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

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

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 $p=1/2$

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

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

The very nearby M/T dwarf binary SCR 1845-6357

The recently discovered star SCR 1845-6357 is the first late M/T dwarf binary discovered. SCR 1845 is a particular object due to its tight orbit (currently around 4 AU) and its proximity to the Sun (3.85 pc). We present spatially resolved VLT/NACO images and low resolution spectra of SCR 1845 in the J, H and K near-infrared bands. Since the T dwarf companion, SCR 1845B, is so close to the primary SCR 1845A, orbital motion is evident even within a year. Following the orbital motion, the binary's mass can be measured accurately within a decade, making SCR 1845B a key T-dwarf mass-luminosity calibrator. The NIR spectra allow for accurate determination of spectral type and also for rough estimates of the object's physical parameters. The spectral type of SCR 1845B is determined by direct comparison of the flux calibrated JHK spectra with T dwarf standard template spectra and also by NIR spectral indices obtained from synthetic photometry. Constrained values for surface gravity, effective temperature and metallicity are derived by comparison with model spectra. Our data prove that SCR 1845B is a brown dwarf of spectral type T6 that is co-moving with and therefore gravitationally bound to the M8.5 primary. Fitting the NIR spectrum of SCR 1845B to model spectra yields an effective temperature of about 950K and a surface gravity log(g)=5.1 (cgs) assuming solar metallicity. Mass and age of SCR 1845B are in the range 40 to 50 Jupiter masses and 1.8 to 3.1 Gyr.Comment: 5 pages, 4 figures, accepted for publication in Astronomy & Astrophysic

Epsilon Indi Ba/Bb: the nearest binary brown dwarf

We have carried out high angular resolution near-infrared imaging and low-resolution (R~1000) spectroscopy of the nearest known brown dwarf, Eps Indi B, using the ESO VLT NAOS/CONICA adaptive optics system. We find it to be a close binary (as also noted by Volk et al. 2003) with an angular separation of 0.732 arcsec, corresponding to 2.65AU at the 3.626pc distance of the Eps Indi system. In our discovery paper (Scholz et al. 2003), we concluded that Eps Indi B was a ~50Mjup T2.5 dwarf: our revised finding is that the two system components (Eps Indi Ba and Eps Indi Bb) have spectral types of T1 and T6, respectively, and estimated masses of 47 and 28Mjup, respectively, assuming an age of 1.3Gyr. Errors in the masses are +/-10 and +/-7Mjup, respectively, dominated by the uncertainty in the age determination (0.8-2Gyr range). This uniquely well-characterised T dwarf binary system should prove important in the study of low-mass, cool brown dwarfs. The two components are bright and relatively well-resolved: Eps Indi B is the only T dwarf binary in which spectra have been obtained for both components. They have a well-established distance and age. Finally, their orbital motion can be measured on a fairly short timescale (nominal orbital period 15 yrs), permitting an accurate determination of the true total system mass, helping to calibrate brown dwarf evolutionary models.Comment: Accepted for publication by Astronomy & Astrophysics main journal. This replacement version includes minor changes made following comments by the referee, along with a reworking of the photometric data and derived quantities using 2MASS catalogue photometry as the basis, with only a minor impact on the final result