Logistic model for stock market bubbles and anti-bubbles

Log-periodic power laws often occur as signatures of impending criticality of hierarchical systems in the physical sciences. It has been proposed that similar signatures may be apparent in the price evolution of financial markets as bubbles and the associated crashes develop. The features of such market bubbles have been extensively studied over the past 20 years, and models derived from an initial discrete scale invariance assumption have been developed and tested against the wealth of financial data with varying degrees of success. In this paper, the equations that form the basis for the standard log-periodic power law model and its higher extensions are compared to a logistic model derived from the solution of the Schroder equation for the renormalization group with nonlinear scaling function. Results for the S&P 500 and Nikkei 225 indices studied previously in the literature are presented and compared to established models, including a discussion of the apparent frequency shifting observed in the S&P 500 index in the 1980s. In the particular case of the Nikkei 225 anti-bubble between 1990 and 2003, the logistic model appears to provide a better description of the large-scale observed features over the whole 13-year period, particularly near the end of the anti-bubble

A computer assisted proof of universality for cubic critical maps of the circle with Golden Mean rotation number

In order to explain the universal metric properties associated with the breakdown of invariant tori in dissipative dynamical systems, Ostlund, Rand, Sethna and Siggia together with Feigenbaum, Kadanoff and Shenker have developed a renormalisation group analysis for pairs of analytic functions that glue together to make a map of the circle. Using a method of Lanford's, we have obtained a proof of the existence and hyperbolicity of a non-trivial fixed point of the renormalisation transformation for rotation number equal to the golden mean (√5 - 1/2). The proof uses numerical estimates obtained rigorously with the aid of a computer. These computer calculations were based on a method of Eckmann, Koch and Wittwer

Low Carbon Futures: confronting electricity challenges on island systems

This paper considers the range of possible long-term futures for an electrically isolated island power system. Emphasis is given to generation investment decisions supportive of low-carbon renewable generation. Ranges of policy interventions are considered for the electrically isolated case study island of São Miguel in the Azores islands in the North Atlantic Ocean. The whole systems methodological approach of System Dynamics is used to bring together key sub-systems relating to, for example, generation adequacy, renewable generation investments and demand-side aspects. In this way, a comprehensive understanding is established with high levels of endogeneity. The model is used to investigate a range of policy scenarios associated with renewable energy growth, electric-vehicle uptake and electricity storage. It is found that policy is most effective when all aspects are addressed simultaneously and in a co-ordinated manner and that policy favouring renewable generation alone is not sufficient to achieve the highest possible penetration of renewables. Finally, the robustness of the observations is addressed via Monte-Carlo based sensitivity testing

Simulating the GB power system frequency during underfrequency events 2018–19

Lightning hit a transmission power line outside London, England on 9 August 2019. There followed a loss of power from a cascade of generator outages that exceeded contingency reserves, leading to an exceptional fall in grid frequency causing widespread transport disruptions and the disconnection of over 1m households. Simulating such events typically involves a system of differential equations representing the overall generation and load present at the time. A standard model based on the swing equation assumes unlimited capacity in aggregated resources, and the availability of these services throughout the duration of the frequency deviation. In simulating the effect of outages on the GB Grid frequency on 2019/8/9, the effect of limiting these services to the capacity of resources engaged during the event is examined. It is shown that by taking these refinements into account the timing and extent of the frequency nadir can be successfully estimated. Insight is gained into the responses of various grid characteristics and how they interact with unplanned generation imbalances. Using this adapted model, further events on the GB grid are examined to validate the influence of these features. With the model’s effectiveness validated, novel mitigating measures to preserve the stability of a low-inertia grid can be evaluated

Change-Point Analysis of Asset Price Bubbles with Power-Law Hazard Function

We present a methodology to identify change-points in financial markets where the governing regime shifts from a constant rate-of-return, i.e. normal growth, to superexponential growth described by a power-law hazard rate. The latter regime corresponds, in our view, to financial bubbles driven by herding behaviour of market participants. Assuming that the time series of log-price returns of a financial index can be modelled by arithmetic Brownian motion, with an additional jump process with power-law hazard function to approximate the superexponential growth, we derive a threshold value of the hazard-function control parameter, allowing us to decide in which regime the market is more likely to be at any given time. An analysis of the Standard \& Poors 500 index over the last 60 years provides evidence that the methodology has merit in identifying when a period of herding behaviour begins, and, perhaps more importantly, when it ends

A short proof of the existence of universal functions

We present a short proof of the existence of universal functions for period-doubling and critical golden-mean circle maps for all degrees of criticality d > 1. The method is based on H. Epstein's Herglotz-function technique

A contraction-mapping proof of Koenigs’ theorem

We give a simple, functional analytic proof of Koenigs’ theorem on the linearisation of a complex analytic function in a neighbourhood of a hyperbolic fixed point. The proof uses the contraction mapping principle in the nonlinearity norm

A Practical Algorithm To Detect Superexponential Behavior In Financial Asset Price Returns

To assist with the detection of bubbles and negative bubbles in financial markets, a criterion is introduced to indicate whether a market is likely to be in a superexponential regime (where growth in such a regime would correspond to an asset price bubble and decline to an negative bubble) as opposed to “normal” exponential behavior typified by a constant rate of growth or decline. The criterion is founded on the Johansen–Ledoit–Sornette model of asset dynamics in a bubble and is derived from a linear fit to observed data with a nonlinear time transformation with parameters distributed uniformly in their permitted ranges. Making use of expected values rather than the underlying distribution, the criterion is straightforward and efficient to compute and can in principle be applied in real time to intra-day markets as well as longer timescales. In some circumstances, the criterion is shown to have certain predictive qualities when applied to a portfolio of stocks, and could be used as input into algorithmic trading strategies. A simple strategy is described which is based on market reversion predictions of a portfolio of stocks and which in back-testing generates notable returns