Monte Carlo methods for the valuation of multiple exercise options

We discuss Monte Carlo methods for valuing options with multiple exercise features in discrete time. By extending the recently developed duality ideas for American option pricing we show how to obtain estimates on the prices of such options using Monte Carlo techniques. We prove convergence of our approach and estimate the error. The methods are applied to options in the energy and interest rate derivative markets

Heavy tails in last passage percolation

We consider last-passage percolation models in two dimensions, in which the underlying weight distribution has a heavy tail of index $\alpha<2$. We prove scaling laws and asymptotic distributions, both for the passage times and for the shape of optimal paths; these are expressed in terms of a family (indexed by $\alpha$) of ``continuous last-passage percolation'' models in the unit square. In the extreme case $\alpha=0$ (corresponding to a distribution with slowly varying tail) the asymptotic distribution of the optimal path can be represented by a random self-similar measure on [0,1], whose multifractal spectrum we compute. By extending the continuous last-passage percolation model to $\mathbb{R}^2$ we obtain a heavy-tailed analogue of the Airy process, representing the limit of appropriately scaled vectors of passage times to different points in the plane. We give corresponding results for a directed percolation problem based on $\alpha$-stable Levy processes, and indicate extensions of the results to higher dimensions

A random hierarchical lattice: the series-parallel graph and its properties

We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability $p$ and $1-p$ respectively. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at $p=1/2$

Random fractal strings: their zeta functions, complex dimensions and spectral asymptotics

In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that using a random recursive self-similar construction it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary

Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries,

We examine a class of fractal graphs which arise from a subclass of finitely ramified fractals. The two-sided heat kernel estimates for these graphs are obtained in terms of an effective resistance metric and they are best possible up to constants. If the graph has symmetry, these estimates can be expressed as the usual Gaussian or sub-Gaussian estimates. However, without symmetry, the off-diagonal terms show different decay in different directions. We also discuss the stability of the sub-Gaussian heat kernel estimates under rough isometries

Direct Detection of Galactic Halo Dark Matter

The Milky Way Galaxy contains a large, spherical component which is believed to harbor a substantial amount of unseen matter. Recent observations indirectly suggest that as much as half of this ``dark matter'' may be in the form of old, very cool white dwarfs, the remnants of an ancient population of stars as old as the Galaxy itself. We conducted a survey to find faint, cool white dwarfs with large space velocities, indicative of their membership in the Galaxy's spherical halo component. The survey reveals a substantial, directly observed population of old white dwarfs, too faint to be seen in previous surveys. This newly discovered population accounts for at least 2% of the halo dark matter. It provides a natural explanation for the indirect observations, and represents a direct detection of Galactic halo dark matter.Comment: 13 pages, 4 figures, 1 table. Note added after Science Express online publication: This text reflects the correction of a few typographical errors in the online version of the table. It also includes the new constraint on the calculation of d_max which accounts for the fact that the survey could not have detected stars with proper motions below 0.33 arcseconds per year. Published online at ScienceExpress www.sciencemag.org 22 March 2001; 10.1126/science.1059954; To appear in Science 27 April 200

Epsilon Indi B: a new benchmark T dwarf

We have identified a new early T dwarf only 3.6pc from the Sun, as a common proper motion companion (separation 1459AU) to the K5V star Epsilon Indi (HD209100). As such, Epsilon Indi B is one of the highest proper motion sources outside the solar system (~4.7 arcsec/yr), part of one of the twenty nearest stellar systems, and the nearest brown dwarf to the Sun. Optical photometry obtained from the SuperCOSMOS Sky Survey was combined with approximate infrared photometry from the 2MASS Quicklook survey data release, yielding colours for the source typical of early T dwarfs. Follow up infrared spectroscopy using the ESO NTT and SOFI confirmed its spectral type to be T2.5+/-0.5. With Ks=11.2, Epsilon Indi B is 1.7 magnitudes brighter than any previously known T dwarf and 4 magnitudes brighter than the typical object in its class, making it highly amenable to detailed study. Also, as a companion to a bright nearby star, it has a precisely known distance (3.626pc) and relatively well-known age (0.8-2Gyr), allowing us to estimate its luminosity as logL/Lsun=-4.67, its effective temperature as 1260K, and its mass as ~40-60Mjup. Epsilon Indi B represents an important addition to the census of the Solar neighbourhood and, equally importantly, a new benchmark object in our understanding of substellar objects.Comment: Accepted by A&A (Letters); 5 pages, 3 figure