Nonlinear Schrödinger approach to European option pricing

Abstract

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.

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

“…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. There is a link between probability function describing the state of the particle and stochastic behavior of the stock price.…”

“…There is a link between probability function describing the state of the particle and stochastic behavior of the stock price. The relations between the quantum and stochastic descriptions of the stock or option prices for linear models is studied in papers (Peña, 2020;Vukovic, 2015;Haven, 2004;Wróblewski, 2017;Wróblewski, 2013).…”

“…These deterministic equations are generated by transforming the stochastic differential equations using Ito's Lemma (Weinan et al, 2019) and portfolio theory. This approach includes also linear Black-Scholes option pricing model (Ivancevic, 2011;Wróblewski, 2017) having the form of the linear parabolic boundary value problem with constant coefficients. The option trading practice indicates, that the assumptions of linear Black-Scholes option pricing model are simplification (Hull et al, 1987;Jankova, 2018) of the real market conditions.…”

Solitary Wave and Singular Wave Solutions for Ivancevic Option Pricing Model

“…Vector fnancial one and two-rogon solutions were found for the coupled nonlinear volatility and option pricing model without embedded w-learning. In 2017, a model augmented by external atomic potentials was proposed to price European call options on a stock index [37]. In [38], a nonzero adaptive market potential was studied.…”

GPU-accelerated solutions of the nonlinear Schrödinger equation for simulating 2D spinor BECs

“…The nonlinear Schrödinger equation (NLSE) is one such mathematical model with widespread applications throughout physics. Notably, this equation explains superfluid and magnetic properties of dilute Bose-Einstein condensates (BECs) [1,2], but it also successfully describes plasma Langmuir waves [3], soliton dynamics [4], the propagation of light in nonlinear media [5][6][7], surface gravity water waves [8] and rogue waves [9], superconductivity [10], and even certain financial situations [11]. The connecting thread between these disparate physical phenomena is the slowly-varying evolution of a weakly nonlinear, complex wave packet in a dispersive environment [10].…”