A robust finite difference scheme for pricing American put options with Singularity-Separating method

Abstract

In this paper we present a stable numerical method for the linear complementary problem arising from American put option pricing. The numerical method is based on a hybrid finite difference spatial discretization on a piecewise uniform mesh and an implicit time stepping technique. The scheme is stable for arbitrary volatility and arbitrary interest rate. We apply some tricks to derive the error estimates for the direct application of finite difference method to the linear complementary problem. We use the Singularity-Separating method to remove the singularity of the non-smooth payoff function. It is proved that the scheme is second-order convergent with respect to the spatial variable. Numerical results support the theoretical results.

American option pricing problem transformed on finite interval

“…Further, we compare the presented l 1 FVM with a more straightforward approach to the original penalized problem on the truncated domain [0, 4E] by centred-space discretization of the first derivative away from the degeneration and one-sided discretization in the degeneration region to ensure the M-matrix property [5] on uniform grid. Again, the l 1 penalization and fully implicit time discretization are used as for the Crank-Nicolson time stepping one may observe oscillations when using centred-space discretization.…”

“…The known function σ (t) characterizes the volatility of the asset, r(t) is the interest rate, d(S, t) is the dividend function and V * is the payoff, given by V (S, T ) = V * (S) = max(S − E, 0) for call option (V * (S) = max(E − S, 0) for put option). Various numerical methods are developed for the LCP (1) [4][5][6]16,18]. Introduced in computational finance by Zvan et al [25], the penalty method is regarded as solid alternative to these popular algorithms.…”

An Alternating-Direction Implicit Difference Scheme for Pricing Asian Options

“…Here, we apply the central difference scheme on a piecewise uniform mesh [16,17] to discrete problem (13) and apply the implicit Euler scheme to discrete problem (14). The matrices associated with discrete operators are M-matrices, which ensure that the scheme is stable for arbitrary volatility and arbitrary interest rate without any extra conditions.…”

“…Furthermore, the uniform mesh on the transformed interval will lead to the originally grid points concentrating around = 0 inappropriately. We have proposed robust difference schemes for the BlackScholes equation, which is based on a central difference spatial discretization on a piecewise uniform mesh and an implicit time stepping technique [16,17].…”

Laplace Transform Homotopy Perturbation Method for the Two Dimensional Black Scholes Model with European Call Option

“…Since the option pricing in a market is dependent on other markets, the multidimensional Black Scholes equation is more efficient than the one dimensional version. There are various methods to find the solution of multidimensional Black Scholes model; for example, a radical basic function (RBF) method [2][3][4][5][6], the Mellin transform method [7], finite different method [8][9][10][11][12], a collection method with the quantic B-spline function [13], and homotopy perturbation method (HPM) [14,15].…”