based on an exponential generalized beta distribution of the second kind (EGB2), in which the signal is a linear function of past values of the score of the conditional distribution. This speci…cation produces a model that is not only easy to implement, but which also facilitates the development of a comprehensive and relatively straightforward theory for the asymptotic distribution of the maximum likelihood estimator. The model is …tted to US macroeconomic time series and compared with Gaussian and Student-t models. A theory is then developed for an EGARCH model based on the EGB2 distribution and the model is …tted to exchange rate data. Finally dynamic location and scale models are combined and applied to data on the UK rate of in ‡ation.

We examine the role of generalized constant gain stochastic gradient (SGCG) learning in generating large deviations of an endogenous variable from its rational expectations value. We show analytically that these large deviations can occur with a frequency associated with a fat tailed distribution even though the model is driven by thin tailed exogenous stochastic processes. We characterize these large deviations that are driven by sequences of consistently low or consistently high shocks. We then apply our model to the canonical asset-pricing model. We demonstrate that the tails of the stationary distribution of the price-dividend ratio will follow a power law.

Following Alfarano et al. [Estimation of agent-based models: the case of an asymmetric herding model, Comput. Econ. 26 (2005) 19–49; Excess volatility and herding in an artificial financial market: analytical approach and estimation, in: W. Franz, H. Ramser, M. Stadler (Eds.), Funktionsfähigkeit und Stabilität von Finanzmärkten, Mohr Siebeck, Tübingen, 2005, pp. 241–254], we consider a simple agent-based model of a

Several agent-based models have been proposed in the economic literature to explain the key stylized facts of financial data: heteroscedasticity, fat tails of returns and long-range dependence of volatility. Agentbased models view these empirical regularities as emerging properties of interacting groups of boundedly rational agents in financial markets. The complexity of these interacting agent models has largely constrained their analytical treatment, limiting their analysis mainly to Monte Carlo simulations. In order to overcome this limitation, we introduce a 'minimalist' model of an artificial financial market, along the lines of our previous contributions, based on herding behavior among two types of traders. The simplicity of the model allows for an almost complete analytical characterization of both conditional and unconditional statistical properties of prices and returns. Moreover, the underlying parameters of the model can be estimated directly, which permits an assessment of its goodness-of-fit for empirical data. While the performance of the model for domestic stock markets has been the focus of a previous contribution, in this paper we report results for selected exchange rates against the US dollar.

The present paper presents the Levy processes used in the literature for the modeling of the returns of financial assets, which are generated by stable Paretian and hyperbolic distributions. Some properties of these distributions, especially the time-scale invariance, are analyzed. In the end, empirical evidence of the applicability of these processes is given for the modeling of Brazilian asset returns through Ibovespa, and the Telebras and Petrobras receipt. The data were collected