A lower bound on the value of an infinite American call option on two assets

A lower bound is derived for the value of an option by applying threshold decision rules based on the ratio of the underlying asset values. When the asset volatility ratio is equal to the correlation coefficient, the lower bound is equal to the maximum value obtained by optimizing a function of two variables represented as the sum of a rapidly converging series.     

论带有趋势变化的变量的相关:数值试验

当计算相关的二个变量都包含有明显的趋势变化成分时,原变量之间的相关特征可能被歪曲(夸大或者缩小).对此问题进行了数值试验,结果表明,变量带有性质相反的趋势变化,会使这二个变量之间的相关系数减小(正相关的数值减小,负相关被夸大).变量带有性质相同的趋势变化,会使这二个变量之间的相关系数增加(正相关被夸大,负相关数值变小).数值试验还表明,趋势变化对相关的影响具有可交换性.只要不改变它们趋势变化的数值,它们叠加的变量互相交换,影响相关系数的后果是一样的;研究还指出,二个变量有相同的变化趋势时,对相关的影响会更大些.给出了实例.     

Investment timing under hybrid stochastic and local volatility

We consider an investment timing problem under a real option model where the instantaneous volatility of the project value is given by a combination of a hidden stochastic process and the project value itself. The stochastic volatility part is given by a function of a fast mean-reverting process as well as a slowly varying process and the local volatility part is a power (the elasticity parameter) of the project value itself. The elasticity parameter controls directly the correlation between the project value and the volatility. Knowing that the project value represents the market price of a real asset in many applications and the value of the elasticity parameter depends on the asset, the elasticity parameter should be treated with caution for investment decision problems. Based on the hybrid structure of volatility, we investigate the simultaneous impact of the elasticity and the stochastic volatility on the real option value as well as the investment threshold.     

A graphical method for valuing switching options

This paper introduces a graphical method for valuing options on real asset investments that allow the investor to switch between different operating modes at a single point-in-time. The technique uses mixtures of truncated exponential functions to approximate both the probability density function for project value and the expressions for option value of each alternative. The distribution for project value is transformed into an expected cash flow function for the option under each mode of operation. After determining an optimal exercise strategy, these functions are used to determine the option value. The graphical method allows the option exercise strategy to be communicated effectively through a graphical representation of the expected cash flow functions. A comparison of this approach to the existing binomial lattice method is presented. The efficiency of the graphical method is comparable to the binomial lattice and in some cases accurate solutions can be obtained with less CPU time.     

Can properly discounted projects follow geometric Brownian motion?

The geometric Brownian motion is routinely used as a dynamic model of underlying project value in real option analysis, perhaps for reasons of analytic tractability. By characterizing a stochastic state variable of future cash flows, this paper considers how transformations between a state variable and cash flows are related to project volatility and drift, and specifies necessary and sufficient conditions for project volatility and drift to be time-varying, a topic that is important for real option analysis because project value and its fluctuation can only seldom be estimated from data. This study also shows how fixed costs can cause project volatility to be mean-reverting. We conclude that the conditions of geometric Brownian motion can only rarely be met, and therefore real option analysis should be based on models of cash flow factors rather than a direct model of project value.     

Estimation of mean and variance of net present value with certain and uncertain project life: a multiperiod CAPM approach

A multiperiod capital asset pricing model has the ability to consider risk by incorporating correlation between project and market parameters. This paper presents expressions for mean and variance of net present value of a project, in a multiperiod capital asset pricing model context, for the cases of certain and uncertain project lives. Both cases consider two types of cash flows, independent and correlated over time. The effects of (i) uncertainty in project life, (ii) correlation amongst subsequent cash flows, and (iii) the elasticity of expectations on the estimates of mean and variance of net present value have been studied through a numerical example.     

Supply chain coordination for the joint determination of order quantity and reorder point using credit option

Decisions regarding order quantity and reorder point are two major challenges in supply chain inventory management. In this paper, a coordination model of the joint determination of these two decision variables is proposed. A decentralized supply chain consisting of one buyer and one supplier in a multi-period setting is investigated. Demand and lead times are uncertain in our model. An incentive scheme based on credit option has been developed to encourage the buyer to participate in the coordination model. In this model, the downstream member has the option of using credit to purchase goods during the credit time, subject to its commitment to a jointly agreed order quantity and reorder point. The credit time is determined in such a way that the two parties have incentives to participate. The proposed incentive scheme can share the benefits of coordination between the two members based on their bargaining power. The proposed model shows that the coordination of the reorder point, together with order quantity, can increase the overall chain profitability as well as each member’s profitability.     

关于协方差、相关系数与相关性的关系

本文利用最小二乘法,导出了协方差与相关系数这两个概念的由来,说明了人们总是用相关系数而不用协方差来判断两个随机变量线性相关程度的原因,阐明了两个普通变量与两个随机变量存在线性关系时,它们的根本区别,并用直观图形展示了两个随机变量线性相关的概率意义。     

How real option disinvestment flexibility augments project NPV

In this article we show how a project’s option value increases with incremental levels of investment and disinvestment flexibility. We do this by presenting two NPV and seven option pricing models in a strict sequence of increasing flexibility. We illustrate each with numerical examples and determine the maximum value that a project option could ever support. We show that managerial consideration of exit options at the time of project initiation can add value.     

The covariance of uncertain variables: definition and calculation formulae

Uncertainty theory as a branch of axiomatic mathematics has been widely used to deal with human uncertainty. The two commonly used numerical characteristics of uncertain variables, the expected value and the variance together with their mathematical properties have been discussed and applied to real optimization problems in an uncertain environment. As a further study, in this paper, we focus on the covariance and correlation coefficient of uncertain variables. The definitions and calculation formulae of covariance and correlation coefficient of two uncertain variables are suggested by means of their inverse distributions. Then we show that the correlation coefficient of uncertain variables is essentially a measure of the relevance of distributions of uncertain variables. Finally, the relation between variance and covariance is analysed and represented with some equalities and inequalities.     

A model and some evidence on pricing compound call options

The most widely accepted option pricing model, derived by Black and Scholes (B-S), studies single priced options. Nevertheless, it has important implications for the relative pricing of compound call options. Compound options are two or more option contracts on a given security with different striking prices but with each expiring on the same day.Studying the relative pricing of compound options provides insight into the efficiency of generally accepted option pricing models. Comparing prices of compound options enables us to analyze factors in option pricing that would remain hidden in studies of single options.We are not primarily concerned with efficiency of option pricing, although some of our results may bear on this issue. Our primary concerns are: (1) to determine the implications of the B-S model for compound options and (2) to explain compound option prices by a number of variables, and thus come to conclusions about option pricing generally.We found difficulty with the B-S model when attempting to explain the relative pricing of compound options. Further, from empirical tests, we found that the most important factor in explaining the relative pricing of compound options is the relative degree of leverage which is operative between the various components of a compound option set.     

Bootstrap confidence interval for a correlation curve

A correlation curve measures the strength of the association between two variables locally at different values of x. The purpose of this study is to obtain point-wise confidence intervals for a correlation curve using wild bootstrap techniques. Empirical coverage probabilities are found to be close to the specified nominal level. Bootstrapping is an attractive alternative to confidence intervals based on asymptotic expressions that have slow rate of convergence.     

Correlation and the time interval in multiple regression models

In this paper we investigate the time interval effect of multiple regression models in which some of the variables are additive and some are multiplicative. The effect on the partial regression and correlation coefficients is influenced by the selected time interval. We find that the partial regression and correlation coefficients between two additive variables approach one-period values as n increases. When one of the variables is multiplicative, they will approach zero in the limit. We also show that the decreasing speed of the n-period correlation coefficients between both multiplicative variables is faster than others, except that a one-period correlation has a higher positive value. The results of this paper can be widely applied in various fields where regression or correlation analyses are employed.     

相关系数矩阵与多元线性相关分析

相关系数指度量两个随机变量间线性关系的无量纲指标,在研究了相关系数矩阵性质及其与多元随机变量线性相关性之间关系的基础上,提出多元线性相关系数的定义,用于衡量多个变鼋间线性相关强弱的无量纲指标.分析表明,所提多元线性相关系数能够较全面地反映变量间的线性相关强度.     

A Test of Multivariate Independence Based on a Single Factor Model

A test of the independence of two sets of variables is developed to have high power against a special family of dependence. In this each set of variables has the structure of a single factor model and the dependence is solely via the correlation γ between the underlying latent variables. This is a model with only one nonzero canonical correlation. It is shown that a test based on the maximum likelihood estimate of γ is appreciably more powerful than that based on r₁, the largest sample canonical correlation. If, however, the model is used, not just as a family of alternatives but as the basis for interpretation, and if substantial cross-correlation is present then the procedure is essentially equivalent to the use of r₁.     

A Discrete-Time American Put Option Model with Fuzziness of Stock Prices

To solve a mathematical model for American put option with uncertainty, we utilize two essentials, i.e., a λ-weighting function and a mean value of fuzzy random variables simultaneously. Estimation of randomness and fuzziness as uncertainty should be important when we deal with a reasonable and natural model extended from the original optimization/decision making. Three kinds of mean values by fuzzy measures, which are based on Possibility, Necessity and Credibility, are demonstrated particularly. We consider the optimal expected price of the American put option by dynamic programming under a reasonable assumption. A numerical example is given to illustrate our idea.     

Properties of game options

A game option is an American option with the added feature that not only the option holder, but also the option writer, can exercise the option at any time. We characterize the value of a perpetual game option in terms of excessive functions, and we use the connection between excessive functions and concave functions to explicitly determine the value in some examples. Moreover, a condition on the two contract functions is provided under which the value is convex in the underlying diffusion value in the continuation region and increasing in the diffusion coefficient.Mathematics Subject Classification (2000) Primary 91A15, Secondary 60G40, 91B28     

基于复杂性和成本的NPD项目权变管理策略研究

本文在项目管理的选择法和学习法的组合模式的基础上,提出同时采用多方案并行和强化学习两种管理策略能有效应对NPD项目的复杂性。在如何决定并行数量和学习强度的问题上,基于效率准则进行了理论分析,提出了主要的权变变量,即复杂性和成本。并基于NK模型进行了仿真模拟,探讨了不同情境下,项目绩效随两种管理策略的变化规律。     

Time aggregation effect on the correlation coefficient: added-systematically sampled framework

The aggregation of financial and economic time series occurs in a number of ways. Temporal aggregation or systematic sampling is the commonly used approach. In this paper, we investigate the time interval effect of multiple regression models in which the variables are additive or systematically sampled. The correlation coefficient changes with the selected time interval when one is additive and the other is systematically sampled. It is shown that the squared correlation coefficient decreases monotonically as the differencing interval increases, approaching zero in the limit. When two random variables are both added or systematically sampled, the correlation coefficient is invariant with time and equal to the one-period values. We find that the partial regression and correlation coefficients between two additive or systematically sampled variables approach one-period values as n increases. When one of the variables is systematically sampled, they will approach zero in the limit. The time interval for the association analyses between variables is not selected arbitrarily or the statistical results are likely affected.     

DG method for numerical pricing of multi-asset Asian options—The case of options with floating strike

Option pricing models are an important part of financial markets worldwide. The PDE formulation of these models leads to analytical solutions only under very strong simplifications. For more general models the option price needs to be evaluated by numerical techniques. First, based on an ideal pure diffusion process for two risky asset prices with an additional path-dependent variable for continuous arithmetic average, we present a general form of PDE for pricing of Asian option contracts on two assets. Further, we focus only on one subclass—Asian options with floating strike—and introduce the concept of the dimensionality reduction with respect to the payoff leading to PDE with two spatial variables. Then the numerical option pricing scheme arising from the discontinuous Galerkin method is developed and some theoretical results are also mentioned. Finally, the aforementioned model is supplemented with numerical results on real market data.