Quasi-Monte Carlo, Discrepancies and Error Estimates

We discuss the problem of defining an estimate for the error in quasi-Monte Carlo integration. The key issue is the definition of an ensemble of quasi-random point sets that, on the one hand, includes a sufficiency of equivalent point sets, and on the other hand uses information on the degree of uniformity of the point set actually used, in the form of a discrepancy or diaphony. A few examples of such discrepancies are given. We derive the distribution of our error estimate in the limit of large number of points. In many cases, Gaussian central limits are obtained. We also present numerical results for the quadratic star-discrepancy for a number of quasi-random sequences

On the structure of Gaussian pricing models and Gaussian Markov functional models

This article investigates the structure of Gaussian pricing models (that is, models in which future returns are normally distributed). Although much is already known about such models, this article differs in that it is based on a formulation of the theory of derivative pricing in which numeraire invariance is manifest, extending earlier work on this subject. The focus on symmetry properties leads to a deeper insight into the structure of these models. The central idea is the construction of the most general class of derived Gaussian tradables given a set of underlying tradables which are themselves Gaussian. These derived tradables are called 'generalized power tradables' and they correspond to portfolios in which the fraction of total value invested in each asset is a deterministic function of time. Applying this theory to Gaussian Heath-Jarrow-Morton models, the new tradables give an explicit description of the interdependence of bonds implicit in such models. Given this structure, a simple condition is derived under which these models allow a description in terms of an M-factor Markov functional model, as introduced by Hunt, Kennedy and Pelsser. Finally, conditions are derived under which these Gaussian Markov functional models are time homogeneous (bond volatilities depending only on the time to maturity). This result is linked to recent results of Bjork and Gombani.Derivatives pricing, Interest rate modelling, Mathematical finance, Exotic options,