High Dimensional Radial Barrier Options

Pricing high dimensional American options is a difficult problem in mathematical finance. Many simulation methods have been proposed, but Monte Carlo is numerically intensive, and therefore slow. We derive an analytic expression for a new type of multi-asset barrier option using Laplace transform methods. The solution is assumed to be radially symmetric in the normalized non dimensional variables, hence the name `Radial Barrier Options'. In the single-asset case our results reduce to published results for American binary barrier options.

Pseudo-steady state solutions for solidification in a wedge

Polubarinova-Kochina's analytical differential equation method is used to determine the pseudo-steady-state solution to problems involving the freezing (solidification) of wedges of liquid which are initially at their fusion temperature. In particular, we consider four distinct problems for wedges which are: freezing with the same constant boundary temperature, freezing with the same constant boundary heat fluxes, freezing with distinct constant boundary temperatures and freezing with distinct constant fluxes at the boundaries. For the last two problems, a Heun's differential equation with an unknown singularity is derived, which in both cases admits a particularly elegant simple solution for the special case when the wedge angle is ?. The moving boundaries obtained are shown pictorially

Designing experiments for multi-variable B-spline models

In a range of practical applications where a response cannot be adequately described by a low order polynomial, B-spline regression models for a single variable have proved useful for prediction. In this paper identifiable models for several explanatory variables are considered which are formulated from B-spline and monomial basis functions of known degree and with specified knots. The use of search methods to find efficient designs under the V-, G- and D-optimality criteria is investigated. Two methods of constructing lists of feasible candidate points are described and compared across a variety of examples

An Asymptotic Analysis of an American Call Option with Small Volatility

In this paper we present an asymptotic analysis of an American call option where the diffusion term (volatility) is small compared to the drift terms (interest rate and continuous dividend yield). We show that in the limit where diffusion is negligible, relative to drift, then, at leading order, the American call’s behaviour is the same as a perpetual American call option (except in a boundary layer about the option’s expiry date).

Fluid flow in the anterior chamber of a human eye

A simple model is presented to analyse fluid flow in the anterior chamber of a human eye. It is shown that under normal conditions such flow inevitably occurs. The flow, whose reduced Reynolds number is small, is viscosity dominated and is driven by buoyancy effects which are present because of the temperature difference between the front and back of the anterior chamber. In cases of severe eye trauma or as a result of certain diseases and medical conditions, particulate matter may be introduced into the anterior chamber. The motion and distribution of such particles is analysed and it is shown that the model is capable of predicting well-established and observed features that may be present in a traumatized eye such as hyphemas, keratic precipitates, hypopyons and Krukenberg spindles