Hongtao Yang

Finite element methods for semilinear elliptic stochastic partial differential equations

We study finite element methods for semilinear stochastic partial differential equations. Error estimates are established. Numerical examples are also presented to examine our theoretical results.

Finite Element Error Estimates for a Nonlocal Problem in American Option Valuation

Based on new exact formulations of American option problems on bounded domains, error estimates are established for finite element approximations of American option prices under admissible regularity. Some numerical results are also presented.

A Front-Fixing Finite Element Method for the Valuation of American Options

A front-fixing finite element method is developed for the valuation of American options on stocks. Stability and solution nonnegativity are established under some appropriate assumptions. Numerical results are presented to examine our method and to compare it with the other methods.

Numerical pricing of American put options on zero-coupon bonds

In this paper we study finite volume methods and finite element methods for American put options on zerocoupon bonds. Stability and convergence are established for both methods. Numerical examples show that our methods converge and provide very accurate options prices and early exercise interest rates for all parameter combinations. We also present an error indicator by which one can examine the accuracy of the approximate option prices and early exercise interest rates actually obtained by a numerical method and determine how fine a grid should be used to achieve the desired accuracy.

A Numerical Analysis of American Options with Regime Switching

A finite element method and a simple lattice method are proposed for numerical valuation of American options under a regime switching model. Their stability estimates are established. Numerical results are presented to compare our methods and to examine their accuracy for various combinations of parameters. The dependency of early exercise prices and option prices on parameters are also investigated numerically.

A front-fixing finite element method for the valuation of American options with regime switching

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.

Numerical solution of diffraction problems by a least-squares finite element method

Consider the diffraction of a time-harmonic wave incident upon a periodic (grating) structure. Under certain assumptions, the diffraction problem may be modelled by a Helmholtz equation with transparent boundary conditions. In this paper, the diffraction problem is formulated as a first-order system of linear equations and solved by a least-squares finite element method. The method follows the general minus one norm approach of Bramble, Lazarov, and Pasciak. Our computational experiments indicate that the method is accurate with the optimal convergence property, and it is capable of dealing with complicated grating structures.

A Least-Squares Finite Element Analysis for Diffraction Problems

The diffraction of a time harmonic wave incident upon a grating (or periodic) structure is treated by a least-squares finite element method that incorporates the jump conditions at interfaces into the objective functional. Two fundamental polarizations are considered. Coercivity of the quadratic functional is established and optimal discretization error estimates are obtained in both cases. The theoretical results indicate that, for sufficiently smooth interfaces, the error estimates are superior to those of standard finite element methods.

Calibration of the Extended CIR Model

In this paper we shall prove that the calibration problem for the extended CIR model in [J. Hull and A. White, Rev. Financial Studies, 3 (1990), pp. 573-592] has a unique solution. The constructive proof leads to a numerical algorithm for computing the approximations of the time-dependent parameters and the zero-coupon bond prices. The results are also extended to multifactor CIR (Cox--Ingersoll--Ross) models. Numerical results are presented to examine the accuracy of our algorithm and to compare the extended CIR model with the Vasicek models.

Optimal convergence of discontinuous Galerkin methods for continuum modeling of supply chain networks

Abstract A discontinuous Galerkin method is considered to simulate materials flow in a supply chain network problem which is governed by a system of conservation laws. By means of a novel interpolation and superclose analysis technique, the optimal and superconvergence error estimates are established under two physically meaningful assumptions on the connectivity matrix. Numerical examples are presented to validate the theoretical results.