Nonlinear dynamical model of regime switching between conventions and business cycles

Abstract

We introduce and study a non-equilibrium continuous-time dynamical model of the price of a single asset traded by a population of heterogeneous interacting agents in the presence of uncertainty and regulatory constraints. The model takes into account (i) the price formation delay between decision and investment by the second-order nature of the dynamical equations, (ii) the linear and nonlinear mean-reversal or their contrarian in the form of speculative price trading, (iii) market friction, (iv) uncertainty in the fundamental value which controls the amplitude of mispricing, (v) nonlinear speculative momentum effects and (vi) market regulations that may limit large mispricing drifts. We find markets with coexisting equilibrium, conventions and business cycles, which depend on (a) the relative strength of value-investing versus momentuminvesting, (b) the level of uncertainty on the fundamental value and (c) the degree of market regulation. The stochastic dynamics is characterized by nonlinear geometric random walk-like processes with spontaneous regime shifts between different conventions or business cycles. This model provides a natural dynamical framework to model regime shifts between different market phases that may result from the interplay between the effects (i-vi).

Rate-induced tipping from periodic attractors: Partial tipping and connecting orbits

“…More details are in Appendix B: note that ξ is a parameter that is determined by solving the BVP: one can think of ξ(r, a) : R 2 → R as a function whose zeros give the desired connections. Figure 4: The two parameter plane of system (9) showing regions of different tracking/tipping behaviour (a) is calculated by directly approximating a collection of initial conditions on the pullback attractor and determining their fate under the dynamics of the system and shows six regions where the system has qualitatively different behaviour (see Figure 5). The curves in (b) are calculated using Lin's method and show the locations of these transitions: r 1,2 are the thresholds of partial and total tipping respectively and r 0…”

“…As initial solution we solve the codimension-zero problem (15,16) and continuing it along r to arrive at a fold where the codimension-one connection exists. Figure 4 illustrates (a, r)-parameter plan for (9) in the case b = 1, ω = 3 and λ max = 8 calculated by Lin's method and compares it with a direct shooting algorithm described in Appendix C. Figure 5 shows the behaviour of (9) in each different region of the parameter plan by looking at a section of the manifolds…”

“…Motivated by studies of climate [3][4][5] , ecological 6,7 , financial 8,9 and biological systems 10 , the importance of tipping points in understanding sudden changes has been a focus of increasing interest in the last few years. Although there is no agreed definition, a tipping point occurs when a system has a sudden, irreversible change in output in response to a small change in input.…”

Super-exponential endogenous bubbles in an equilibrium model of fundamentalist and chartist traders

“…In some phases, chartists are prone to herding, while at other times, they are more incoherently disorganized "noise" traders. This captures in our dynamical framework the phenomenon of regime switching (Hamilton, 1989;Lux, 1995;Hamilton and Raj, 2002;Yukalov et al, 2009;Binder and Gross, 2013;Fischer and Seidl, 2013;Kadilli, 2013), where successive phases are characterized by changing values of the herding propensity. In this respect, we follow the model approach of Harras et al (2012) developed in a similar context and assume that the strength Ä of social imitation and momentum influence slowly varies in time.…”

Predicting Financial Market Crashes Using Ghost Singularities

“…[1][2][3][4][5][6]), other models attempt to capture the effects of feedbacks between financial information and investment strategies using various stochastic non-linear processes (see e.g. [7][8][9][10]). These references can be conceptually linked to the pioneering work of [11][12][13][14][15][16] describing the dynamical behaviour of heterogeneous markets with many trader types using dynamical system concepts, including limit cycles as the large type limit of interaction agents, bifurcation routes to instability and strange attractors in evolutionary financial market models.…”