A sliced-finite difference method for the American option

It is generally accepted that the Black-Scholes model for American option pricing is a free-boundary problem involving a partial differential equation. Approximate solutions may be obtained using numerical methods, but the precision and stability of these solutions is hard to control since they are significantly affected by the singularity at the exercise boundary near the expiration date. We propose a Sliced-Finite Difference (SFD) numerical method, to solve the pricing problem of American put options. The proposed method combines the advantages of the semi-analytical method and the sliced-fixed boundary finite difference method and overcomes their deficiencies. Using the SFD method, we can resolve the problem that results from the singularity near the optimal exercise boundary. Numerical experiments show that the SFD method is more accurate than other numerical methods. The proposed method is targeted at American put options but it is also applicable to other types of options.     

分数布朗运动环境下亚式期权定价的新方法

本文考虑分数布朗运动环境下几何平均型亚式期权定价问题,给出了一种基于可靠性数学思想的期权定价的新方法.首先,通过It5公式推导出亚式期权所满足的概率密度转移函数.其次,类比可靠性数学中求平均寿命的方法,用无套利定价方法给出了分数布朗运动环境下几何平均型亚式看涨期权定价公式.结果表明该方法同样适用于其它类型欧式期权.     

Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models

The stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.     

A convenient solver for solving optimal control problems

Abstract

This paper focuses on the development of a solver for solving optimal control problems. A developed numerical optimal control module integrated with the Sequential Quadratic Programming method is introduced. An optimal control problem solver based on the proposed method is implemented to solve optimal control problems efficiently in engineering applications. In addition, a systematic procedure for solving optimal control problems by using the optimal control problem solver is also proposed. A time‐optimal benchmark problem presented in the literature is used to illustrate for the capability and facility of solving optimal control problems. The numerical results demonstrate the proposed method and the procedure suggested in this paper are helpful to engineers in solving optimal control problems in a systematic and efficient manner.     

The boundary element-linear complementarity method for the Signorini problem

The boundary element-linear complementarity method for solving the Laplacian Signorini problem is presented in this paper. Both Green's formula and the fundamental solution of the Laplace equation have been used to solve the boundary integral equation. By imposing the Signorini constraints of the potential and its normal derivative on the boundary, the discrete integral equation can be written into a standard linear complementarity problem (LCP). In the LCP, the unique variable to be affected by the Signorini boundary constraints is the boundary potential variable. A projected successive over-relaxation (PSOR) iterative method is employed to solve the LCP, and some numerical results are presented to illustrate the efficiency of this method.     

Solving subclasses of multi-player stochastic games via linear complementarity problem formulations—a survey and some new results

We briefly survey some results on the orderfield property of 2-player and multi-player stochastic games and their mixtures. Some of these classes of stochastic games can be solved by formulating them as a Linear Complementarity Problem (LCP) or (Generalized) Vertical Linear Complementarity Problem (VLCP). We discuss some of these results and prove that certain new subclasses and mixtures of multi-player (or n-person) stochastic games can be solved via LCP formulations. Mohan, Neogy and Parthasarathy (in Proceedings of the International Conference on Complementarity Problems, 1997) proposed an LCP formulation of ??-discounted (multi-player) polystochastic games where the transitions are controlled by one player, and proved that this LCP is processible by Lemke??s algorithm. Using this formulation repeatedly, we prove that we can solve a subclass of ??-discounted switching control polystochastic games. As our proof is constructive, we have an algorithm for solving this subclass. This algorithm only involves iteratively solving different LCPs and hence, it follows that this subclass has the orderfield property, a?question left open in the paper on orderfield property of mixtures of stochastic games by Krishnamurthy et al., (Indian J. Stat. 72:246?C275, 2010). Furthermore, we use results from Krishnamurthy et al., Indian J. Stat. 72:246?C275, (2010) to solve some mixture classes using LCP (or VLCP) formulations. We also propose two different VLCP formulations for ??-discounted zero-sum perfect information stochastic games, the underlying matrices of both formulations being R ₀. As a result, we also have an alternative proof of the orderfield property of such games.     

Tri-Diagonal Preconditioner for Toeplitz Systems from Finance

We consider a nonsymmetric Toeplitz system which arises in the discretization of a partial integro-differential equation in option pricing problems. The preconditioned conjugate gradient method with a tri-diagonal preconditioner is used to solve this system. Theoretical analysis shows that under certain conditions the tri-diagonal preconditioner leads to a superlinear convergence rate. Numerical results exemplify our theoretical analysis.     

An efficient high-precision recursive dynamic algorithm for closed-loop multibody systems

As most closed-loop multibody systems do not have independent generalized coordinates, their dynamic equations are differential/algebraic equations (DAEs). In order to accurately solve DAEs, a usual method is using generalized α-class numerical methods to convert DAEs into difference equations by differential discretization and solve them by the Newton iteration method. However, the complexity of this method is O(n²) or more in each iteration, since it requires calculating the complex Jacobian matrix. Therefore, how to improve computational efficiency is an urgent problem. In this paper, we modify this method to make it more efficient. The first change is in the phase of building dynamic equations. We use the spatial vector note and the recursive method to establish dynamic equations (DAEs) of closed-loop multibody systems, which makes the Jacobian matrix have a special sparse structure. The second change is in the phase of solving difference equations. On the basis of the topology information of the system, we simplify this Jacobian matrix by proper matrix processing and solve the difference equations recursively. After these changes, the algorithm complexity can reach O(n) in each iteration. The algorithm proposed in this paper is not only accurate, which can control well the position/velocity constraint errors, but also efficient. It is suitable for chain systems, tree systems, and closed-loop systems.     

A hybrid SVM and supervised learning approach to fuzzy min‐max hyperbox classifiers

Abstract

In this paper, a fuzzy min‐max hyperbox classifier is designed to solve M‐class classification problems using a hybrid SVM and supervised learning approach. In order to solve a classification problem, a set of training patterns is gathered from a considered classification problem. However, the training set may include several noisy patterns. In order to delete the noisy patterns from the training set, the support vector machine is applied to find the noisy patterns so that the remaining training patterns can describe the behavior of the considered classification system well. Subsequently, a supervised learning method is proposed to generate fuzzy min‐max hyperboxes for the remaining training patterns so that the generated fuzzy min‐max hyperbox classifier has good generalization performance. Finally, the Iris data set is considered to demonstrate the good performance of the proposed approach for solving this classification problem.     

A simulated annealing algorithm for balancing the assembly line type II problem with sequence-dependent setup times between tasks

This paper addresses the general assembly line balancing problem where the simple version is enriched by considering sequence-dependent setup times between tasks. Recently, Andres et al. (Andres, C., Miralles, C., and Pastor, R., 2008. Balancing and scheduling tasks in assembly lines with sequence-dependent setup times. European Journal of Operational Research, 187, (3), 1212–1223.) proposed the type I general assembly line balancing problem with setups (GALBPS-I) and developed a mathematical model and several algorithms for solving the problem. In a similar vein, we scrutinised the GALBPS type II problem where the challenge is to find the minimum cycle time for a predefined number of work stations. To solve the problem, we develop a mathematical model and a novel simulated annealing (SA) algorithm to solve such an NP-hard problem. We then employed the Taguchi method as an optimisation technique to extensively tune different parameters of our algorithm and make the classical SA algorithm more efficient in terms of running time and solution quality. Computational results reflected the high efficiency of the SA algorithm in both aspects.     

基于线性互补问题的多体系统接触/碰撞动力学研究

多体系统往往包含大量的接触/碰撞行为,这些非光滑事件的存在造成了动力学方程的不连续或分段连续,给数值求解带来了很大的困难。为综合考虑平顺接触与碰撞,采用基于线性互补问题的非光滑动力学方法,首先在当前时刻为起点的短时间内对对缝隙函数进行均匀化,然后与法向接触力建立标准线性互补方程,最后将线性互补方程改造为一组非线性代数方程,通过非线性迭代可直接求解接触力。算例结果表明,采用该方法不需在接触状态发生改变时切换模型,且严格满足互补关系,保证了接触力的非负性。研究成果可成为求解多柔体系统接触/碰撞问题的新途径。     

A DRBEM model for harbor oscillation with the effect of energy dissipation

Abstract

In this study, the numerical scheme of dual reciprocity boundary element method (DRBEM) is adopted to investigate the resonant problem in a harbor while considering the effect of energy dissipation. The numerical model employed the mild slope equation as a basic equation. To avoid complicated procedures for solving the equation, DRBEM is used to improve numerical efficiency. Computation results are compared with the existing experimental data and other theoretical results. It shows that the present model is valid and effective to solve the harbor oscillation problem.     

Space-Time Adaptive Fup Multi-Resolution Approach for Boundary-Initial Value Problems

The space-time Adaptive Fup Collocation Method (AFCM) for solving boundary-initial value problems is presented. To solve the one-dimensional initial boundary value problem, we convert the problem into a two-dimensional boundary value problem. This quasi-boundary value problem is then solved simultaneously in the space-time domain with a collocation technique and by using atomic Fup basis functions. The proposed method is a generally meshless methodology because it requires only the addition of collocation points and basis functions over the domain, instead of the classical domain discretization and numerical integration. The grid is adapted progressively by setting the threshold as a direct measure of the solution accuracy at a given resolution level. At higher resolution levels, collocation points are only added in the space-time sub-domains where the solution correction is greater than the prescribed threshold. In contrast to the classical time-stepping schemes, in which globally accumulated errors can arise and which are not easily adapted to multiple time steps, the space-time AFCM covers all space and time multiple scales, while global error is strictly controlled in time by an a priori threshold.     

Implicit implementation of the stabilized non-ordinary state-based peridynamic model

This paper presents a stabilized non-ordinary state-based peridynamic model, in which the numerical instability problems induced by the zero-energy mode are overcome. The implicit discretization formulation of this model is proposed. In order to depict the progressively damaging process in coarse discretization conditions, a bilinear damage model based on the influence function is developed. An implicit implementation of the stabilized non-ordinary state-based peridynamic model is presented, in which an iterative procedure based on the secant stiffness method is used to solve the nonlinear problem. This method does not need to introduce a damping term in solving static problems, and relatively large load steps are desirable. Five numerical examples are analyzed to demonstrate the effectiveness of the present method for quantitatively simulating the quasi-static crack propagation problems.     

CALCULATION OF THE STRESS INTENSITY FACTOR AND ESTIMATION OF ITS ERROR BY A SHAPE SENSITIVITY ANALYSIS

Abstract— The precision with which the stress intensity factor (SIF) can be calculated from a finite element solution depends essentially on the extraction method and on the discretization error. In this paper, the influence of the discretization error in the SIF calculation was studied and a method for estimating the resulting error was developed. The SIF calculation method used is based on a shape design sensitivity analysis; this assures that the resulting error in the extracted SIF depends solely on the global discretization error present in the finite element solution. Moreover, this method allows us to extend the Zienkiewicz-Zhu discretization error estimator to the SIF calculation. The reliability of the proposed method was analysed solving a two-dimensional problem using an h -adaptive process. Also the convergence of the error with the h -adaptive refinement was studied.     

混合指数跳扩散模型下基于 FST 方法的期权定价

混合指数跳扩散模型因其能够近似任意分布被广泛应用于刻画股价实际变动趋势．本文利用傅里叶空间时间步长（Fourier Space Time-stepping, FST）方法研究混合指数跳扩散模型下的欧式期权定价．通过傅里叶变换和特征指数将模型所满足的偏积分-微分方程转换为常微分方程并求解,得到了欧式期权价格．数值结果表明 FST 方法精度高、运行时间短．然后,通过搜集市场交易数据,利用非线性最小二乘方法,将得到的期权价格应用于模型校正,得到符合实际市场的模型参数．通过检验跳参数对隐含波动率图像的影响,发现混合指数跳扩散模型能够很好地体现资产收益的“波动率微笑”等特征．     

Flexible job-shop scheduling with flexible workdays,preemption, overlapping in operations and satisfaction criteria: an industrial application

This paper addresses a real scheduling problem, namely, a complex flexible job-shop scheduling problem (FJSP) with special characteristics (flexible workdays, preemption and overlapping in operations), where the objective is to maximise a satisfaction criterion defined through goal programming. To allow for flexible workdays, the solution representation of the classical FJSP is extended to consider overtime decisions and a sequence of time-cell states, which is used to model resource capability. A new temporal-constraint-handling method is proposed to solve the problem of overlapping in operations in a flexible-workday environment. Three solution methods are proposed to solve this scheduling problem: a heuristic method based on priority rules, a goal-guided tabu search (GGTS) and an extended genetic algorithm (EGA). In the GGTS, the neighbourhood functions are defined based on elimination approaches, and five possible neighbourhood functions (N0???N1???N2???N3???N4) are presented. The effectiveness and efficiency of the three solution methods are verified using dedicated benchmark instances. Computational simulations and comparisons indicate that the proposed N4-based GGTS demonstrates performance competitive with that of the EGA and the GGTSs based on the other neighbourhood functions (N0, N1, N2 and N3) for solving the scheduling problem.     

Cubic spline approximation based on half-step discretization for 2D quasilinear elliptic equations

Abstract

We report a new cubic spline approximation based on half-step discretization of order 2 in y- and order 4 in x-directions, for 2D quasi-linear elliptic PDEs. We use only two extra half-step points in x-direction and a central point. The cubic spline method is directly obtained from the continuity of first derivative terms and is applicable to elliptic problems irrespective of coordinates, which is the main advantage of our work. The error analysis of a model problem is discussed in details. Some benchmark problems are solved in order to test the numerical stability and accuracy of the method.     

A boundary integral equation method for plate flexure with conditions inside the domain

A method for solving boundary value problems for thin plate flexure as described by Kirchhoff's theory is proposed. An integral formulation leads to a system of boundary integral equations involving values of deflection, slope, bending moment and transverse shear force along the edge. A discretization leading to a matrix formulation is proposed. To solve problems with inner conditions in the plate domain, an elimination of boundary unknowns proves successful. The degenerate case where the boundary is free (which leads to a non-invertible matrix) is investigated. Three examples illustrate the efficiency of the method.     

标的资产由分数维布朗运动驱动的亚式期权定价及套期保值

在标的资产价格由分数维布朗运动驱动的假设下,文章研究了一种亚式期权的定价.我们利用Numeraire变换与复制首先将亚式期权定价转变为类似的欧式期权定价,然后运用Merton对冲风险的思想得到亚式期权的定价,最后运用Malliavin分析与一般地Clark公式给出亚式期权的套期保值策略.