Abstract: The goal of option pricing theory is to help the investors
to manage their money, enhance returns and control their financial
future by theoretically valuing their options. However, most of the
option pricing models have no analytical solution. Furthermore,
not all the numerical methods are efficient to solve these models
because they have nonsmoothing payoffs or discontinuous derivatives
at the exercise price. In this paper, we solve the American option
under jump diffusion models by using efficient time-dependent
numerical methods. several techniques are integrated to reduced
the overcome the computational complexity. Fast Fourier Transform
(FFT) algorithm is used as a matrix-vector multiplication solver,
which reduces the complexity from O(M2) into O(M logM).
Partial fraction decomposition technique is applied to rational
approximation schemes to overcome the complexity of inverting
polynomial of matrices. The proposed method is easy to implement
on serial or parallel versions. Numerical results are presented to prove
the accuracy and efficiency of the proposed method.
Abstract: The goal of option pricing theory is to help the
investors to manage their money, enhance returns and control their
financial future by theoretically valuing their options. Modeling
option pricing by Black-School models with jumps guarantees to
consider the market movement. However, only numerical methods
can solve this model. Furthermore, not all the numerical methods
are efficient to solve these models because they have nonsmoothing
payoffs or discontinuous derivatives at the exercise price. In this
paper, the exponential time differencing (ETD) method is applied
for solving partial integrodifferential equations arising in pricing
European options under Merton’s and Kou’s jump-diffusion models.
Fast Fourier Transform (FFT) algorithm is used as a matrix-vector
multiplication solver, which reduces the complexity from O(M2)
into O(M logM). A partial fraction form of Pad`e schemes is used
to overcome the complexity of inverting polynomial of matrices.
These two tools guarantee to get efficient and accurate numerical
solutions. We construct a parallel and easy to implement a version
of the numerical scheme. Numerical experiments are given to show
how fast and accurate is our scheme.