Bifurcations and Chaos for 2D Discontinuous Dynamical Model of Financial Markets

2017 ◽  
Vol 27 (12) ◽  
pp. 1750185 ◽  
Author(s):  
En-Guo Gu
Keyword(s):  
Financial Markets ◽  
Piecewise Linear ◽  
Critical Value ◽  
Market Model ◽  
Analytical Treatment ◽  
Piecewise Linear Map ◽  
Bear Market ◽  
Basin Boundary ◽  
Model Dynamics ◽  
Excess Volatility

We develop a financial market model with interacting chartists and fundamentalists and chase sellers, the model dynamics is driven by a two-dimensional discontinuous piecewise linear map. Assume that the fixed point on the left side of border is restricted to regular saddle, we provide a more or less complete analytical treatment of the model dynamics by characterizing its possible outcomes in parameter space. The interpretation of structure for basin boundary and chaotic attractor is given by using contact bifurcation resulting from the contact between invariant set and the border. The critical value of occurring boundary crisis is given. In addition, we show that quite different scenarios can trigger real world phenomena such as bull and bear market dynamics and excess volatility.

On the Existence of Chaos in a Discontinuous Area-Preserving Map Arising in Financial Markets

2018 ◽  
Vol 28 (14) ◽  
pp. 1850177 ◽  
Author(s):  
En-Guo Gu
Keyword(s):  
Financial Markets ◽  
Complex Dynamics ◽  
Piecewise Linear ◽  
Market Model ◽  
Parameter Setting ◽  
Smale Horseshoe ◽  
Model Dynamics ◽  
Internal Change ◽  
Area Preserving

In this paper, we further study a discontinuous piecewise-linear financial market model established in our previous paper. The model dynamics is driven by a two-dimensional discontinuous area-preserving map. We exploit its complex dynamics from the viewpoint of open conservative systems. We first give the classification of fixed points, and then theoretically present the existence of periodic saddle orbit and a homoclinic chaos for some parameter setting. We finally apply the Conley–Moser conditions to verify the existence of Smale horseshoe-like dynamics and chaos for another parameter setting. The result is helpful for understanding the internal change rule of the finance market.

scholarly journals Beta Coefficients of Polish Blue Chip Companies in the Period Of 2005–2011

2012 ◽  
Vol 12 (2) ◽  
pp. 90-102 ◽  
Author(s):  
Wiesław Dębski ◽  
Ewa Feder-Sempach
Keyword(s):  
Financial Markets ◽  
Strong Evidence ◽  
Stock Exchange ◽  
Good News ◽  
Market Model ◽  
Bear Market ◽  
The World ◽  
Warsaw Stock Exchange ◽  
Financial Decision ◽  
Blue Chip

Abstract Risk plays a significant role in various aspects of financial decision throughout the world financial markets. Beta parameter is one of the commonly used coefficient to estimate the systematic risk associated with stocks. Beta is mostly calculated using single index market model by W. Sharpe. This study examined the beta parameter under bull and bear market conditions on the Warsaw Stock Exchange (WSE). This paper analyses the beta responses for bad and good news for 44 stocks (14 stocks from the WIG20 index and 30 stocks from the mWIG40 index) over the last six years of trading at the WSE. Beta was calculated using monthly returns over the period 2005-2011, separately for the bull and the bear market. Our analysis finds strong evidence that beta is different in bull and bear market phase.

BCB Curves and Contact Bifurcations in Piecewise Linear Discontinuous Map Arising in a Financial Market

2019 ◽  
Vol 29 (02) ◽  
pp. 1950022 ◽  
Author(s):  
En-Guo Gu ◽  
Jun Guo
Keyword(s):  
Fixed Points ◽  
Financial Market ◽  
Structural Changes ◽  
Piecewise Linear ◽  
Invariant Set ◽  
Market Model ◽  
Stable And Unstable Manifolds ◽  
Model Dynamics ◽  
Discontinuous Map ◽  
Unstable Manifolds

In this paper, we further study a financial market model established in our earlier paper. The model dynamics is driven by a two-dimensional piecewise linear discontinuous map, which is investigated analytically and numerically for one-sided fixed points being flip saddle and two-sided fixed points being attractors. The existence of chaotic orbit is explained by using the theory of homoclinic intersection between stable and unstable manifolds of the flip saddle invariant set. The structure of chaotic attractor is disclosed. It consists of finite segments rooted on both sides of the [Formula: see text]-axis which are unstable manifolds of flip saddle invariant set. The basins and their structural changes of bounded attractors and coexisting attractors are presented by contact bifurcation theory and numerical simulations. The border collision bifurcation (BCB for short) curves are calculated and coexisting multiattractors are disclosed by overlapping periodicity regions. The results can deepen our understanding of financial markets and dynamical systems.

scholarly journals How Many Inflections are There in the Lyapunov Spectrum?

2021 ◽  
Author(s):  
O. Jenkinson ◽  
M. Pollicott ◽  
P. Vytnova
Keyword(s):  
Piecewise Linear ◽  
Lyapunov Spectrum ◽  
Stat Phys ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Expanding Map ◽  
Lyapunov Spectra ◽  
Linear Maps ◽  
Inflection Points ◽  
Piecewise Linear Maps

AbstractIommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

scholarly journals International Liquidity and Exchange Rate Dynamics *

2015 ◽  
Vol 130 (3) ◽  
pp. 1369-1420 ◽  
Author(s):  
Xavier Gabaix ◽  
Matteo Maggiori
Keyword(s):  
Exchange Rate ◽  
Financial Markets ◽  
Exchange Rates ◽  
Capital Flows ◽  
Risk Sharing ◽  
Market Model ◽  
International Liquidity ◽  
Balance Sheets ◽  
Macroeconomic Analysis ◽  
International Pricing

Abstract We provide a theory of the determination of exchange rates based on capital flows in imperfect financial markets. Capital flows drive exchange rates by altering the balance sheets of financiers that bear the risks resulting from international imbalances in the demand for financial assets. Such alterations to their balance sheets cause financiers to change their required compensation for holding currency risk, thus affecting both the level and volatility of exchange rates. Our theory of exchange rate determination in imperfect financial markets not only helps rationalize the empirical disconnect between exchange rates and traditional macroeconomic fundamentals, it also has real consequences for output and risk sharing. Exchange rates are sensitive to imbalances in financial markets and seldom perform the shock absorption role that is central to traditional theoretical macroeconomic analysis. Our framework is flexible; it accommodates a number of important modeling features within an imperfect financial market model, such as nontradables, production, money, sticky prices or wages, various forms of international pricing-to-market, and unemployment.

Critical Value-Based Power Options Pricing Problems in Uncertain Financial Markets

2021 ◽  
pp. 2150002
Author(s):  
Guimin Yang ◽  
Yuanguo Zhu
Keyword(s):  
Financial Markets ◽  
Financial Market ◽  
Numerical Experiments ◽  
Critical Value ◽  
The Other ◽  
Options Pricing ◽  
Underlying Asset ◽  
Pricing Problems ◽  
Fractional Differential ◽  
The One

Compared with investing an ordinary options, investing the power options may possibly yield greater returns. On the one hand, the power option is the best choice for those who want to maximize the leverage of the underlying market movements. On the other hand, power options can also prevent the financial market changes caused by the sharp fluctuations of the underlying assets. In this paper, we investigate the power option pricing problem in which the price of the underlying asset follows the Ornstein–Uhlenbeck type of model involving an uncertain fractional differential equation. Based on critical value criterion, the pricing formulas of European power options are derived. Finally, some numerical experiments are performed to illustrate the results.

scholarly journals The Chaotic Saddle in the Lozi Map, Autonomous and Nonautonomous Versions

2015 ◽  
Vol 25 (13) ◽  
pp. 1550184 ◽  
Author(s):  
Carlos Lopesino ◽  
Francisco Balibrea-Iniesta ◽  
Stephen Wiggins ◽  
Ana M. Mancho
Keyword(s):  
Piecewise Linear ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
The Third ◽  
Autonomous Version ◽  
Area Preserving ◽  
Chaotic Saddle

In this paper, we prove the existence of a chaotic saddle for a piecewise-linear map of the plane, referred to as the Lozi map. We study the Lozi map in its orientation and area preserving version. First, we consider the autonomous version of the Lozi map to which we apply the Conley–Moser conditions to obtain the proof of a chaotic saddle. Then we generalize the Lozi map on a nonautonomous version and we prove that the first and the third Conley–Moser conditions are satisfied, which imply the existence of a chaotic saddle. Finally, we numerically demonstrate how the structure of this nonautonomous chaotic saddle varies as parameters are varied.

DIGITAL CONTROL AS SOURCE OF CHAOTIC BEHAVIOR

2010 ◽  
Vol 20 (05) ◽  
pp. 1365-1378 ◽  
Author(s):  
GÁBOR CSERNÁK ◽  
GÁBOR STÉPÁN
Keyword(s):  
Mechanical System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Linear Control ◽  
Control Loop ◽  
Digital Implementation ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Chaotic Vibrations ◽  
Processing Delay

In the present paper, we introduce and analyze a mechanical system, in which the digital implementation of a linear control loop may lead to chaotic behavior. The amplitude of such oscillations is usually very small, this is why these are called micro-chaotic vibrations. As a consequence of the digital effects, i.e. the sampling, the processing delay and the round-off error, the behavior of the system can be described by a piecewise linear map, the micro-chaos map. We examine a 2D version of the micro-chaos map and prove that the map is chaotic.

The Dynamics of a Piecewise Linear Map and its Smooth Approximation

1997 ◽  
Vol 07 (02) ◽  
pp. 351-372 ◽  
Author(s):  
D. Aharonov ◽  
R. L. Devaney ◽  
U. Elias
Keyword(s):  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Invariant Curves ◽  
Smooth Approximation ◽  
Linear Map ◽  
Island Structure ◽  
The Real ◽  
Piecewise Linear Map ◽  
Real Analytic ◽  
Area Preserving

The paper describes the dynamics of a piecewise linear area preserving map of the plane, F: (x, y) → (1 - y - |x|, x), as well as that portion of the dynamics that persists when the map is approximated by the real analytic map Fε: (x, y) → (1 - y - fε(x), x), where fε(x) is real analytic and close to |x| for small values of ε. Our goal in this paper is to describe in detail the island structure and the chaotic behavior of the piecewise linear map F. Then we will show that these islands do indeed persist and contain infinitely many invariant curves for Fε, provided that ε is small.

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