Search

Abstract: A compact finite difference method is designed to obtain quick and accurate solutions to partial differential equation problems. The problem of pricing an American option can be cast as a partial differential equation. Using the compact finite difference method this problem can be recast as an ordinary differential equation initial value problem. The complicating factor for American options is the existence of an optimal exercise boundary which is jointly determined with the value of the option. In this article we develop three ways of combining compact finite difference methods for American option price on a single asset with methods for dealing with this optimal exercise boundary. Compact finite difference method one uses the implicit condition that solutions of the transformed partial differential equation be nonnegative to detect the optimal exercise value. This method is very fast and accurate even when the spatial step size h is large . Compact difference method two must solve an algebraic nonlinear equation obtained by Pantazopoulos (1998) at every time step. This method can obtain second order accuracy for space x and requires a moderate amount of time comparable with that required by the Crank Nicolson projected successive over relaxation method. Compact finite difference method three refines the free boundary value by a method developed by Barone-Adesi and Lugano [The saga of the American put, 2003], and this method can obtain high accuracy for space x. The last two of these three methods are convergent, moreover all the three methods work for both short term and long term options. Through comparison with existing popular methods by numerical experiments, our work shows that compact finite difference methods provide an exciting new tool for American option pricing. [Copyright &y& Elsevier]

Abstract: In this paper, we give a fourth-order compact finite difference scheme for the general forms of two point boundary value problems and two-dimensional elliptic partial differential equations (PDE’s). By decomposing the coefficient matrix into a sum of several matrixes after we discretize the original problems, we can obtain a lower bound for the smallest eigenvalue of the coefficient matrix. Thus we prove the compact finite difference scheme converges with the fourth order of accuracy. To solve the discretized block tri-diagonal matrix equations of the two-dimensional elliptic PDE’s, we develop an efficient iterative method: Full Multigrid Method. In our numerical experiments, we compare the compact finite difference method with the finite element method and Crank–Nicolson finite difference method. The results show that the compact finite difference scheme is a highly efficient and accurate method. [Copyright &y& Elsevier]

Abstract: In this paper, we give sixth order compact finite difference formula for second order integro-differential equations (IDE) with different boundary conditions, and both of error estimates and numerical experiments confirm our compact finite difference method can get fifth order of accuracy. We also adjust compact finite difference method for first order IDE and a system of IDE and give numerical experiments for them. Our algorithm even can solve nonlinear IDE and unsplit kernel of IDE. The most advantages of compact finite difference method for IDE are that it obtains high order of accuracy, while the time complexity to solve the matrix equations after we use compact finite difference method on IDE is O(N), and it can solve very general case of IDE. [Copyright &y& Elsevier]