Records 
Author 
Contreras, M.; Pena, J.P. 
Title 
The quantum dark side of the optimal control theory 
Type 

Year 
2019 
Publication 
Physica AStatistical Mechanics And Its Applications 
Abbreviated Journal 
Physica A 
Volume 
515 
Issue 

Pages 
450473 
Keywords 
Optimal control theory; Pontryagin's equations; HamiltonJacobiBellman equation; Constrained systems; Dirac's method; Quantum mechanics 
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 HamiltonJacobiBellman 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 HamiltonJacobiBellman equation, which implies that the cost functional J(+)(x, t) must necessarily satisfy the usual HamiltonJacobiBellman equation. Thus, the Bellman theory effectively corresponds to a quantum view of the optimal control problem. (C) 2018 Elsevier B.V. All rights reserved. 
Address 
[Contreras G, Mauricio] Univ Adolfo Ibanez, Fac Ingn & Ciencias, 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:000452941100042 
Approved 

Call Number 
UAI @ eduardo.moreno @ 
Serial 
952 
Permanent link to this record 



Author 
Contreras, M.; Echeverria, J.; Pena, J.P.; Villena, M. 
Title 
Resonance phenomena in option pricing with arbitrage 
Type 

Year 
2020 
Publication 
Physica AStatistical Mechanics And Its Applications 
Abbreviated Journal 
Physica A 
Volume 
540 
Issue 

Pages 
21 pp 
Keywords 
BlackScholes model; Option pricing; Arbitrage; Barrier options 
Abstract 
In this paper, we want to report an interesting resonance phenomena that appears in option pricing, when the presence of arbitrage is incorporated explicitly into the BlackScholes model. In Contreras et al. (2010), the authors after analyse empirical financial data, determines that the mispricing between the empirical and the BlackScholes prices can be described by Heaviside type function (called an arbitrage bubble there). These bubbles are characterised by a finite time span and an amplitude which measures the price deviation from the BlackScholes model. After that, in Contreras et al. (2010), the BlackScholes equation is generalised to incorporates explicitly these arbitrage bubbles, which generates an interaction potential that changes the usual BlackScholes free dynamics completely. However, an interesting phenomena appears when the amplitude of the arbitrage bubble is equal to the volatility parameter of the BlackScholes model: in that case, the potential becomes infinite, and option pricing decrease abruptly to zero. We analyse this limit behaviour for two situations: a European and a barrier option. Also, we perform an analytic study of the propagator in each case, to understand the cause of the resonance. We think that it resonance phenomena could to help to understand the origin of certain financial crisis in the option pricing area. (C) 2019 Elsevier B.V. All rights reserved. 
Address 
[Contreras, M.; Pena, J. P.] Univ Andres Bello, Dept Ciencias Fis, Sazie 2212, Chile, Email: mauriccio1965@gmail.com; 
Corporate Author 

Thesis 

Publisher 
Elsevier 
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:000506711900078 
Approved 

Call Number 
UAI @ eduardo.moreno @ 
Serial 
1095 
Permanent link to this record 