Automatic Markov Chain Monte Carlo Procedures for Sampling from Multivariate Distributions

Generating samples from multivariate distributions efficiently is an important task in Monte Carlo integration and many other stochastic simulation problems. Markov chain Monte Carlo has been shown to be very efficient compared to "conventional methods", especially when many dimensions are involved. In this article we propose a Hit-and-Run sampler in combination with the Ratio-of-Uniforms method. We show that it is well suited for an algorithm to generate points from quite arbitrary distributions, which include all log-concave distributions. The algorithm works automatically in the sense that only the mode (or an approximation of it) and an oracle is required, i.e., a subroutine that returns the value of the density function at any point x. We show that the number of evaluations of the density increases slowly with dimension. (author's abstract)Series: Preprint Series / Department of Applied Statistics and Data Processin

Optimization in Quasi-Monte Carlo Methods for Derivative Valuation

Computational complexity in financial theory and practice has seen an immense rise recently. Monte Carlo simulation has proved to be a robust and adaptable approach, well suited for supplying numerical solutions to a large class of complex problems. Although Monte Carlo simulation has been widely applied in the pricing of financial derivatives, it has been argued that the need to sample the relevant region as uniformly as possible is very important. This led to the development of quasi-Monte Carlo methods that use deterministic points to minimize the integration error. A major disadvantage of low-discrepancy number generators is that they tend to lose their ability of homogeneous coverage as the dimensionality increases. This thesis develops a novel approach to quasi-Monte Carlo methods to evaluate complex financial derivatives more accurately by optimizing the sample coordinates in such a way so as to minimize the discrepancies that appear when using lowdiscrepancy sequences. The main focus is to develop new methods to, optimize the sample coordinate vector, and to test their performance against existing quasi-Monte Carlo methods in pricing complicated multidimensional derivatives. Three new methods are developed, the Gear, the Simulated Annealing and the Stochastic Tunneling methods. These methods are used to evaluate complex multi-asset financial derivatives (geometric average and rainbow options) for dimensions up to 2000. It is shown that the two stochastic methods, Simulated Annealing and Stochastic Tunneling, perform better than existing quasi-Monte Carlo methods, Faure' and Sobol'. This difference in performance is more evident in higher dimensions, particularly when a low number of points is used in the Monte Carlo simulations. Overall, the Stochastic Tunneling method yields the smallest percentage root mean square relative error and requires less computational time to converge to a global solution, proving to be the most promising method in pricing complex derivativesImperial Users onl

Adaptive radial-based direction sampling; Some flexible and robust Monte Carlo integration methods

Adaptive radial-based direction sampling (ARDS) algorithms are specified for Bayesian analysis of models with nonelliptical, possibly, multimodal target distributions.A key step is a radial-based transformation to directions and distances. After the transformations a Metropolis-Hastings method or, alternatively, an importance sampling method is applied to evaluate generated directions. Next, distances are generated from the exact target distribution by means of the numerical inverse transformation method. An adaptive procedure is applied to update the initial location and covariance matrix in order to sample directions in an efficient way. Tested on a set of canonical mixture models that feature multimodality, strong correlation, and skewness, the ARDS algorithms compare favourably with the standard Metropolis-Hastings and importance samplers in terms of flexibility and robustness. The empirical examples include a regression model with scale contamination and a mixture model for economic growth of the USA.Markov chain Monte Carlo;importance sampling;radial coordinates

Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals

Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing data problem where the diffusion bridges in between discrete-time observations are missing. The computational problem can then be dealt with using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown parameters appear in the diffusion coefficient, direct implementation of data-augmentation results in a Markov chain that is reducible. Furthermore, data-augmentation requires efficient sampling of diffusion bridges, which can be difficult, especially in the multidimensional case. We present a general framework to deal with with these problems that does not rely on discretisation. The construction generalises previous approaches and sheds light on the assumptions necessary to make these approaches work. We define a random-walk type Metropolis-Hastings sampler for updating diffusion bridges. Our methods are illustrated using guided proposals for sampling diffusion bridges. These are Markov processes obtained by adding a guiding term to the drift of the diffusion. We give general guidelines on the construction of these proposals and introduce a time change and scaling of the guided proposal that reduces discretisation error. Numerical examples demonstrate the performance of our methods