Pricing High-Dimensional American Options Using Local Consistency Conditions

We investigate a new method for pricing high-dimensional American options. The method is of finite difference type but is also related to Monte Carlo techniques in that it involves a representative sampling of the underlying variables.An approximating Markov chain is built using this sampling and linear programming is used to satisfy local consistency conditions at each point related to the infinitesimal generator or transition density.The algorithm for constructing the matrix can be parallelised easily; moreover once it has been obtained it can be reused to generate quick solutions for a large class of related problems.We provide pricing results for geometric average options in up to ten dimensions, and compare these with accurate benchmarks.option pricing;inequality;markov chains

An Irregular Grid Approach for Pricing High-Dimensional American Options

We propose and test a new method for pricing American options in a high-dimensional setting.The method is centred around the approximation of the associated complementarity problem on an irregular grid.We approximate the partial differential operator on this grid by appealing to the SDE representation of the underlying process and computing the root of the transition probability matrix of an approximating Markov chain.Experimental results in five dimensions are presented for four different payoff functions.option pricing;inequality;markov chains

All-at-once preconditioning in PDE-constrained optimization

The optimization of functions subject to partial differential equations (PDE) plays an important role in many areas of science and industry. In this paper we introduce the basic concepts of PDE-constrained optimization and show how the all-at-once approach will lead to linear systems in saddle point form. We will discuss implementation details and different boundary conditions. We then show how these system can be solved efficiently and discuss methods and preconditioners also in the case when bound constraints for the control are introduced. Numerical results will illustrate the competitiveness of our techniques

Static output-feedback stabilization of discrete-time Markovian jump linear systems: a system augmentation approach

This paper studies the static output-feedback (SOF) stabilization problem for discrete-time Markovian jump systems from a novel perspective. The closed-loop system is represented in a system augmentation form, in which input and gain-output matrices are separated. By virtue of the system augmentation, a novel necessary and sufficient condition for the existence of desired controllers is established in terms of a set of nonlinear matrix inequalities, which possess a monotonic structure for a linearized computation, and a convergent iteration algorithm is given to solve such inequalities. In addition, a special property of the feasible solutions enables one to further improve the solvability via a simple D-K type optimization on the initial values. An extension to mode-independent SOF stabilization is provided as well. Compared with some existing approaches to SOF synthesis, the proposed one has several advantages that make it specific for Markovian jump systems. The effectiveness and merit of the theoretical results are shown through some numerical example