A Note on Numerical Solution of a Linear Black-Scholes Model

Uddin, Md. Kazi Salah; Ahmed, Mostak; Bhowmilk, Samir Kumar

doi:10.3329/ganit.v33i0.17664

Cited by 7 publications

(10 citation statements)

References 13 publications

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“…Uddin, Ahmed and Bhowmik in [15], consider solution methods for the Black Scholes model with European options, by studying a weighted average method using different weights numerical approximations, and as such approximate the model using finite difference scheme. Algliardi, Popivanov and Slavova [16,17] Considering the solutions of linear and nonlinear Black-Scholes equations, other methods-Adomian Decomposition Method (ADM), modified ADM (MADM), modified VIM (MVIM), homotopy analysis method (HAM) and modified HAM (MHAM) are applied [23][24][25].…”

Section: T DX T G T X T Dt G T X T Dw T Rmentioning

confidence: 99%

Analytical Solutions of the Black–Scholes Pricing Model for European Option Valuation via a Projected Differential Transformation Method

Edeki

Ugbebor

Owoloko

2015

Entropy

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Abstract:In this paper, a proposed computational method referred to as Projected Differential Transformation Method (PDTM) resulting from the modification of the classical Differential Transformation Method (DTM) is applied, for the first time, to the Black-Scholes Equation for European Option Valuation. The results obtained converge faster to their associated exact solution form; these easily computed results represent the analytical values of the associated European call options, and the same algorithm can be followed for European put options. It is shown that PDTM is more efficient, reliable and better than the classical DTM and other semi-analytical methods since less computational work is involved. Hence, it is strongly recommended for both linear and nonlinear stochastic differential equations (SDEs) encountered in financial mathematics.

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Section: T DX T G T X T Dt G T X T Dw T Rmentioning

confidence: 99%

Analytical Solutions of the Black–Scholes Pricing Model for European Option Valuation via a Projected Differential Transformation Method

Edeki

Ugbebor

Owoloko

2015

Entropy

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“…A great attention of many scholars to mathematical models utilised for pricing complex financial instruments, in recent years, has suggested the interesting research area of financial mathematics in which Black-Scholes model (BS) is a prominent example. Indeed, there is a large number of studies on this equation in the solution and application context with the employment of diverse numerical schemes like explicit difference schemes [13], finite difference scheme [45], multivariate padé approximation scheme [46] and Cauchy Euler method [32]. Nevertheless, applications of monotone finite difference scheme of second-order approximation for pricing models in finance are still very few.…”

Section: Discussionmentioning

confidence: 99%

Monotone Finite-Difference Schemes With Second Order Approximation Based on Regularization Approach for the Dirichlet Boundary Problem of the Gamma Equation

2020

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We investigate the initial boundary value problem for the Gamma equation transformed from the nonlinear Black-Scholes equation for pricing option to a quasilinear parabolic equation of second derivative. Furthermore, two-side estimates for the exact solution are also provided. By using regularization principle, the unconditionally monotone second order approximation finite-difference scheme on uniform and nonuniform grids is generalized, in that the maximum principle is satisfied without depending on relations of the coefficients and grid parameters. By using the difference maximum principle, we acquired two-side estimates for difference solution for the arbitrary non-sign-constant input data. Finally, we also provide a proof for a priori estimate. It can be confirmed that the two-side estimates for difference solution are completely consistent with the differential problem. Otherwise, the maximal and minimal values of the difference solution is independent from the diffusion and convection coefficients.

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“…e option popularly known as European option can only be applicable at the expiration time T [7,9,10,12,13], whereas American option can be exercised at any time until the expiration time T [4,7,9,10]. It is true that there is exibility in American option.…”

Section: Introductionmentioning

confidence: 99%

“…Basically, the BS model represents a PDE [4,9,13,16,17], and it is also one kind of parabolic equation [1,7].…”

Section: Introductionmentioning

confidence: 99%

High‐Accurate Numerical Schemes for Black–Scholes Models with Sensitivity Analysis

Bhowmik

Khan

2022

Journal of Mathematics

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The significance of both the linear and nonlinear Black-Scholes partial differential equation model is huge in the field of financial analysis. In most cases, the exact solution to such a nonlinear problem is very hard to obtain, and in some cases, it is impossible to get an exact solution to such models. In this study, both the linear and the nonlinear Black-Scholes models are investigated. This research mainly focuses on the numerical approximations of the Black-Scholes (BS) model with sensitivity analysis of the parameters. It is to note that most applied researchers use finite difference and finite element-based schemes to approximate the BS model. Thus, an urge for a high accurate numerical scheme that needs fewer grids/nodes is huge. In this study, we aim to approximate and analyze the models using two such higher-order schemes. To be specific, the Chebyshev spectral method and the differential quadrature method are employed to approximate the BS models to see the efficiency of such highly accurate schemes for the option pricing model. First, we approximate the model using the mentioned methods. Then, we move on to use the numerical results to analyze different aspects of stock market through sensitivity analysis. All the numerical schemes have been illustrated through some graphics and relevant discussions. We finish the study with some concluding remarks.

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A Note on Numerical Solution of a Linear Black-Scholes Model

Cited by 7 publications

References 13 publications

Analytical Solutions of the Black–Scholes Pricing Model for European Option Valuation via a Projected Differential Transformation Method

Analytical Solutions of the Black–Scholes Pricing Model for European Option Valuation via a Projected Differential Transformation Method

Monotone Finite-Difference Schemes With Second Order Approximation Based on Regularization Approach for the Dirichlet Boundary Problem of the Gamma Equation

High‐Accurate Numerical Schemes for Black–Scholes Models with Sensitivity Analysis

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