Time domain decomposition for European options in financial modelling

Finance is one of the fastest growing areas in modern applied mathematics with real world applications. The interest of this branch of applied mathematics is best described by an example involving shares. Shareholders of a company receive dividends which come from the profit made by the company. The proceeds of the company, once it is taken over or wound up, will also be distributed to shareholders. Therefore shares have a value that reflects the views of investors about the likely dividend payments and capital growth of the company. Obviously such value will be quantified by the share price on stock exchanges. Therefore financial modelling serves to understand the correlations between asset and movements of buy/sell in order to reduce risk. Such activities depend on financial analysis tools being available to the trader with which he can make rapid and systematic evaluation of buy/sell contracts. There are other financial activities and it is not an intention of this paper to discuss all of these activities. The main concern of this paper is to propose a parallel algorithm for the numerical solution of an European option. This paper is organised as follows. First, a brief introduction is given of a simple mathematical model for European options and possible numerical schemes of solving such mathematical model. Second, Laplace transform is applied to the mathematical model which leads to a set of parametric equations where solutions of different parametric equations may be found concurrently. Numerical inverse Laplace transform is done by means of an inversion algorithm developed by Stehfast. The scalability of the algorithm in a distributed environment is demonstrated. Third, a performance analysis of the present algorithm is compared with a spatial domain decomposition developed particularly for time-dependent heat equation. Finally, a number of issues are discussed and future work suggested.

A distributed algorithm for European options with nonlinear volatility

A distributed algorithm is developed to solve nonlinear Black-Scholes equations in the hedging of portfolios. The algorithm is based on an approximate inverse Laplace transform and is particularly suitable for problems that do not require detailed knowledge of each intermediate time steps.

Semi-Lagrange time integration for PDE models of asian options

Semi-Lagrange time integration is used with the finite difference method to provide accurate stable prices for Asian options, with or without early exercise. These are combined with coordinate transformations for computational efficiency and compared with published results

Numerical solutions of certain nonlinear models in European options on a distributed computing environment

Financial modelling in the area of option pricing involves the understanding of the correlations between asset and movements of buy/sell in order to reduce risk in investment. Such activities depend on financial analysis tools being available to the trader with which he can make rapid and systematic evaluation of buy/sell contracts. In turn, analysis tools rely on fast numerical algorithms for the solution of financial mathematical models. There are many different financial activities apart from shares buy/sell activities. The main aim of this chapter is to discuss a distributed algorithm for the numerical solution of a European option. Both linear and non-linear cases are considered. The algorithm is based on the concept of the Laplace transform and its numerical inverse. The scalability of the algorithm is examined. Numerical tests are used to demonstrate the effectiveness of the algorithm for financial analysis. Time dependent functions for volatility and interest rates are also discussed. Applications of the algorithm to non-linear Black-Scholes equation where the volatility and the interest rate are functions of the option value are included. Some qualitative results of the convergence behaviour of the algorithm is examined. This chapter also examines the various computational issues of the Laplace transformation method in terms of distributed computing. The idea of using a two-level temporal mesh in order to achieve distributed computation along the temporal axis is introduced. Finally, the chapter ends with some conclusions.

[BSHM meetings] Topics in the history of financial mathematics: early commerce to chaos in modern stock markets

This account provides an overview of the study day, entitled 'Topics in the History of Financial Mathematics: Early commerce to chaos in modern stock markets,' held by the British Society for the History of Mathematics jointly with Gresham College, at Gresham College, London on 25th April 2008. The series of talks explored the development of mathematics and mathematical techniques in a commercial and financial context.

Options to renew leases: time for legislative relief?

Argues that England should follow the example of Australia and New Zealand and give the courts statutory power to grant a tenant relief against forfeiture of the right to exercise an option to renew a lease by reason of the tenant being in breach of covenant. Suggests the legislative provision introducing this power should adopt wording mirroring that in the Law of Property Act 1925 s.146(2).

Borehole Optimisation System (BOS) - a case study assessing options for abstraction of urban groundwater in Nottingham, UK

The recognition that urban groundwater is a potentially valuable resource for potable and industrial uses due to growing pressures on perceived less polluted rural groundwater has led to a requirement to assess the groundwater contamination risk in urban areas from industrial contaminants such as chlorinated solvents. The development of a probabilistic risk based management tool that predicts groundwater quality at potential new urban boreholes is beneficial in determining the best sites for future resource development. The Borehole Optimisation System (BOS) is a custom Geographic Information System (GIs) application that has been developed with the objective of identifying the optimum locations for new abstraction boreholes. BOS can be applied to any aquifer subject to variable contamination risk. The system is described in more detail by Tait et al. [Tait, N.G., Davison, J.J., Whittaker, J.J., Lehame, S.A. Lerner, D.N., 2004a. Borehole Optimisation System (BOS) - a GIs based risk analysis tool for optimising the use of urban groundwater. Environmental Modelling and Software 19, 1111-1124]. This paper applies the BOS model to an urban Permo-Triassic Sandstone aquifer in the city centre of Nottingham, UK. The risk of pollution in potential new boreholes from the industrial chlorinated solvent tetrachloroethene (PCE) was assessed for this region. The risk model was validated against contaminant concentrations from 6 actual field boreholes within the study area. In these studies the model generally underestimated contaminant concentrations. A sensitivity analysis showed that the most responsive model parameters were recharge, effective porosity and contaminant degradation rate. Multiple simulations were undertaken across the study area in order to create surface maps indicating areas of low PCE concentrations, thus indicating the best locations to place new boreholes. Results indicate that northeastern, eastern and central regions have the lowest potential PCE concentrations in abstraction groundwater and therefore are the best sites for locating new boreholes. These locations coincide with aquifer areas that are confined by low permeability Mercia Mudstone deposits. Conversely southern and northwestern areas are unconfined and have shallower depth to groundwater. These areas have the highest potential PCE concentrations. These studies demonstrate the applicability of BOS as a tool for informing decision makers on the development of urban groundwater resources. (c) 2007 Elsevier Ltd. All rights reserved.