On some nonlinear boundary value problems related to a Black-Scholes model with transaction costs

Figueroa, Rubén; Grossinho, Maria do Rosário

doi:10.1186/s13661-015-0410-9

Cited by 3 publications

(3 citation statements)

References 4 publications

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“…In the stationary case, the BS model taking into account the presence of transaction costs is a nonlinear second-order Dirichlet boundary problem [9,22]:…”

Section: Nonlinear Bs Model and Discretizationmentioning

confidence: 99%

“…In this paper, we are interested in the study of a stationary BS model [9,22], which is a nonlinear secondorder differential equation that models the valuation of a call option in presence of transaction costs. These stationary solutions give the option value V as a function of the stock price, which can be interesting when dealing with a model where the time does not play a relevant role such as, for instance, in perpetual options.…”

Section: Introductionmentioning

confidence: 99%

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Numerical simulation of a nonlinear model in finance by Broyden’s method

Zeng,

Liu,

Song

2024

ScienceAsia

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In this paper, we study the stationary Black-Scholes model arising in finance with transaction costs. This model becomes interesting when the time does not play a role such as, for instance, in perpetual options. The equation describing this model is a nonlinear second-order boundary value problem and there is no analytic solutions in closed form for such a nonlinear equation. After discretization via the centered finite difference formula we have to solve a nonlinear algebraic system which would be a serious problem when we use a small discretization mesh. We solve this nonlinear system by the residual-based Broyden's method, which is an efficient quasi-Newton method and is convenient to implement by a desk computer. We give a convergence analysis of the Broyden's method by assuming a lower and upper bound of the converged solution of the Black-Scholes model. Numerical results are given to show that the convergence rate of the method is robust with respect to the discretization mesh and the problem parameters.

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“…In the stationary case, the BS model taking into account the presence of transaction costs is a nonlinear second-order Dirichlet boundary problem [9,22]:…”

Section: Nonlinear Bs Model and Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical simulation of a nonlinear model in finance by Broyden’s method

Zeng,

Liu,

Song

2024

ScienceAsia

View full text Add to dashboard Cite

show abstract

“…A solution V = V (S, t) represents the price of an option if the price of the underlying asset is S > 0 at the time t ∈ [0, T ]. If we include transaction costs in model (1.2) then we can obtain (see [2,12,18]) the following nonlinear version of (1.2):…”

Section: Introductionmentioning

confidence: 99%

Generalized Jacobi Reproducing Kernel Method in Hilbert Spaces for Solving the Black-Scholes Option Pricing Problem Arising in Financial Modelling

Foroutan

Ebadian

Fazli

2018

Mathematical Modelling and Analysis

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Based on the reproducing kernel Hilbert space method, a new approach is proposed to approximate the solution of the Black-Scholes equation with Dirichlet boundary conditions and introduce the reproducing kernel properties in which the initial conditions of the problem are satisfied. Based on reproducing kernel theory, reproducing kernel functions with a polynomial form will be constructed in the reproducing kernel spaces spanned by the generalized Jacobi basis polynomials. Some new error estimates for application of the method are established. The convergence analysis is established theoretically. The proposed method is successfully used for solving an option pricing problem arising in financial modelling. The ideas and techniques presented in this paper will be useful for solving many other problems.

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The exact traveling wave solutions of a class of generalized Black-Scholes equation

Gao¹,

2017

AIMS Mathematics

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On some nonlinear boundary value problems related to a Black-Scholes model with transaction costs

Cited by 3 publications

References 4 publications

Numerical simulation of a nonlinear model in finance by Broyden’s method

Numerical simulation of a nonlinear model in finance by Broyden’s method

Generalized Jacobi Reproducing Kernel Method in Hilbert Spaces for Solving the Black-Scholes Option Pricing Problem Arising in Financial Modelling

The exact traveling wave solutions of a class of generalized Black-Scholes equation

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