Pricing of Claims in Discrete Time with Partial Information

We consider the pricing problem of a seller with delayed price information. By using Lagrange duality, a dual problem is derived, and it is proved that there is no duality gap. This gives a characterization of the seller’s price of a contingent claim. Finally, we analyze the dual problem, and compare the prices offered by two sellers with delayed and full information respectively.     

Electricity swing option pricing by stochastic bilevel optimization: A survey and new approaches

We demonstrate how the problem of determining the ask price for electricity swing options can be considered as a stochastic bilevel program with asymmetric information. Unlike as for financial options, there is no way for basing the pricing method on no-arbitrage arguments. Two main situations are analyzed: if the seller has strong market power he/she might be able to maximize his/her utility, while in fully competitive situations he/she will just look for a price which makes profit and has acceptable risk. In both cases the seller has to consider the decision problem of a potential buyer – the valuation problem of determining a fair value for a specific option contract – and anticipate the buyer’s optimal reaction to any proposed strike price. We also discuss some methods for finding numerical solutions of stochastic bilevel problems with a special emphasis on using duality gap penalizations.     

The infinite time horizon dynamic pricing problem with multi-unit demand

This paper studies a dynamic pricing problem for a monopolist selling multiple identical items to potential buyers arriving over time, where the time horizon is infinite, the goods are imperishable and the buyers’ arrival follows a renewal process. Each potential buyer has some private information about his purchasing will, and this private information is unknown to the seller and therefore characterized as a random variable in this paper. Thus, the buyers may have multi-unit demand. Meanwhile, the seller needs to determine the optimal posted price such that his expected discounted revenue is maximized. This problem is formulated as a stochastic dynamic programming in this paper and then how to obtain the solution is explored. A numerical study shows that the optimal posted price performs better than that of optimal fixed price, and this advantage becomes obvious as the interest rate and/or the number of initial items increases.     

A note on pricing with risk aversion

We consider the pricing problem of a risk-averse seller facing uncertain demand. Demand uncertainty stems from buyers’ valuations being privately observed. By imposing very mild restrictions on the distribution of buyers’ valuations (an increasing generalized failure rate distribution) and the Bernoulli utility function, we show that a risk-averse seller will unambiguously post a lower price than a risk-neutral counterpart.     

Pricing and hedging in the presence of extraneous risks

Given an underlying complete financial market, we study contingent claims whose payoffs may depend on the occurrence of nonmarket events. We first investigate the almost-sure hedging of such claims. In particular, we obtain new representations of the hedging prices and provide necessary and sufficient conditions for a claim to be marketed. The analysis of various examples then leads us to investigate alternative pricing rules. We choose to embed the pricing problem into the agent’s portfolio decision and study reservation prices. We establish the existence and consistency of this pricing rule in a semimartingale model. We characterize the nonlinear dependence of the reservation price with respect to both the agent’s initial capital and the size of her position. The fair price arises as a limiting case.     

Sequential buying policies

This paper deals with the problem of a decision maker trying to purchase a certain number of units of an item at the lowest total purchasing cost within a given number of time periods. At the beginning of every period, the decision maker must decide whether or not to pay a fee to search for a seller. Once a seller is found, he offers a price to him under the realization that the price may be rejected. Assume that a maximum of one unit of item can be purchased every period. This paper develops a dynamic model which provides decision rules that consist of pricing policies and search rules for the decision maker.     

Inventory control by different service levels

This paper examines the multiple period inventory control problem of a single product with multiple (two) prices, depending on service level, in which optimal pricing and ordering decisions are made in each period. Traditional inventory and pricing models consider only single products, single prices, and single service levels. However, this research paper finds that a seller can improve inventory control and revenue by offering multiple prices depending on service level. This research considers a single product with multiple (two) pricing policies corresponding to service level as follows: if the customer is willing to delay the shipment, he/she will be offered a lower regular price. Otherwise, the customer will pay the regular price plus extra charges for express service. In this paper, I show the following: (1) there is an optimal pricing and replenishment policy that can control inventory and (2) there exists a finite threshold for inventory levels such that if the inventory level at the beginning of each period is higher than the threshold, the customer will be offered the express service at the regular price, without any extra charge.     

A capacitated firm’s pricing strategies for strategic consumers with different search costs

This paper studies pricing strategies of a seller with budget constraints facing two types of strategic consumers with different search costs, and proposes three pricing strategies to motivate all consumers to visit his shop. These are the basic price strategy, differentiated compensation strategy and an improved differentiated compensation strategy. Based on the rational expectations paradigm, we characterize the rational expectations equilibrium in the game and propose a basic pricing strategy. In order to address the interplay between price and demand, we further propose a differentiated compensation strategy to improve the basic model. We then compare the differentiated compensation strategy to the basic pricing strategy when both are feasible. We find that selection of the optimal strategy is independent of composition of consumers but is dependent on the seller’s budget level and the difference between the two search costs. If the budget is large enough and the difference between the search costs is small enough, a differentiated compensation strategy can further improve the seller’s profitability. In addition to these findings, we first propose an improved differentiated compensation strategy to further enhance the firm’s profit. We find that the optimal strategy is to implement the improved differentiated compensation strategy when all three strategies are feasible. Interestingly, the firm may benefit from paying a high compensation to the consumers.     

Utility-based indifference pricing in regime-switching models

In this paper, we study utility-based indifference pricing and hedging of a contingent claim in a continuous-time, Markov, regime-switching model. The market in this model is incomplete, so there is more than one price kernel. We specify the parametric form of price kernels so that both market risk and economic risk are taken into account. The pricing and hedging problem is formulated as a stochastic optimal control problem and is discussed using the dynamic programming approach. A verification theorem for the Hamilton-Jacobi-Bellman (HJB) solution to the problem is given. An issuer’s price kernel is obtained from a solution of a system of linear programming problems and an optimal hedged portfolio is determined.     

Equilibrium pricing bounds on option prices

We consider the problem of valuing European options in a complete market but with incomplete data. Typically, when the underlying asset dynamics is not specified, the martingale probability measure is unknown. Given a consensus on the actual distribution of the underlying price at maturity, we derive an upper bound on the call option price by putting two kinds of restrictions on the pricing probability measure. First, we put a restriction on the second risk-neutral moment of the underlying asset terminal value. Second, from equilibrium pricing arguments one can put a monotonicity restriction on the Radon-Nikodym density of the pricing probability with respect to the true probability measure. This density is restricted to be a nonincreasing function of the underlying price at maturity. The bound appears then as the solution of a constrained optimization problem and we adopt a duality approach to solve it. Explicit bounds are provided for the call option. Finally, we provide a numerical example.     

基于古诺博弈策略的报童问题的模型及分析

通过建立和分析基于古诺博弈策略的报童问题的数学模型,寻求两个销售商有重叠销售区域时的竞争价格策略优化方法.模型的主要特点是:基于蜂窝理论刻画每个销售商的营销区域呈正六边形,分析了相邻销售商有重叠销售区域时,如何确定最优竞争价格策略,保证其利润最大;其次,该模型把需求量看成价格的函数,将消费者与销售商的距离看成是影响实际消费行为的重要因素纳入到消费者密度函数.     

Dynamic pricing of airline tickets with competition

Competition has a huge influence on customer buying behaviour and will impact on the optimal price that companies should charge for goods or services. To date, many dynamic pricing models have not modelled competition explicitly. In this paper, we introduce pricing strategies that maximize revenue when selling an inventory of identical items by a fixed time and where there is a competing seller. The model used incorporates a probabilistic formulation of customer demand, which is influenced by the prices offered by the company and the competitor, and the time remaining until the end of the selling period. Calculus of variations is used to solve the problem and simple conditions are given that ensure the uniqueness of a solution. Illustrative examples are included. A practical implementation that uses dynamic updating is proposed and tested using simulated data, showing the effectiveness of the method.     

考虑线下渠道时销售商在团购网站上的最优销售策略研究

基于团购网站和销售商的典型合作模式,考虑了团购网站和线下市场的相互影响,用非线性优化理论为基础以销售商制定团购上限和团购项目定价为决策变量,考虑团购价格和最低团购数量的约束条件,建立模型优化,求解出销售商推出团购项目的最优策略。考虑到团购网站下限和团购网站销售成本以及网上销售的广告效用,分析了销售商是否应该制定团购上限,如何结合团购项目定价制定团购上限。探讨了在线低价限量销售的优势,以及顾客转移购买率和团购下限对销售商策略选择的影响。通过和单一线下渠道的最优销售策略相比较,得出销售商推出团购项目的前提条件。同时为团购网站运营商如何引导销售商推出低价团购项目提供了管理启示。     

A pricing problem under Monge property

We study a pricing problem where buyers with non-uniform demand purchase one of many items. Each buyer has a known benefit for each item and purchases the item that gives the largest utility, which is defined to be the difference between the benefit and the price of the item. The optimization problem is to decide on the prices that maximize total revenue of the seller. This problem is also called the optimal product line design problem in the absence of competition.

Even though the general problem is known to be NP-hard, it can be solved efficiently under some natural assumptions on customer benefits. In this paper we study properties of optimal solutions and present a dynamic programming algorithm when customer benefits satisfy the Monge property. The same algorithm can also be used to solve the problem under the additional requirement that all buyers should be served.     

Pricing and hedging of American contingent claims in incomplete markets

0.IntroductionandSummaryThecelebratedpapersof[2]and[3],pavedthewayforpricingoptionsonstocks,onthebasisofthefollowingprinciple:inacompletemarket(suchastheoneinSection1.5),everycontingentclaimcanbeattainedexactlybyinvestinginthemarketandstartingwithala...     

中介企业的服务定价研究

定价问题是中介企业的核心问题之一。本主要目的是探讨中介企业如何对自己提供的中介服务进行定价。定性研究表明：中介对买卖双方收取的服务费用与以下三个因素有关1．双方在选择中介交易方式下节约的交易成本；2．中介企业自身的成本；3．买卖双方面临的风险成本。并在此基础上，给出了中介企业定价的基本模型。     

Auctions with an asking price

This paper studies a sales mechanism, prevalent in housing markets, where the seller does not reveal or commit to a reserve price but instead publicly announces an asking price. We show that the seller sets an asking price such that, in equilibrium, buyers of certain types would accept it with positive probability. We also show that this sales mechanism, with an optimally chosen asking price set prior to the seller learning her value, does better than any standard auction with a reserve price equal to the seller’s reservation value. We then extend the analysis to the case where the asking price reveals information about the seller’s reservation value. We show that in this case there is a separating equilibrium with fully-revealing asking prices, which is revenue-equivalent to a standard auction with a reserve price set at the seller’s reservation value.     

Utility maximization problem with random endowment and transaction costs: when wealth may become negative

In this article, we study the problem of maximizing expected utility from the terminal wealth with proportional transaction costs and random endowment. In the context of the existence of consistent price systems, we consider the duality between the primal utility maximization problem and the dual one, which is set up on the domain of finitely additive measures. In particular, we prove duality results for utility functions supporting possibly negative values. Moreover, we construct a shadow market by the dual optimal process and consider the utility-based pricing for random endowment.     

Asymptotic asset pricing and bubbles

We define the concept of asymptotic superreplication, and prove a duality principle of asset pricing for sequences of financial markets (e.g., weakly converging financial markets and large financial markets) based on contiguous sequences of equivalent local martingale measures. This provides a pricing mechanism to calculate the fundamental value of a financial asset in the asymptotic market. We introduce the notion of asymptotic bubbles by showing that this fundamental value can be strictly lower than the current price of the asset. In the case of weakly converging markets, we show that this fundamental value is equal to an expectation of the terminal value of the asset in the weak-limit market. From a practical perspective, we relate the asymptotic superreplication price to a limit of quantile-hedging prices. This shows that even when a price process is a true martingale, it can have properties similar to a bubble, up to a set of small probability. For practical applications, we give examples of weakly converging discrete-time models (e.g. some GARCH models) and large financial models that present bubbles.     

Investigation of Nash Equilibrium existence involving complementarity-constrained pricing models

The goal of this paper is to study Nash Equilibrium (NE) existence of some game-theoretic pricing models. In Soon et al. [17], deterministic pricing models incorporating a complete demand system were proposed. As in those models, the demand function is defined via a Nonlinear Complementarity Problem (NCP), the models’ pricing constraints include complementarity conditions. When incorporated within a game, the best response problem facing each seller is a Mathematical Program with Equilibrium Constraints. A randomized version of this pricing problem will be introduced in this work and the issue of NE existence will be discussed for both the deterministic and random pricing games.