
Bustamante, M., & Contreras, M. (2016). Multiasset BlackScholes model as a variable second class constrained dynamical system. Physica A, 457, 540–572.
Abstract: In this paper, we study the multiasset BlackScholes model from a structural point of view. For this, we interpret the multiasset BlackScholes equation as a multidimensional Schrodinger one particle equation. The analysis of the classical Hamiltonian and Lagrangian mechanics associated with this quantum model implies that, in this system, the canonical momentums cannot always be written in terms of the velocities. This feature is a typical characteristic of the constrained system that appears in the highenergy physics. To study this model in the proper form, one must apply Dirac's method for constrained systems. The results of the Dirac's analysis indicate that in the correlation parameters space of the multi assets model, there exists a surface (called the Kummer surface Sigma(K), where the determinant of the correlation matrix is null) on which the constraint number can vary. We study in detail the cases with N = 2 and N = 3 assets. For these cases, we calculate the propagator of the multiasset BlackScholes equation and show that inside the Kummer Sigma(K) surface the propagator is well defined, but outside Sigma(K) the propagator diverges and the option price is not well defined. On Sigma(K) the propagator is obtained as a constrained path integral and their form depends on which region of the Kummer surface the correlation parameters lie. Thus, the multiasset BlackScholes model is an example of a variable constrained dynamical system, and it is a new and beautiful property that had not been previously observed. (C) 2016 Elsevier B.V. All rights reserved.



Contreras, G. M. (2014). Stochastic volatility models at rho = +/ 1 as second class constrained Hamiltonian systems. Physica A, 405, 289–302.
Abstract: The stochastic volatility models used in the financial world are characterized, in the continuoustime 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 FokkerPlanck 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 bidimensional BlackScholes 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 FokkerPlanck equation or the bidimensional BlackScholes 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 pathintegral (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 semiclassical 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.



Contreras, M., & Hojman, S. A. (2014). Option pricing, stochastic volatility, singular dynamics and constrained path integrals. Physica A, 393, 391–403.
Abstract: Stochastic volatility models have been widely studied and used in the financial world. The Heston model (Heston, 1993) [7] is one of the best known models to deal with this issue. These stochastic volatility models are characterized by the fact that they explicitly depend on a correlation parameter p which relates the two Brownian motions that drive the stochastic dynamics associated to the volatility and the underlying asset. Solutions to the Heston model in the context of option pricing, using a path integral approach, are found in Lemmens et al. (2008) [21] while in Baaquie (2007,1997) [12,13] propagators for different stochastic volatility models are constructed. In all previous cases, the propagator is not defined for extreme cases rho = +/ 1. It is therefore necessary to obtain a solution for these extreme cases and also to understand the origin of the divergence of the propagator. In this paper we study in detail a general class of stochastic volatility models for extreme values rho = +/ 1 and show that in these two cases, the associated classical dynamics corresponds to a system with second class constraints, which must be dealt with using Dirac's method for constrained systems (Dirac, 1958,1967) [22,23] in order to properly obtain the propagator in the form of a Euclidean Hamiltonian path integral (Henneaux and Teitelboim, 1992) [25]. After integrating over momenta, one gets an Euclidean Lagrangian path integral without constraints, which in the case of the Heston model corresponds to a path integral of a repulsive radial harmonic oscillator. In all the cases studied, the price of the underlying asset is completely determined by one of the second class constraints in terms of volatility and plays no active role in the path integral. (C) 2013 Elsevier B.V. All rights reserved.



Contreras, M., Montalva, R., Pellicer, R., & Villena, M. (2010). Dynamic option pricing with endogenous stochastic arbitrage. Physica A, 389(17), 3552–3564.
Abstract: Only few efforts have been made in order to relax one of the key assumptions of the BlackScholes model: the noarbitrage assumption. This is despite the fact that arbitrage processes usually exist in the real world, even though they tend to be shortlived. 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 tstep 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 BS model to that new trajectory even when the arbitrage already started. (C) 2010 Elsevier B.V. All rights reserved.



Contreras, M., Pellicer, R., & Villena, M. (2017). Dynamic optimization and its relation to classical and quantum constrained systems. Physica A, 479, 12–25.
Abstract: We study the structure of a simple dynamic optimization problem consisting of one state and one control variable, from a physicist's point of view. By using an analogy to a physical model, we study this system in the classical and quantum frameworks. Classically, the dynamic optimization problem is equivalent to a classical mechanics constrained system, so we must use the Dirac method to analyze it in a correct way. We find that there are two secondclass constraints in the model: one fix the momenta associated with the control variables, and the other is a reminder of the optimal control law. The dynamic evolution of this constrained system is given by the Dirac's bracket of the canonical variables with the Hamiltonian. This dynamic results to be identical to the unconstrained one given by the Pontryagin equations, which are the correct classical equations of motion for our physical optimization problem. In the same Pontryagin scheme, by imposing a closedloop lambdastrategy, the optimality condition for the action gives a consistency relation, which is associated to the HamiltonJacobiBellman equation of the dynamic programming method. A similar result is achieved by quantizing the classical model. By setting the wave function Psi (x, t) = e(is(x,t)) in the quantum Schrodinger equation, a nonlinear partial equation is obtained for the S function. For the righthand side quantization, this is the HamiltonJacobiBellman equation, when S(x, t) is identified with the optimal value function. Thus, the HamiltonJacobiBellman equation in Bellman's maximum principle, can be interpreted as the quantum approach of the optimization problem. (C) 2017 Elsevier B.V. All rights reserved.



Contreras, M., Pellicer, R., Villena, M., & Ruiz, A. (2010). A quantum model of option pricing: When BlackScholes meets Schrodinger and its semiclassical limit. Physica A, 389(23), 5447–5459.
Abstract: The BlackScholes 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, shortterm volatility, extreme discontinuities, or serial correlations; the classical nonarbitrage assumption of the BlackScholes model is violated, implying a nonriskfree portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the BlackScholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new BlackScholesSchrodinger 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 BlackScholes 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 BlackScholes 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.

