Realizing stock market crashes: stochastic cusp catastrophe model of returns under the time-varying volatility

This paper develops a two-step estimation methodology, which allows us to apply catastrophe theory to stock market returns with time-varying volatility and model stock market crashes. Utilizing high frequency data, we estimate the daily realized volatility from the returns in the first step and use stochastic cusp catastrophe on data normalized by the estimated volatility in the second step to study possible discontinuities in markets. We support our methodology by simulations where we also discuss the importance of stochastic noise and volatility in deterministic cusp catastrophe model. The methodology is empirically tested on almost 27 years of U.S. stock market evolution covering several important recessions and crisis periods. Due to the very long sample period we also develop a rolling estimation approach and we find that while in the first half of the period stock markets showed marks of bifurcations, in the second half catastrophe theory was not able to confirm this behavior. Results suggest that the proposed methodology provides an important shift in application of catastrophe theory to stock markets

Fitting the Cusp Catastrophe in R: A cusp Package Primer

Of the seven elementary catastrophes in catastrophe theory, the âÂÂcuspâ model is the most widely applied. Most applications are however qualitative. Quantitative techniques for catastrophe modeling have been developed, but so far the limited availability of flexible software has hindered quantitative assessment. We present a package that implements and extends the method of Cobb (Cobb and Watson'80; Cobb, Koppstein, and Chen'83), and makes it easy to quantitatively fit and compare different cusp catastrophe models in a statistically principled way. After a short introduction to the cusp catastrophe, we demonstrate the package with two instructive examples.

Non-linear Dynamics and Leadership Emergence

The process by which leaders emerge from leaderless groups is well-documented, but not nearly as well understood. This article describes how non-linear dynamical systems concepts of attractors, bifurcations, and self-organization culminate in a swallowtail catastrophe model for the leadership emergence process, and presents the experimental results that the model has produced thus far for creative problem solving, production, and coordination-intensive groups. Several control variables have been identified that vary in their function depending on what type of group is involved, e.g. creative problem solving, production, and coordination-intensive groups. The exposition includes the relevant statistical strategies that are based on non-linear regression along with some directions for new research questions that can be explored through this non-linear model

Predicting Discontinuity in the Decision to Allocate Funds to Credit Memes with a Fokker-Planck Equation Based Model

The model is one theoretical approach within a broader research program that could verify the nonlinear conjectures made to quantify and predict potential discontinuous behaviour. In this case, the crisis behaviour associated with financial funds reallocation among various credit instruments, described as memes with the sense of Dawkins, is shown to be of discontinuous nature stemming from a logistic penetration into the behaviour niche. A Fokker-Planck equation description results in a stationary solution having a bifurcation like the solution with evolution trajectories on a ‘cusp’ type catastrophe that may describe discontinuous decision behaviour.nonlinear models, decision, financial crisis

Caustic Skeleton & Cosmic Web

We present a general formalism for identifying the caustic structure of an evolving mass distribution in an arbitrary dimensional space. For the class of Hamiltonian fluids the identification corresponds to the classification of singularities in Lagrangian catastrophe theory. Based on this we develop a theoretical framework for the formation of the cosmic web, and specifically those aspects that characterize its unique nature: its complex topological connectivity and multiscale spinal structure of sheetlike membranes, elongated filaments and compact cluster nodes. The present work represents an extension of the work by Arnol'd et al., who classified the caustics for the 1- and 2-dimensional Zel'dovich approximation. His seminal work established the role of emerging singularities in the formation of nonlinear structures in the universe. At the transition from the linear to nonlinear structure evolution, the first complex features emerge at locations where different fluid elements cross to establish multistream regions. The classification and characterization of these mass element foldings can be encapsulated in caustic conditions on the eigenvalue and eigenvector fields of the deformation tensor field. We introduce an alternative and transparent proof for Lagrangian catastrophe theory, and derive the caustic conditions for general Lagrangian fluids, with arbitrary dynamics, including dissipative terms and vorticity. The new proof allows us to describe the full 3-dimensional complexity of the gravitationally evolving cosmic matter field. One of our key findings is the significance of the eigenvector field of the deformation field for outlining the spatial structure of the caustic skeleton. We consider the caustic conditions for the 3-dimensional Zel'dovich approximation, extending earlier work on those for 1- and 2-dimensional fluids towards the full spatial richness of the cosmic web

Fitting the Cusp Catastrophe in R: A cusp Package Primer

Of the seven elementary catastrophes in catastrophe theory, the "cusp" model is the most widely applied. Most applications are however qualitative. Quantitative techniques for catastrophe modeling have been developed, but so far the limited availability of flexible software has hindered quantitative assessment. We present a package that implements and extends the method of Cobb (Cobb and Watson 1980; Cobb, Koppstein, and Chen 1983), and makes it easy to quantitatively fit and compare different cusp catastrophe models in a statistically principled way. After a short introduction to the cusp catastrophe, we demonstrate the package with two instructive examples