A Nonstandard Finite Difference Method for a Generalized Black–Scholes Equation

Khalsaraei, Mohammad Mehdizadeh; Rashidi, Mohammad Mehdi; Shokri, Ali; Ramos, Higinio; Khakzad, Pari

doi:10.3390/sym14010141

Cited by 5 publications

(3 citation statements)

References 32 publications

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“…W 1 t and W 2 t are two independent Brownian motion processes, and ρ represents the instantaneous correlation between the return process and the volatility process. Heston class of models is subject to many extensions and research (Ankudinova, 2008;El Hajaji, 2015;Almeida, 2022;Mehdizadeh, 2022). Among others, in (Ankudinova, 2008) the non-linear option pricing equation for European and American options is numerically solved with Barle's and Soner's volatility model with transaction costs.…”

Section: Non-linear Black-scholes Modelsmentioning

confidence: 99%

“…These models accounts for stock prices following the geometric Lévy process. A non standard implicit finite difference method for numerical solving a generalized Black-Scholes equation taking into account its symmetry is developed in (Mehdizadeh, 2022). Viscosity solution of a Delta Greek non-linear Black-Scholes equation is studied in (Almeida, 2022).…”

Section: Non-linear Black-scholes Modelsmentioning

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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Section: Non-linear Black-scholes Modelsmentioning

confidence: 99%

Section: Non-linear Black-scholes Modelsmentioning

confidence: 99%

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

Wróblewski

Myśliński

2023

Preprint

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“…Apart from predicting the behaviour of the dynamical system correctly, the NSFD method is known to preserve the dynamical properties of an epidemic model and is less difficult to implement when compared with the aforementioned numerical methods (Qui et al, 2014). Applications of NSFD method are found in financial theory (Mehdizadeh et al, 2022;Mehdizadeh et al, 2023), epidemiology (ur Rehman et al, 2023, Butt et al, 2023, enzymology (Miller & O'Riordan, 2020;Zafar et al, 2023), pharmacology (Egbelowo, 2018;Ebgelowo & Hoang, 2021), immunology (Costa et al, 2023;Elaiw et al, 2023). The purpose of this study is to apply the NSFD scheme to solve a mathematical model presented in Ibrahim (2023).…”

Section: Introductionmentioning

confidence: 99%

Application of Non-Standard Finite Difference Method on Covid-19 Mathematical Model With Fear of Infection

Usman,

Ibrahim,

Isah

et al. 2023

FJS

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This study presents a novel application of Non -Standard Finite Difference (NSFD) Method to solve a COVID-19 epidemic mathematical model with the impact of fear due to infection. The mathematical model is governed by a system of first-order non-linear ordinary differential equations and is shown to possess a unique positive solution that is bounded. The proposed numerical scheme is used to obtain an approximate solution for the COVID-19 model. Graphical results were displayed to show that the solution obtained by NSFD agrees well with those obtained by the Runge-Kutta-Fehlberg method built-in Maple 18.

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Analytical and Numerical Solution for the Time Fractional Black-Scholes Model Under Jump-Diffusion

2023

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In this work, we study the numerical solution for time fractional Black-Scholes model under jump-diffusion involving a Caputo differential operator. For simplicity of the analysis, the model problem is converted into a time fractional partial integro-differential equation with a Fredholm integral operator. The L1 discretization is introduced on a graded mesh to approximate the temporal derivative. A second order central difference scheme is used to replace the spatial derivatives and the composite trapezoidal approximation is employed to discretize the integral part. The stability results for the proposed numerical scheme are derived with a sharp error estimation. A rigorous analysis proves that the optimal rate of convergence is obtained for a suitable choice of the grading parameter. Further, we introduce the Adomian decomposition method to find out an analytical approximate solution of the given model and the results are compared with the numerical solutions. The main advantage of the fully discretized numerical method is that it not only resolves the initial singularity occurred due to the presence of the fractional operator, but it also gives a higher rate of convergence compared to the uniform mesh. On the other hand, the Adomian decomposition method gives the analytical solution as well as a numerical approximation of the solution which does not involve any mesh discretization. Furthermore, the method does not require a large amount of computer memory and is free of rounding errors. Some experiments are performed for both methods and it is shown that the results agree well with the theoretical findings. In addition, the proposed schemes are investigated on numerous European option pricing jump-diffusion models such as Merton’s jump-diffusion and Kou’s jump-diffusion for both European call and put options.

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A Nonstandard Finite Difference Method for a Generalized Black–Scholes Equation

Cited by 5 publications

References 32 publications

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

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

Application of Non-Standard Finite Difference Method on Covid-19 Mathematical Model With Fear of Infection

Analytical and Numerical Solution for the Time Fractional Black-Scholes Model Under Jump-Diffusion

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