Stochastic competitive entries and dynamic pricing

How should firms price new products when they do not know the timing, nor the nature of the next competitive entry? To guide managers’ pricing decisions in such contexts, we propose a dynamic pricing model with two types of randomly timed entry, i.e. imitative and innovative. The characterization of the equilibrium strategies reveals how optimal prices vary with the manager’s knowledge about the timing of future competitive entries. We show that price skimming is not always optimal when entry dates are unknown to managers. Everything else equal, we demonstrate that the randomness of competitive entries make forward looking managers to choose constant prices, even though the characteristics of the market would have justified skimming the demand in the normal course. Moreover, we show that the constant pricing policy remains optimal even when the incumbent’s optimal pricing strategy influences the probability of facing a competitive entry. Finally, we find that uncertainty does not necessarily hurt firms’ profits.     

电力行业碳排放权资源影子价格模型:以海南为例

在电力行业碳排放受限情况下,碳排放权成为电力企业生产必须获取的一种资源。碳排放权资源影子价格可以成为排放权交易定价机制构建和碳税税率制订的重要参考。本文构建了以系统发电总成本最小化为目标的电力系统数学规划模型,利用对偶原理求解碳排放权的影子价格。以海南电网为案例,得到电力系统碳排放权的影子价格并分析了其主要影响因素。研究结果表明,碳排放权影子价格随着碳排放总量上限的降低呈现阶梯状增长走势。碳排放强度高的煤电和排放强度低的风电对排放权的影子价格影响截然相反。     

Minimizing the probability of lifetime ruin under stochastic volatility

We assume that an individual invests in a financial market with one riskless and one risky asset, with the latter’s price following a diffusion with stochastic volatility. Given the rate of consumption, we find the optimal investment strategy for the individual who wishes to minimize the probability of going bankrupt. To solve this minimization problem, we use techniques from stochastic optimal control.     

A new distributionally robust p-hub median problem with uncertain carbon emissions and its tractable approximation method

The p-hub median problem is to determine the optimal location for p hubs and assign the remaining nodes to hubs so as to minimize the total transportation costs. Under the carbon cap-and-trade policy, we study this problem by addressing the uncertain carbon emissions from the transportation, where the probability distributions of the uncertain carbon emissions are only partially available. A novel distributionally robust optimization model with the ambiguous chance constraint is developed for the uncapacitated single allocation p-hub median problem. The proposed distributionally robust optimization problem is a semi-infinite chance-constrained optimization model, which is computationally intractable for general ambiguity sets. To solve this hard optimization model, we discuss the safe approximation to the ambiguous chance constraint in the following two types of ambiguity sets. The first ambiguity set includes the probability distributions with the bounded perturbations with zero means. In this case, we can turn the ambiguous chance constraint into its computable form based on tractable approximation method. The second ambiguity set is the family of Gaussian perturbations with partial knowledge of expectations and variances. Under this situation, we obtain the deterministic equivalent form of the ambiguous chance constraint. Finally, we validate the proposed optimization model via a case study from Southeast Asia and CAB data set. The numerical experiments indicate that the optimal solutions depend heavily on the distribution information of carbon emissions. In addition, the comparison with the classical robust optimization method shows that the proposed distributionally robust optimization method can avoid over-conservative solutions by incorporating partial probability distribution information. Compared with the stochastic optimization method, the proposed method pays a small price to depict the uncertainty of probability distribution. Compared with the deterministic model, the proposed method generates the new robust optimal solution under uncertain carbon emissions.     

Admission and pricing optimization of on-street parking with delivery bays

We analyze a parking lot, modeled as a loss queue, with passenger and delivery vehicles. The arrival process of delivery vehicles is exogenous, while that of passenger vehicles is a function of the parking price rate and accessibility. The aim of the parking operator is to maximize the revenue generated from passenger vehicles while providing a sufficient service level for delivery vehicles, in terms of their probability to find an available parking spot. Two levels of control are exercised: pricing and admission. From a Markov decision process approach, we prove that the optimal policy is a state-dependent reservation threshold policy that randomizes in at most one state. When some parking spots should be reserved for delivery vehicles, the price rate is selected to saturate the service level constraint, whereas when it is optimal not to restrict the parking lot accessibility, the price can also be selected as the unique local maximum of the revenue or to incentivize all potential passenger vehicles to arrive. Pricing should be used as a primary tool to control the flow of passenger vehicles. In complement, admission control is exercised with a limited use of reservation only when the service level guarantee for delivery vehicles is high.     

Pricing catastrophe options in discrete operational time

We employ a doubly-binomial process as in Gerber [Gerber, H.U., 1988. Mathematical fun with the compound binomial process. ASTIN Bull. 18, 161-168] to discretize and generalize the continuous “randomized operational time” model of Chang et al. ([Chang, C.W., Chang, J.S.K., Yu, M.T., 1996. Pricing catastrophe insurance futures call spreads: A randomized operational time approach. J. Risk Insurance 63, 599-616] and CCY hereafter) from a complete-market continuous-time setting to an incomplete-market discrete-time setting, so as to price a richer set of catastrophe (CAT) options. For futures options, we derive the equivalent martingale probability measures by benchmarking to the shadow price of a bond to span arrival uncertainty, and the underlying futures price to span price uncertainty. With a time change from calendar time to the operational transaction-time dimension, we derive CCY as a limiting case under risk-neutrality when both calendar-time and transaction-time intervals shrink to zero. For a cash option with non-traded underlying loss index, we benchmark to the market reinsurance premiums to span claim uncertainty, and with a time change to claim time, we derive the cash option price as a binomial sum of claim-time binomial Asian option prices under the martingale measures.     

The effect of supply uncertainty in price-setting newsvendor models

We consider a price-setting newsvendor model in which a firm needs to make joint inventory and pricing decisions before the selling season. The supply process is uncertain such that the received quantity is the product of the order quantity and a random yield rate. Two cost structures are investigated, the in-house production case in which the firm pays for the input quantity and the procurement case in which the firm pays for the quantity received only. Our objective is to investigate the effect of yield randomness on optimal decisions and expected profit. By using the theory of stochastic comparisons, we find that under both cost structures, a less variable yield rate leads to a lower optimal price and a higher expected profit. Moreover, we show that in the in-house production case, a stochastically larger yield rate also results in a lower optimal price and a higher profit, but this is not true in the procurement case. Examples show that the effect of supply uncertainty on optimal order quantity is not universal.     

The average shadow price for MILPs with integral resource availability and its relationship to the marginal unit shadow price

The economic significance of the average shadow price for integer and mixed integer linear programming (MILP) problems has been established by researchers [Kim and Cho, Eur. J. Operat. Res. 37 (1988) 328; Crema Eur. J. Operat. Res. 85 (1995) 625]. In this paper we introduce a valid shadow price (ASPIRA) for integer programs where the right-hand side resource availability can only be varied in discrete steps. We also introduce the concept of marginal unit shadow price (MUSP). We show that for integer programs, a sufficient condition for the marginal unit shadow price to equal the average shadow price is that the Law of Diminishing Returns should hold. The polyhedral structures that will guarantee this equivalence have been explored. Identification of the problem classes for which the equivalence holds complements the existing procedure for determining shadow price for such integer programs. The concepts of ASPIRA and MUSP introduced in this paper can play a vital role in resource acquisition plans and in defining efficient market clearing prices in the presence of indivisibilities.     

Pricing catastrophe options in discrete operational time

We employ a doubly-binomial process as in Gerber [Gerber, H.U., 1988. Mathematical fun with the compound binomial process. ASTIN Bull. 18, 161–168] to discretize and generalize the continuous “randomized operational time” model of Chang et al. ([Chang, C.W., Chang, J.S.K., Yu, M.T., 1996. Pricing catastrophe insurance futures call spreads: A randomized operational time approach. J. Risk Insurance 63, 599–616] and CCY hereafter) from a complete-market continuous-time setting to an incomplete-market discrete-time setting, so as to price a richer set of catastrophe (CAT) options. For futures options, we derive the equivalent martingale probability measures by benchmarking to the shadow price of a bond to span arrival uncertainty, and the underlying futures price to span price uncertainty. With a time change from calendar time to the operational transaction-time dimension, we derive CCY as a limiting case under risk-neutrality when both calendar-time and transaction-time intervals shrink to zero. For a cash option with non-traded underlying loss index, we benchmark to the market reinsurance premiums to span claim uncertainty, and with a time change to claim time, we derive the cash option price as a binomial sum of claim-time binomial Asian option prices under the martingale measures.     

A comment on “solving the puzzles of structural efficiency”

In a recent paper by Li and Cheng [Li,S.K., Cheng, Y.S., 2007. Solving the puzzles of structural efficiency. European Journal of Operational Research 180(2), 713–722], they developed the shadow price model to solve the existing puzzles of structural efficiency theoretically. However, we observe that the optimal shadow price vector in the shadow price model by Li and Cheng (2007) is not always unique. As a result, the decomposition of the structural efficiency is arbitrarily generated, depending on the shadow price vector we choose. Finally, an example with multiple inputs and outputs is used to illustrate the phenomenon.     

Generalized average shadow prices and bottlenecks

Usually some of the constraints of a 0-1-Mixed Integer Linear Programming problem correspond to resources and in this paper we suppose that they may be redefined. For the availability of the resources the average shadow price is the maximum price that the decision maker is willing to pay for an additional unit of the package (i.e. a combination) of resources defined by some direction. In this paper we present a generalization of the average shadow price and its relation with bottlenecks including the analysis relative to the coefficients matrix of resource constraints. The generalization presented does not have some limitations of the usual average shadow price. A mathematical programming approach to find the strategy for investment in resources is presented.     

乘客策略行为下考虑需求学习的航空客运动态定价策略研究

研究航空公司在需求学习下的动态定价策略。通过假设乘客到达率不确定以及具有策略等待行为,运用贝叶斯理论和博弈论对航空公司需求学习下的多周期动态定价问题进行建模,探讨了机票最优定价策略的充分条件,并通过分析航空公司收益函数的性质,得到了最优定价随时间和已出售机票数量的变化趋势。最后应用算例分析了需求学习的效果,得出:需求学习能够缓解需求不确定带来的损失,但不能完全消除;乘客策略程度越大,需求学习效果越明显。     

影子价格及其在资源配置中的应用研究

为了应用影子价格实现资源在全社会的最优配置,本文通过线性规划的对偶理论和非线性优化问题的Kuhn-Tucker条件揭示了影子价格的本质,在资源配置优化问题中线性规划模型中的影子价格就是其对偶问题的最优解,非线性规划模型中的影子价格就是与最优解相对应的拉格朗日乘数。根据松紧定理解释了资源影子价格与资源限量之间的关系,还对线性规划模型与非线性规划模型中影子价格的不同表现进行了分析。最后阐明了影子价格在资源配置中的应用。     

线性规划影子价格的求解判断方法探讨

针对线性规划对偶问题最优解不唯一时,在已有文献提出的对偶最优解不唯一的充要条件定理基础上,结合线性规划灵敏度分析,提出影子价格的求解判断的简单准则及其命题,并进行证明.最后,用算例加以分析,指出该判断方法简单易行,也可作为通过计算软件求解影子价格的准确判断方法.     

基于C-STGARCH模型的大庆原油现货价格波动研究

引用Dueker等(2011)提出的同期门槛平滑转换广义自回归条件异方差(C-STGARCH)模型对我国大庆原油现货价格的波动状态进行了实证分析,以求对大庆原油现货价格的波动有一个新的、更深刻的量度.研究显示:第一,大庆原油现货价格的波动是不稳定的,并且存在显著的非对称和非线性现象;第二,CSTGARCH模型能很好地刻画大庆原油现货价格波动的这些现象,并且发现油价的波动以3.738%为门槛点存在高波动区和低波动区两种状态,低波动区的波动持续性比高波动区强,然而,对平滑转换持续性的影响方面,高波动区要略大于低波动区.     

On the regularity of American options with regime-switching uncertainty

We study the regularity of the stochastic representation of the solution of a class of initial–boundary value problems related to a regime-switching diffusion. This representation is related to the value function of a finite-horizon optimal stopping problem such as the price of an American-style option in finance. We show continuity and smoothness of the value function using coupling and time-change techniques. As an application, we find the minimal payoff scenario for the holder of an American-style option in the presence of regime-switching uncertainty under the assumption that the transition rates are known to lie within level-dependent compact sets.     

Pointwise adaptive estimation of a multivariate function

In this paper, we address the problem of pointwise estimation in the Gaussian white noise model. We propose a new data-driven procedure that achieves (up to a multiplicative logarithmic term) the minimax rate of convergence over a scale of anisotropic Hölder spaces. Moreover we present a general criterion in order to define what should be an “optimal” estimation procedure and we prove that our procedure satisfies this criterion. The extra logarithmic term can thus be viewed as an unavoidable price to pay for adaptation.     

Production decisions under joint price and production uncertainty

While production decisions in the presence of price uncertainty have been extensively studied, this is not so for the case in which the level of production is itself uncertain. In this paper,we provide a decision analysis under multiplicative production uncertainty, both with and without price uncertainty. We depict equilibrium and obtain comparative statics results with the aid of a diagram based on the difference between expected price and marginal cost. Comparative statics results are obtained for the model with production uncertainty alone and also for simultaneous price and production uncertainty (including two special cases). We first derive results based on the Arrow–Pratt coefficients of risk aversion, and then supplement these with the Ross measure of relative risk aversion, since this proves useful in the presence of multiple sources of uncertainty. We find that increases in risk (both price and production) or input prices reduce expected output. However, expected output supply is an increasing function of (expected) price only for “low” levels of risk aversion, and in general the relationship is ambiguous.     

Determining optimal selling price and lot size with a varying rate of deterioration and exponential partial backlogging

In this paper, a deterministic inventory model for deteriorating items with price-dependent demand is developed. The demand and deterioration rates are continuous and differentiable function of price and time, respectively. In addition, we allow for shortages and the unsatisfied demand is partially backlogged at a negative exponential rate with the waiting time. Under these assumptions, for any given selling price, we first develop the criterion for the optimal solution for the replenishment schedule, and prove that the optimal replenishment policy not only exists but also is unique. If the criterion is not satisfied, the inventory system should not be operated. Next, we show that the total profit per unit time is a concave function of price when the replenishment schedule is given. We then provide a simple algorithm to find the optimal selling price and replenishment schedule for the proposed model. Finally, we use numerical examples to illustrate the algorithm.     

基于期权博弈的中国风电投资分析

本文运用期权博弈的思想建立模型,将风电特许权投资项目看作不完全信息下的抢滩博弈问题,讨论在现有的特许权机制下,引入碳排放交易机制对风电投资的影响。模型分析了在未来碳价格存在不确定性的情况下,风电投资竞价机制会对投资者的竞价行为会产生什么样的影响,投资者应如何确定自己的最优投标价格,其他竞标者的策略对竞标者的影响将如何体现以及不同因素变化时对投资者投资行为的影响。