Predicting Financial Contagion and Crisis by Using Jones, Alexander Polynomial and Knot Theory

Topological methods are rapidly developing and are becoming more used in physics, biology and chemistry. One area of topology has showed its immense potential in explaining potential financial contagion and financial crisis in financial markets. The aforementioned method is knot theory. The movement of stock price has been marked and braids and knots have been noted. By analysing the knots and braids using Jones polynomial, it is tried to find if there exists an untrivial knot equal to unknot? After thorough analysis, possible financial contagion and financial crisis prediction are analysed by using instruments of knot theory pertaining in that sense to Jones, Laurent and Alexander polynomial. It is proved that it is possible to predict financial disruptions by observing possible knots in the graphs and finding appropriate polynomials. In order to analyse knot formation, the following approach is used: “Knot formation in three-dimensional space is considered and the equations about knot forming and its disentangling are considered”. After having defined the equations in three-dimensional space, the definition of Brownian bridge concerning formation of knots in three-dimensional space is defined. Using analogy method, the notion of Brownian bridge is translated into 2-dimensional space and the foundations for the application of knot theory in 2-dimensional space have been set up. At the same time, the aforementioned approach is innovative and it could be used in accordance with stochastic analysis and quantum finance.

Price distortions and municipal bonds premiums:evidence from Switzerland

This study examines the pricing of municipal bonds before and after a currency shock in Switzerland.Two approaches are used to decompose the municipal to treasuries bond spreads into liquidity,maturity,and default risk premiums.The first approach is the model of the cross-sectional instrumental variables,and the second approach is the model of the instrumental variables with panel data.This study examines the composition of spreads for both approaches,in three scenarios:before,throughout,and after the currency shock.The study performed Durbin-Wu-Hausman tests for each decisive model to verify endogeneity issues,including the Lagrangian Multiplier test,the Cragg-Donald Wald F statistic to confirm the relationship of instrumental and endogenous variables,and the structural break test(Bai-Perron test)to determine the existence of structural breaks in bond distortions.This study finds that the currency price distortions of the Swiss franc in January 2015 made long-run changes in the composition of the municipal bond spreads.This research contributes to the understanding of municipal bond pricing by showing that default risk accounts for a large portion of the municipal bond spread,while maturity risk plays a lesser role.According to our empirical findings,unexpected large currency price shocks may have long-term implications on the municipal bond spreads.

Time Optimal Control in Time Series Movement

The paper analyses time series that exhibit equilibrium states. It analyses the formation of equilibrium and how the system can return to the aforementioned equilibrium. The tool that is used in the aforementioned analysis is time optimal control in the phase plane. It is proved that equilibrium state is sustainable if initial state is not too far from the equilibrium as well as control vector is large enough. On the other hand, if initial state is one standard deviation away from equilibrium state, it is proved that equilibrium cannot be reached. It is the same case with control vector. If it is unbounded, time optimal control cannot be applied. The approach that is introduced represents unconventional method of analysing equilibrium in time series.

On the Application of Fokker-Planck Equation to Psychological Future Time

This paper tries to make a comparison and connection between Fokker-Planck or forward Kolmogorov equation and psychological future time which is based on quantum mechanics. It will be showed that in quantum finance forward interest rate model can be further improved by noting that the predicted correlation structure for field theory models depends only on variable where t is present time and x is future time. On the other side, forward Kolmogorov equation is a parabolic partial differential equation, requiring international conditions at time t and to be solved for . The aforementioned equation is to be used if there are some special states now and it is necessary to know what can happen later. It will be tried to establish the connection between these two equations. It is proved that the psychological future time if applied and implemented in Fokker-Planck equation is unstable and is changeable so it is not easily predictable. Some kinds of nonlinear functions can be applied in order to establish the notion of psychological future time, however it is unstable and it should be continuously changed.

Existence and Smoothness of Solution of Navier-Stokes Equation on R³

Navier-Stokes equation has for a long time been considered as one of the greatest unsolved problems in three and more dimensions. This paper proposes a solution to the aforementioned equation on R3. It introduces results from the previous literature and it proves the existence and uniqueness of smooth solution. Firstly, the concept of turbulent solution is defined. It is proved that turbulent solutions become strong solutions after some time in Navier-Stokes set of equations. However, in order to define the turbulent solution, the decay or blow-up time of solution must be examined. Differential inequality is defined and it is proved that solution of Navier-Stokes equation exists in a finite time although it exhibits blow-up solutions. The equation is introduced that establishes the distance between the strong solutions of Navier-Stokes equation and heat equation. As it is demonstrated, as the time goes to infinity, the distance decreases to zero and the solution of heat equation is identical to the solution of N-S equation. As the solution of heat equation is defined in the heat-sphere, after its analysis, it is proved that as the time goes to infinity, solution converges to the stationary state. The solution has a finite τ time and it exists when τ → ∞ that implies that it exists and it is periodic. The aforementioned statement proves the existence and smoothness of solution of Navier-Stokes equation on R3 and represents a major breakthrough in fluid dynamics and turbulence analysis.