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.

The two best ways to derive the Black–Scholes PDE

2019 ◽  
Vol 10 (2) ◽  
pp. 168-174
Author(s):  
Paul Wilmott
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Design Methodology ◽  
Content Type ◽  
Technical Requirements ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Business Perspective ◽  
Big Picture ◽  
Complex Methods

Purpose The purpose of this paper is to explain the Black–Scholes model with minimal technical requirements and to illustrate its impact from a business perspective. Design/methodology/approach The paper employs simple accounting concepts and an argument part based on business need. Findings The Black–Scholes partial differential equation can be derived in many ways, some easy to understand, some hard, some useful and others not. The two methods in this paper are extremely insightful. Originality/value The paper takes a big-picture view of derivatives valuation. As such, it is a simple accompaniment to more complex methods and aims to keep modelling grounded in reality.

STABILITY ANALYSIS OF A COLLOCATION METHOD FOR SOLVING THE RETARDED POTENTIAL INTEGRAL EQUATION

2005 ◽  
Vol 13 (02) ◽  
pp. 287-299 ◽  
Author(s):  
P. J. HARRIS ◽  
H. WANG ◽  
R. CHAKRABARTI ◽  
D. HENWOOD
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Stability Analysis ◽  
Boundary Element Method ◽  
Numerical Solution ◽  
Boundary Element ◽  
Collocation Method ◽  
Stability Problem ◽  
Numerical Scheme ◽  
Type Boundary

This paper deals with the numerical solution of the retarded potential integral equation using a collocation type boundary element method. This method is widely used in practice but often suffers from stability problems. The purpose of the paper is to carry out a stability analysis of the numerical scheme and examine how any instability arises. This paper will then propose a method for overcoming this stability problem. A comparison with an exact solution demonstrates that the approach proposed here is effective for the case of a sphere.

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