More on hedging American options under model uncertainty

The purpose of this note is to reconcile two different results concerning the model-free upper bound on the price of an American option, given a set of European option prices. Neuberger (2007, `Bounds on the American option') and Hobson and Neuberger (2016, `On the value of being American') argue that the cost of the cheapest super-replicating strategy is equal to the highest model-based price, where we search over all models which price correctly the given European options. Bayraktar, Huang and Zhou (2015, `On hedging American options under model uncertainty', SIAM J. Financial Math ematics) argue that the cost of the cheapest super-replicating strategy can strictly exceed the highest model-based price. We show that the reason for the difference in conclusion is that Bayraktar et al do not search over a rich enough class of models

On the value of being American

The virtue of an American option is that it can be exercised at any time. This right is particularly valuable when there is model uncertainty. Yet almost all the extensive literature on American options assumes away model uncertainty. This paper quantifies the potential value of this flexibility by identifying the supremum on the price of an American option when no model is imposed on the data, but rather any model is required to be consistent with a family of European call prices. The bound is enforced by a hedging strategy involving these call options which is robust to model error

Dynamics of a spherical object of uniform density in an expanding universe

We present Newtonian and fully general-relativistic solutions for the evolution of a spherical region of uniform interior density \rho_i(t), embedded in a background of uniform exterior density \rho_e(t). In both regions, the fluid is assumed to support pressure. In general, the expansion rates of the two regions, expressed in terms of interior and exterior Hubble parameters H_i(t) and H_e(t), respectively, are independent. We consider in detail two special cases: an object with a static boundary, H_i(t)=0; and an object whose internal Hubble parameter matches that of the background, H_i(t)=H_e(t). In the latter case, we also obtain fully general-relativistic expressions for the force required to keep a test particle at rest inside the object, and that required to keep a test particle on the moving boundary. We also derive a generalised form of the Oppenheimer-Volkov equation, valid for general time-dependent spherically-symmetric systems, which may be of interest in its own right.Comment: 22 pages, 4 figures, submitted to PR

Logolinear series expansions with applications to primordial cosmology

We develop a method for computing series expansions for solutions to ordinary differential equations when the asymptotic form contains both linear and logarithmic terms. Such situations are common in primordial cosmology when considering series expansions out of a singularity in the equations arising from a pre-inflationary phase of the universe. We develop mathematical techniques for generating these series expansions, and apply them to polynomial and Starobinsky inflationary potentials with kinetic initial conditions. Code for analytic and numerical computation of logolinear series is provided on GitHub.Comment: 12 pages, 4 figure

Localizing the Energy and Momentum of Linear Gravity

A framework is developed which quantifies the local exchange of energy and momentum between matter and the linearized gravitational field. We derive the unique gravitational energy-momentum tensor consistent with this description, and find that this tensor only exists in the harmonic gauge. Consequently, nearly all the gauge freedom of our framework is naturally and unavoidably removed. The gravitational energy-momentum tensor is then shown to have two exceptional properties: (a) it is gauge-invariant for gravitational plane-waves, (b) for arbitrary transverse-traceless fields, the energy-density is never negative, and the energy-flux is never spacelike. We analyse in detail the local gauge invariant energy-momentum transferred between the gravitational field and an infinitesimal point-source, and show that these invariants depend only on the transverse-traceless components of the field. As a result, we are led to a natural gauge-fixing program which at last renders the energy-momentum of the linear gravitational field completely unambiguous, and additionally ensures that gravitational energy is never negative nor flows faster than light. Finally, we calculate the energy-momentum content of gravitational plane-waves, the linearized Schwarzschild spacetime (extending to arbitrary static linear spacetimes) and the gravitational radiation outside two compact sources: a vibrating rod, and an equal-mass binary.Comment: 20 pages, 3 figures, published in Phys. Rev.

First-order adiabatic perturbations of a perfect fluid about a general FLRW background using the 1+3 covariant and gauge-invariant formalism

An analysis of adiabatic perturbations of a perfect fluid is performed to first-order about a general FLRW background using the 1+3 covariant and gauge-invariant formalism. The analog of the Mukhanov-Sasaki variable and the canonical variables needed to quantise respectively the scalar and tensor perturbations in a general FLRW background space-time are identified. The dynamics of the vector perturbations is also discussed.Comment: 13 page

Spherically-symmetric solutions in general relativity

We present a tetrad-based method for solving the Einstein field equations for spherically-symmetric systems and compare it with the widely-used Lema\^itre-Tolman-Bondi (LTB) model. In particular, we focus on the issues of gauge ambiguity and the use of comoving versus 'physical' coordinate systems. We also clarify the correspondences between the two approaches, and illustrate their differences by applying them to the classic examples of the Schwarzschild and Friedmann-Robertson-Walker spacetimes. We demonstrate that the tetrad-based method does not suffer from the gauge freedoms inherent to the LTB model, naturally accommodates non-zero pressure and has a more transparent physical interpretation. We further apply our tetrad-based method to a generalised form of 'Swiss cheese' model, which consists of an interior spherical region surrounded by a spherical shell of vacuum that is embedded in an exterior background universe. In general, we allow the fluid in the interior and exterior regions to support pressure, and do not demand that the interior region be compensated. We pay particular attention to the form of the solution in the intervening vacuum region and verify the validity of Birkhoff's theorem at both the metric and tetrad level. We then reconsider critically the original theoretical arguments underlying the so-called $R_h = ct$ cosmological model, which has recently received considerable attention. These considerations in turn illustrate the interesting behaviour of a number of 'horizons' in general cosmological models.Comment: 21 pages, 3 figures, submitted to Physical Review

Ghost and tachyon free Poincar\'e gauge theories: a systematic approach

A systematic method is presented for determining the conditions on the parameters in the action of a parity-preserving gauge theory of gravity for it to contain no ghost or tachyon particles. The technique naturally accommodates critical cases in which the parameter values lead to additional gauge invariances. The method is implemented as a computer program, and is used here to investigate the particle content of parity-conserving Poincar\'e gauge theory, which we compare with previous results in the literature. We find 450 critical cases that are free of ghosts and tachyons, and we further identify 10 of these that are also power-counting renormalizable, of which four have only massless tordion propagating particles and the remaining six have only a massive tordion propagating mode.Comment: 16 pages, 2 figure

An alternative approach to modelling a cosmic void and its effect on the cosmic microwave background

We apply our tetrad-based approach for constructing spherically-symmetric solutions in general relativity to modelling a void, and compare it with the standard Lema\^itre-Tolman-Bondi (LTB) formalism. In particular, we construct models for the void observed in the direction of Draco in the WISE-2MASS galaxy survey, and a corresponding cosmic microwave background (CMB) temperature decrement in the Planck data in the same direction. We find that the present-day density and velocity profiles of the void are not well constrained by the existing data, so that void models produced from the two approaches can differ substantially while remaining broadly consistent with the observations. We highlight the importance of considering the velocity as well as the density profile in constraining voids.Comment: 12 pages, 14 figures, submitted to MNRA

Bayesian model selection without evidences: application to the dark energy equation-of-state

A method is presented for Bayesian model selection without explicitly computing evidences, by using a combined likelihood and introducing an integer model selection parameter $n$ so that Bayes factors, or more generally posterior odds ratios, may be read off directly from the posterior of $n$. If the total number of models under consideration is specified a priori, the full joint parameter space $(\theta, n)$ of the models is of fixed dimensionality and can be explored using standard Markov chain Monte Carlo (MCMC) or nested sampling methods, without the need for reversible jump MCMC techniques. The posterior on $n$ is then obtained by straightforward marginalisation. We demonstrate the efficacy of our approach by application to several toy models. We then apply it to constraining the dark energy equation-of-state using a free-form reconstruction technique. We show that $\Lambda$CDM is significantly favoured over all extensions, including the simple $w(z){=}{\rm constant}$ model.Comment: Published in MNRAS. Article is 13 pages long including 12 figures, 3 tables and 2 appendice