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Author Contreras, M.; Pena, J.P.
Title The quantum dark side of the optimal control theory Type
Year 2019 Publication Physica A-Statistical Mechanics And Its Applications Abbreviated Journal Physica A
Volume 515 Issue Pages 450-473
Keywords Optimal control theory; Pontryagin's equations; Hamilton-Jacobi-Bellman 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 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.
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 (down) Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0378-4371 ISBN Medium
Area Expedition Conference
Notes WOS:000452941100042 Approved
Call Number UAI @ eduardo.moreno @ Serial 952
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Author Contreras, M.; Echeverria, J.; Pena, J.P.; Villena, M.
Title Resonance phenomena in option pricing with arbitrage Type
Year 2020 Publication Physica A-Statistical Mechanics And Its Applications Abbreviated Journal Physica A
Volume 540 Issue Pages 21 pp
Keywords Black-Scholes 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 Black-Scholes model. In Contreras et al. (2010), the authors after analyse empirical financial data, determines that the mispricing between the empirical and the Black-Scholes 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 Black-Scholes model. After that, in Contreras et al. (2010), the Black-Scholes equation is generalised to incorporates explicitly these arbitrage bubbles, which generates an interaction potential that changes the usual Black-Scholes free dynamics completely. However, an interesting phenomena appears when the amplitude of the arbitrage bubble is equal to the volatility parameter of the Black-Scholes 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 (down) Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0378-4371 ISBN Medium
Area Expedition Conference
Notes WOS:000506711900078 Approved
Call Number UAI @ eduardo.moreno @ Serial 1095
Permanent link to this record