Matthias Ehrhardt

Discrete transparent boundary conditions for the Schrödinger equation

This paper is concerned with transparent boundary conditions for the one dimensional time–dependent Schrodinger equation. They are used to restrict the original PDE problem that is posed on an unbounded domain onto a finite interval in order to make this problem feasible for numerical simulations. The main focus of this article is on the appropriate discretization of such transparent boundary conditions in conjunction with some chosen discretization of the PDE (usually Crank–Nicolson finite differences in the case of the Schrodinger equation). The presented discrete transparent boundary conditions yield an unconditionally stable numerical scheme and are completely reflection–free at the boundary.

A Fast, Stable and Accurate Numerical Method for the Black-Scholes Equation of American Options

In this work we improve the algorithm of Han and Wu [SIAM J. Numer. Anal. 41 (2003), 2081–2095] for American Options with respect to stability, accuracy and order of computational effort. We derive an exact discrete artificial boundary condition (ABC) for the Crank–Nicolson scheme for solving the Black–Scholes equation for the valuation of American options. To ensure stability and to avoid any numerical reflections we derive the ABC on a purely discrete level.Since the exact discrete ABC includes a convolution with respect to time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate ABCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove a simple stability criteria for the approximated artificial boundary conditions.Finally, we illustrate the efficiency and accuracy of the proposed method on several benchmark examples and compare it to previously obtained discretized ABCs of Mayfield and Han and Wu.

A nonstandard finite difference scheme for convection-diffusion equations having constant coefficients

In this note we derive, using the subequation method, a new nonstandard finite difference scheme (NSFD) for a class of convection-diffusion equations having constant coefficients. Despite the fact that this scheme has nonlinear denominator functions of the step sizes (even for linear PDEs), it has a couple of favourable properties: it is explicit and due to its construction it reproduces important properties of the solution of the parabolic PDE. This proposed method conserves, by construction, the positivity of the solution if one choses a right combination of spatial and temporal step sizes and hence it is perfectly suited for solving for example air pollution problems or the Black-Scholes equation for the valuation of standard options, since it avoids negative values for the calculated prices. Finally, we illustrate the usefulness of this newly proposed method on a classical benchmark example from the literature.

Characteristic boundary conditions in the lattice Boltzmann method for fluid and gas dynamics

For numerically solving fluid dynamics problems efficiently one is often facing the problem of having to confine the computational domain to a small domain of interest introducing so-called non-reflecting boundary conditions (NRBCs). In this work we address the problem of supplying NRBCs in fluid simulations in two space dimensions using the lattice Boltzmann method (LBM): so-called characteristic boundary conditions are revisited and transferred to the framework of lattice Boltzmann simulations. Numerical tests show clearly that the unwanted unphysical reflections can be reduced significantly by applying our newly developed methods. Hereby the key idea is to transfer and generalize Thompsons boundary conditions originally developed for the nonlinear Euler equations of gas dynamics to the setting of lattice Boltzmann methods. Finally, we give strong numerical evidence that the proposed methods possess a long-time stability property.

A high-order compact method for nonlinear Black-Scholes option pricing equations of American options

Due to transaction costs, illiquid markets, large investors or risks from an unprotected portfolio the assumptions in the classical Black–Scholes model become unrealistic and the model results in nonlinear, possibly degenerate, parabolic diffusion–convection equations. Since in general, a closed-form solution to the nonlinear Black–Scholes equation for American options does not exist (even in the linear case), these problems have to be solved numerically. We present from the literature different compact finite difference schemes to solve nonlinear Black–Scholes equations for American options with a nonlinear volatility function. As compact schemes cannot be directly applied to American type options, we use a fixed domain transformation proposed by Ševčovič and show how the accuracy of the method can be increased to order four in space and time.

Modelling the dynamics of the students’ academic performance in the German region of the North Rhine-Westphalia: an epidemiological approach with uncertainty

Student academic underachievement is a concern of paramount importance in Europe, where around 15% of the students in the last high school courses do not achieve the minimum knowledge academic requirement. In this paper, we propose a model based on a system of differential equations to study the dynamics of the students’ academic performance in the German region of the North Rhine-Westphalia. This approach is supported by the idea that both good and bad study habits, are a mixture of personal decisions and influence of classmates. This model allows us to forecast the student academic performance by means of confidence intervals over the next few years.

BILATERAL COUNTERPARTY RISK VALUATION OF CDS CONTRACTS WITH SIMULTANEOUS DEFAULTS

We analyze the general risk-neutral valuation for counterparty risk embedded in a Credit Default Swap (CDS) contract by adapting the recent findings of Brigo and Capponi (2009) to allow for simultaneous defaults among the two parties and the underlying reference credit, while the counterparty risk is considered bilaterally. For the default intensities, we employ a Markov copula model allowing for the possibility of a simultaneous default. The dependence between defaults of three names in a CDS contract and the wrong-way risk will thus be represented by the possibility of simultaneous defaults.We investigate numerically the effect of considering simultaneous defaults on the counterparty risk valuation of a CDS contract. Finally, we study a CDS contract between Royal Dutch Shell and British Airways based on Lehman Brothers applying this methodology, illustrating the bilateral adjustments with the possibility of simultaneous defaults in concrete crisis situations.

Novel methods in computational finance

It is with pleasure that we offer the readers of the International Journal of Computer Mathematics this special issue consisting of some of the most significant contributions to computational and mathematical methods with advanced applications in finance presented at the International Conference on Mathematical Modelling in Engineering & Human Behaviour 2013, held at the Instituto Universitario de Matemática, Multidisciplinar, Polytechnic City of Innovation in Valencia, Spain, September 4–6, 2013, cf. http://jornadas. imm.upv.es/2013/. Since its founding the International Conference on Mathematical Modelling in Engineering & Human Behaviour has been a truly multi-disciplinary conference, covering all aspects of applied mathematics in a very broad field of areas of science and engineering with its increasing level of complexity. The aim of this conference series is to encourage cross-fertilization between different disciplines and to gain new insights into the emerging research trends in mathematical modelling and engineering methods. The first paper of this special issue, Kilianová and Trnovská [6], analyses a problem of dynamic stochastic portfolio optimization modelled by a fully non-linear Hamilton–Jacobi– Bellman equation. The authors provide an application to robust portfolio optimization for the German DAX30 Index. The article The operation of infimal/supremal convolution in mathematical economics, by Bayón et al. [1], considers the infimal convolution operation arising in the analysis of several problems of mathematical economics. Further, the authors present a new application: the analytical solution of the utility maximization problem obtained by applying the supremal convolution operation. The third article, Optimal allocation-consumption problem for a portfolio with an illiquid asset [2], considers an optimization problem for a portfolio with an illiquid, a risky and a riskfree asset. The authors study two different distributions of the liquidation time of the illiquid asset – a classical exponential distribution and a more practically relevant Weibull distribution. The research by Calvo-Garrido and Vázquez [3] deals with the valuation of fixed-rate mortgages including prepayment and default options, where the underlying stochastic factors are the house price and the interest rate. The pricing model is a free boundary problem associated with a partial differential equation (PDE). Appropriate numerical methods based on a Lagrange– Galerkin discretization of the PDE, an augmented Lagrangian active set method and a Newton iteration scheme are proposed. In their work, On splitting-based numerical methods for nonlinear models of European options, Koleva and Vulkov [7], study a large class of non-linear models of European options as parabolic equations with quasi-linear diffusion and fully non-linear hyperbolic part. The

Modelling and Numerical Simulation

This field of the research network deals with the development of methods based on mathematical modelling, system theory and numerical analysis, in order to analyse, improve and simulate numerically in an efficient and robust way the system “Reacting Atmosphere” on sub and overall model level. A transdis- ciplinary research approach is inevitable, combining modelling and simulation expertise of mathematics, scientific computing, atmospheric physics and chemistry, economics, and social sciences available in this research network. The aim is to construct a tool for illustrating the broad range of interdependencies in system “humans-atmosphere-air quality-climate” and their impacts on air quality and climate.

Mathematical Modelling and Numerical Simulation of Oil Pollution Problems

Written by outstanding experts in the fields of marine engineering, atmospheric physics and chemistry, fluid dynamics and applied mathematics, the contributions in this book cover a wide range of subjects, from pure mathematics to real-world applications in the oil spill engineering business. Offering a truly interdisciplinary approach, the authors present both mathematical models and state-of-the-art numerical methods for adequately solving the partial differential equations involved, as well as highly practical experiments involving actual cases of ocean oil pollution. It is indispensable that different disciplines of mathematics, like analysis and numerics,together with physics, biology, fluid dynamics, environmental engineering and marine science, join forces to solve todays oil pollution problems.The book will be of great interest to researchers and graduate students in the environmental sciences, mathematics and physics, showing the broad range of techniques needed in order to solve these pollution problems; and to practitioners working in the oil spill pollution industry, offering them a professional reference resource.