BOUNDARY EVOLUTION EQUATIONS FOR AMERICAN OPTIONS

We consider the problem of finding optimal exercise policies for American options, both under constant and stochastic volatility settings. Rather than work with the usual equations that characterize the price exclusively, we derive and use boundary evolution equations that characterize the evolution of the optimal exercise boundary. Using these boundary evolution equations we show how one can construct very efficient computational methods for pricing American options that avoid common sources of error. First, we detail a methodology for standard static grids and then describe an improvement that defines a grid that evolves dynamically while solving the problem. When integral representations are available, as in the Black–Scholes setting, we also describe a modified integral method that leverages on the representation to solve the boundary evolution equations. Finally we compare runtime and accuracy to other popular numerical methods. The ideas and methodology presented herein can easily be extended to other optimal stopping problems.     

ON THE AMERICAN OPTION PROBLEM

We show how the change-of-variable formula with local time on curves derived recently in Peskir (2002) can be used to prove that the optimal stopping boundary for the American put option can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium representation. This settles the question raised in Myneni (1992) and dating back to McKean (1965) .     

THE EARLY EXERCISE PREMIUM FOR THE AMERICAN PUT UNDER DISCRETE DIVIDENDS

We derive an integral equation for the early exercise boundary of an American put option under Black–Scholes dynamics with discrete dividends at fixed times during the lifetime of the option. Our result is a generalization of the results obtained by Carr, Jarrow, and Myneni; Jacka; and Kim for the case without discrete dividends, and it requires a careful study of Snell envelopes for semimartingales with discontinuities.     

NONCONVEXITY OF THE OPTIMAL EXERCISE BOUNDARY FOR AN AMERICAN PUT OPTION ON A DIVIDEND‐PAYING ASSET

We prove that when the dividend rate of the underlying asset following a geometric Brownian motion is slightly larger than the risk‐free interest rate, the optimal exercise boundary of the American put option is not convex.     

THE RANGE OF TRADED OPTION PRICES

Suppose we are given a set of prices of European call options over a finite range of strike prices and exercise times, written on a financial asset with deterministic dividends which is traded in a frictionless market with no interest rate volatility. We ask: when is there an arbitrage opportunity? We give conditions for the prices to be consistent with an arbitrage-free model (in which case the model can be realized on a finite probability space). We also give conditions for there to exist an arbitrage opportunity which can be locked in at time zero. There is also a third boundary case in which prices are recognizably misspecified, but the ability to take advantage of an arbitrage opportunity depends upon knowledge of the null sets of the model.     

American options on assets with dividends near expiry

Explicit expressions valid near expiry are derived for the values and the optimal exercise boundaries of American put and call options on assets with dividends. The results depend sensitively on the ratio of the dividend yield rate D to the interest rate r . For D > r the put boundary near expiry tends parabolically to the value rK / D where K is the strike price, while for D ≤ r the boundary tends to K in the parabolic-logarithmic form found for the case D =0 by Barles et al. (1995) and by Kuske and Keller (1998) . For the call, these two behaviors are interchanged: parabolic and tending to rK / D for D < r , as was shown by Wilmott, Dewynne, and Howison (1993) , and parabolic-logarithmic and tending to K for D ≥ r . The results are derived twice: once by solving an integral equation, and again by constructing matched asymptotic expansions.     

网络型产业自然垄断边界的判定

随着对自然垄断特征认识的不断变化 ,经济学家对网络型产业的自然垄断边界的判定也不断地深化。文章结合自然垄断理论的演进规律 ,把影响自然垄断边界的因素分为成本的和非成本两个方面 ,对网络型产业的自然垄断边界的判定进行了探讨。并且指出 ,网络效应不但影响到垄断性产业的成本 ,并且改变了其供给技术和市场需求 ,对垄断边界判定和反垄断产生了重要影响。     

Exercise Regions And Efficient Valuation Of American Lookback Options

This paper presents an efficient method to compute the values and early exercise boundaries of American fixed strike lookback options. The method reduces option valuation to a single optimal stopping problem for standard Brownian motion and an associated path-dependent functional, indexed by one parameter in the absence of dividends and by two parameters in the presence of a dividend rate. Numerical results obtained by this method show that, after a space-time transformation, the stopping boundaries are well approximated by certain piecewise linear functions with a few pieces, leading to fast and accurate approximations for American lookback option values. An explicit decomposition formula for American lookback options is derived and applied not only to the development of these approximations but also to the asymptotic analysis of the early exercise boundary near the expiration date.     

基于系统观视角的团队边界工作研究评介

有关团队边界的研究表明团队有效性是团队内外部因素共同作用的结果,识别团队边界所承担的多种角色,采取系统观,强调团队边界工作的内部导向与外部导向维度有助于整合团队过程研究与团队开放观研究。团队边界工作是团队所从事的旨在建立和维持团队边界,以及管理跨团队边界的互动行为。本文在对团队边界研究内部观与外部观分析基础上,提出了基于系统观指导下团队边界工作的研究同时重视内部导向与外部导向团队边界工作的发展趋势;梳理团队边界工作的内涵与分类,考察了团队边界工作的影响效应及其背景影响因素,进一步提出了团队边界工作的综合影响模型,指出内部导向与外部导向协调发展的团队边界工作对于组织管理实践的启迪以及未来的研究方向。     

The Valuation of American Options on Multiple Assets

In this paper we provide valuation formulas for several types of American options on two or more assets. Our contribution is twofold. First, we characterize the optimal exercise regions and provide valuation formulas for a number of American option contracts on multiple underlying assets with convex payoff functions. Examples include options on the maximum of two assets, dual strike options, spread options, exchange options, options on the product and powers of the product, and options on the arithmetic average of two assets. Second, we derive results for American option contracts with nonconvex payoffs, such as American capped exchange options. For this option we explicitly identify the optimal exercise boundary and provide a decomposition of the price in terms of a capped exchange option with automatic exercise at the cap and an early exercise premium involving the benefits of exercising prior to reaching the cap. Besides generalizing the current literature on American option valuation our analysis has implications for the theory of investment under uncertainty. A specialization of one of our models also provides a new representation formula for an American capped option on a single underlying asset.     

MULTIDIMENSIONAL PORTFOLIO OPTIMIZATION WITH PROPORTIONAL TRANSACTION COSTS

We provide a computational study of the problem of optimally allocating wealth among multiple stocks and a bank account, to maximize the infinite horizon discounted utility of consumption. We consider the situation where the transfer of wealth from one asset to another involves transaction costs that are proportional to the amount of wealth transferred. Our model allows for correlation between the price processes, which in turn gives rise to interesting hedging strategies. This results in a stochastic control problem with both drift-rate and singular controls, which can be recast as a free boundary problem in partial differential equations. Adapting the finite element method and using an iterative procedure that converts the free boundary problem into a sequence of fixed boundary problems, we provide an efficient numerical method for solving this problem. We present computational results that describe the impact of volatility, risk aversion of the investor, level of transaction costs, and correlation among the risky assets on the structure of the optimal policy. Finally we suggest and quantify some heuristic approximations.     

Spanning Boundaries in Offshore Outsourcing Firms： A Case Study on KM System Initiatives in Wipro Technologies

Despite the increased attention on boundary spanning in organiz- ations, we still know surprisingly little about how boundaries are formed and what spanning activities can be applied to the context of offshore outsourcing firms （OOFs）. This paper examines the antecedents of boundary formation in OOFs and proposes that spanning activities can help OOFs to realize return from their IT investment in an effective and efficient manner. Our findings show that two theoretical models have been identified in this paper. The first model proposed that three boundaries are formed in the OOFs, namely structure, knowledge and subcultural boundaries. We explored the formation process and detailed with steps. The second model therefore is proposed to span these three boundaries and we have expiored detailed mechanisms in this paper.     

SIMULATION-BASED PORTFOLIO OPTIMIZATION FOR LARGE PORTFOLIOS WITH TRANSACTION COSTS

We consider a portfolio optimization problem where the investor's objective is to maximize the long-term expected growth rate, in the presence of proportional transaction costs. This problem belongs to the class of stochastic control problems with singular controls , which are usually solved by computing solutions to related partial differential equations called the free-boundary Hamilton–Jacobi–Bellman (HJB) equations . The dimensionality of the HJB equals the number of stocks in the portfolio. The runtime of existing solution methods grow super-exponentially with dimension, making them unsuitable to compute optimal solutions to portfolio optimization problems with even four stocks. In this work we first present a boundary update procedure that converts the free boundary problem into a sequence of fixed boundary problems. Then by combining simulation with the boundary update procedure, we provide a computational scheme whose runtime, as shown by the numerical tests, scales polynomially in dimension. The results are compared and corroborated against existing methods that scale super-exponentially in dimension. The method presented herein enables the first ever computational solution to free-boundary problems in dimensions greater than three.     

企业市场边界的管理学思考

市场边界是影响企业边界的重要变量。当前社会经济环境下,由于企业规模和进入市场能力的差异、人为的进入市场障碍以及不同国家或地区市场发育成熟度不同等原因,企业只能在有限的市场进行经营活动,即企业只能在其市场边界内从事经营活动。发现或确认企业的市场边界对企业的高效运作具有重大意义。现实企业运作中存在大量无效或低效的经营活动,如超越企业自身能力的广告宣传活动等。企业的发展取决于企业生产能力的提高和市场边界扩张之间的互动。如果企业生产能力的提高是可控的,那么不可控的市场边界扩张及市场边界界定、选择就是企业运作成功与否的关键因素。市场边界的隐性假设在现实环境下应该是显性的,企业是在其市场边界内而不是在一个无边界的市场中开展经营活动。对市场边界的研究,包括市场边界的类型、影响因素、演变规律和基于市场边界考虑的经营决策特点等,应该成为当代管理学尤其是市场营销学研究发展的重点。     

ALTERNATIVE CHARACTERIZATIONS OF AMERICAN PUT OPTIONS

We derive alternative representations of the McKean equation for the value of the American put option. Our main result decomposes the value of an American put option into the corresponding European put price and the early exercise premium. We then represent the European put price in a new manner. This representation allows us to alternatively decompose the price of an American put option into its intrinsic value and time value, and to demonstrate the equivalence of our results to the McKean equation.     

On American VIX options under the generalized 3/2 and 1/2 models

In this paper, we extend the 3/2 model for VIX studied by Goard and Mazur and introduce the generalized 3/2 and 1/2 classes of volatility processes. Under these models, we study the pricing of European and American VIX options, and for the latter, we obtain an early exercise premium representation using a free‐boundary approach and local time‐space calculus. The optimal exercise boundary for the volatility is obtained as the unique solution to an integral equation of Volterra type. We also consider a model mixing these two classes and formulate the corresponding optimal stopping problem in terms of the observed factor process. The price of an American VIX call is then represented by an early exercise premium formula. We show the existence of a pair of optimal exercise boundaries for the factor process and characterize them as the unique solution to a system of integral equations.     

基于累积直方图的视频镜头边界检测方法

提出了一种基于累积直方图的视频镜头边界检测方法,以累积直方图来代表视频帧图像的特征,其帧差充分反映了视频帧图像间的差异性;结合滑动窗的局部阈值分割处理,获得镜头边界检测对物体/摄像机的运动和光线变化的不敏感性。实验结果表明,本方法在镜头突变边界检测中达到95.97%的查全率和96.75%的查准率。     

The optimal method for pricing Bermudan options by simulation

Least‐squares methods enable us to price Bermudan‐style options by Monte Carlo simulation. They are based on estimating the option continuation value by least‐squares. We show that the Bermudan price is maximized when this continuation value is estimated near the exercise boundary, which is equivalent to implicitly estimating the optimal exercise boundary by using the value‐matching condition. Localization is the key difference with respect to global regression methods, but is fundamental for optimal exercise decisions and requires estimation of the continuation value by iterating local least‐squares (because we estimate and localize the exercise boundary at the same time). In the numerical example, in agreement with this optimality, the new prices or lower bounds (i) improve upon the prices reported by other methods and (ii) are very close to the associated dual upper bounds. We also study the method's convergence.     

ON THE EXISTENCE OF THE ENDOGENOUS MORTGAGE RATE PROCESS

The mortgage rate is a major factor in the refinancing decision. The refinancing behavior influences cash flow and, therefore, mortgage price. The prices of mortgage instruments drives the mortgage rates. We consider a problem of the existence of a dynamic mortgage rate process which resolves this circular dependence. The existence is proved by constructing a solution using a newly proposed level set method.     

行业协会组织边界与组织能力模型的构建研究——基于价值网络的分析

本文通过借鉴企业研究领域中组织边界理论,将行业协会组织边界分为实体、行业和关系边界三类,并从价值网络的角度分析了三类组织边界形成机理,在此基础上进行行业协会组织能力模块化,构建行业协会组织能力的S-O-R模型。