Back to basics: historical option pricing revisited

We reconsider the problem of option pricing using historical probability distributions. We first discuss how the risk-minimisation scheme proposed recently is an adequate starting point under the realistic assumption that price increments are uncorrelated (but not necessarily independent) and of arbitrary probability density. We discuss in particular how, in the Gaussian limit, the Black-Scholes results are recovered, including the fact that the average return of the underlying stock disappears from the price (and the hedging strategy). We compare this theory to real option prices and find these reflect in a surprisingly accurate way the subtle statistical features of the underlying asset fluctuations.Comment: 14 pages, 2 .ps figures. Proceedings, to appear in Proc. Roy. So

Filtering of combustion products in airbags

A mathematical model was developed to primarily investigate gas flow through a filter and to examine how particles are trapped in the filter. Using the model the pressure drop across the filter and the deposition of particles and consequent clogging of the filter could be predicted

High speed video capture for mobile phone cameras

We consider an electromechanical model for the operation of a voice coil motor in a mobile phone camera, with the aim of optimizing how a lens can be moved to a desired focusing motion. Although a methodology is developed for optimizing lens shift, there is some concern about the experimentally-determined model parameters that are at our disposal. Central to the model is the value of the estimated magnetic force constant, Kf: its value determines how far it is actually possible to move lens, but it appears that, from the value given, it would not be possible to shift the lens through the displacements desired. Furthermore, earlier experiments have also estimated the value of the back EMF constant, Kg , to be roughly five times greater than Kf, even though we present two theoretical arguments that show that Kf = Kg: a conclusion supported by readily-available manufacturers’ data

Customer Focused Price Optimisation

Tesco want to better understand how to set online prices for their general merchandise (i.e. not groceries or clothes) in the UK. Because customers can easily compare prices from different retailers we expect they will be very sensitive to price, so it is important to get it right. There are four aspects of the problem. • Forecasting: Estimating the customer demand as a function of the price chosen (especially hard for products with no sales history or infrequent sales). • Objective function: What exactly should Tesco aim to optimise? Sales volume? Profit? Profit margin? Conversion rates? • Optimisation: How to choose prices for many related products to optimise the chosen objective function. • Evalution: How to demonstrate that the chosen prices are optimal, especially to people without a mathematical background. Aggregate sales data was provided for about 400 products over about 2 years so that quantitive approaches could be tested. For some products competitors’ prices were also provided

Brewing of filter coffee

We report progress on mathematical modelling of coffee grounds in a drip filter coffee machine. The report focuses on the evolution of the shape of the bed of coffee grounds during extraction with some work also carried out on the chemistry of extraction. This work was sponsored by Philips who are interested in understanding an observed correlation between the final shape of the coffee grounds and the quality of the coffee. We used experimental data gathered by Philips and ourselves to identify regimes in the coffee brewing process and relevant regions of parameter space. Our work makes it clear that a number of separate processes define the shape of the coffee bed depending on the values of the parameters involved e.g. the size of the grains and the speed of fluid flow during extraction. We began work on constructing mathematical models of the redistribution of the coffee grounds specialised to each region and on a model of extraction. A variety of analytic and numerical tools were used. Furthermore our research has progressed far enough to allow us to begin to exploit connections between this problem and other areas of science, in particular the areas of sedimentology and geomorphology, where the processes we have observed in coffee brewing have been studied