Volume 2001http://edoc.hu-berlin.de/18452/3582023-03-22T06:38:23Z2023-03-22T06:38:23ZMartingale pricing measures in incomplete markets via stochastic programming duality in the dual of L ∞King, Alan J.Korf, Lisa A.http://edoc.hu-berlin.de/18452/89162020-03-07T04:14:08Z2001-11-13T00:00:00ZMartingale pricing measures in incomplete markets via stochastic programming duality in the dual of L ∞
King, Alan J.; Korf, Lisa A.
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
We propose a new framework for analyzing pricing theory for incomplete markets and contingent claims, using conjugate duality and optimization theory. Various statements in the literature of the fundamental theorem of asset pricing give conditions under which an essentially arbitrage-free market is equivalent to the existence of an equivalent martingale measure, and a formula for the fair price of a contingent claim as an expectation with respect to such a measure. In the setting of incomplete markets, the fair price is not attainable as such a particular expectation, but rather as a supremum over an infinite set of equivalent martingale measures. Here, we consider the problem as a stochastic program and derive pricing results for quite general discrete time processes. It is shown that in its most general form, the martingale pricing measure is attainable if it is permitted to be finitely additive. This setup also gives rise to a natural way of analyzing models with risk preferences, spreads and margin constraints, and other problem variants. We consider a discrete time, multi-stage, infinite probability space setting and derive the basic results of arbitrage pricing in this framework.
2001-11-13T00:00:00ZAdapting an approximate level method to the two-stage stochastic programming problemFábián, Csaba I.http://edoc.hu-berlin.de/18452/89152020-03-07T04:14:08Z2001-11-05T00:00:00ZAdapting an approximate level method to the two-stage stochastic programming problem
Fábián, Csaba I.
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
We present a decomposition method for the solution of stwo-stage stochastic programming problems. This is an approximate method that can handle problems with large number scenarios. At the beginning, only rough approximation of the objective function is required. As the optimum is gradually approached, more and more accurate data are computed. The required accuracy is known at each step, hence efforts can be coordinated. The present framwork enables the application of interior-point methods because the convergence proof does not rely on basic solutions. Moreover, the classic discretization methods and stochastic approximation schemes naturally fit into the present framework.
2001-11-05T00:00:00ZRisk measures for income streamsPflug, Georg Ch.Ruszczynski, Andrzejhttp://edoc.hu-berlin.de/18452/89142020-03-07T04:14:08Z2001-10-04T00:00:00ZRisk measures for income streams
Pflug, Georg Ch.; Ruszczynski, Andrzej
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
A new measure of risk is introduced for a sequence of random incomes adapted to some filtration. This measure is formulated as the optimal net present value of a stream of adaptively planned commitments for consumption. The calculation of the new measure is done by solving a stochastic dynamic linear optimization problem, which, in case of a finite filtration, reduces to a simple deterministic linear program. We show properties of the new measure by exploiting the convexity and duality structure of the stochastic dynamic linear problem. The measure depends on the full distribution of the income process (not only on its marginal distribution) as well as on the filtration, which is interpreted as the available information about the future.
2001-10-04T00:00:00ZTree-sparse convex programsSteinbach, Marchttp://edoc.hu-berlin.de/18452/89132020-03-07T04:14:08Z2001-09-23T00:00:00ZTree-sparse convex programs
Steinbach, Marc
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
Dynamic stochastic programs are prototypical for optimization problems with an inherent tree structure including characteristic sparsity patterns in the KKT systems of interior methods. We propose an integrated modeling and solution approach for such tree-sparse programs. Three closely related natural formulations are theoretically analyzed from a control-theoretic perspective and compared to each other. Associated KKT system solution algorithms with linear complexity are developed and comparisons to other interior approaches and related problem formulations are discussed.
2001-09-23T00:00:00Z