No abstract

One of the most widely used methods to build yield curve models is to use principal components analysis on the correlation matrix of the innovations. R. Litterman and J. Scheinkman found that three factors are enough to explain most of the moves in the case of the US treasury curve. These factors are level, steepness and curvature. Working in the context of commodity futures, G. Cortazar and E. Schwartz found that the spectral structure of the correlation matrices is strikingly similar to those found by R. Litterman and J. Scheinkman. We observe that in both cases the correlation between two different contracts maturing at times t and s is roughly of the form ρ|t-s|, for a certain (fixed) 0 ≤ ρ ≤ 1. Assuming this correlation structure we prove that the observed factors are perturbations of cosine waves and we extend the analysis to multiple curves.

This article presents a family of term structure models that can be applied to value contingent claims in multicommodity and seasonal markets. We apply the framework to the futures contracts on crude and heating oils trading on NYMEX. We show how to deal with the problem of having to value products depending on the "whole" market, such as spread options on contracts on a single commodity maturing at different times (timespreads) or spread options on the added value of the products derived from the raw commodity (crack spreads). Also, we show how to build term structure models for a commodity that experiences seasonality, such as heating oil.

Electrical impedance tomography (EIT) is used to find the conductivity distribution inside a region using electrostatic measurements collected on the boundary of the region. We study a simple version of the general EIT problem, in which the medium has constant conductivity but might contain a buried object of unknown shape and different, but also constant, conductivity. Our linearization about an approximate shape of the buried object follows Kaup and Santosa and has the advantage that it is valid for large contrast in the conductivity. We present a procedure to reconstruct the object boundary in the case where we know the conductivities and the centre and radius of a good circular approximation of the object boundary, using analytic solutions to the forward problem for circular objects with known conductivity. Assuming that the unknown object boundary is star-shaped with respect to the centre of the circle and a small perturbation of the approximating circle, we develop a linearized relation between the output voltage data that result from fixed input currents, entering as parameters, and the interface, entering as variables. This relation is used to find the Fourier coefficients of the perturbation of the interface. At least two measurements are needed to determine all coefficients, and more can be used for a least-squares fit. The quality of the recovered perturbation depends on the input frequency and the frequencies of the perturbation. Low frequencies work best as input, and are most easily recovered, in that case even where the amplitude of the perturbation is not very small. Several dipoles, corresponding to very close pairs of electrodes to induce the current, can be used successfully as input as well.