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

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

Irregular grid methods for pricing high-dimensional American options

This thesis proposes and studies numerical methods for pricing high-dimensional American options; important examples being basket options, Bermudan swaptions and real options. Four new methods are presented and analysed, both in terms of their application to various test problems, and in terms of their theoretical stability and convergence properties. A method using matrix roots (Chapter 2) and a method using local consistency conditions (Chapter 4) are found to be stable and to give accurate solutions, in up to ten dimensions for the latter case. A method which uses local quadratic functions to approximate the value function (Chapter 3) is found to be vulnerable to instabilities in two dimensions, and thus not suitable for high-dimensional problems. A proof of convergence related to these methods is provided in Chapter 6. Finally, a method based on interpolation of the value function (Chapter 5) is found to be effective in pricing Bermudan swaptions.

Irregular Grid Methods for Pricing High-Dimensional American Options.

This thesis proposes and studies numerical methods for pricing high-dimensional American options; important examples being basket options, Bermudan swaptions and real options. Four new methods are presented and analysed, both in terms of their application to various test problems, and in terms of their theoretical stability and convergence properties. A method using matrix roots (Chapter 2) and a method using local consistency conditions (Chapter 4) are found to be stable and to give accurate solutions, in up to ten dimensions for the latter case. A method which uses local quadratic functions to approximate the value function (Chapter 3) is found to be vulnerable to instabilities in two dimensions, and thus not suitable for high-dimensional problems. A proof of convergence related to these methods is provided in Chapter 6. Finally, a method based on interpolation of the value function (Chapter 5) is found to be effective in pricing Bermudan swaptions.

An Irregular Grid Approach for Pricing High Dimensional American Options

AMS classifications: 35R35; 60G40; 65D15; 90C33;

Using Localised Quadratic Functions on an Irregular Grid for Pricing High-Dimensional American Options

We propose a method for pricing high-dimensional American options on an irregular grid; the method involves using quadratic functions to approximate the local effect of the Black-Scholes operator.Once such an approximation is known, one can solve the pricing problem by time stepping in an explicit or implicit manner.We study stability of the method in two dimensions, and find that the grid structure is important in providing a stable approximation to the operator.

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

An Irregular Grid Approach for Pricing High Dimensional American Options

AMS classifications: 35R35; 60G40; 65D15; 90C33;American options;high-dimensional problems;free boundary problems;optimal stopping;variational inequalities;numerical methods;unstructured mesh;Markov chain approximation

The geometry of spontaneous spiking in neuronal networks

The mathematical theory of pattern formation in electrically coupled networks of excitable neurons forced by small noise is presented in this work. Using the Freidlin-Wentzell large deviation theory for randomly perturbed dynamical systems and the elements of the algebraic graph theory, we identify and analyze the main regimes in the network dynamics in terms of the key control parameters: excitability, coupling strength, and network topology. The analysis reveals the geometry of spontaneous dynamics in electrically coupled network. Specifically, we show that the location of the minima of a certain continuous function on the surface of the unit n-cube encodes the most likely activity patterns generated by the network. By studying how the minima of this function evolve under the variation of the coupling strength, we describe the principal transformations in the network dynamics. The minimization problem is also used for the quantitative description of the main dynamical regimes and transitions between them. In particular, for the weak and strong coupling regimes, we present asymptotic formulas for the network activity rate as a function of the coupling strength and the degree of the network. The variational analysis is complemented by the stability analysis of the synchronous state in the strong coupling regime. The stability estimates reveal the contribution of the network connectivity and the properties of the cycle subspace associated with the graph of the network to its synchronization properties. This work is motivated by the experimental and modeling studies of the ensemble of neurons in the Locus Coeruleus, a nucleus in the brainstem involved in the regulation of cognitive performance and behavior