Search

This paper deals with the efficient valuation of American options. We adopt Heston's approach for a model of stochastic volatility, leading to a generalized Black–Scholes equation called Heston's equation. Together with appropriate boundary conditions, this can be formulated as a parabolic boundary value problem with a free boundary, the optimal exercise price of the option. For its efficient numerical solution, we employ, among other multiscale methods, a monotone multigrid method based on linear finite elements in space and display corresponding numerical experiments. [ABSTRACT FROM PUBLISHER]

American option problems under regime-switching model are considered in this paper. The conjectures in [H. Yang, A numerical analysis of American options with regime switching, J. Sci. Comput. 44 (2010), pp. 69–91] about the position of early exercise prices are proved, which generalize the results in [F. Yi, American put option with regime-switching volatility (finite time horizon) – Variational inequality approach, Math. Methods. Appl. Sci. 31 (2008), pp. 1461–1477] by allowing the interest rates to be different in two states. A front-fixing finite element method for the free boundary problems is proposed and implemented. Its stability is established under reasonable assumptions. Numerical results are given to examine the rate of convergence of our method and compare it with the usual finite element method. [ABSTRACT FROM PUBLISHER]

In this paper, we present a highly efficient approach for numerically solving the Black–Scholes equation in order to price European and American basket options. Therefore, hardware features of contemporary high performance computer architectures such as non-uniform memory access and hardware-threading are exploited by a hybrid parallelization using MPI and OpenMP which is able to drastically reduce the computing time. In this way, we achieve very good speed-ups and are able to price baskets with up to six underlyings. Our approach is based on a sparse grid discretization with finite elements and makes use of a sophisticated adaption. The resulting linear system is solved by a conjugate gradient method that uses a parallel operator for applying the system matrix implicitly. Since we exploit all levels of the operator's parallelism, we are able to benefit from the compute power of more than 100 cores. Several numerical examples as well as an analysis of the performance for different computer architectures are provided. [ABSTRACT FROM PUBLISHER]