Ömür Uğur

An Introduction to Computational Finance

Although there are several publications on similar subjects, this book mainly focuses on pricing of options and bridges the gap between Mathematical Finance and Numerical Methodologies. The author collects the key contributions of several monographs and selected literature, values and displays their importance, and composes them here to create a work which has its own characteristics in content and style.

On optimization, dynamics and uncertainty: A tutorial for gene-environment networks

An emerging research area in computational biology and biotechnology is devoted to mathematical modeling and prediction of gene-expression patterns; to fully understand its foundations requires a mathematical study. This paper surveys and mathematically expands recent advances in modeling and prediction by rigorously introducing the environment and aspects of errors and uncertainty into the genetic context within the framework of matrix and interval arithmetic. Given the data from DNA microarray experiments and environmental measurements we extract nonlinear ordinary differential equations which contain parameters that are to be determined. This is done by a generalized Chebychev approximation and generalized semi-infinite optimization. Then, time-discretized dynamical systems are studied. By a combinatorial algorithm which constructs and follows polyhedra sequences, the region of parametric stability is detected. Finally, we analyze the topological landscape of gene-environment networks in terms of structural stability. This pioneering work is practically motivated and theoretically elaborated; it is directed towards contributing to applications concerning better health care, progress in medicine, a better education and more healthy living conditions.

An algorithmic approach to analyse genetic networks and biological energy production: an introduction and contribution where OR meets biology†

An emerging research area in computational biology and biotechnology is devoted to modelling and prediction of gene-expression patterns. In this article, after a short review of recent achievements we deepen and extend them, especially, by emphasizing and analysing the elegant means of matrix algebra. Based on experimental data, ordinary differential equations with nonlinearities on the right-hand side and a generalized treatment of the absolute shift term, representing the environmental effects, are investigated. Then, the genetic process is studied by a time-discretization, in particular, Runge–Kutta type discretization. By a utilization of the combinatorial algorithm of Brayton and Tong, which is based on the orbits of polyhedra, the possibility of detecting stability and instability regions has been shown. The time-continuous and -discrete systems can be represented by means of matrices allowing biological implications, such as thresholds, and interpretations; which are motivated by our gene-environment networks. A specific contribution of this article consists of a careful but rigorous integration of the environment into modelling and dynamics, and in further new sights. Relations to the parameter estimation within modelling, especially, by using optimization, are indicated, and future research is addressed. †With gratitude dedicated to our dear teacher and friend Prof. Dr Alexander Rubinov who passed away in 2006.

Variational iteration method for Sturm-Liouville differential equations

In this article, Hes variational iteration method is applied to linear Sturm-Liouville eigenvalue and boundary value problems, including the harmonic oscillator. In this method, solutions of the problems are approximated by a set of functions that may include possible constants to be determined from the boundary conditions. By computing variations, the Lagrange multipliers are derived and the generalised expressions of variational iterations are constructed. Numerical results show that the method is simple, however powerful and effective.

Numerical method for optimizing stirrer configurations

A numerical approach for the numerical optimization of stirrer configurations is presented. The methodology is based on a parametrized grid generator, a flow solver, and a mathematical optimization tool, which are integrated into an automated procedure. The flow solver is based on the discretization of the Navier-Stokes equations by means of the finite-volume method for block-structured, boundary-fitted grids with multi-grid acceleration and parallelization by grid partitioning. The optimization tool is an implementation of a trust region based derivative-free method. It is designed to minimize smooth functions whose evaluations are considered expensive and whose derivatives are not available or not desirable to approximate. An exemplary application illustrates the functionality and the properties of the proposed method.

An Eigenfunction Expansion for the Schrödinger Equation with Arbitrary Non-Central Potentials

An eigenfunction expansion for the Schrödinger equation for a particle moving in an arbitrary non-central potential in the cylindrical polar coordinates is introduced, which reduces the partial differential equation to a system of coupled differential equations in the radial variable r. It is proved that such an orthogonal expansion of the wavefunction into the complete set of Chebyshev polynomials is uniformly convergent on any domain of (r,θ). As a benchmark application, the bound states calculations of the quartic oscillator show that both analytical and numerical implementations of the present method are quite satisfactory.

Derivative free optimization methods for optimizing stirrer configurations

In this paper a numerical approach for the optimization of stirrer configurations is presented. The methodology is based on a flow solver, and a mathematical optimization tool, which are integrated into an automated procedure. The flow solver is based on the discretization of the incompressible Navier-Stokes equations by means of a fully conservative finite-volume method for block-structured, boundary-fitted grids, for allowing a flexible discretization of complex stirrer geometries. Two derivative free optimization algorithms, the DFO and CONDOR are considered, they are implementations of trust region based derivative-free methods using multivariate polynomial interpolation. Both are designed to minimize smooth functions whose evaluations are considered to be expensive and whose derivatives are not available or not desirable to approximate. An exemplary application for a standard stirrer configuration illustrates the functionality and the properties of the proposed methods. It also gives a comparison of the two optimization algorithms.

Solution of initial and boundary value problems by the variational iteration method

The Variational Iteration Method (VIM) is an iterative method that obtains the approximate solution of differential equations. In this paper, it is proven that whenever the initial approximation satisfies the initial conditions, VIM obtains the solution of Initial Value Problems (IVPs) with a single iteration. By using this fact, we propose a new algorithm for Boundary Value Problems (BVPs): linear and nonlinear ones. Main advantage of the present method is that it does not use Greens function, however, it has the same effect that it produces the exact solution to linear problems within a single, but simpler, integral. In order to show the effectiveness of the method we give some examples including linear and nonlinear BVPs.

On the single name CDS price under structural modeling

Regulators, banks and other market participants realized that true assessment of the credit risk is more critical and complex than their ex-ante appraisals after the US Credit Crunch. They have turned their attention to complex credit risk models and credit instruments such as credit derivatives. Credit default swap contracts (CDSs) are the most common credit derivatives used for speculation and hedging purposes in the credit markets. Thus, in this paper we fundamentally study the pricing of a single name CDS via the discounted cash flow method with survival probability functions of two pioneer structural credit risk models, Merton model and Black-Cox model with constant barrier. Hence, this approach is not only a new one, but also provides a practical technique to price CDSs using publicly available data of equity returns.

On the methods of pricing American options: case study

In this study, a comparative analysis of numerical and approximation methods for pricing American options is performed. Binomial and finite difference approximations are discussed; furthermore, Roll-Geske-Whaley, Barone-Adesi and Whaley and Bjerksund-Stensland analytical approximations as well as the least-squares Monte Carlo method of Longstaff and Schwartz are presented. Applicability and efficiency in almost all circumstances, numerical solutions of the corresponding free boundary problem is emphasized. Methods used in pricing American options are also compared on dividend and non-dividend paying assets; and their pros and cons are discussed along with numerical experiments.