对一道不等式习题的探究性学习

人教版教材高中数学第二册 (上 ) (必修 )第30页有这样一道习题 :已知a >b>c ,求证 :1a -b 1b -c 1c -a>0 .这样一道看似普通的不等式习题 ,却蕴涵着丰富的教学功能 .笔者在教学中从这道习题出发 ,引导学生开展了一次数学探究活动 .探究 1　 变题题 1　已知a>b >c,求证 1a -b     

中考试题是怎样炼成的

中考中许多试题都是来自课本习题的改编,如果我们在平时学习中能对这些习题多加研究,掌握它的多种变化,在考场上,我们就能以不变应万变.下面以一道课本习题为例,看看中考试题是如何变化出来的.     

善思多变,活力无限

一、问题提出 笔者在进行等差数列习题教学中选用了这样一道习题：已知等差数列{an}的前n项和为Sn,且S10=100,S100=10,则S110=.这是一道让人觉得很简单的问题,然而这个问题确实是一道有深刻思想内涵的数学问题,下面是我和学生的探究过程.     

源于新教材习题的说题案例及教学启示

通过选择一道教材习题参加说题比赛,从说题目来源,说题目考查的核心素养,说解题方法,说结论应用,说变式拓展,说教学价值等方面展示说题过程.由于题目来源于教材,结合教学实践,提高对教材习题重要性的认识,熟悉习题设置,布置作业有的放矢,对重点习题适当拓展研究,针对习题完成情况及时讲评和评价等方面浅谈对新教材习题的认识.     

巧用题源 多向变题——一道教材习题的改编及反思

本文以一道习题为题源,探讨编制新试题的具体思考过程及做法.     

重期望公式的误用

指出[2]中一道习题的错解,并分析出现错误的原因.     

一道概率题的三种解法

通过求和运算、条件概率、解方程三种方法求解一道概率习题.     

以教材上一道习题为例谈习题的适切性

习题设计,是教师日常教学工作的重要组成部分.与学情适切的习题对教学目标的达成和学生能力的提升都是十分有利的,因而,很多教师在教材之外的习题设计上花了不少的时间,却忽视了对教材上习题的研究,拿来即用,导致出现"教学意外".本文就从课本上一道"拓广探索"习题的"教学意外"入手,谈谈笔者对习题设计的看法,希望能引发大家的思考.     

一道求导运算题的多种解法与推广

讨论一道多项式函数求导习题的多种解法并推广到一般的情形.     

纠正一道习题的解答

针对《近世代数习题解》中一道习题的错解,深入分析了错误的原因,利用一个概率问题的解答给出了正确的解答,并得到了第二类Stirling数的解析表达式.     

Valuing American options under the CEV model by Laplace-Carson transforms

This paper investigates American option pricing under the constant elasticity of variance (CEV) model. Taking the Laplace-Carson transform (LCT) to the corresponding free-boundary problem enables the determination of the optimal early exercise boundary to be separated from the valuation procedure. Specifically, a functional equation for the LCT of the early exercise boundary is obtained. By applying Gaussian quadrature formulas, an efficient method is developed to compute the early exercise boundary, American option price and Greeks under the CEV model.     

A dual approach to multiple exercise option problems under constraints

This paper considers the pricing of multiple exercise options in discrete time. This type of option can be exercised up to a finite number of times over the lifetime of the contract. We allow multiple exercise of the option at each time point up to a constraint, a feature relevant for pricing swing options in energy markets. It is shown that, in the case where an option can be exercised an equal number of times at each time point, the problem can be reduced to the case of a single exercise possibility at each time. In the general case there is not a solution of this type. We develop a dual representation for the problem and give an algorithm for calculating both lower and upper bounds for the prices of such multiple exercise options.     

A simple approximation formula for calculating the optimal exercise boundary of American puts

In this paper, a simple algorithm to improve the computational accuracy of the analytical approximation for the value of American put options and their optimal exercise boundary proposed by Zhu (Int. J. Theor. Appl. Finance 9(7):1141–1177, 2006) is presented. In the current approach, Zhu’s simple approximation formula is used as an initial guess for the optimal exercise boundary of American put options. The determination of an improved optimal exercise boundary is then achieved by setting a null value of the Theta of option on the optimal exercise boundary. Numerical test results show that the improvement in accuracy is indeed significant in determining the optimal exercise boundary.     

American options under uncertain volatility

The uncertain volatility approach to financial derivatives is extended to American options (which allow early exercise before expiry). The requirement to model at the portfolio level made necessary by the non-linearity of the approach is found to lead to a recursive structure to the exercise possibilities across options. Other novel features include: the optimality sometimes of partial exercise; an interesting resolution to the issues surrounding short options whose exercise is controlled by a buyer counterparty; and the occurrence of a simple game structure for portfolios containing both long and short options. It is demonstrated that the exercise strategies resulting can significantly alter measured uncertain volatility risk. Contrary to the set of attributes for sensible risk measures put forward by Artzner, Delbaen, Eber and Heath, this risk need not be homogenous in portfolio size- forming a convincing argument for weakening this particular requirement.     

Valuation of employee stock options using the exercise multiple approach and life tables

Employee stock options (ESOs) are common in performance-based employee remuneration. Financial reporting standards such as IFRS2 and AASB2 require public corporations to report on the cost of providing ESOs, and mandate the incorporation of voluntary and involuntary early exercise. In this paper we extend the exercise multiple approach of Hull and White (2004) and decompose the attrition unadjusted voluntary exercise ESO into a gap call option and two partial-time barrier options. We use exit probabilities obtained from empirically determined multiple decrement or life tables to model involuntary early exercise or forfeiture. We provide a new analytic valuation formula which expresses the ESO value in terms of a portfolio of exotic European bivariate power options and which correctly accounts for both voluntary exercise and employee attrition. Recent approaches seek to model employee attrition using a constant hazard rate. Our approach uses an empirically driven actuarial method for incorporating employee attrition in the valuation.     

Game options with gradual exercise and cancellation under proportional transaction costs

ABSTRACT

Game (Israeli) options in a multi-asset market model with proportional transaction costs are studied in the case when the buyer is allowed to exercise the option and the seller has the right to cancel the option gradually at a mixed (or randomized) stopping time, rather than instantly at an ordinary stopping time. Allowing gradual exercise and cancellation leads to increased flexibility in hedging, and hence tighter bounds on the option price as compared to the case of instantaneous exercise and cancellation. Algorithmic constructions for the bid and ask prices, and the associated superhedging strategies and optimal mixed stopping times for both exercise and cancellation are developed and illustrated. Probabilistic dual representations for bid and ask prices are also established.     

The early exercise region for Bermudan options on two underlyings

This paper investigates the early exercise region for Bermudan options on two underlying assets. We present a set of analytical validation results for the early exercise region which can be used as a means of validating pricing techniques. When all strike prices are identical we show the existence of an intersection point such that for any asset price pair below this point early exercise is always optimal. We develop an approximation to this point in the two asset put case. When the strike prices are not all equal, we show that three separate cases exist for the early exercise region. For a Bermudan put on two assets we present these cases and show that there exists a critical point in which the boundaries of the two asset early exercise region bifurcate. Comparisons are drawn between the Bermudan results presented and the corresponding American option results.     

American Call Options Under Jump‐Diffusion Processes – A Fourier Transform Approach

We consider the American option pricing problem in the case where the underlying asset follows a jump‐diffusion process. We apply the method of Jamshidian to transform the problem of solving a homogeneous integro‐partial differential equation (IPDE) on a region restricted by the early exercise (free) boundary to that of solving an inhomogeneous IPDE on an unrestricted region. We apply the Fourier transform technique to this inhomogeneous IPDE in the case of a call option on a dividend paying underlying to obtain the solution in the form of a pair of linked integral equations for the free boundary and the option price. We also derive new results concerning the limit for the free boundary at expiry. Finally, we present a numerical algorithm for the solution of the linked integral equation system for the American call price, its delta and the early exercise boundary. We use the numerical results to quantify the impact of jumps on American call prices and the early exercise boundary.     

Teaching mathematical modelling in microeconomics

Three modelling excercises from microeconomics are presented, both as examples of a methodology for teaching economics to mathematically inclined students, and as a vehicle for teaching modelling as a general skill. The first exercise, a price adjustment model for perfect competition, is given in detail within a formulation/solution/generalization/validation scheme with suggestions for an interactive teaching strategy. The results of giving the exercise to a second‐year BSc class in Mathematics and Computing are reported. Outlines of two further exercises relating to oligopoly and related models are given, and an Appendix provides a series of graduated problems for the first of these exercises.