A Pratical Approach to Financial Crisis Indicators Based on Random Matrices

URL des Documents de travail : http://centredeconomiesorbonne.univ-paris1.fr/documents-de-travail/Documents de travail du Centre d'Economie de la Sorbonne 2015.49 - ISSN : 1955-611XThe aim of this work is to build financial crisis indicators based on market data time series. After choosing an optimal size for a rolling window, the market data is seen every trading day as a random matrix from which a covariance and correlation matrix is obtained. Our indicators deal with the spectral properties of these covariance and correlation matrices. Our basic financial intuition is that correlation and volatility are like the heartbeat of the financial market: when correlations between asset prices increase or develop abnormal patterns, when volatility starts to increase, then a crisis event might be around the corner. Our indicators will be mainly of two types. The first one is based on the Hellinger distance, computed between the distribution of the eigenvalues of the empirical covariance matrix and the distribution of the eigenvalues of a reference covariance matrix. As reference distribution we will use the theoretical Marchenko Pastur distribution and, mainly, simulated ones using a random matrix of the same size as the empirical rolling matrix and constituted of Gaussian or Student-t coefficients with some simulated correlations. The idea behind this first type of indicators is that when the empirical distribution of the spectrum of the covariance matrix is deviating from the reference in the sense of Hellinger, then a crisis may be forthcoming. The second type of indicators is based on the study of the spectral radius and the trace of the covariance and correlation matrices as a mean to directly study the volatility and correlations inside the market. The idea behind the second type of indicators is the fact that large eigenvalues are a sign of dynamic instability.Le but de ce travail de recherche est la construction d'indicateurs de crises financières basés sur des données de marché. Après avoir choisi la taille optimale d'une fenêtre roulante, les données de marchés seront vues comme une matrice aléatoire à partir de laquelle une matrice de covariance et une matrice de corrélation seront obtenues. Nos indicateurs exploitent les propriétés spectrales de cette matrice de covariance et de cette matrice de corrélation. Notre intuition financière de base est que la corrélation et la volatilité sont le pouls d'un marché financier : quand les corrélations entre les actifs augmentent ou développent des comportements anormaux, quand la volatilité commence à augmenter, alors un évènement de crise est peut être sur le point de se produire. Nos indicateurs seront essentiellement de deux types. Le premier type est basé sur la distance de Hellinger, calculée entre la distribution des valeurs propres de la matrice de covariance empirique et la distribution des valeurs propres d'une matrice de covariance de référence. Comme distribution de référence nous utiliserons la distribution théorique de Marchenko Pasur et aussi, essentiellement, des distributions simulées en utilisant une matrice aléatoire de même taille que la matrice de covariance roulante empirique et constituée de coefficients suivant une loi Gaussienne ou t-student et présentant des corrélations. L'idée derrière ce premier type d'indicateurs est que quand la distribution empirique du spectre de la matrice de covariance commence à dévier au sens de Hellinger de la référence, alors une crise est probablement sur le point de se produire. Le second type d'indicateurs est basé sur l'étude du rayon spectral et de la trace de la matrice de covariance et de la matrice de corrélation, dans le but d'étudier directement la volatilité et la corrélation à l'intérieur du marché. L'idée derrière ce second type d'indicateurs est que de grandes valeurs propres sont un signe d'instabilité dynamique

Worldwide Genotyping in the Planktonic Foraminifer Globoconella inflata: Implications for Life History and Paleoceanography

The planktonic foraminiferal morpho-species Globoconella inflata is widely used as a stratigraphic and paleoceanographic index. While G. inflata was until now regarded as a single species, we show that it rather constitutes a complex of two pseudo-cryptic species. Our study is based on SSU and ITS rDNA sequence analyses and genotyping of 497 individuals collected at 49 oceanic stations covering the worldwide range of the morpho-species. Phylogenetic analyses unveil the presence of two divergent genotypes. Type I inhabits transitional and subtropical waters of both hemispheres, while Type II is restricted to the Antarctic subpolar waters. The two genetic species exhibit a strictly allopatric distribution on each side of the Antarctic Subpolar Front. On the other hand, sediment data show that G. inflata was restricted to transitional and subtropical environments since the early Pliocene, and expanded its geographic range to southern subpolar waters ∼700 kyrs ago, during marine isotopic stage 17. This datum may correspond to a peripatric speciation event that led to the partition of an ancestral genotype into two distinct evolutionary units. Biometric measurements performed on individual G. inflata from plankton tows north and south of the Antarctic Subpolar Front indicate that Types I and II display slight but significant differences in shell morphology. These morphological differences may allow recognition of the G. inflata pseudo-cryptic species back into the fossil record, which in turn may contribute to monitor past movements of the Antarctic Subpolar Front during the middle and late Pleistocene

Surface flows of granular materials: A short introduction to some recent models

We present a short review of recent theoretical descriptions of flows occuring at the surface of granular piles, and focus mainly on two models: the phenomenological ``BCRE'' model and the hydrodynamic model, based on Saint-Venant equations. Both models distinguish a ``static phase'' and a ``rolling'' phase inside the granular packing and write coupled equations for the evolutions of the height of each of these phases, which prove similar in both approaches. The BCRE description provides a very intuitive picture of the flow, whereas the Saint-Venant hydrodynamic description establishes a general and rigorous framework for granular flow studies.Comment: 10 pages, 3 figures, published in a special issue of C. R. Physique (Paris) on granular matte

Shape Self-Regulation in Early Lung Morphogenesis

The arborescent architecture of mammalian conductive airways results from the repeated branching of lung endoderm into surrounding mesoderm. Subsequent lung’s striking geometrical features have long raised the question of developmental mechanisms involved in morphogenesis. Many molecular actors have been identified, and several studies demonstrated the central role of Fgf10 and Shh in growth and branching. However, the actual branching mechanism and the way branching events are organized at the organ scale to achieve a self-avoiding tree remain to be understood through a model compatible with evidenced signaling. In this paper we show that the mere diffusion of FGF10 from distal mesenchyme involves differential epithelial proliferation that spontaneously leads to branching. Modeling FGF10 diffusion from sub-mesothelial mesenchyme where Fgf10 is known to be expressed and computing epithelial and mesenchymal growth in a coupled manner, we found that the resulting laplacian dynamics precisely accounts for the patterning of FGF10-induced genes, and that it spontaneously involves differential proliferation leading to a self-avoiding and space-filling tree, through mechanisms that we detail. The tree’s fine morphological features depend on the epithelial growth response to FGF10, underlain by the lung’s complex regulatory network. Notably, our results suggest that no branching information has to be encoded and that no master routine is required to organize branching events at the organ scale. Despite its simplicity, this model identifies key mechanisms of lung development, from branching to organ-scale organization, and could prove relevant to the development of other branched organs relying on similar pathways

Managing the Downside of Active and Passive Strategies: Convexity and Fragilities

International audienceQuestion of the day: how to manage a large (or small) portfolio in low interest rate conditions, while equity markets bear significant draw-down risk? More generally, how to build an "antifragile" portfolio that can weather the most extreme market scenarios without impacting long-term performances? Do active strategies systematically create or increase already existing market instabilities? By analyzing in depth markets behavior during past speculative bubbles and credit crises, we aim at addressing these questions. Our goal is to describe as faithfully as possible the major mechanisms at stake, avoiding the trap of mapping the complexity of financial markets into a single mathematical model, which would necessarily be wrong at some point. Starting from Minsky's "Financial Instability Hypothesis", we try to disentangle the complex relation between dynamics and randomness, including the presence of "fat tails". We provide methods to monitor the evolving probability of a forthcoming crisis through the measurement of "market instability". Scalable investment strategies result from the application of these methods.

Modèles mathématiques et crise financière

National audienceLorsque le monde de la finance s'est écroulé en août 2007 lors de la "crise des subprimes", puis en septembre 2008, suite à la faillite de Lehman Brothers, de nombreux doigts se sont pointés vers les mathématiciens. L'auteur de cet article propose une analyse fine et objective des causes de la crise. Il présente la théorie de l'équilibre général ainsi que la théorie de l'arbitrage, des modèles mathématiques sources des mathématiques financières

Yield Curve Smoothing and Residual Variance of Fixed Income Positions

URL des Documents de travail : http://centredeconomiesorbonne.univ-paris1.fr/documents-de-travail/Documents de travail du Centre d'Economie de la Sorbonne 2014.91 - ISSN : 1955-611XWe model the yield curve in any given country as an object lying in an infinite-dimensional Hilbert space, the evolution of which is driven by what is known as a cylindrical Brownian motion. We assume that volatilities and correlations do not depend on rates (which hence are Gaussian). We prove that a principal component analysis (PCA) can be made. These components are called eigenmodes or principal deformations of the yield curve in this space. We then proceed to provide the best approximation of the curve evolution by a Gaussian Heath-Jarrow-Morton model that has a given finite number of factors. Finally, we describe a method, based on finite elements, to compute the eigenmodes using historical interest rate data series and show how it can be used to compute approximate hedges which optimise a criterion depending on transaction costs and residual variance

Capital Adequacy, Pro-cyclicality and Systemic Risk

International audienc

A Non-cyclical Capital Adequacy Rule and the Aversion of Systemic Risk

Contenu de l'ouvrage présenté lors du Forum européen de la finance de Strasbourg le 2 septembre 2013International audienc