A quantum model of option pricing: When Black–Scholes meets Schrödinger and its semi-classical limit

Contreras, Mauricio; Pellicer, Rely; Villena, Marcelo; Ruiz, Aaron

doi:10.1016/j.physa.2010.08.018

Cited by 36 publications

(39 citation statements)

References 24 publications

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“…The existence results for di erent types of linear Schrödinger equations can be found in book [22]. Stock options pricing models based on linear Schrödinger equations and their relation to Black-Scholes models are reported in many papers [23][24][25][26][27][28][29]. Among others in the author's previous paper [29], the European call option price based on the linear Schrödinger equation has been calculated.…”

Section: U(t S(t)) = Max{ S(t) − K}mentioning

confidence: 99%

“…Functions (27) and (28) denote a general solution for m ∈ [ , ) and the so-called dark soliton solution for m = , respectively. We expect the function ψ(S, t) to be real [30].…”

Section: β|ψ(S T)| ψ(S T) =mentioning

confidence: 99%

“…The function presented in equation (27) also contains an imaginary part. Using the splitting formula of any complex function into real and imaginary parts, i.e., exp ix = cos x+ i sin x and considering that β > , we see that the real part of equation (27) is given by:…”

Section: β|ψ(S T)| ψ(S T) =mentioning

confidence: 99%

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Nonlinear Schrödinger approach to European option pricing

Wróblewski

2017

Open Physics

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This paper deals with numerical option pricing methods based on a Schrödinger model rather than the Black-Scholes model. Nonlinear Schrödinger boundary value problems seem to be alternatives to linear models which better re ect the complexity and behavior of real markets. Therefore, based on the nonlinear Schrödinger option pricing model proposed in the literature, in this paper a model augmented by external atomic potentials is proposed and numerically tested. In terms of statistical physics the developed model describes the option in analogy to a pair of two identical quantum particles occupying the same state. The proposed model is used to price European call options on a stock index. the model is calibrated using the Levenberg-Marquardt algorithm based on market data. A Runge-Kutta method is used to solve the discretized boundary value problem numerically. Numerical results are provided and discussed. It seems that our proposal more accurately models phenomena observed in the real market than do linear models.

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Section: U(t S(t)) = Max{ S(t) − K}mentioning

confidence: 99%

“…Functions (27) and (28) denote a general solution for m ∈ [ , ) and the so-called dark soliton solution for m = , respectively. We expect the function ψ(S, t) to be real [30].…”

Section: β|ψ(S T)| ψ(S T) =mentioning

confidence: 99%

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Nonlinear Schrödinger approach to European option pricing

Wróblewski

2017

Open Physics

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“…It was noted that while the Black-Scholes Hamiltonian was anti-Hermitian causing the eigenvalues to be complex, the Schrödinger Hamiltonian was Hermitian. It was further showed that the Black-Scholes equation can be derived from the Schrödinger equation via the application of quantum mechanics tools [12,13]. The facts incorporated include the points that: the Schrödinger equation requires a complex state function while the BlackScholes equation is a real PDE that yields a real valued expression for the option price at all time.…”

Section: Introductionmentioning

confidence: 99%

Analytical Solutions of the Ivancevic Option Pricing Model With a Nonzero Adaptive Market Potential

Edeki¹,

Ugbebor²,

'alez-Gaxiola³

2017

Int. J. of Pure and Appl. Math.

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“…The object of this paper is to show that the answer is positive and to develop a methodology for extracting the arbitrage bubble f from the empirical financial data through the analysis of the option mispricing. In order to do that, one will need to use some result of semi-classical approximations applied to option pricing as develop in [2]. There, an approximate solution for the non equilibrium Black-Scholes equation in the presence of an arbitrary arbitrage bubble was constructed.…”

Section: Introductionmentioning

confidence: 99%

Calibration and Simulation of Arbitrage Effects in a Non-Equilibrium Quantum Black-Scholes Model by Using Semi-Classical Methods

Contreras¹,

Pellicer²,

Santiagos³

et al. 2016

JMF

Self Cite

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An interacting Black-Scholes model for option pricing, where the usual constant interest rate r is replaced by a stochastic time dependent rate r(t) of the form r(t) = r + f (t)Ẇ (t), accounting for market imperfections and prices non-alignment, was developed in [1]. The white noise amplitude f (t), called arbitrage bubble, generates a time dependent potential U (t) which changes the usual equilibrium dynamics of the traditional Black-Scholes model. The purpose of this article is to tackle the inverse problem, that is, is it possible to extract the time dependent potential U (t) and its associated bubble shape f (t) from the real empirical financial data? In order to give an answer to this question, the interacting Black-Scholes equation must be interpreted as a quantum Schrödinger equation with hamiltonian operator H = H 0 + U (t), where H 0 is the equilibrium Black-Scholes hamiltonian and U (t) is the interaction term. If the U (t) term is small enough, the interaction potential can be thought as a perturbation, so one can compute the solution of the interacting Black-Scholes equation in an approximate form by perturbation theory. In [2] by applying the semi-classical considerations, an approximate solution of the non equilibrium Black-Scholes equation for an arbitrary bubble shape f (t) was developed. Using this semi-classical solution and the knowledge about the mispricing of the financial data, one can determinate an equation, which solutions permit obtain the functional form of the potential term U (t) and its associated bubble f (t). In all the studied cases, the non equilibrium model performs a better estimation of the real data than the usual equilibrium model. It is expected that this new and simple methodology for calibrating and simulating option pricing solutions in the presence of market imperfections, could help to improve option pricing estimations.

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A quantum model of option pricing: When Black–Scholes meets Schrödinger and its semi-classical limit

Cited by 36 publications

References 24 publications

Nonlinear Schrödinger approach to European option pricing

Nonlinear Schrödinger approach to European option pricing

Analytical Solutions of the Ivancevic Option Pricing Model With a Nonzero Adaptive Market Potential

Calibration and Simulation of Arbitrage Effects in a Non-Equilibrium Quantum Black-Scholes Model by Using Semi-Classical Methods

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