A convex duality approach for pricing contingent claims under partial information and short selling constraints

We consider the pricing problem facing a seller of a contingent claim. We assume that this seller has some general level of partial information, and that he is not allowed to sell short in certain assets. This pricing problem, which is our primal problem, is a constrained stochastic optimization problem. We derive a dual to this problem by using the conjugate duality theory introduced by Rockafellar. Furthermore, we give conditions for strong duality to hold. This gives a characterization of the price of the claim involving martingale- and super-martingale conditions on the optional projection of the price processes.     

Dual sourcing using capacity reservation and spot market: Optimal procurement policy and heuristic parameter determination

This contribution focuses on the cost-effective management of the combined use of two procurement options: the short-term option is given by a spot market with random price, whereas the long-term alternative is characterized by a multi period capacity reservation contract with fixed purchase price and reservation level. A reservation cost, proportional with the reservation level, has to be paid for the option of receiving any amount per period up to the reservation level. A long-term decision has to be made regarding the reserved capacity level, and then it has to be decided – period by period – which quantities to procure from the two sources. Considering the multi-period problem with stochastic demand and spot price, the structure of the optimal combined purchasing policy is derived using stochastic dynamic programming. Exploiting these structural properties, an advanced heuristic is developed to determine the respective policy parameters. This heuristic is compared with two rolling-horizon approaches which use the one-period and two-period optimal solution. A comprehensive numerical study reveals that the approaches based on one-period and two-period solutions have considerable drawbacks, while the advanced heuristic performs very well compared to the optimal solution. Finally, by exploiting our numerical results we give some insights into the system’s behavior under problem parameter variations.     

Infinite reload options: Pricing and analysis

Infinite reload options allow the user to exercise his reload right as often as he chooses during the lifetime of the contract. Each time a reload occurs, the owner receives new options where the strike price is set to the current stock price. We consider a modified version of the infinite reload option contract where the strike price of the new options received by the owner is increased by a certain percentage; we refer to this new contract as an increased reload option. The pricing problem for this modified contract is characterized as an impulse control problem resulting in a Hamilton–Jacobi–Bellman equation. We use fully implicit timestepping and prove that the discretized equations are monotone, stable and consistent, implying convergence to the viscosity solution. We also derive a globally convergent iterative method for solving the non-linear discrete equations. Numerical examples show that both the exercise policy and the option value are very sensitive to the percentage increase in the reload strike.     

下层独立的一主多从双层随机线性规划问题研究

Herminia I.Calvete等研究了一主多从双层确定性线性规划问题,证明了这类问题等价于一类常规的双层线性规划问题.本文在此基础上,推广确定型的问题到随机型优化情况,考虑了一类下层优化相互独立的一主多从双层随机优化问题(SLBMFP).在特定的随机变量分布条件下,理论上证明了该类问题可以转化为一主一从双层确定性优化问题.本文的研究对于求解一主多从双层随机优化模型,解决此类模型在实际应用中的问题具有一定的意义.     

Reduction of the bilevel stochastic optimization problem with quantile objective function to a mixed‐integer problem

The paper is devoted to the stochastic optimistic bilevel optimization problem with quantile criterion in the upper level problem. If the probability distribution is finite, the problem can be transformed into a mixed‐integer nonlinear optimization problem. We formulate assumptions guaranteeing that an optimal solution exists. A production planning problem is used to illustrate usefulness of the model. Copyright © 2017 John Wiley & Sons, Ltd.     

Electricity swing options: Behavioral models and pricing

Electricity swing options are supply contracts for power, which give the owner the right to change the required delivery on short time notice. It gives more flexibility than fixed base load or peak load contracts. The name “option” is a bit misleading, since it gives the owner multiple exercise rights at many different time horizons with exercise amounts on a continuous scale. We look at the problem to determine a rational ask price for such a contract from the viewpoint of the contract seller. The pricing of these contracts differs drastically from the pricing of financial options. First, peculiar properties arise from the non-storability of the underlying (the energy) and therefore the impossibility to hedge with the underlying, hedging is only possible with some future contracts. Second, the behavior of the owner plays an important role. Based on some behavioral model for the option holder, we develop a game-theoretic model, which allows to identify the equilibrium price. Besides some theoretical results, we present some numerical results which clarify the dependence of the asked price on the amount of flexibility offered in the swing option.     

Natural gas cash-out problem: Bilevel stochastic optimization approach

A stochastic formulation of the natural gas cash-out problem is given in a form of a bilevel multi-stage stochastic programming model with recourse. After reducing the original formulation to a bilevel linear problem, a stochastic scenario tree is defined by its node events, and time series forecasting is used to produce stochastic values for data of natural gas price and demand. Numerical experiments were run to compare the stochastic solution with the perfect information solution and the expected value solutions.     

Generalised soft binomial American real option pricing model (fuzzy–stochastic approach)

The stochastic discrete binomial models and continuous models are usually applied in option valuation. Valuation of the real American options is solved usually by the numerical procedures. Therefore, binomial model is suitable approach for appraising the options of American type. However, there is not in several situations especially in real option methodology application at to disposal input data of required quality. Two aspects of input data uncertainty should be distinguished; risk (stochastic) and vagueness (fuzzy). Traditionally, input data are in a form of real (crisp) numbers or crisp-stochastic distribution function. Therefore, hybrid models, combination of risk and vagueness could be useful approach in option valuation. Generalised hybrid fuzzy–stochastic binomial American real option model under fuzzy numbers (T-numbers) and Decomposition principle is proposed and described. Input data (up index, down index, growth rate, initial underlying asset price, exercise price and risk-free rate) are in a form of fuzzy numbers and result, possibility-expected option value is also determined vaguely as a fuzzy set. Illustrative example of equity valuation as an American real call option is presented.     

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.     

Computing arbitrage upper bounds on basket options in the presence of bid–ask spreads

We study the problem of computing the sharpest static-arbitrage upper bound on the price of a European basket option, given the bid–ask prices of vanilla call options in the underlying securities. We show that this semi-infinite problem can be recast as a linear program whose size is linear in the input data size. These developments advance previous related results, and enhance the practical value of static-arbitrage bounds as a pricing technique by taking into account the presence of bid–ask spreads. We illustrate our results by computing upper bounds on the price of a DJX basket option. The MATLAB code used to compute these bounds is available online at www.andrew.cmu.edu/user/jfp/arbitragebounds.html.     

The traveling purchaser problem with stochastic prices: Exact and approximate algorithms

The paper formulates an extension of the traveling purchaser problem where multiple types of commodities are sold at spatially distributed locations with stochastic prices (each following a known probability distribution). A purchaser’s goal is to find the optimal routing and purchasing strategies that minimize the expected total travel and purchasing costs needed to purchase one unit of each commodity. The purchaser reveals the actual commodity price at a seller upon arrival, and then either purchases the commodity at the offered price, or rejects the price and visits a next seller. In this paper, we propose an exact solution algorithm based on dynamic programming, an iterative approximate algorithm that yields bounds for the minimum total expected cost, and a greedy heuristic for fast solutions to large-scale applications. We analyze the characteristics of the problem and test the computational performance of the proposed algorithms. The numerical results show that the approximate and heuristic algorithms yield near-optimum strategies and very good estimates of the minimum total cost.     

Optimal monopoly pricing with congestion and random utility via partial mass transport

We consider a bilevel optimization framework corresponding to a monopoly spatial pricing problem: the price for a set of given facilities maximizes the profit (upper level problem) taking into account that the demand is determined by consumers' cost minimization (lower level problem). In our model, both transportation costs and congestion costs are considered, and the lower level problem is solved via partial transport mass theory. The partial transport aspect of the problem comes from the fact that each consumer has the possibility to remain out of the market. We also generalize the model and our variational analysis to the stochastic case where utility involves a random term.     

Sequential Discrete p-Facility Models for Competitive Location Planning

Two new models for duopolistic competitive discrete location planning with sequential acting and variable delivered prices are introduced. If locations and prices are assumed to be set once and for all by the players, the resulting bilevel program is nonlinear. Under the assumption that further price adjustments are possible, i.e., that a Nash equilibrium in prices is reached, the model can be simplified to a linear discrete bilevel formulation. It is shown that in either situation players should not share any locations or markets if they strive for profit-maximization.For the situation with price adjustments, a heuristic solution procedure is suggested. In addition, the bilevel models are shown to serve as a basis from which different well-known location models – as, for example, the p-median problem, the preemptive location problem and the maximum covering problem – can be derived as special cases.     

模糊集理论在随机波动率期权定价中的应用

目的是对基于随机波动率模型的期权定价问题应用模糊集理论.主要思想是把波动率的概率表示转换为可能性表示,从而把关于股票价格的带随机波动率的随机过程简化为带模糊参数的随机过程.然后建立非线性偏微分方程对欧式期权进行定价.     

An interval approach based on expectation optimization for fuzzy random bilevel linear programming problems

This paper considers a class of bilevel linear programming problems in which the coefficients of both objective functions are fuzzy random variables. The main idea of this paper is to introduce the Pareto optimal solution in a multi-objective bilevel programming problem as a solution for a fuzzy random bilevel programming problem. To this end, a stochastic interval bilevel linear programming problem is first introduced in terms of α-cuts of fuzzy random variables. On the basis of an order relation of interval numbers and the expectation optimization model, the stochastic interval bilevel linear programming problem can be transformed into a multi-objective bilevel programming problem which is solved by means of weighted linear combination technique. In order to compare different optimal solutions depending on different cuts, two criterions are given to provide the preferable optimal solutions for the upper and lower level decision makers respectively. Finally, a production planning problem is given to demonstrate the feasibility of the proposed approach.     

A generalized optimization method for night vision devices design considering stochastic external surveillance conditions

The paper describes a generalized optimization method for night vision devices (NVDs) design taking into account stochastic external surveillance conditions – ambient light illumination, atmosphere transmittance, contrast between target and background and target type. The main idea of the presented method is optimizing of the NVD design process by optimal choice of the optoelectronic channel modules – objective, image intensifier tube, ocular and electrical battery power supply while considering the influence of the external surveillance conditions uncertainty. For that goal a stochastic nonlinear mixed-integer multicriteria optimization problem is formulated and solved. The problem solution gives the optimal modules combination and some preliminary estimation about the operational characteristics of the designed NVD – working range (consistent with the uncertainty of the external surveillance conditions), weight, price and electrical batteries power supply operational duration. An illustrative numerical example based on the optimal choice from sets of objectives, image intensifier tubes, oculars and electrical batteries types is solved. The described method increases the effectiveness of the NVD design.     

Mathematical structure of a bilevel strategic pricing model

This paper is concerned with the characterization of optimal strategies for a service firm acting in an oligopolistic environment. The decision problem is formulated as a leader–follower game played on a transportation network, where the leader firm selects a revenue-maximizing price schedule that takes explicitly into account the rational behavior of the customers. In the context of our analysis, the follower’s problem is associated with a competitive network market involving non atomic customer groups. The resulting bilevel model can therefore be viewed as a model of product differentiation subject to structural network constraints.     

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.     

The valuation of options on capacity with cost and demand uncertainty

Options contracts can provide trading partners with enhanced flexibility to respond to uncertain market conditions and allow for superior capacity planning thanks to early information on future demand. We develop an analytical framework to value options on capacity for production of non-storable goods or dated services. The market consists of a sequence of contract and spot market. Reservations are made during the contract market session in period 0, where the buyer’s future demand, the seller’s future marginal costs as well as the future spot price are uncertain, the latter being impacted neither by the buyer nor the seller. During the spot market session in period 1, the buyer may execute his options or satisfy his entire or additional demand from a competing seller in the spot market. The seller allocates reserved capacity now being called and attempts to sell remaining capacity into the spot market. Analytical expressions for the buyer’s optimal reservation quantity and the seller’s tariff are derived, making explicit the risk-sharing benefits of options contracts. The combination of an options contract and a spot market is demonstrated to be Pareto improving as compared to alternative market schemes. An analysis of the determinants of the efficiency gain characterizes industries particularly suitable to the options approach.     

Buyer behavior adaptation based on a fuzzy logic controller and prediction techniques

The current form of Web provides numerous product resources available to users. Users can rely on intelligent agents for purchase actions. These actions are taken in specific environments such as Electronic Markets (EMs). In this paper, we study the interaction process between buyers and sellers and focus on the buyer side. Each buyer has the opportunity to interact with a number of sellers trying to buy the most appropriate products. This interaction can be modeled as a finite horizon Bargaining Game (BG). In this game, players have opposite goals concerning the product price. We adopt a number of techniques in the buyer side trying to give the appropriate level of efficiency in the buyer decision process. The buyer uses a prediction mechanism in combination with the use of Fuzzy Logic (FL) theory in order to be able to predict the upcoming seller proposal and, thus, understand the seller pricing policy. Based on this, he/she can adapt his/her behavior when trying to purchase products. The buyer adaptation mechanism produces the belief that the buyer has about the seller pricing policy and a parameter that indicates his/her own pricing policy which yields the buyer offers in the upcoming rounds. Moreover, the buyer is based on FL system that derives the appropriate actions at every round of the BG. Our results show that the combination of Fuzzy Logic (FL) with the above-mentioned techniques provides an efficient decision mechanism in the buyer side that in specific scenarios outperforms an optimal stopping model.