A robust numerical scheme for a time-fractional Black-Scholes partial differential equation describing stock exchange dynamics

2021 ◽  
Vol 145 ◽  
pp. 110753
Author(s):  
Samuel M Nuugulu ◽  
Frednard Gideon ◽  
Kailash C Patidar
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Numerical Scheme ◽  
Stock Exchange ◽  
Partial Differential ◽  
Black Scholes

A family of positivity preserving schemes for numerical solution of Black–Scholes equation

2016 ◽  
Vol 03 (04) ◽  
pp. 1650025
Author(s):  
M. Mehdizadeh Khalsaraei ◽  
R. Shokri Jahandizi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Numerical Solution ◽  
Numerical Scheme ◽  
Initial Vector ◽  
Positivity Preserving ◽  
Fully Implicit ◽  
Black Scholes ◽  
Spurious Oscillations

When one solves the Black–Scholes partial differential equation, it is of great important that numerical scheme to be free of spurious oscillations and satisfy the positivity requirement. With positivity, we mean, the component non-negativity of the initial vector, is preserved in time for the exact solution. Numerically, such property for fully implicit scheme is not always satisfied by approximated solutions and they generate spurious oscillations in the presence of discontinuous payoff. In this paper, by using the nonstandard discretization strategy, we propose a new scheme that is free of spurious oscillations and satisfies the positivity requirement.

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.

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.

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