A. Q. M. Khaliq

The numerical approximation of nonlinear Black–Scholes model for exotic path-dependent American options with transaction cost

In this paper, a new second-order exponential time differencing (ETD) method based on the Cox and Matthews approach is developed and applied for pricing American options with transaction cost. The method is seen to be strongly stable and highly efficient for solving the nonlinear Black–Scholes model. Furthermore, it does not incur unwanted oscillations unlike the ETD–Crank–Nicolson method for exotic path-dependent American options. The computational efficiency and reliability of the new method are demonstrated by numerical examples and by comparing it with the existing methods.

Stabilized explicit Runge-Kutta methods for multi-asset American options

American derivatives have become very popular instruments in financial markets. However, they are more complicated to price than European options since at each time level we have to determine not only the option value but also whether or not it should be exercised. Several procedures have been proposed to dissolve these difficulties, but they usually involve the solution of nonlinear partial differential equations (PDEs). In the case of multi-dimensional problems, solving these equations is a very challenging task.In this paper we propose Stabilized Explicit Runge-Kutta (SERK) methods to solve this kind of problems. They can easily be applied to many different classes of problems with large dimensions and they have low memory demand. Since these methods are explicit, they do not require algebra routines to solve large nonlinear systems associated to ODEs (as, for example, LAPACK and BLAS packages or multigrid or iterative methods applied together with Newton-type algorithms) and are especially well-suited for the method of lines (MOL) discretizations of parabolic PDEs.

Pricing American options under multi-state regime switching with an efficient L-stable method

An efficient second-order method based on exponential time differencing approach for solving American options under multi-state regime switching is developed and analysed for stability and convergence. The method is seen to be strongly stable (L-stable) in each regime. The implicit predictor–corrector nature of the method makes it highly efficient in solving nonlinear systems of partial differential equations arising from multi-state regime switching model. Stability and convergence of the method are examined. The impact of regime switching on option prices for different jump rates and volatility is illustrated. A general framework for multi-state regime switching in multi-asset American option has been provided. Numerical experiments are performed on one and two assets to demonstrate the performance of the method with convex as well as non-convex payoffs. The method is compared with some of the existing methods available in the literature and is found to be reliable, accurate and efficient.

Fourth-order compact schemes for the numerical simulation of coupled Burgers’ equation☆

Abstract This paper introduces two new modified fourth-order exponential time differencing Runge–Kutta (ETDRK) schemes in combination with a global fourth-order compact finite difference scheme (in space) for direct integration of nonlinear coupled viscous Burgers’ equations in their original form without using any transformations or linearization techniques. One scheme is a modification of the Cox and Matthews ETDRK4 scheme based on ( 1 , 3 ) -Pade approximation and other is a modification of Krogstad’s ETDRK4-B scheme based on ( 2 , 2 ) -Pade approximation. Efficient versions of the proposed schemes are obtained by using a partial fraction splitting technique of rational functions. The stability properties of the proposed schemes are studied by plotting the stability regions, which provide an explanation of their behavior for dispersive and dissipative problems. The order of convergence of the schemes is examined empirically and found that the modification of ETDRK4 converges with the expected rate even if the initial data are nonsmooth. On the other hand, modification of ETDRK4-B suffers with order reduction if the initial data are nonsmooth. Several numerical experiments are carried out in order to demonstrate the performance and adaptability of the proposed schemes. The numerical results indicate that the proposed schemes provide better accuracy than other schemes available in the literature. Moreover, the results show that the modification of ETDRK4 is reliable and yields more accurate results than modification of ETDRK4-B, while solving problems with nonsmooth data or with high Reynolds number.

A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations

A fourth-order implicit-explicit time-discretization scheme based on the exponential time differencing approach with a fourth-order compact scheme in space is proposed for space fractional nonlinear Schrödinger equations. The stability and convergence of the compact scheme are discussed. It is shown that the compact scheme is fourth-order convergent in space and in time. Numerical experiments are performed on single and coupled systems of two and four fractional nonlinear Schrödinger equations. The results demonstrate accuracy, efficiency, and reliability of the scheme. A linearly implicit conservative method with the fourth-order compact scheme in space is also considered and used on the system of space fractional nonlinear Schrödinger equations.

Split-step Adams–Moulton Milstein methods for systems of stiff stochastic differential equations

In this paper we discuss new split-step methods for solving systems of Itô stochastic differential equations (SDEs). The methods are based on a L-stable (strongly stable) second-order split Adams–Moulton Formula for stiff ordinary differential equations in collusion with Milstein methods for use on SDEs which are stiff in both the deterministic and stochastic components. The L-stability property is particularly useful when the drift components are stiff and contain widely varying decay constants. For SDEs wherein the diffusion is especially stiff, we consider balanced and modified balanced split-step methods which posses larger regions of mean-square stability. Strong order convergence one is established and stability regions are displayed. The methods are tested on problems with one and two noise channels. Numerical results show the effectiveness of the methods in the pathwise approximation of stiff SDEs when compared to some existing split-step methods.

The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations

An efficient local extrapolation of the exponential operator splitting scheme is introduced to solve the multi-dimensional space-fractional nonlinear Schrödinger equations. Stability of the scheme is examined by investigating its amplification factor and by plotting the boundaries of the stability regions. Empirical convergence analysis and calculation of the local truncation error exhibit the second-order accuracy of the proposed scheme. The performance and reliability of the proposed scheme are tested by implementing it on two- and three-dimensional space-fractional nonlinear Schrödinger equations including the space-fractional Gross-Pitaevskii equation, which is used to model optical solitons in graded-index fibers.

Exponential time differencing Crank–Nicolson method with a quartic spline approximation for nonlinear Schrödinger equations

Abstract This paper studies a central difference and quartic spline approximation based exponential time differencing Crank – Nicolson (ETD-CN) method for solving systems of one-dimensional nonlinear Schrodinger equations and two-dimensional nonlinear Schrodinger equations. A local extrapolation is employed to achieve a fourth order accuracy in time. The numerical method is proven to be highly efficient and stable for long-range soliton computations. Numerical examples associated with Dirichlet, Neumann and periodic boundary conditions are provided to illustrate the accuracy, efficiency and stability of the method proposed.

On the monotonicity of an adaptive splitting scheme for two-dimensional singular reaction-diffusion equations

This paper studies a fully adaptive splitting method for the numerical solution of two-dimensional quenching differential equations. The non-uniform adaptive mesh is established in space and time via standard arc-length formulations. We concentrate on the monotonic property of the numerical algorithm developed since the issue is fundamental for this type of nonlinear singular differential equations. Numerical examples are given to illustrate our computational results as well as the quenching phenomena approximated.

Fourth-order methods for space fractional reaction–diffusion equations with non-smooth data

ABSTRACT We propose two fourth-order methods in time for one-dimensional space fractional reaction–diffusion equations. The methods are based on fourth-order Exponential Time Differencing Runge–Kutta method. Padé approximations of matrix exponential functions are used to construct an L-stable and an A-stable method. Partial fraction splitting technique is applied to construct more reliable and computationally efficient versions of the methods. Solution profiles as well as convergence rates in time are presented for fractional enzyme kinetics equation and fractional Fisher equation. The L-stable method performs well in the presence of non-smooth mismatched initial-boundary data while the A-stable method is more economical for smooth matched initial-boundary data.