Optimal procurement strategies for online spot markets

Spot markets have emerged for a broad range of commodities, and companies have started to use them in addition to their traditional, long-term procurement contracts (forward contracts). In comparison to forward contracts, spot markets offer products at essentially negligible lead time, but typically command a higher expected price for this added flexibility while also exhibiting substantial price uncertainty. In our research, we analyze the resulting procurement challenge and quantify the benefits of using spot markets from a supply chain perspective. We develop and solve mathematical models that determine the optimal order quantity to purchase via forward contracts and the optimal quantity to purchase via spot markets. We analyze the most general situation where commodities can be both bought and sold via a spot market and derive closed-form results for this case. We compare the obtained results to the reference scenario of pure contract sourcing and we include results for situations where the use of spot markets is restricted to either buying or selling only. Our approaches can be used by decision makers to determine optimal procurement strategies based on key parameters such as, demand and spot price volatilities, correlation between demand and spot prices, and risk aversion. The results of our analysis demonstrate that significant profit improvements can be achieved if a moderate fraction of the commodity demand is procured via spot markets. The results also show that companies who use spot markets can offer a higher expected service level, but that they might experience a higher variability in profits than companies who do not use spot markets. We illustrate our analytical results with numerical examples throughout the paper.     

Valuation of guaranteed minimum maturity benefits in variable annuities with surrender options

We present a numerical approach to the pricing of guaranteed minimum maturity benefits embedded in variable annuity contracts in the case where the guarantees can be surrendered at any time prior to maturity that improves on current approaches. Surrender charges are important in practice and are imposed as a way of discouraging early termination of variable annuity contracts. We formulate the valuation framework and focus on the surrender option as an American put option pricing problem and derive the corresponding pricing partial differential equation by using hedging arguments and Itô’s Lemma. Given the underlying stochastic evolution of the fund, we also present the associated transition density partial differential equation allowing us to develop solutions. An explicit integral expression for the pricing partial differential equation is then presented with the aid of Duhamel’s principle. Our analysis is relevant to risk management applications since we derive an expression of the delta for the sensitivity analysis of the guarantee fees with respect to changes in the underlying fund value. We provide algorithms for implementing the integral expressions for the price, the corresponding early exercise boundary and the delta of the surrender option. We quantify and assess the sensitivity of the prices, early exercise boundaries and deltas to changes in the underlying variables including an analysis of the fair insurance fees.     

Valuation of guaranteed annuity options using a stochastic volatility model for equity prices

Guaranteed annuity options are options providing the right to convert a policyholder’s accumulated funds to a life annuity at a fixed rate when the policy matures. These options were a common feature in UK retirement savings contracts issued in the 1970’s and 1980’s when interest rates were high, but caused problems for insurers as the interest rates began to fall in the 1990’s. Currently, these options are frequently sold in the US and Japan as part of variable annuity products. The last decade the literature on pricing and risk management of these options evolved. Until now, for pricing these options generally a geometric Brownian motion for equity prices is assumed. However, given the long maturities of the insurance contracts a stochastic volatility model for equity prices would be more suitable. In this paper explicit expressions are derived for prices of guaranteed annuity options assuming stochastic volatility for equity prices and either a 1-factor or 2-factor Gaussian interest rate model. The results indicate that the impact of ignoring stochastic volatility can be significant.     

Market selection decisions for inventory models with price-sensitive demand

In the majority of classical inventory theory literature, demand arises from exogenous sources upon which the firm has little or no control. In many practical contexts, however, aggregate demand is comprised of individual demands from a number of distinct customers or markets. This introduces new dimensions to supply chain planning problems involving the selection of markets or customers to include in the demand portfolio. We present a nonlinear, combinatorial optimization model to address planning decisions in both deterministic and stochastic settings, where a firm constructs a demand portfolio from a set of potential markets having price-sensitive demands. We first consider a pricing strategy that dictates a single price throughout all markets and provide an efficient algorithm for maximizing total profit. We also analyze the model under a market-specific pricing policy and describe its optimal solution. An extensive computational study characterizes the effects of key system parameters on the optimal value of expected profit, and provides some interesting insights on how a given market’s characteristics can affect optimal pricing decisions in other markets.     

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。     

The stochastic guaranteed service model with recourse for multi-echelon warehouse management

The guaranteed service model (GSM) computes optimal order-points in multi-echelon inventory control under the assumptions that delivery times can be guaranteed and the demand is bounded. Our new stochastic guaranteed service model (SGSM) with Recourse covers also scenarios that violate these assumptions. Simulation experiments on real-world data of a large German car manufacturer show that policies based on the SGSM dominate GSM-policies.     

Flow auctions

Flow goods (like electricity) are sold through auctions in a dynamic framework. An important design question is the frequency of such auctions. We use a simple dynamic auction model in continuous time to answer this question. We focus on the relationship between the persistency of bidders’ valuations and the optimal choice of frequency. If the seller focuses on the equilibrium in which bidders follow a repeated static Nash strategy, then the frequency of auctions should typically increase when persistency declines. However, accounting for the fact that bidders can follow different equilibria that are collusive in nature, the comparative statics are reversed, forcing the seller to reduce the frequency when bidders’ valuations are less persistent. The argument builds on the fact that high frequency auctions are more conducive to collusion among bidders.     

收入管理中多数量需求的动态分批拍卖机制研究

分析了可运用于收入管理的定价及分配存量的动态分批拍卖机制,传统拍卖机制假设竞标者是单一需求,与实际情况不相符合.本文研究的模型中一个卖方在有限时间限制T内采用分批拍卖的方式销售商品出售C件产品,每个时期的竞标者有着多数量的产品需求,并对所需求产品有统一的,独立的私有价值.为使得整个拍卖收益最大化,研究了最优的分配方案和每个时期应该出售的最优产品数量kt*(x),并且运用改进的多需求第二价格拍卖模型实现最优分配机制.     

Ein allgemeines Modell zur Analyse und Bewertung von Guaranteed Minimum Benefits in Fondspolicen

Variable Annuities with embedded guarantees are very popular in the US-market. There exists a great variety of products with both, guaranteed minimum death benefits (GMDB) and guaranteed minimum living benefits (GMLB). Although several approaches for pricing some of the corresponding guarantees have been proposed in the academic literature, there is no general framework in which the existing variety of such guarantees can be priced consistently. The present paper fills this gap by introducing a model, which permits a consistent and extensive analysis of all types of guarantees currently offered within Variable Annuity contracts. Aside from a valuation assuming that the policyholder follows a given strategy with respect to surrender and withdrawals, we are able to price the contract under optimal policyholder behavior. Using both, Monte-Carlo methods and a generalization of a finite mesh discretization approach proposed by Tanskanen and Lukkarinen (2004), we find that some guarantees are overpriced, whereas others, e.g. guaranteed annuities within guaranteed minimum income benefits (GMIB), are offered significantly below their risk-neutral value. We identify a variety of ‘‘new risks’’ associated with such products.     

A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB)

In this paper, we outline an impulse stochastic control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB) assuming the policyholder is allowed to withdraw funds continuously. We develop a numerical scheme for solving the Hamilton–Jacobi–Bellman (HJB) variational inequality corresponding to the impulse control problem. We prove the convergence of our scheme to the viscosity solution of the continuous withdrawal problem, provided a strong comparison result holds. The scheme can be easily generalized to price discrete withdrawal contracts. Numerical experiments are conducted, which show a region where the optimal control appears to be non-unique.     

Valuing variable annuities with guaranteed minimum lifetime withdrawal benefits

Variable annuities with guaranteed minimum lifetime withdrawal benefits (VA/GLWB) offer retirees longevity protection, exposure to equity markets, and access to flexible withdrawals in emergencies. We model how risk-averse retirees optimally withdraw from the products, balancing returns and the embedded longevity protection. Assuming reasonable individual preferences, the resulting cash flow generates a Money’s Worth Ratio of around 0.9 for our stylized VA/GLWB in the post-crisis product, considerably lower than what was offered prior to 2008. Sensitivity analyses with respect to portfolio choice, mortality, fees, and guaranteed withdrawal rates show that MWRs range from 0.80 to 1.0, with the portfolio choice making the biggest difference. For most parameter choices, the utility value of the VA/GLWB exceeds that of a similar mutual fund, but it is less than for a fixed annuity. Interestingly, VA/GLWB withdrawals in early retirement can optimally exceed contract maximum withdrawals, despite the fact that this reduces future withdrawal guarantees.     

The design of equity-indexed annuities

There is a rich variety of tailored investment products available to the retail investor in every developed economy. These contracts combine upside participation in bull markets with downside protection in bear markets. Examples include equity-linked contracts and other types of structured products. This paper analyzes these contracts from the investor’s perspective rather than the issuer’s using concepts and tools from financial economics. We analyze and critique their current design and examine their valuation from the investor’s perspective. We propose a generalization of the conventional design that has some interesting features. The generalized contract specifications are obtained by assuming that the investor wishes to maximize end of period expected utility of wealth subject to certain constraints. The first constraint is a guaranteed minimum rate of return which is a common feature of conventional contracts. The second constraint is new. It provides the investor with the opportunity to outperform a benchmark portfolio with some probability. We present the explicit form of the optimal contract assuming both constraints apply and we illustrate the nature of the solution using specific examples. The paper focusses on equity-indexed annuities as a representative type of such contracts but our approach is applicable to other types of equity-linked contracts and structured products.     

Robust Management and Pricing of Liquefied Natural Gas Contracts with Cancelation Options

Liquefied Natural Gas contracts offer cancelation options that make their pricing difficult, especially if many gas storages need to be taken into account. We develop a valuation mechanism from the buyer’s perspective, a large gas company whose main interest in these contracts is to provide to clients a reliable supply of gas. The approach combines valuation with hedging, taking into account that price-risk is driven by international markets, while volume-risk depends on local weather and is stage-wise dependent. The methodology is based on setting risk-averse stochastic mixed 0-1 programs, for different contract configurations. These difficult problems are solved with light computational effort, thanks to a robust rolling-horizon approach. The resulting pricing mechanism not only shows how a specific set of contracts will impact the company business, but also provides the manager with alternative contract configurations to counter-propose to the contract seller.     

统计信息下在线反向拍卖的竞争分析

对于投标具有统计特征的在线反向拍卖问题,利用在线算法与平均情形竞争分析相结合的方法,讨论了单一定价策略的平均情形最优单一定价及其竞争性能,提出了无限可分商品在线反向拍卖的平均情形竞争分析策略,基于此策略建立了具有均匀分布特征的在线反向拍卖模型,通过对模型求解得到了采购商的竞争需要曲线。与不考虑投标的统计信息、只是利用常规的最坏情形竞争分析得到的在线反向拍卖的竞争策略进行对比分析,发现统计信息的利用提高了在线反向拍卖策略的竞争性能。     

考虑海外代购的品牌商定价模型研究

随着海外代购体量的日趋增大,代购带来的低价威胁对于在不同国家不同市场销售产品的公司来说已经成为一个日益严重的问题。同时,代购渠道中假货的问题也愈发严重。因此,在海外代购背景下探究产品定价模型具有必要性。以往研究普遍认为这种未经授权的销售会削减品牌方的利润,但实则不然。基于这一发现,本研究为在两个不同市场销售相同产品但面临代购低价威胁的公司制定考虑代购的市场定价模型。由公司制定两个市场的价格,消费者选择是否从包括代购在内的三个渠道购买产品。推出两个授权市场的最优价格,分析各参数变化对最优价格的影响,并校验最优价格对消费者需求和总利润的影响。模型分析表明,高价市场中有部分消费者需求转向海外代购,同时低价市场的消费者需求也受到了影响,且在一定条件下,提高高价市场的产品定价能够扩大低价市场的需求,从代购的角度解释了现实中需求曲线向上倾斜的现象。此外,两个独立市场之间的价格差距对代购市场的销售也产生了积极影响,并且在某些条件下,增大价格差距可以提高公司的收益水平。随后讨论了一种极端模型和三种扩展模型,通过模型分析表示,扩展后的定价模型也显示出与基础市场模型相似的灵敏度分析结果,同样得到两个市场的价差扩大会导致代购市场的销售额增加的结论,并且在一定条件下,公司的利润更高,增加了结论的可信度。     

考虑码头内外堆场竞争的集装箱堆存定价模型

针对由于集装箱吞吐量增加造成翻箱成本增加与码头堆场拥堵,考虑内外堆场竞争,构建集装箱堆存定价模型,研究内外堆场竞争下的堆存定价决策和货主移箱决策,分析码头堆场操作成本、场外堆场运输成本、货主需求参数和码头堆场收益的内在关系。算例结果表明:一旦过了免费期立即移箱到场外堆场,货主成本最小。随着场外堆场运输成本的增加,码头堆场的堆存定价和箱量增加,提高码头堆场收益。随着码头堆场操作成本降低,码头堆场堆存价格随之下降,但码头堆存箱量增加,提高码头堆场收益。此外降低堆场定价对货主需求的影响参数以及提高运输成本对货主需求的影响系数,有利于提高码头堆场收益。     

A literature review on dynamic pricing of electricity

Revenue management and dynamic pricing are concepts that have immense possibilities for application in the energy sector. Both can be considered as demand-side management tools that can facilitate the offering of different prices at different demand levels. This paper studies literature on various topics related to the dynamic pricing of electricity and lists future research avenues in pricing policies, consumers’ willingness to pay and market segmentation in this field. Demand and price forecasting play an important role in determining prices and scheduling load in dynamic pricing environments. This allows different forms of dynamic pricing policies to different markets and customers depending on customers’ willingness to pay. Consumers’ willingness to pay for electricity services is also necessary in setting price limits depending on the demand and demand response curve. Market segmentation can enhance the effects of such pricing schemes. Appropriate scheduling of electrical load enhances the consumer response to dynamic tariffs.     

Pricing and clearing combinatorial markets with singleton and swap orders

In this article we consider combinatorial markets with valuations only for singletons and pairs of buy/sell-orders for swapping two items in equal quantity. We provide an algorithm that permits polynomial time market-clearing and -pricing. The results are presented in the context of our main application: the futures opening auction problem. Futures contracts are an important tool to mitigate market risk and counterparty credit risk. In futures markets these contracts can be traded with varying expiration dates and underlyings. A common hedging strategy is to roll positions forward into the next expiration date, however this strategy comes with significant operational risk. To address this risk, exchanges started to offer so-called futures contract combinations, which allow the traders for swapping two futures contracts with different expiration dates or for swapping two futures contracts with different underlyings. In theory, the price is in both cases the difference of the two involved futures contracts. However, in particular in the opening auctions price inefficiencies often occur due to suboptimal clearing, leading to potential arbitrage opportunities. We present a minimum cost flow formulation of the futures opening auction problem that guarantees consistent prices. The core ideas are to model orders as arcs in a network, to enforce the equilibrium conditions with the help of two hierarchical objectives, and to combine these objectives into a single weighted objective while preserving the price information of dual optimal solutions. The resulting optimization problem can be solved in polynomial time and computational tests establish an empirical performance suitable for production environments.     

On the existence of supply function equilibria

Supply function equilibria are used in the analysis of divisible good auctions with a large number of identical objects to be sold or bought. An important example occurs in wholesale electricity markets. Despite the substantial literature on supply function equilibria the existence of a pure strategy Nash equilibria for a uniform price auction in asymmetric cases has not been established in a general setting. In this paper we prove the existence of a supply function equilibrium for a duopoly with asymmetric firms having convex non-decreasing marginal costs, with decreasing concave demand subject to an additive demand shock, provided that the second derivative of the demand function is small enough and not increasing. The proof is constructive and also gives insight into the structure of the equilibrium solutions.