M. Yousuf

Power Line Communications: An Overview - Part I

We give an overview of the power line communications (PLC) technology, its importance, its standards and an overview of the HomePlug standards associated with it. This is done is two parts due to publication constraints. In this part, we will concentrate on the PLC applications and the technical issues regarding it. We will also see the layers and methods that are needed to make it work.

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.

Power Line Communications: An Overview - Part II

Following up on our previous work, power line communications: An Overview - Part I, we discuss the standards involved in PLC technology and name the established standards. We then concentrate on the HomePlug standard in detail. Our prime focus is on these standards: HomePlug 1.0 and HomePlug AV as these have become the basis for the worlds most widely deployed PLC systems. An overview of the HomePlug standards is presented which discusses the technology in the Physical and MAC layers. It also discusses the QoS issues and compares the performance of HomePlug AV with HomePlug 1.0.

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.

Efficient L-stable method for parabolic problems with application to pricing American options under stochastic volatility

Efficient L-stable numerical method for semilinear parabolic problems with nonsmooth initial data is proposed and implemented to solve Hestons stochastic volatility model based PDE for pricing American options under stochastic volatility. The proposed new method is also used to solve two asset American options pricing problem. Cox and Matthews [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, Journal of Computational Physics 176 (2002) 430-455] developed a class of exponential time differencing Runge-Kutta schemes (ETDRK) for nonlinear parabolic problems. Kassam and Trefethen [A.K. Kassam, L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM Journal on Scientific Computing 26 (4) (2005) 1214-1233] showed that while computing certain functions involved in the Cox-Matthews schemes, severe cancelation errors can occur which affect the accuracy and stability of the schemes. Kassam and Trefethen proposed complex contour integration technique to implement these schemes in a way that avoids these cancelation errors. But this approach creates new difficulties in choosing and evaluating the contour integrals for larger problems. We modify the ETDRK schemes using positivity preserving Pade approximations of the matrix exponential functions and construct computationally efficient parallel version using splitting technique. As a result of this approach it is required only to solve several backward Euler linear problems in serial or parallel.

On the class of high order time stepping schemes based on Padé approximations for the numerical solution of Burgers’ equation

Abstract Numerical solution of Burgers’ equation is presented using finite difference methods in space and positivity preserving Pade approximations in time. A class of high order time stepping schemes is introduced. For practical purposes, first, second, third, and fourth order schemes are constructed. Efficient parallel versions of these schemes are given using a splitting technique of rational functions. Accuracy of the schemes is demonstrated by solving a test problem and comparing numerical results with the exact solution. Time evolution graphs show the physical phenomenon of the problem. Convergence tables are given to verify the theoretical order of convergence.

A fourth-order smoothing scheme for pricing barrier options under stochastic volatility

A discontinuity is introduced each time a barrier is applied, and standard smoothing schemes do not work well when used to solve problems with non-smooth payoff. An improved smoothing strategy was introduced in [M. Yousuf, Efficient Smoothing of Crank–Nicolson Method for Pricing Barrier Options Under Stochastic Volatility, Published Online: Jul 10, 2008, DOI: 10.1002/pamm.200700249, 1081101–1081102.] for smoothing A-stable Cranck–Nicolson scheme at each time a barrier is applied. In the present work, we are extending the results to develop a fourth order smoothing scheme for pricing barrier options under stochastic volatility. Partial differential equation approach is utilized for the valuation of complex option pricing models under stochastic volatility which brings major mathematical and computational challenges for estimation of stability of the estimates. Efficient parallel version of the scheme is constructed using splitting technique. Convergence table and graphs are given to demonstrate the performance of the smoothing scheme.

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.

Effects of parameter values and noise on PSO-based predictive control: An empirical study

In this paper, a Particle Swarm Optimization (PSO) based Model Predictive Control (MPC) scheme is studied through a variety of tests to better understand its behavior and characteristics. The technique has already been presented in the literature. Here, the PSO and MPC parameters are varied to study the effects on the quality of control and system dynamics. Model mismatch and noise are also introduced to test the controller performance. The results from various tests are compared and conclusions are drawn.

High-order time stepping scheme for pricing American option under Bates model

ABSTRACT Pricing of European and American options under Bates model give rise to a partial integro-differential equation. In this paper a strongly stable fourth-order implicit predictor–corrector time stepping method based on exponential time differencing) is proposed for solving such problems. We provide stability, and convergence of the proposed method, and study the impact of the jump intensity, penalty and other parameters on convergence and solution accuracy. The American option constraint is enforced by using a penalty method. Spatial derivatives are approximated using second-order finite central differences which leads to block tridiagonal systems. The integral term is evaluated using simple quadrature where the non-locality of the jump term in such models leads to dense matrix. We treat the approximated integral term and nonlinear penalty term explicitly in time. Numerical experiments are demonstrated by discussing the efficiency, accuracy and reliability of the proposed method.