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Author 
Contreras, M.; Pellicer, R.; Villena, M. 


Title 
Dynamic optimization and its relation to classical and quantum constrained systems 
Type 


Year 
2017 
Publication 
Physica AStatistical Mechanics And Its Applications 
Abbreviated Journal 
Physica A 


Volume 
479 
Issue 

Pages 
1225 


Keywords 
Dynamic optimization; Constrained systems; Dirac's method; Quantum mechanics 


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. 


Address 
[Contreras, Mauricio; Pellicer, Rely; Villena, Marcelo] Univ Adolfo Ibanez, Fac Engn & Sci, Santiago, Region Metropol, Chile, Email: mauricio.contreras@uai.cl 


Corporate Author 

Thesis 



Publisher 
Elsevier Science Bv 
Place of Publication 

Editor 



Language 
English 
Summary Language 

Original Title 



Series Editor 

Series Title 

Abbreviated Series Title 



Series Volume 

Series Issue 

Edition 



ISSN 
03784371 
ISBN 

Medium 



Area 

Expedition 

Conference 



Notes 
WOS:000400213800002 
Approved 



Call Number 
UAI @ eduardo.moreno @ 
Serial 
731 

Permanent link to this record 




Author 
Contreras, M.; Montalva, R.; Pellicer, R.; Villena, M. 


Title 
Dynamic option pricing with endogenous stochastic arbitrage 
Type 


Year 
2010 
Publication 
Physica AStatistical Mechanics And Its Applications 
Abbreviated Journal 
Physica A 


Volume 
389 
Issue 
17 
Pages 
35523564 


Keywords 
BlackScholes model; Arbitrage; Option pricing 


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. 


Address 
[Contreras, Mauricio; Montalva, Rodrigo; Pellicer, Rely; Villena, Marcelo] Univ Adolfo Ibanez, Fac Sci & Engn, Vina Del Mar, Chile, Email: mauricio.contreras@uai.cl 


Corporate Author 

Thesis 



Publisher 
Elsevier Science Bv 
Place of Publication 

Editor 



Language 
English 
Summary Language 

Original Title 



Series Editor 

Series Title 

Abbreviated Series Title 



Series Volume 

Series Issue 

Edition 



ISSN 
03784371 
ISBN 

Medium 



Area 

Expedition 

Conference 



Notes 
WOS:000280118100023 
Approved 



Call Number 
UAI @ eduardo.moreno @ 
Serial 
91 

Permanent link to this record 




Author 
Contreras, M.; Pellicer, R.; Villena, M.; Ruiz, A. 


Title 
A quantum model of option pricing: When BlackScholes meets Schrodinger and its semiclassical limit 
Type 


Year 
2010 
Publication 
Physica AStatistical Mechanics And Its Applications 
Abbreviated Journal 
Physica A 


Volume 
389 
Issue 
23 
Pages 
54475459 


Keywords 
BlackScholes model; Arbitrage; Option pricing; Quantum mechanics; Semiclassical methods 


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. 


Address 
[Contreras, Mauricio; Pellicer, Rely; Villena, Marcelo; Ruiz, Aaron] Adolfo Ibanez Univ, Fac Sci & Engn, Santiago, Chile, Email: mauricio.contreras@uai.cl 


Corporate Author 

Thesis 



Publisher 
Elsevier Science Bv 
Place of Publication 

Editor 



Language 
English 
Summary Language 

Original Title 



Series Editor 

Series Title 

Abbreviated Series Title 



Series Volume 

Series Issue 

Edition 



ISSN 
03784371 
ISBN 

Medium 



Area 

Expedition 

Conference 



Notes 
WOS:000283904000012 
Approved 



Call Number 
UAI @ eduardo.moreno @ 
Serial 
116 

Permanent link to this record 