Abstract: Recent empirical findings suggest that societies have become more polarized in various countries. That is, the median voter of today represents a smaller fraction of society compared to two decades ago and yet, the mechanisms underlying this phenomenon are not fully understood. Since interactions between influential actors ("activists'') and voters play a major role in opinion formation, e.g. through social media, we develop a macroscopic opinion model in which competing activists spread their political ideas in specific groups of society. These ideas spread further to other groups in declining strength. While unilateral spreading shifts the opinion distribution, competition of activists leads to additional phenomena: Small heterogeneities among competing activists cause them to target different groups in society, which amplifies polarization. For moderate heterogeneities, we obtain target cycles and further amplification of polarization. In such cycles, the stronger activist differentiates himself from the weaker one, while the latter aims to imitate the stronger activist. (C) 2019 Elsevier B.V. All rights reserved.

Abstract: The stochastic volatility models used in the financial world are characterized, in the continuous-time case, by a set of two coupled stochastic differential equations for the underlying asset price S and volatility sigma. In addition, the correlations of the two Brownian movements that drive the stochastic dynamics are measured by the correlation parameter rho (-1 <= rho <= 1). This stochastic system is equivalent to the Fokker-Planck equation for the transition probability density of the random variables S and sigma. Solutions for the transition probability density of the Heston stochastic volatility model (Heston, 1993) were explored in Dragulescu and Yakovenko (2002), where the fundamental quantities such as the transition density itself, depend on rho in such a manner that these are divergent for the extreme limit rho = +/- 1. The same divergent behavior appears in Hagan et al. (2002), where the probability density of the SABR model was analyzed. In an option pricing context, the propagator of the bi-dimensional Black-Scholes equation was obtained in Lemmens et al. (2008) in terms of the path integrals, and in this case, the propagator diverges again for the extreme values rho = +/- 1. This paper shows that these similar divergent behaviors are due to a universal property of the stochastic volatility models in the continuum: all of them are second class constrained systems for the most extreme correlated limit rho = +/- 1. In this way, the stochastic dynamics of the rho = +/- 1 cases are different of the rho (1 <= rho <= 1) case, and it cannot be obtained as a continuous limit from the rho not equal +/- 1 regimen. This conclusion is achieved by considering the Fokker-Planck equation or the bi-dimensional Black-Scholes equation as a Euclidean quantum Schrodinger equation. Then, the analysis of the underlying classical mechanics of the quantum model, implies that stochastic volatility models at rho = +/- 1 correspond to a constrained system. To study the dynamics in an appropriate form, Dirac's method for constrained systems (Dirac, 1958, 1967) must be employed, and Dirac's analysis reveals that the constraints are second class. In order to obtain the transition probability density or the option price correctly, one must evaluate the propagator as a constrained Hamiltonian path-integral (Henneaux and Teitelboim, 1992), in a similar way to the high energy gauge theory models. In fact, for all stochastic volatility models, after integrating over momentum variables, one obtains an effective Euclidean Lagrangian path integral over the volatility alone. The role of the second class constraints is determining the underlying asset price S completely in terms of volatility, so it plays no role in the path integral. In order to examine the effect of the constraints on the dynamics for both extreme limits, the probability density function is evaluated by using semi-classical arguments, in an analogous manner to that developed in Hagan et al. (2002), for the SABR model. (C) 2014 Elsevier B.V. All rights reserved.

Abstract: In a recent article, a generic optimal control problem was studied from a physicist's point of view (Contreras et al. 2017). Through this optic, the Pontryagin equations are equivalent to the Hamilton equations of a classical constrained system. By quantizing this constrained system, using the right ordering of the operators, the corresponding quantum dynamics given by the Schrodinger equation is equivalent to that given by the Hamilton-Jacobi-Bellman equation of Bellman's theory. The conclusion drawn there were based on certain analogies between the equations of motion of both theories. In this paper, a closer and more detailed examination of the quantization problem is carried out, by considering three possible quantization procedures: right quantization, left quantization, and Feynman's path integral approach. The Bellman theory turns out to be the classical limit h -> 0 of these three different quantum theories. Also, the exact relation of the phase S(x, t) of the wave function Psi(x, t) = e(i/hS(x,t)) of the quantum theory with Bellman's cost function J(+)(x, t) is obtained. In fact, S(x, t) satisfies a 'conjugate' form of the Hamilton-Jacobi-Bellman equation, which implies that the cost functional J(+)(x, t) must necessarily satisfy the usual Hamilton-Jacobi-Bellman equation. Thus, the Bellman theory effectively corresponds to a quantum view of the optimal control problem. (C) 2018 Elsevier B.V. All rights reserved.

Abstract: Only few efforts have been made in order to relax one of the key assumptions of the Black-Scholes model: the no-arbitrage assumption. This is despite the fact that arbitrage processes usually exist in the real world, even though they tend to be short-lived. The purpose of this paper is to develop an option pricing model with endogenous stochastic arbitrage, capable of modelling in a general fashion any future and underlying asset that deviate itself from its market equilibrium. Thus, this investigation calibrates empirically the arbitrage on the futures on the S&P 500 index using transaction data from September 1997 to June 2009, from here a specific type of arbitrage called “arbitrage bubble”, based on a t-step function, is identified and hence used in our model. The theoretical results obtained for Binary and European call options, for this kind of arbitrage, show that an investment strategy that takes advantage of the identified arbitrage possibility can be defined, whenever it is possible to anticipate in relative terms the amplitude and timespan of the process. Finally, the new trajectory of the stock price is analytically estimated for a specific case of arbitrage and some numerical illustrations are developed. We find that the consequences of a finite and small endogenous arbitrage not only change the trajectory of the asset price during the period when it started, but also after the arbitrage bubble has already gone. In this context, our model will allow us to calibrate the B-S model to that new trajectory even when the arbitrage already started. (C) 2010 Elsevier B.V. All rights reserved.

Abstract: The Black-Scholes equation can be interpreted from the point of view of quantum mechanics, as the imaginary time Schrodinger equation of a free particle. When deviations of this state of equilibrium are considered, as a product of some market imperfection, such as: Transaction cost, asymmetric information issues, short-term volatility, extreme discontinuities, or serial correlations; the classical non-arbitrage assumption of the Black-Scholes model is violated, implying a non-risk-free portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the Black-Scholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new Black-Scholes-Schrodinger model based on the endogenous arbitrage option pricing formulation introduced by Contreras et al. (2010) [2]. Hence, we derive a more general quantum model of option pricing, that incorporates arbitrage as an external time dependent force, which has an associated potential related to the random dynamic of the underlying asset price. This new resultant model can be interpreted as a Schrodinger equation in imaginary time for a particle of mass 1/sigma(2) with a wave function in an external field force generated by the arbitrage potential. As pointed out above, this new model can be seen as a more general formulation, where the perfect market equilibrium state postulated by the Black-Scholes model represent a particular case. Finally, since the Schrodinger equation is in place, we can apply semiclassical methods, of common use in theoretical physics, to find an approximate analytical solution of the Black-Scholes equation in the presence of market imperfections, as it is the case of an arbitrage bubble. Here, as a numerical illustration of the potential of this Schrodinger equation analogy, the semiclassical approximation is performed for different arbitrage bubble forms (step, linear and parabolic) and compare with the exact solution of our general quantum model of option pricing. (C) 2010 Elsevier B.V. All rights reserved.