Soliton theory and modulation instability analysis: The Ivancevic option pricing model in economy

In this projected paper, we study on the Ivancevic option pricing model. We apply two important methods, namely, rational sine-Gordon expansion method which is recently developed, and secondly, modified exponential method. Via these methods, we obtain some important properties of Ivancevic option pricing model. We extract many solutions such as complex, periodic, dark bright, mixed dark-bright, singular, travelling and hyperbolic functions. We investigate the option price wave functions of dependent variable, and also, observe the modulation instability analysis in detail. Furthermore, we report the strain conditions for the valid solutions under the family conditions, as well. We simulate the 2D, 3D and counter plots by choosing the suitable values of the parameters involved. Finally, we present the top and low points of pricing in the mentioned intervals via contour simulations. (C) 2022 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University

Polarization modulation instability in a Manakov fiber system

The Manakov model is the simplest multicomponent model of nonlinear wave theory: It describes elementary stable soliton propagation and multisoliton solutions, and it applies to nonlinear optics, hydrodynamics, and Bose-Einstein condensates. It is also of fundamental interest as an asymptotic model in the context of the widely used wavelength-division-multiplexed optical fiber transmission systems. However, although its physical relevance was confirmed by the experimental observation of Manakov (vector) solitons in a planar waveguide in 1996, there have in fact been no quantitative experiments confirming its validity for nonlinear dynamics other than soliton formation. Here, we report experiments in optical fiber that provide evidence of passband and baseband polarization modulation instabilities in a defocusing Manakov system. In the spontaneous regime, we also reveal a unique saturation effect as the pump power increases. We anticipate that such observations may impact the application of this minimal model to describe and understand more complicated phenomena in nature, such as the formation of extreme waves in multicomponent systems

Adaptive-Wave Alternative for the Black-Scholes Option Pricing Model

A nonlinear wave alternative for the standard Black-Scholes option-pricing model is presented. The adaptive-wave model, representing 'controlled Brownian behavior' of financial markets, is formally defined by adaptive nonlinear Schr\"odinger (NLS) equations, defining the option-pricing wave function in terms of the stock price and time. The model includes two parameters: volatility (playing the role of dispersion frequency coefficient), which can be either fixed or stochastic, and adaptive market potential that depends on the interest rate. The wave function represents quantum probability amplitude, whose absolute square is probability density function. Four types of analytical solutions of the NLS equation are provided in terms of Jacobi elliptic functions, all starting from de Broglie's plane-wave packet associated with the free quantum-mechanical particle. The best agreement with the Black-Scholes model shows the adaptive shock-wave NLS-solution, which can be efficiently combined with adaptive solitary-wave NLS-solution. Adjustable 'weights' of the adaptive market-heat potential are estimated using either unsupervised Hebbian learning, or supervised Levenberg-Marquardt algorithm. In the case of stochastic volatility, it is itself represented by the wave function, so we come to the so-called Manakov system of two coupled NLS equations (that admits closed-form solutions), with the common adaptive market potential, which defines a bidirectional spatio-temporal associative memory. Keywords: Black-Scholes option pricing, adaptive nonlinear Schr\"odinger equation, market heat potential, controlled stochastic volatility, adaptive Manakov system, controlled Brownian behavior

Finite-Dimensional Quantum Model For The Stock Market, Discrete Nature Of The Quantities Used In Finance, Spectral Representation Of Bessel Processes With Constant Drift, Credit Spreads, And Stochastic Volatility In Finance, Quantum Structure In Cognition

Following structural and syllogistical confederational concatenation is studied with concomitant and consummative properties: Finite-Dimensional Quantum Model For The Stock Market, Discrete Nature Of The Quantities Used In Finance, Spectral Representation Of Bessel Processes With Constant Drift, Credit Spreads, And Stochastic Volatility In Finance, Quantum Structure In Cognition, Non Classicality Of The Membership Weight Structure, Minimal Length Uncertainty And The Quantum Model For The Stock Market, Superpositions Of Probability Distributions, Quantum Finance: The Finite Dimensional Case, Foreign Exchange Market As A Lattice Gauge Theory, Classical Logical Versus Quantum Conceptual Thought, Adaptive-Wave Alternative For The Black-Scholes Option Pricing Model Key words: Spectral Representation Of Bessel Processes, Quantum Conceptual Thought, Black-Scholes Option Pricing Model, Black-Scholes Option Pricing Model, Quantum Structure In Cognition The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-648791-654990.pd