Coupling Control Variates for Markov Chain Monte Carlo

We show that Markov couplings can be used to improve the accuracy of Markov chain Monte Carlo calculations in some situations where the steady-state probability distribution is not explicitly known. The technique generalizes the notion of control variates from classical Monte Carlo integration. We illustrate it using two models of nonequilibrium transport

Static Hedging of Multivariate Derivatives by Simulation

We propose an approximate static hedging procedure for multivariate derivatives. The hedging portfolio is composed of statically held simple univariate options, optimally weighted minimizing the variance of the difference between the target claim and the approximate replicating portfolio. The method uses simulated paths to estimate the weights of the hedging portfolio and is related to Monte Carlo control variates techniques. We report numerical results showing the performance of this static hedging procedure on bivariate options on the maximum of two assets and on 2- and 7-dimensional portfolio options. It is shown that, in the presence of transaction costs, Value at Risk and Expected Shortfall of the dynamically hedged positions can be higher than the ones obtained by a static hedge.Monte Carlo methods, option pricing, static and dynamic hedging

Error Estimation of Bathymetric Grid Models Derived from Historic and Contemporary Data Sets

The past century has seen remarkable advances in technologies associated with positioning and the measurement of depth. Lead lines have given way to single beam echo sounders, which in turn are being replaced by multibeam sonars and other means of remotely and rapidly collecting dense bathymetric datasets. Sextants were replaced by radio navigation, then transit satellite, GPS and now differential GPS. With each new advance comes tremendous improvement in the accuracy and resolution of the data we collect. Given these changes and given the vastness of the ocean areas we must map, the charts we produce are mainly compilations of multiple data sets collected over many years and representing a range of technologies. Yet despite our knowledge that the accuracy of the various technologies differs, our compilations have traditionally treated each sounding with equal weight. We address these issues in the context of generating regularly spaced grids containing bathymetric values. Gridded products are required for a number of earth sciences studies and for generating the grid we are often forced to use a complex interpolation scheme due to the sparseness and irregularity of the input data points. Consequently, we are faced with the difficult task of assessing the confidence that we can assign to the final grid product, a task that is not usually addressed in most bathymetric compilations. Traditionally the hydrographic community has considered each sounding equally accurate and there has been no error evaluation of the bathymetric end product. This has important implications for use of the gridded bathymetry, especially when it is used for generating further scientific interpretations. In this paper we approach the problem of assessing the confidence of the final bathymetry gridded product via a direct-simulation Monte Carlo method. We start with a small subset of data from the International Bathymetric Chart of the Arctic Ocean (IBCAO) grid model [Jakobsson et al., 2000]. This grid is compiled from a mixture of data sources ranging from single beam soundings with available metadata, to spot soundings with no available metadata, to digitized contours; the test dataset shows examples of all of these types. From this database, we assign a priori error variances based on available meta-data, and when this is not available, based on a worst-case scenario in an essentially heuristic manner. We then generate a number of synthetic datasets by randomly perturbing the base data using normally distributed random variates, scaled according to the predicted error model. These datasets are next re-gridded using the same methodology as the original product, generating a set of plausible grid models of the regional bathymetry that we can use for standard deviation estimates. Finally, we repeat the entire random estimation process and analyze each run’s standard deviation grids in order to examine sampling bias and standard error in the predictions. The final products of the estimation are a collection of standard deviation grids, which we combine with the source data density in order to create a grid that contains information about the bathymetric model’s reliability