Valuing American Put Options Using Chebyshev Polynomial Approximation

This paper suggests a simple valuation method based on Chebyshev approximation at Chebyshev nodes to value American put options. It is similar to the approach taken in Sullivan (2000), where the option`s continuation region function is estimated by using a Chebyshev polynomial. However, in contrast to Sullivan (2000), the functional is fitted by using Chebyshev nodes. The suggested method is flexible, easy to program and efficient, and can be extended to price other types of derivative instruments. It is also applicable in other fields, providing efficient solutions to complex systems of partial differential equations. The paper also describes an alternative method based on dynamic programming and backward induction to approximate the option value in each time period

Integration-Based Method as an Alternative Way to Estimate Parameters in the IV Bolus Compartment Model

An alternative method of integration-based parameter estimation applied in pharmacokinetics problems is proposed here. The method, introduced by Holder and Rodrigo, is used to estimate the rate of drug elimination and distribution when it enters the body via intravenous bolus. The estimation results are then compared with the classical method, the least squares method for the one-compartment model, and the residual method for the two-compartment model. Graphical simulations of drug concentration versus time are also performed in this article to view not only the dynamics of drug delivery in the body, but also the comparisons between the approximate solutions and the arbitrarily generated data points. Comparisons are also presented when the data points take into account noise in the form of random values. Based on the estimation and simulation results, the integration-based method gives good results and even better than the classical method although when noise is applied to the data points

Solution of the determinantal assignment problem using the Grassmann matrices

The paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation (Formula presented.) where (Formula presented.) is an n −dimensional vector space over (Formula presented.) which is an integral part of the solution of DAP. New criteria for existence of solution and their computation based on the properties of structured matrices are referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of (Formula presented.), and it is in turn characterised by the set of quadratic Plücker relations (QPRs) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector (Formula presented.) are given in terms of the rank properties of the Grassmann matrix, (Formula presented.) of the vector (Formula presented.), which is constructed by the coordinates of (Formula presented.). It is shown that the exterior equation is solvable ((Formula presented.) is decomposable), if and only if (Formula presented.) where (Formula presented.); the solution space for a decomposable (Formula presented.), is the space (Formula presented.). This provides an alternative linear algebra characterisation of the decomposability problem and of the Grassmann variety to that defined by the QPRs. Further properties of the Grassmann matrices are explored by defining the Hodge–Grassmann matrix as the dual of the Grassmann matrix. The connections of the Hodge–Grassmann matrix to the solution of exterior equations are examined, and an alternative new characterisation of decomposability is given in terms of the dimension of its image space. The framework based on the Grassmann matrices provides the means for the development of a new computational method for the solutions of the exact DAP (when such solutions exist), as well as computing approximate solutions, when exact solutions do not exist

Resolution of the COBE Earth sensor anomaly

Since its launch on November 18, 1989, the Earth sensors on the Cosmic Background Explorer (COBE) have shown much greater noise than expected. The problem was traced to an error in Earth horizon acquisition-of-signal (AOS) times. Due to this error, the AOS timing correction was ignored, causing Earth sensor split-to-index (SI) angles to be incorrectly time-tagged to minor frame synchronization times. Resulting Earth sensor residuals, based on gyro-propagated fine attitude solutions, were as large as plus or minus 0.45 deg (much greater than plus or minus 0.10 deg from scanner specifications (Reference 1)). Also, discontinuities in single-frame coarse attitude pitch and roll angles (as large as 0.80 and 0.30 deg, respectively) were noted several times during each orbit. However, over the course of the mission, each Earth sensor was observed to independently and unexpectedly reset and then reactivate into a new configuration. Although the telemetered AOS timing corrections are still in error, a procedure has been developed to approximate and apply these corrections. This paper describes the approach, analysis, and results of approximating and applying AOS timing adjustments to correct Earth scanner data. Furthermore, due to the continuing degradation of COBE's gyroscopes, gyro-propagated fine attitude solutions may soon become unavailable, requiring an alternative method for attitude determination. By correcting Earth scanner AOS telemetry, as described in this paper, more accurate single-frame attitude solutions are obtained. All aforementioned pitch and roll discontinuities are removed. When proper AOS corrections are applied, the standard deviation of pitch residuals between coarse attitude and gyro-propagated fine attitude solutions decrease by a factor of 3. Also, the overall standard deviation of SI residuals from fine attitude solutions decrease by a factor of 4 (meeting sensor specifications) when AOS corrections are applied

Haar wavelet method for solving generalized Burgers–Huxley equation

In this paper, an efficient numerical method for the solution of nonlinear partial differential equations based on the Haar wavelets approach is proposed, and tested in the case of generalized Burgers–Huxley equation. Approximate solutions of the generalized Burgers–Huxley equation are compared with exact solutions. The proposed scheme can be used in a wide class of nonlinear reaction–diffusion equations. These calculations demonstrate that the accuracy of the Haar wavelet solutions is quite high even in the case of a small number of grid points. The present method is a very reliable, simple, small computation costs, flexible, and convenient alternative method. © 201

A Novel Representation of the Exact Solution for Differential Algebraic Equations System Using Residual Power-Series Method

We implement a relatively new analytic iterative technique to get approximate solutions of differential algebraic equations system based on generalized Taylor series formula. The solution methodology is based on generating the residual power series expansion solution in the form of a rapidly convergent series with easily computable components. The residual power series method (RPSM) can be used as an alternative scheme to obtain analytical approximate solution of different types of differential algebraic equations system applied in mathematics. Simulations and test problems were analyzed to demonstrate the procedure and confirm the performance of the proposed method, as well as to show its potentiality, generality, viability, and simplicity. The results reveal that the proposed method is very effective, straightforward, and convenient for solving different forms of such systems