Explicit solution of a nonlinear Black-Scholes partial differential equation: tanh method

2019 ◽  
Vol 13 (7) ◽  
pp. 339-346
Author(s):  
Purity J. Kiptum ◽  
Joseph Esekon ◽  
Rhoda Ong'awa Esilaba
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Explicit Solution ◽  
Partial Differential ◽  
Black Scholes ◽  
Tanh Method

scholarly journals Numerical Solution of European and American Option with Dividends using Finite Difference Methods

2020 ◽  
Vol 13 (13) ◽  
pp. 55-61
Author(s):  
Kedar Nath Uprety ◽  
Ganesh Prasad Panday
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Analytical Formula ◽  
Partial Differential ◽  
Dividend Payments ◽  
Fully Implicit ◽  
Black Scholes ◽  
Nicolson Scheme

Numerical methods form an important part of the pricing of financial derivatives where there is no closed form analytical formula. Black-Scholes equation is a well known partial differential equation in financial mathematics. In this paper, we have studied the numerical solutions of the Black-Scholes equation for European options (Call and Put) as well as American options with dividends. We have used different approximate to discretize the partial differential equation in space and explicit (Forward Euler’s), fully implicit with projected Successive Over-Relaxation (SOR) algorithm and Crank-Nicolson scheme for time stepping. We have implemented and tested the methods in MATLAB. Finally, some numerical results have been presented and the effects of dividend payments on option pricing have also been considered.

scholarly journals Explicit Solution for Cylindrical Heat Conduction

2016 ◽  
Vol 13 (2) ◽  
Author(s):  
Kaitlyn Parsons ◽  
Tyler Reichanadter ◽  
Andi Vicksman ◽  
Harvey Segur
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Analytical Method ◽  
Explicit Solution ◽  
Separation Of Variables ◽  
Partial Differential ◽  
Forcing Function

The heat equation is a partial differential equation that elegantly describes heat conduction or other diffusive processes. Primary methods for solving this equation require time-independent boundary conditions. In reality this assumption rarely has any validity. Therefore it is necessary to construct an analytical method by which to handle the heat equation with time-variant boundary conditions. This paper analyzes a physical system in which a solid brass cylinder experiences heat flow from the central axis to a heat sink along its outer rim. In particular, the partial differential equation is transformed such that its boundary conditions are zero which creates a forcing function in the transform PDE. This transformation constructs a Green’s function, which admits the use of variation of parameters to find the explicit solution. Experimental results verify the success of this analytical method. KEYWORDS: Heat Equation; Bessel-Fourier Decomposition; Cylindrical; Time-dependent Boundary Conditions; Orthogonality; Partial Differential Equation; Separation of Variables; Green’s Functions

A Fast Parallel Implicit Method for Solving Black-Scholes Partial Differential Equation

Information ◽  
2020 ◽  
Vol 23 (3) ◽  
pp. 159-192
Author(s):  
Ikuya Uematsu ◽  
◽  
Lei Li ◽  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Linear Equations ◽  
Financial Derivatives ◽  
Efficient Computation ◽  
Partial Differential ◽  
Black Scholes ◽  
The Finite Difference Method

The Option is well known as one of the typical financial derivatives. In order to determine the price of this option, the finite difference method is used, which must be calculated using the Black―Scholes partial differential equation. In this paper, efficient computation is performed for tridiagonal Toeplitz linear equations which is needed when solving Black―Scholes partial differential equation. Let size of discretization with time is n, and size of discretization for property's value is m, we propose a method to find the solution with the required number of parallel steps of 4n log m, and the required number of processors m + log m.

scholarly journals Ulam-Hyers stability of a parabolic partial differential equation

2019 ◽  
Vol 52 (1) ◽  
pp. 475-481 ◽  
Author(s):  
Daniela Marian ◽  
Sorina Anamaria Ciplea ◽  
Nicolaie Lungu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Stability Result ◽  
Parabolic Partial Differential Equation ◽  
Ulam Stability ◽  
Partial Differential ◽  
Black Scholes ◽  
Rassias Stability

AbstractThe goal of this paper is to give an Ulam-Hyers stability result for a parabolic partial differential equation. Here we present two types of Ulam stability: Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability. Some examples are given, one of them being the Black-Scholes equation.

Comparing between G′/G expansion method and tanh-method

2014 ◽  
Vol 4 (4) ◽  
Author(s):  
Zainab Ayati
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Expansion Method ◽  
Partial Differential ◽  
Tanh Method

AbstractIn this paper, G′/G-expansion and tanh-methods, as two well known methods, for solving partial differential equations are compared. It has been shown that these two methods are the same, for solving partial Differential equation in special conditions. For illustration and more explanation of the idea, two examples are provided.

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