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## Additive Damages, Fat-Tailed Climate Dynamics, and Uncertain Discounting

Weitzman, Martin L.
Fonte: Economics Publicador: Economics
Tipo: Artigo de Revista Científica
Português
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This paper in applied theory argues that there is a loose chain of reasoning connecting the following three basic links in the economics of climate change: 1) additive disutility damages may be appropriate for analyzing some impacts of global warming; 2) an uncertain feedback-forcing coefficient, which might be near one with infinitesimal probability, can cause the distribution of the future time trajectory of global temperatures to have fat tails and a high variance; 3) when high-variance additive damages are discounted at an uncertain rate of pure time preference, which might be near zero with infinitesimal probability, it can make expected present discounted disutility very large. Some possible implications for welfare analysis and climate-change policy are briefly noted.; Economics

## Fat Tails and the Social Cost of Carbon

Weitzman, Martin L.
Fonte: American Economic Association Publicador: American Economic Association
Tipo: Artigo de Revista Científica
Português
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At high enough greenhouse gas concentrations, climate change might conceivably cause catastrophic damages with small but non-negligible probabilities. If the bad tail of climate damages is sufficiently fat, and if the coefficient of relative risk aversion is greater than one, the catastrophe-reducing insurance aspect of mitigation investments could in theory have a strong influence on raising the social cost of carbon. In this paper I exposit the influence of fat tails on climate change economics in a simple stark formulation focused on the social cost of carbon. I then attempt to place the basic underlying issues within a balanced perspective.; Economics

## An Eigenfunction Approach for Volatility Modeling.

MEDDAHI, Nour
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Artigo de Revista Científica Formato: 2523603 bytes; application/pdf
Português
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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.; Dans cet article...

## Three Essays on Asset Pricing

Wang, Zhiguang
Fonte: FIU Digital Commons Publicador: FIU Digital Commons
Tipo: Artigo de Revista Científica Formato: application/pdf
Português
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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...

## Option pricing using path integrals.

Bonnet, Frederic D. R.
Fonte: Universidade de Adelaide Publicador: Universidade de Adelaide
Tipo: Tese de Doutorado
Publicado em //2010 Português
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It is well established that stock market volatility has a memory of the past, moreover it is found that volatility correlations are long ranged. As a consequence, volatility cannot be characterized by a single correlation time in general. Recent empirical work suggests that the volatility correlation functions of various assets actually decay as a power law. Moreover it is well established that the distribution functions for the returns do not obey a Gaussian distribution, but follow more the type of distributions that incorporate what are commonly known as fat–tailed distributions. As a result, if one is to model the evolution of the stock price, stock market or any financial derivative, then standard Brownian motion models are inaccurate. One must take into account the results obtained from empirical studies and work with models that include realistic features observed on the market. In this thesis we show that it is possible to derive the path integral for a non-Gaussian option pricing model that can capture fat–tails. However we find that the path integral technique can only be used on a very small set of problems, as a number of situations of interest are shown to be intractable.; Thesis (Ph.D.) -- University of Adelaide...

## The origin of fat tailed distributions in financial time series

Viswanathan, G. M.; Fulco, U. L.; Lyra, M. L.; Serva, M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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A classic problem in physics is the origin of fat tailed distributions generated by complex systems. We study the distributions of stock returns measured over different time lags $\tau.$ We find that destroying all correlations without changing the $\tau = 1$ d distribution, by shuffling the order of the daily returns, causes the fat tails almost to vanish for $\tau>1$ d. We argue that the fat tails are caused by known long-range volatility correlations. Indeed, destroying only sign correlations, by shuffling the order of only the signs (but not the absolute values) of the daily returns, allows the fat tails to persist for $\tau >1$ d.; Comment: minor corrections

## Where Do Thin Tails Come From?

Taleb, Nassim Nicholas
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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The literature of heavy tails (typically) starts with a random walk and finds mechanisms that lead to fat tails under aggregation. We follow the inverse route and show how starting with fat tails we get to thin-tails when deriving the probability distribution of the response to a random variable. We introduce a general dose-response curve and argue that the left and right-boundedness or saturation of the response in natural things leads to thin-tails, even when the "underlying" random variable at the source of the exposure is fat-tailed.

## Common Origin of Power-law Tails in Income Distributions and Relativistic Gases

Modanese, G.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 05/09/2015 Português
Relevância na Pesquisa
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Power-law tails are ubiquitous in income distributions and in the energy distributions of diluted relativistic gases. We analyze the conceptual link between these two cases. In economic interactions fat tails arise because the richest individuals enact some protection mechanisms ("saving propensity") which allow them to put at stake, in their interactions, only a small part of their wealth. In high-energy particle collisions something similar happens, in the sense that when particles with very large energy collide with slow particles, then as a sole consequence of relativistic kinematics (mass dilation), they tend to exchange only a small part of their energy; processes like the frontal collision of two identical particles, where the exchanged energy is 100%, are very improbable, at least in a diluted gas. We thus show how in two completely different systems, one of socio-economic nature and one of physical nature, a certain feature of the binary microscopic interactions leads to the same consequence in the macroscopic distribution for the income or respectively for the energy.; Comment: 9 pages, 2 figures. To appear in Phys. Lett. A

## Leverage Causes Fat Tails and Clustered Volatility

Thurner, Stefan; Farmer, J. Doyne; Geanakoplos, John
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We build a simple model of leveraged asset purchases with margin calls. Investment funds use what is perhaps the most basic financial strategy, called "value investing", i.e. systematically attempting to buy underpriced assets. When funds do not borrow, the price fluctuations of the asset are normally distributed and uncorrelated across time. All this changes when the funds are allowed to leverage, i.e. borrow from a bank, to purchase more assets than their wealth would otherwise permit. During good times competition drives investors to funds that use more leverage, because they have higher profits. As leverage increases price fluctuations become heavy tailed and display clustered volatility, similar to what is observed in real markets. Previous explanations of fat tails and clustered volatility depended on "irrational behavior", such as trend following. Here instead this comes from the fact that leverage limits cause funds to sell into a falling market: A prudent bank makes itself locally safer by putting a limit to leverage, so when a fund exceeds its leverage limit, it must partially repay its loan by selling the asset. Unfortunately this sometimes happens to all the funds simultaneously when the price is already falling. The resulting nonlinear feedback amplifies large downward price movements. At the extreme this causes crashes...

## Self-similar solutions with fat tails for Smoluchowski's coagulation equation with locally bounded kernels

Niethammer, Barbara; Velazquez, Juan J. L.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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The existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation has so far only been established for the solvable and the diagonal kernel. In this paper we prove the existence of such self-similar solutions for continuous kernels $K$ that are homogeneous of degree $\gamma \in [0,1)$ and satisfy $K(x,y) \leq C (x^{\gamma} + y^{\gamma})$. More precisely, for any $\rho \in (\gamma,1)$ we establish the existence of a continuous weak self-similar profile with decay $x^{-(1{+}\rho)}$ as $x \to \infty$.

## Extreme values and fat tails of multifractal fluctuations

Muzy, Jean-Francois; Bacry, Emmanuel; Kozhemyak, Alexey
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 14/09/2005 Português
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In this paper we discuss the problem of the estimation of extreme event occurrence probability for data drawn from some multifractal process. We also study the heavy (power-law) tail behavior of probability density function associated with such data. We show that because of strong correlations, standard extreme value approach is not valid and classical tail exponent estimators should be interpreted cautiously. Extreme statistics associated with multifractal random processes turn out to be characterized by non self-averaging properties. Our considerations rely upon some analogy between random multiplicative cascades and the physics of disordered systems and also on recent mathematical results about the so-called multifractal formalism. Applied to financial time series, our findings allow us to propose an unified framemork that accounts for the observed multiscaling properties of return fluctuations, the volatility clustering phenomenon and the observed inverse cubic law'' of the return pdf tails.

## From short to fat tails in financial markets: A unified description

Cortines, A. A. G.; Riera, R.; Anteneodo, C.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 21/01/2008 Português
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In complex systems such as turbulent flows and financial markets, the dynamics in long and short time-lags, signaled by Gaussian and fat-tailed statistics, respectively, calls for a unified description. To address this issue we analyze a real dataset, namely, price fluctuations, in a wide range of temporal scales to embrace both regimes. By means of Kramers-Moyal (KM) coefficients evaluated from empirical time series, we obtain the evolution equation for the probability density function (PDF) of price returns. We also present consistent asymptotic solutions for the timescale dependent equation that emerges from the empirical analysis. From these solutions, new relationships connecting PDF characteristics, such as tail exponents, to parameters of KM coefficients arise. The results reveal a dynamical path that leads from Gaussian to fat-tailed statistics, furnishing insights on other complex systems where akin crossover is observed.; Comment: 11 pages, 5 figures

## Correlation Structure and Fat Tails in Finance: a New Mechanism

Airoldi, Marco
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 30/07/2001 Português
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Fat tails in financial time series and increase of stocks cross-correlations in high volatility periods are puzzling facts that ask for new paradigms. Both points are of key importance in fundamental research as well as in Risk Management (where extreme losses play a key role). In this paper we present a new model for an ensemble of stocks that aims to encompass in a unitary picture both these features. Equities are modelled as quasi random walk variables, where the non-Brownian components of stocks movements are leaded by the market trend, according to typical trader strategies. Our model suggests that collective effects may play a very important role in the characterization of some significantly statistical properties of financial time series.; Comment: 7 pages, 4 figures, submitted to Physical Review E on 27 July 2001

## Fat Tails Quantified and Resolved: A New Distribution to Reveal and Characterize the Risk and Opportunity Inherent in Leptokurtic Data

Thorne, Lawrence R.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 29/10/2011 Português
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I report a new statistical distribution formulated to confront the infamous, long-standing, computational/modeling challenge presented by highly skewed and/or leptokurtic ("fat- or heavy-tailed") data. The distribution is straightforward, flexible and effective. Even when working with far fewer data points than are routinely required, it models non-Gaussian data samples, from peak center through far tails, within the context of a single probability density function (PDF) that is valid over an extremely broad range of dispersions and probability densities. The distribution is a precision tool to characterize the great risk and the great opportunity inherent in fat-tailed data.; Comment: 26 pages, 9 figures & 3 tables

## On Rational Bubbles and Fat Tails

Lux, Thomas; Sornette, D.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 08/10/1999 Português
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This paper addresses the statistical properties of time series driven by rational bubbles a la Blanchard and Watson (1982), corresponding to multiplicative maps, whose study has recently be revived recently in physics as a mechanism of intermittent dynamics generating power law distributions. Using insights on the behavior of multiplicative stochastic processes, we demonstrate that the tails of the unconditional distribution emerging from such bubble processes follow power-laws (exhibit hyperbolic decline). More precisely, we find that rational bubbles predict a 'fat' power tail for both the bubble component and price differences with an exponent smaller than 1, implying absence of convergence of the mean. The distribution of returns is dominated by the same power-law over an extended range of large returns. Although power-law tails are a pervasive feature of empirical data, these numerical predictions are in disagreement with the usual empirical estimates of an exponent between 2 and 4. It, therefore, appears that exogenous rational bubbles are hardly reconcilable with some of the stylized facts of financial data at a very elementary level.; Comment: 20 pages, 3 figures, submitted to the Journal of Monetary Economics

## Multi-dimensional Rational Bubbles and fat tails: application of stochastic regression equations to financial speculation

Malevergne, Y.; Sornette, D.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 24/01/2001 Português
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We extend the model of rational bubbles of Blanchard and of Blanchard and Watson to arbitrary dimensions d: a number d of market time series are made linearly interdependent via d times d stochastic coupling coefficients. We first show that the no-arbitrage condition imposes that the non-diagonal impacts of any asset i on any other asset j different from i has to vanish on average, i.e., must exhibit random alternative regimes of reinforcement and contrarian feedbacks. In contrast, the diagonal terms must be positive and equal on average to the inverse of the discount factor. Applying the results of renewal theory for products of random matrices to stochastic recurrence equations (SRE), we extend the theorem of Lux and Sornette (cond-mat/9910141) and demonstrate that the tails of the unconditional distributions associated with such d-dimensional bubble processes follow power laws (i.e., exhibit hyperbolic decline), with the same asymptotic tail exponent mu<1 for all assets. The distribution of price differences and of returns is dominated by the same power-law over an extended range of large returns. This small value mu<1 of the tail exponent has far-reaching consequences in the non-existence of the means and variances. Although power-law tails are a pervasive feature of empirical data...

## Multicanonical Simulations of the Tails of the Order-Parameter Distribution of the Two-Dimensional Ising Model

Hilfer, Rudolf; Biswal, Bibhu; Mattutis, Hans-Georg; Janke, Wolfhard
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/02/2005 Português
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We report multicanonical Monte Carlo simulations of the tails of the order-parameter distribution of the two-dimensional Ising model for fixed boundary conditions. Clear numerical evidence for "fat" stretched exponential tails is found below the critical temperature, indicating the possible presence of fat tails at the critical temperature.; Comment: 4 pages, elsart3.cls (included), 5 postscript figures, author information under http://www.physik.uni-leipzig.de/index.php?id=22

## The Origin of Fat Tails

Gremm, Martin
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We propose a random walk model of asset returns where the parameters depend on market stress. Stress is measured by, e.g., the value of an implied volatility index. We show that model parameters including standard deviations and correlations can be estimated robustly and that all distributions are approximately normal. Fat tails in observed distributions occur because time series sample different stress levels and therefore different normal distributions. This provides a quantitative description of the observed distribution including the fat tails. We discuss simple applications in risk management and portfolio construction.; Comment: 17 Pages, 11 Figures, typos corrected

## Self-similar solutions with fat tails for Smoluchowski's coagulation equation with singular kernels

Niethammer, Barbara; Throm, Sebastian; Velázquez, Juan J. L.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 06/11/2014 Português
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We show the existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation for homogeneous kernels satisfying $C_1 \left(x^{-a}y^{b}+x^{b}y^{-a}\right)\leq K\left(x,y\right)\leq C_2\left(x^{-a}y^{b}+x^{b}y^{-a}\right)$ with $a>0$ and $b<1$. This covers especially the case of Smoluchowski's classical kernel $K(x,y)=(x^{1/3} + y^{1/3})(x^{-1/3} + y^{-1/3})$. For the proof of existence we first consider some regularized kernel $K_{\epsilon}$ for which we construct a sequence of solutions $h_{\epsilon}$. In a second step we pass to the limit $\epsilon\to 0$ to obtain a solution for the original kernel $K$. The main difficulty is to establish a uniform lower bound on $h_{\epsilon}$. The basic idea for this is to consider the time-dependent problem and choosing a special test function that solves the dual problem.

## The Future Has Thicker Tails than the Past: Model Error As Branching Counterfactuals

Taleb, Nassim N.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 11/09/2012 Português
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Ex ante forecast outcomes should be interpreted as counterfactuals (potential histories), with errors as the spread between outcomes. Reapplying measurements of uncertainty about the estimation errors of the estimation errors of an estimation leads to branching counterfactuals. Such recursions of epistemic uncertainty have markedly different distributial properties from conventional sampling error. Nested counterfactuals of error rates invariably lead to fat tails, regardless of the probability distribution used, and to powerlaws under some conditions. A mere .01% branching error rate about the STD (itself an error rate), and .01% branching error rate about that error rate, etc. (recursing all the way) results in explosive (and infinite) higher moments than 1. Missing any degree of regress leads to the underestimation of small probabilities and concave payoffs (a standard example of which is Fukushima). The paper states the conditions under which higher order rates of uncertainty (expressed in spreads of counterfactuals) alters the shapes the of final distribution and shows which a priori beliefs about conterfactuals are needed to accept the reliability of conventional probabilistic methods (thin tails or mildly fat tails).