Browsing by Subject "Stochastic programming."

Now showing 1 - 4 of 4

Open Access
A critical review of the approaches to optimization problems under uncertainty
(2001) Gürtuna, Filiz
In this study, the issue of uncertainty in optimization problems is studied. First of all, the meaning and sources of uncertainty are explained and then possible ways of its representation are analyzed. About the modelling process, different approaches as sensitivity analysis, parametric programming, robust optimization, stochastic programming, fuzzy programming, multiobjective programming and imprecise optimization are presented with advantages and disadvantages from different perspectives. Some extensions of the concepts of imprecise optimization are also presented.
Open Access
Pricing and hedging of contingent claims in incomplete markets
(2010) Camcı, Ahmet
In this thesis, we analyze the problem of pricing and hedging contingent claims in the multi-period, discrete time, discrete state case. We work on both European and American type contingent claims. For European contingent claims, we analyze the problem using the concept of a “λ gain-loss ratio opportunity”. Pricing results which are somewhat different from, but reminiscent of, the arbitrage pricing theorems of mathematical finance are obtained. Our analysis provides tighter price bounds on the contingent claim in an incomplete market, which may converge to a unique price for a specific value of a gain-loss preference parameter imposed by the market while the hedging policies may be different for different sides of the same trade. The results are obtained in the simpler framework of stochastic linear programming in a multiperiod setting. They also extend to markets with transaction costs. Until now, determining the buyer’s price for American contingent claims (ACC) required solving an integer program unlike European contingent claims for which solving a linear program is sufficient. We show that a relaxation of the integer programming problem which is a linear program, can be used to get the buyer’s price for an ACC. We also study the problem of computing the lower hedging price of an American contingent claim in a market where proportional transaction costs exist. We derive a new mixed-integer linear programming formulation for calculating the lower hedging price. We also present and discuss an alternative, aggregate formulation with similar properties. Our results imply that it might be optimal for the holder of several identical American claims to exercise portions of the portfolio at different time points in the presence of proportional transaction costs while this incentive disappears in their absence. We also exhibit some counterexamples for some new ideas based on our work. We believe that these counterexamples are important in determining the direction of research on the subject.
Open Access
Pricing and hedging of contingent claims in incopmplete markets by modeling losses as conditional value at risk in (formula)-gain loss opportunities
(2009) Aydın, Zeynep
We combine the principles of risk aversion and no-arbitrage pricing and propose an alternative way for pricing and hedging contingent claims in incomplete markets. We re-consider the pricing problem under the condition that losses are modeled by the measure of CVaR in the concept of λ gain-loss opportunities. The proposed model enables investors to specify their preferences by putting restrictions on the parameter λ that stands for risk aversion. Using CVaR as a measure of risk enables us to account for extreme losses and yield a conservative result. The pricing problem is studied in discrete time, multi-period, stochastic linear optimization environment with a finite probability space. We extend our model to include the perspectives of writers and buyers of the contingent claims. We use duality to establish a pricing interval of the contingent claims excluding CVaR-λ gain-loss opportunities in the market. Duality results also provide a way for passing to appropriate martingale measures and we express the pricing interval also in terms of martingale measures. This pricing interval is shown to be tighter than the no-arbitrage bounds. We also present a numerical study of our work with respect to the risk aversion parameter λ and in various levels of confidence. We compute prices of the the writers and buyers of 48 European call and put options on the S&P500 index on September 10, 2002 using the remaining options as market traded assets. It is possible to say that our proposed model yields good bounds as most of the bounds we obtained are very close to the true bid and ask values.
Open Access
Stochastic lot sizing problems under monopoly
(2009) Yanıkoğlu, İhsan
In this thesis, we study stochastic lot sizing problems under monopoly. We consider production planning of a single item using uncapacitated resources over a multi-period time horizon. The demand uncertainty is modeled via a scenario tree structure. Each node of the tree corresponds to a scenario of demand realization with an associated probability. We first consider the stochastic lot sizing problem under monopoly (SLS), which addresses the period based production plan of a manufacturer with uncertain demands and a monopolistic supplier. We propose an exact dynamic programming algorithm to solve the SLS problem in polynomial time. The second problem we consider, the stochastic lot sizing problem with extra ordering (SLSE), is based on two-stage stochastic programming. In addition to the period based production decision variables of the SLS model, there exist scenario based extra ordering decision variables in the problem setting of SLSE. We develop two families of valid inequalities for the feasible region of the introduced SLSE model. The required separation algorithms of both valid inequalities are presented along with their implementations with branch-and-cut algorithm in solving SLSE. An extensive computational analysis with branch-and-cut algorithms shows the effectiveness of these inequalities.