Lok Pati Tripathi

A cubic B-spline collocation method for a numerical solution of the generalized Black–Scholes equation

Abstract In this paper, the uniform cubic B-spline collocation method is implemented to find the numerical solution of the generalized Black–Scholes partial differential equation. We use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of a θ -method, θ ∈ [ 1 / 2 , 1 ] ( θ = 1 corresponds to the back-ward Euler method and θ = 1 / 2 corresponds to the Crank–Nicolson method), and a cubic B-spline collocation method on uniform meshes, respectively. The method corresponding to θ = 1 is shown to be unconditionally stable and first order accurate with respect to the time variable and second order accurate with respect to the space variable while the method corresponding to θ = 1 / 2 is shown to be unconditionally stable and second order accurate with respect to both the variables. Finally, the numerical examples demonstrate the stability and accuracy of the method.

A radial basis functions based finite differences method for wave equation with an integral condition

The hyperbolic partial differential equation, which contains integral condition in place of classical boundary condition arises in many application of modern physics and technologies. In this article, we propose a numerical method to solve the hyperbolic equation with nonlocal boundary condition using radial basis function based finite difference method. Several numerical experiments are presented and compared with some existing method to demonstrate the efficiency of the proposed method.

Second Order Accurate IMEX Methods for Option Pricing Under Merton and Kou Jump-Diffusion Models

In this paper three implicit-explicit (IMEX) time semi-discrete methods, namely IMEX-BDF1, IMEX-BDF2 and CN-LF, are developed for solving parabolic partial integro-differential equations which arise in option pricing theory when the underlying asset follows a jump diffusion process. It is shown that IMEX-BDF2 and CN-LF are stable and second order accurate, whereas IMEX-BDF1 is stable but only first order accurate. After time semi-discretization, the resulting linear differential equations are solved by using a cubic B-spline collocation method. The methods so developed have computational complexity of

Application of radial basis function with L-stable Padé time marching scheme for pricing exotic option

O(MNlog_{2}(M))

An Error Analysis of a Finite Element Method with IMEX-Time Semidiscretizations for Some Partial Integro-differential Inequalities Arising in the Pricing of American Options

O(MNlog2(M)) for Merton model and of

A robust nonuniform B-spline collocation method for solving the generalized Black–Scholes equation

O(MN)