A robust upwind difference scheme for pricing perpetual American put options under stochastic volatility

In this paper, we present an upwind difference scheme for the valuation of perpetual American put options, using Heston's stochastic volatility model. The matrix associated with the discrete operator is an M-matrix, which ensure that the scheme is stable. We apply the maximum principle to the discrete linear complementarity problem in two mesh sets and derive the error estimates. Numerical results support the theoretical results.     

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.     

Front-Tracking Finite Difference Methods for the Valuation of American Options

This paper is concerned with the numerical solution of the American option valuation problem formulated as a parabolic free boundary/initial value model. We introduce and analyze a front-tracking finite difference method and compare it with other commonly used techniques. The numerical experiments performed indicate that the front-tracking method considered is an efficient alternative for approximating simultaneously the option value and free boundary functions associated with the valuation problem.     

High-accuracy finite-difference methods for the valuation of options

A finite-difference scheme often employed for the valuation of options from the Black–Scholes equation is the Crank–Nicolson (CN) scheme. The CN scheme is second order in both time and asset. For a rapid valuation with a reasonable resolution of the option price curve, it requires extremely small steps in both time and asset. In this paper, we present high-accuracy finite-difference methods for the Black–Scholes equation in which we employ the fourth-order L-stable Simpson-type (LSIMP) time integration schemes developed earlier and the well-known Numerov method for discretization in the asset direction. The resulting schemes, called LSIMP–NUM, are fourth order in both time and asset. The LSIMP–NUM schemes obtained can provide a rapid, stable and accurate resolution of option prices, allowing for relatively large steps in both time and asset. We compare the computational efficiency of the LSIMP–NUM schemes with the CN and Douglas schemes by considering valuation of European options and American options via the linear complementarity approach.     

A predictor-corrector scheme based on the ADI method for pricing American puts with stochastic volatility

In this paper, we introduce a new numerical scheme, based on the ADI (alternating direction implicit) method, to price American put options with a stochastic volatility model. Upon applying a front-fixing transformation to transform the unknown free boundary into a known and fixed boundary in the transformed space, a predictor-corrector finite difference scheme is then developed to solve for the optimal exercise price and the option values simultaneously. Based on the local von Neumann stability analysis, a stability requirement is theoretically obtained first and then tested numerically. It is shown that the instability introduced by the predictor can be damped, to some extent, by the ADI method that is used in the corrector. The results of various numerical experiments show that this new approach is fast and accurate, and can be easily extended to other types of financial derivatives with an American-style exercise.Another key contribution of this paper is the proposition of a set of appropriate boundary conditions, particularly in the volatility direction, upon realizing that appropriate boundary conditions in the volatility direction for stochastic volatility models appear to be controversial in the literature. A sound justification is also provided for the proposed boundary conditions mathematically as well as financially.     

Moving boundary transformation for American call options with transaction cost: finite difference methods and computing

The pricing of American call option with transaction cost is a free boundary problem. Using a new transformation method the boundary is made to follow a certain known trajectory in time. The new transformed problem is solved by various finite difference methods, such as explicit and implicit schemes. Broyden's and Schubert's methods are applied as a modification to Newton's method in the case of nonlinearity in the equation. An alternating direction explicit method with second-order accuracy in time is used as an example in this paper to demonstrate the technique. Numerical results demonstrate the efficiency and the rate of convergence of the methods.     

GPU acceleration of the stochastic grid bundling method for early-exercise options

In this work, a parallel graphics processing units (GPU) version of the Monte Carlo stochastic grid bundling method (SGBM) for pricing multi-dimensional early-exercise options is presented. To extend the method's applicability, the problem dimensions and the number of bundles will be increased drastically. This makes SGBM very expensive in terms of computational costs on conventional hardware systems based on central processing units. A parallelization strategy of the method is developed and the general purpose computing on graphics processing units paradigm is used to reduce the execution time. An improved technique for bundling asset paths, which is more efficient on parallel hardware is introduced. Thanks to the performance of the GPU version of SGBM, a general approach for computing the early-exercise policy is proposed. Comparisons between sequential and GPU parallel versions are presented.     

Convergence and stability of the balanced methods for stochastic differential equations with jumps

This paper deals with the balanced methods which are implicit methods for stochastic differential equations with Poisson-driven jumps. It is shown that the balanced methods give a strong convergence rate of at least 1/2 and can preserve the linear mean-square stability with the sufficiently small stepsize. Weak variants are also considered and their mean-square stability analysed. Some numerical experiments are given to demonstrate the conclusions.     

Study of a stochastic model of the“dangling spider” problem with an infinite previous history and poisson switchings

To study the stability of the stochastic“dangling spider” model, the second Lyapunov method is substantiated for stochastic functional differential equations with the entire previous history. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 79–105, July–August, 2000.     

A new superconvergent method for systems of nonlinear singular boundary value problems

A new superconvergent method based on a sextic spline is described and analysed for the solution of systems of nonlinear singular two-point boundary value problems (BVPs). It is well known that the optimal orders of convergence could not be achieved using standard formulation of a sextic spline for the solution of BVPs. Based on the method used in our earlier research papers [J. Rashidinia and M. Ghasemi, B-spline collocation for solution of two-point boundary value problems, J. Comput. Appl. Math. 235 (2011), pp. 2325–2342; J. Rashidinia, M. Ghasemi, and R. Jalilian, An o(h ⁶) numerical solution of general nonlinear fifth-order two point boundary value problems, Numer. Algorithms 55(4) (2010), pp. 403–428], we construct a new O(h ⁸) locally superconvergent method for the solution of general nonlinear two-point BVPs up to order 6. The error bounds and the convergence properties of the method have been proved theoretically. Then, the method is extended to solve the system of nonlinear two-point BVPs. Some test problems are given to demonstrate the applicability and the superconvergent properties of the proposed method numerically. It is shown that the method is very efficient and applicable for stiff BVPs too.     

Finite-element methods for analysis of the dynamics and control of Czochralski crystal growth

Numerical methods are presented for solution of the complex moving-boundary problem described by a thermal-capillary model for Czochralski crystal growth, which accounts for conduction through melt, crystal, and crucible and radiation between diffuse-gray body surfaces. Transients are included that are caused by energy transport, by changes in the shapes of the melt-crystal, melt-ambient phase boundaries and the moving crystal, and by the batchwise decrease of the melt volume in the crucible. Finite-element discretizations are used to approximate the moving boundaries and the energy equation in each phase. A two-level, implicit integration algorithm is presented for transient calculations. The temperature fields and moving boundaries are advanced in time by a trapezoid rule approximation with modified Newton's iterations to solve algebraic systems for effective ambient temperatures computed with diffuse-gray radiation. The implicit coupling between radiative exchange, interface shapes, and the temperature field is necessary for preserving the second-order accuracy of the integration method and is achieved by successive iterations between the radiation calculation and solution of the thermal capillary model. Analysis of a quasi-steady-state model (QSSM) demonstrates the inherent stability of the CZ process. Including either diffuse-gray radiation among crystal, melt, and crucible or a simple controller for maintaining constant radius can lead to oscillations in the crystal radius. The effects of these oscillations on batchwise crystal growth are addressed.     

An efficient numerical technique for the solution of nonlinear singular boundary value problems

In this work, a new technique based on Green’s function and the Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems (SBVPs) is proposed. The technique relies on constructing Green’s function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on the ADM, the proposed technique avoids solving a sequence of transcendental equations for the undetermined coefficients. It approximates the solution in the form of a series with easily computable components. Additionally, the convergence analysis and the error estimate of the proposed method are supplemented. The reliability and efficiency of the proposed method are demonstrated by several numerical examples. The numerical results reveal that the proposed method is very efficient and accurate.     

Comparative analysis of methods for determining the clothing surface temperature (tcl) in order to provide a balance between man and the environment

The PMV (Predicted Mean Vote) is an index which shows the thermal sensation of a large group of people exposed to the same environment. However, discrepancies are found between the PMV model and thermal sensation responses obtained in field studies. One of the components for the calculation of PMV is clothing surface temperature (t_cl), which can be a factor which contributes towards these discrepancies. Therefore, the aim of this article was to show t_cl influence on the PMV index. For this, the calculation of t_cl was done in two ways: the first was through the algorithm presented in the Annex D of ISO 7730 (2005) (Algorithm 1) and the second way was through Newton's Method (Algorithm 2), on a group of welders of a metal-mechanic industry. After calculating t_cl in these two ways, (algorithms 1 and 2), the two values obtained for t_cl were compared and the results were not the same. Consequently, different values for t_cl generate different results for the PMV. Newton's Method, having quadratic convergence, is more precise for the calculation; therefore, the suggestion is to use this method to determine t_cl since this is a variable with direct influence in determining the PMV.Relevancy to industryWhenever it is possible to determine the PMV and when this result is closer to people's responses to thermal sensation, one can provide a more appropriate environment to users in order to have Thermal Comfort and, principally, to avoid Thermal Stress.     

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.     

A new method for computing a solution of the Cauchy problem with polynomial data for the system of crystal optics

A new analytical method for solving an initial value problem (IVP) for the system of crystal optics with polynomial data and a polynomial inhomogeneous term is suggested. The found solution of the IVP is a polynomial. Theoretical and computational analysis of polynomial solutions and their comparison with non-polynomial solutions corresponding to smooth data are given. The applicability of polynomial solutions to physical processes is discussed. An implementation of this method has been made by symbolic computations in Maple 10.     

Extended version with the analysis of dynamic system for iterative refinement of solution

The extended version with the analysis of dynamic system for Wilkinson's iteration improvement of solution is presented in this paper. It turns out that the iteration improvement can be viewed as applying explicit Euler method with step size h=1 to a dynamic system which has a unique globally asymptotically stable equilibrium point, that is, the solution x*=A ^?1 b of linear system Ax=b with non-singular matrix A. As a result, an extended iterative improvement process for solving ill-conditioned linear system of algebraic equations with non-singular coefficients matrix is proposed by following the solution curve of a linear system of ordinary differential equations. We prove the unconditional convergence and derive the roundoff results for the extended iterative refinement process. Several numerical experiments are given to show the effectiveness and competition of the extended iteration refinement in comparison with Wilkinson's.