Non-linear Black-Scholes Option Pricing Model based on Quantum Dynamics

Wróblewski, Marcin; Myśliński, Andrzej

doi:10.5220/0011066000003197

Cited by 1 publication

(42 citation statements)

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“…Recently, a new class of option pricing models and methods inspired by the quantum mechanics approach (Haven, 2004;Vukovic, 2015;Wróblewski, 2017;Kartono, 2020;Wróblewski, 2022) has appeared. This option pricing approach is based on similarity between the description of the micro world provided by quantum dynamics (Einstein, 1925;Wróblewski, 2017;Wróblewski, 2013) and the description of stocks and the associated options price evolution and prediction in terms of the stochastic processes.…”

Section: Quantum Approach For Option Pricingmentioning

confidence: 99%

“…Analytical solutions to the non-linear Schrödinger equation (Ivancevic, 2011;Ivancevic, 2010) with different non-linear potentials, i.e., dark solitons, have been used in (Wróblewski, 2017) to calibrate the model based on the market option prices. Using reverse rather than forward transformation as in (Wróblewski, 2017), the extended nonlinear Black-Scholes option pricing model is formulated by authors in (Wróblewski, 2022). Originally, non-linear Schrödinger equation is considered in complex Hilbert space (Schneider et al, 2017).…”

Section: Quantum Approach For Option Pricingmentioning

confidence: 99%

“…Originally, non-linear Schrödinger equation is considered in complex Hilbert space (Schneider et al, 2017). This new model has been formulated in (Wróblewski, 2022) by transforming non-linear Schrödinger equation variables into real Euclidean space and using Itô differentiation formula. In comparison with standard non-linear Black-Scholes model, this model contains additional non-linear third order term depending on a coupling parameter.…”

Section: Quantum Approach For Option Pricingmentioning

confidence: 99%

“…In comparison with standard non-linear Black-Scholes model, this model contains additional non-linear third order term depending on a coupling parameter. The method of lines (Sanchez, 2017) combined with Runge-Kutta method (Kutta, 1901) to solve ordinary differential equations is applied in (Wróblewski, 2022) to find numerical solution of this model for European call options.…”

Section: Quantum Approach For Option Pricingmentioning

confidence: 99%

“…These discrete models assumes perfectly efficient markets and can price the option at each point of a specified time interval. In these models the underlaying asset price is assumed (Wróblewski, 2022) either to increase or to decrease with the same probability when moving from one instant time to the other. The other approach is based on the use of partial differential equations governing the evolution of option contract price (Peszek, 1995).…”

Section: Introductionmentioning

confidence: 99%

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Quantum Dynamics approach for non-linear Black-Scholes option pricing

Wróblewski

Myśliński

2023

Preprint

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Option pricing models are formulated based on mathematical theories. They are applied to estimate the fair value of an option. Among different pricing models, the linear Black-Scholes equation is very frequently used as option pricing model. Since assumptions of this linear model do not match precisely the real market conditions and do not allow to estimate the option price precisely there are developed more complicated non-linear Black-Scholes option pricing models. In these models volatility of underlying asset price or market risk free interest rate are assumed stochastic or time dependent. Moreover transaction costs are also taken into account. Quantum mechanics provides other approach to calculate option values using non-linear models. This approach is based on the similarity between the evolution of elementary particles in space and the volatility of the stock prices. In previous paper the authors have proposed non-linear Black-Scholes model to calculate option price based on quantum dynamics approach. This model has been obtained by suitable transformation of variables in non-linear Schrödinger equation with the external potential term. In this paper the non-linear quantum based option pricing model is numerically tested and verified. Based on this model, the calculation of European, Asian and American call option prices for market data is provided. The model parameters, especially the adaptive market potential, have been estimated based on market prices of European call options listed on Warsaw Stock Exchange as well as American call options prices based on the selected NYSE stock prices. The sensitivity of this model with respect to risk free interest rate has been investigated. For the sake of comparison non-linear Heston option pricing model has been also solved and calibrated using Monte Carlo method. The comparison of option pricing using the developed quantum based model, linear Black-Scholes model, Heston model, indicates higher precision and lower computational costs of the proposed enhanced non-linear model.

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