Abdul Q.M. Khaliq

NEW NUMERICAL SCHEME FOR PRICING AMERICAN OPTION WITH REGIME-SWITCHING

This paper is concerned with regime-switching American option pricing. We develop new numerical schemes by extending the penalty method approach and by employing the θ-method. With regime-switching, American option prices satisfy a system of m free boundary value problems, where m is the number of regimes considered for the market. An (optimal) early exercise boundary is associated with each regime. Straightforward implementation of the θ-method would result in a system of nonlinear equations requiring a time-consuming iterative procedure at each time step. To avoid such complications, we implement an implicit approach by explicitly treating the nonlinear terms and/or the linear terms from other regimes, resulting in computationally efficient algorithms. We establish an upper bound condition for the time step size and prove that under the condition the implicit schemes satisfy a discrete version of the positivity constraint for American option values. We compare the implicit schemes with a tree model that generalizes the Cox-Ross-Rubinstein (CRR) binomial tree model, and with an analytical approximation solution for two-regime case due to Buffington and Elliott. Numerical examples demonstrate the accuracy and stability of the new implicit schemes.

High-order compact scheme for solving nonlinear Black-Scholes equation with transaction cost

In this paper, an unconditionally stable high-order compact finite difference scheme is proposed. The compact scheme is fourth-order accurate in both the temporal and spatial dimensions. The new method computes both the option price and the hedging delta ∂ V/∂ S simultaneously. Two numerical examples are presented to demonstrate the accuracy and efficiency of the proposed scheme.

The locally extrapolated exponential time differencing LOD scheme for multidimensional reaction-diffusion systems

This paper introduces the local extrapolation of first order locally one-dimensional exponential time differencing scheme for numerical solution of multidimensional nonlinear reaction-diffusion systems. This novel scheme has the benefit of solving multidimensional problems in locally one dimensional fashion by implementing sequences of tridiagonal matrix solvers instead of solving a banded system. The storage size needed for solving systems in higher dimensions with this scheme is similar to that needed for one spatial dimension systems. The stability, monotonicity, and convergence of the locally extrapolated exponential time differencing scheme have been examined. Stability analysis shows that the scheme is strongly stable ( L -stable) and is particularly beneficial to nonlinear partial differential equations with irregular initial data or discontinuity involving initial and boundary conditions due to its ability to damp spurious oscillations caused by high frequency components in the solution. The performance of the novel scheme has been investigated by testing it on a two-dimensional Schnakenberg model, two and three-dimensional Brusselator models, and a three-dimensional enzyme kinetics of Michaelis-Menten type reaction-diffusion model. Numerical experiments demonstrate the efficiency, accuracy, and reliability of the scheme. Locally extrapolated LOD exponential time differencing scheme is developed.Multidimensional nonlinear reaction-diffusion systems are considered.Monotonicity and stability are analyzed.The order of convergence is examined.

Higher order exponential time differencing scheme for system of coupled nonlinear Schrödinger equations

The coupled nonlinear Schrodinger equations are highly used in modeling the various phenomena in nonlinear fiber optics, like propagation of pulses. Efficient and reliable numerical schemes are required for analysis of these models and for improvement of the fiber communication system. In this paper, we introduce a new version of the Cox and Matthews third order exponential time differencing Runge-Kutta (ETD3RK) scheme based on the (1,2)-Pade approximation to the exponential function. In addition, we present its local extrapolation form to improve temporal accuracy to the fourth order. The developed scheme and its extrapolation are seen to be strongly stable, which have ability to damp spurious oscillations caused by high frequency components in the solution. A computationally efficient algorithm of the new scheme, based on a partial fraction splitting technique is presented. In order to investigate the performance of the novel scheme we considered the system of two and four coupled nonlinear Schrodinger equations and performed several numerical experiments on them. The numerical experiments showed that the developed numerical scheme provide an efficient and reliable way for computing long-range solitary solutions given by coupled nonlinear Schrodinger equations and conserved the conserved quantities mass and energy exactly, to at least five decimal places.

A parallel time stepping approach using meshfree approximations for pricing options with non-smooth payoffs

In this paper we consider a meshfree radial basis function approach for the valuation of pricing options with non-smooth payoffs. By taking advantage of parallel architecture, a strongly stable and highly accurate time stepping method is developed with computational complexity comparable to the implicit Euler method implemented concurrently on each processor. This, in collusion with the radial basis function approach, provides an efficient and reliable valuation of exotic options, such as American digital options.

A linearly implicit predictor-corrector method for reaction-diffusion equations

Abstract A novel linearly implicit predictor-corrector scheme is developed for the numerical solution of reaction-diffusion equations. Iterative processes are avoided by treating the nonlinear reaction terms explicitly, while maintaining superior accuracy and stability properties compared to the well-known θ methods and linearly implicit Runge-Kutta methods. The proposed method allows the opportunity of solving large systems of reaction-diffusion equations by alleviating the necessity of solving the accompanying large linear systems of algebraic equations due to the natural parallelism which surfaces across the system. Numerical results confirm the enhanced stability, accuracy and efficiency of the method when applied to reaction-diffusion equations arising in biochemistry and population ecology.

Local RBF method for multi-dimensional partial differential equations

Abstract In this paper, a local meshless differential quadrature collocation method is utilized to solve multi-dimensional reaction–convection–diffusion PDEs numerically. In some cases, global version of the meshless method is considered as well. The meshless methods approximate solution on scattered and uniform nodes in both local and global sense. In the case of convection-dominated PDEs, the local meshless method is coupled with an upwind technique to avoid spurious oscillations. For this purpose, a physically motivated local domain is utilized in the flow direction. Both regular and irregular geometries are taken into consideration. Numerical experiments are performed to demonstrate effective applications and accuracy of the meshless method on regular and irregular domains.

A real distinct poles Exponential Time Differencing scheme for reaction-diffusion systems

A second order Exponential Time Differencing (ETD) method for reaction-diffusion systems which uses a real distinct poles discretization method for the underlying matrix exponentials is developed. The method is established to be stable and second order convergent. It is demonstrated to be robust for problems involving non-smooth initial and boundary conditions and steep solution gradients. We discuss several advantages over competing second order ETD schemes.

Split-step Milstein methods for multi-channel stiff stochastic differential systems

Abstract We consider split-step Milstein methods for the solution of stiff stochastic differential equations with an emphasis on systems driven by multi-channel noise. We show their strong order of convergence and investigate mean-square stability properties for different noise and drift structures. The stability matrices are established in a form convenient for analyzing their impact arising from different deterministic drift integrators. Numerical examples are provided to illustrate the effectiveness and reliability of these methods.

Trapezoidal scheme for time–space fractional diffusion equation with Riesz derivative

Abstract In this paper, a finite difference scheme is proposed to solve time–space fractional diffusion equation which has second-order accuracy in both time and space direction. The time and space fractional derivatives are considered in the senses of Caputo and Riesz, respectively. First, the centered difference approach is used to approximate the Riesz fractional derivative in space. Then, the obtained fractional ordinary differential equations are transformed into equivalent Volterra integral equations. And then, the trapezoidal rule is utilized to approximate the Volterra integral equations. The stability and convergence of our scheme are proved via mathematical induction method. Finally, numerical experiments are performed to confirm the high accuracy and efficiency of our scheme.