Predicting financial market crashes using ghost singularities

Abstract

We analyse the behaviour of a non-linear model of coupled stock and bond prices exhibiting periodically collapsing bubbles. By using the formalism of dynamical system theory, we explain what drives the bubbles and how foreshocks or aftershocks are generated. A dynamical phase space representation of that system coupled with standard multiplicative noise rationalises the log-periodic power law singularity pattern documented in many historical financial bubbles. The notion of ‘ghosts of finite-time singularities’ is introduced and used to estimate the end of an evolving bubble, using finite-time singularities of an approximate normal form near the bifurcation point. We test the forecasting skill of this method on different stochastic price realisations and compare with Monte Carlo simulations of the full system. Remarkably, the approximate normal form is significantly more precise and less biased. Moreover, the method of ghosts of singularities is less sensitive to the noise realisation, thus providing more robust forecasts.

Functional observability and subspace reconstruction in nonlinear systems

“…In both cases, high reconstruction errors stem from estimating slow variables from measures of fast variables. Future works can formally explore this interesting relation, complementing the analysis for linear systems [75], by extending the notion of functional observability to (nonlinear) differential-algebraic systems of form ẋ1 = f 1 (x 1 , x 2 ), 0 = f 2 (x 1 , x 2 ), (42) where a strong timescale separation arises from a quasisteady-state assumption ( ẋ2 ≈ 0). Second, our analysis of the Epileptor model shows a potential relation between the system's observability and its bifurcation points.…”

“…In practice, even if the original state space is not entirely observable (reconstructible), one may focus on particular subspaces (e.g., state variables) that are relevant to the considered applications. Examples include the estimation of the phase variable of nonlinear oscillators for synchronization analysis of chaotic systems [21,39,40], modeling of climate dynamics [41], and forecasting of financial crashes [42]; the positioning and tracking of a particular spatial coordinate (e.g., altitude) in autonomous aerial vehicles from indirect measurements [43]; or the inference of control variables (which dictate how close a system is to a bifurcation) for the early warning of transitions from healthy to disease states in atrial fibrillation [44] and epileptic seizures [45].…”

Network percolation provides early warnings of abrupt changes in coupled oscillatory systems: An explanatory analysis

“…The occurrence of sudden shifts between radically different dynamical regimes is a striking phenomenon displayed by a variety of complex systems, including climatic [1][2][3][4], ecological [5,6], financial [7,8], and physiological [9] ones. These so-called critical transitions or regime shifts can have large impacts, as they bring about a drastic change in the function and structure of the systems undergoing them.…”