Muhammad Yousuf

A new family of numerical schemes for inhomogeneous parabolic partial dieren tial equations is de... more A new family of numerical schemes for inhomogeneous parabolic partial dieren tial equations is developed. The new family of algorithms utilizes diagonal Pad e schemes combined with positivity{preserving Pad e schemes as damping devices. We also utilize partial fraction decomposition to address problems with accuracy and computational eciency in solving the higher order methods and to implement the algorithms in

Smoothing with positivity-preserving Pad� schemes for parabolic problems with nonsmooth data

Numerical Methods for Partial Differential Equations, 2005

ABSTRACT We introduce a new class of higher order numerical schemes for parabolic partial differe... more ABSTRACT We introduce a new class of higher order numerical schemes for parabolic partial differential equations that are more robust than the well-known Rannacher schemes. The new family of algorithms utilizes diagonal Padé schemes combined with positivity-preserving Padé schemes instead of first subdiagonal Padé schemes. We utilize a partial fraction decomposition to address problems with accuracy and computational efficiency in solving the higher order methods and to implement the algorithms in parallel. Optimal order convergence for nonsmooth data is proved for the case of a self-adjoint operator in Hilbert space as well as in Banach space for the general case. Numerical experiments support the theorems, including examples in pricing options with nonsmooth payoff in financial mathematics. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

An efficient ETD method for pricing American options under stochastic volatility with nonsmooth payoffs

Numerical Methods for Partial Differential Equations, 2013

ABSTRACT Based on Cox and Matthews Exponential Time Differencing (ETD) approach, a fourth–order s... more ABSTRACT Based on Cox and Matthews Exponential Time Differencing (ETD) approach, a fourth–order strongly–stable method having real distinct poles is developed and applied to solve American options under stochastic volatility with nonsmooth payoffs. A computationally efficient version of the method is constructed using partial fraction splitting technique. This approach requires to solve several backward Euler-type linear systems at each time step. Numerical experiments are presented to demonstrate the computational efficiency, accuracy, and reliability of the method.

Smoothing schemes for reaction-diffusion systems with nonsmooth data

Journal of Computational and Applied Mathematics, 2009

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On smoothing of the Crank–Nicolson scheme and higher order schemes for pricing barrier options

Journal of Computational and Applied Mathematics, 2007

Most option pricing problems have nonsmooth payoffs or discontinuous derivatives at the exercise ... more Most option pricing problems have nonsmooth payoffs or discontinuous derivatives at the exercise price. Discrete barrier options have not only nonsmooth payoffs but also time dependent discontinuities. In pricing barrier options, certain aspects are triggered if the asset price becomes too high or too low. Standard smoothing schemes used to solve problems with nonsmooth payoff do not work well for

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

International Journal of Computer Mathematics, 2012

In this paper, a new second-order exponential time differencing (ETD) method based on the Cox and... more 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