May 5, 2006
Department of Computer Science
University of Toronto
Toronto, Ontario, Canada

Programme (with abstracts of talks)
All activities in Room B 149 Earth Sciences Building (5 Bancroft Ave., south side), except otherwise stated.

Friday 5 May 2006
8:30	Refreshments
9:25	Opening remarks
9:30	Xiaofang Ma, University of Toronto Loss Distribution Evaluation for Synthetic CDOs
	Efficient numerical methods for a synthetic CDO pool loss distribution evaluation are necessary for both pricing and risk management. In this talk, we study two approaches to obtaining the loss distribution of a synthetic CDO pool. The first approach computes the loss distribution exactly provided that each name's recovery-adjusted notional value sits on a common lattice. A new efficient and stable recursive method is proposed for this problem. The second approach evaluates the loss distribution approximately. The normal power and improved compound Poisson approximations are considered for this problem. Accuracy and performance comparisons between the methods proposed in this talk and the methods proposed by other researchers show that our methods are applicable to pricing and risk management of synthetic CDOs.
9:55	Wei Xu, McMaster University A Fast Symmetric SVD Algorithm for Hankel Matrices
	We present a fast SVD algorithm for Hankel Matrices. It computes all the singular values and singular vectors of an n-by-n Hankel matrix in O(n^2 log n) operations. It combines the Lanczos tridiagonalization incorporated with the FFT for fast Hankel matrix-vector multiplication, the implicit QR method for computing the singular values, and the twisted factorization method for computing the singular vectors. To our best knowledge, this is the first fast SVD algorithm for Hankel matrices.
10:20	Simon S. Clift, University of Waterloo Numerical Solution of Two-Factor, Jump Diffusion Models for Option Pricing
	The evolution of the price of financial assets can be usefully modelled as a series of random jumps superimposed on a diffusion process. The fair value of an option contract written on two such assets, or written on one asset with stochastic volatility, is given by solving a two--dimensional, parabolic, partial integro-differential equation (PIDE). This talk presents an implicit difference method for solving these PIDE's. The approach avoids a dense linear system solution by combining a fixed point iteration scheme with an FFT. The method prices both American and European style contracts independent of the option payoff and distribution of jumps, and is shown to be practical, convergent and efficient.
10:45	Coffee Break
11:05	Paul Muir, St. Mary's University A User-Friendly Fortran 90/95 Boundary Value ODE Solver
	Boundary value ordinary differential equations (BVODEs) are systems of ODEs that have solution constraints imposed at two or more distinct points within the problem domain. After presenting BVODES arising in several applications, we will briefly survey a well-known software package for the numerical solution of these problems. The success of these solvers depends critically on efficient high order discretization techniques and powerful adaptive mesh refinement algorithms, and it has become apparent that these same tools can be employed in the development of packages for the numerical solution of systems of 1D parabolic PDEs. We will briefly review some recent work in this direction. The main focus of the talk will consider a new user-friendly Fortran 90/95 software package for the numerical solution of BVODEs. Based on an earlier Fortran 77 package, this new solver takes advantage of a number of language features of Fortran 90/95 to provide a substantially simpler experience for the user, while at the same time employing the underlying robust, high quality algorithms of the original code. Several examples are presented to demonstrate the capabilities of the new solver. This solver was developed jointly with Larry Shampine, Southern Methodist University.
11:50	Hossein Zivari Piran, University of Toronto DDVERK90: A first step toward a user-friendly DDE modeling package
	Delay differential equations(DDEs) provide powerful and rich mathematical models that have been proven to be useful in simulating some real life problems. Ideally a DDE modelling package should provide facilities for approximating the solution, performing a sensitivity analysis and estimating unknown parameters. As a first step we have developed a new Fortran 90 DDE solver DDVERK90 that aims to combine the effectiveness of the Fortran 77 DDE solver DDVERK and the user-friendliness of the MATLAB ODE solvers. We have added the capability of embedded event location to facilitate many tasks that arise in applications. We will illustrate, through several experiments, the ease of use and effectiveness of this method.
12:15	Lunch Break
1:45	Thomas Coleman, University of Waterloo Practical optimization for finance
	Optimization methodology has played a role in addressing important problems in finance for many years; however, recently there has been a surge of new interest in applying the optimization viewpoint. We discuss this new interest touching on finance applications such as hedging, insurance applications, index tracking, and robust portfolio optimization. We will point out both benefits and pitfalls.
2:30	Cheng Liu, York University A New Instantaneous Frequency Estimator using Time-Frequency Distribution
	Accurate estimation of instantaneous frequency (IF) provides a useful insight on time-varying frequency characteristics of a non-stationary signal. Time-frequency distribution (TFD) of a signal concentrates signal energy at and around IF in the time-frequency plane. Hence, one can estimate IF by detecting the maxima of its TFD. However, multi-component non-stationary signals often present many challenges in IF estimation. Here, we develop a new IF estimator based on the TFD given by the Stockwell transform (ST). As a linear transform with frequency-dependent resolution in time-frequency plane, the ST can be effective in estimating IF, particularly for multi-component non-stationary signals. We demonstrate the performance of the ST-based IF estimator through numerical simulations and comparison with the commonly used estimators.
2:55	Kimia Ghobadi, McMaster University Discretization and Optimization of a Heat-Transfer Problem
	We consider a heat-transfer problem, where a one-dimensional bar is heated at its ends. The temperature at each point is bounded below by a given function. The initial conditions are fixed, while the boundary conditions control the temperature, such that it is above its lower-bound. Our goal is to minimize the energy needed to satisfy it. If we derive the optimality conditions and then optimize the system of equations, then this optimization problem is not solvable with traditional optimal control methods. However, by discretizing it in time and space, we translate this problem into a convex-quadratic problem. The related coefficient matrices are large and very sparse. A good optimization solver, for example Mosek, can solve this problem efficiently, even for very fine discretizations. In this talk, we explore the properties of the convex-quadratic problem, and in particular, its duality properties and the sparsity structure of the matrices that arise.
3:20	Break
3:40	Richard Pancer, University of Toronto The Numerical Solution of Almost Block Diagonal or Bordered Almost Block Diagonal Linear Systems Arising in Numerical Methods for BVODEs
	In this talk we discuss three algorithms for solving Almost Block Diagonal (ABD) or Bordered Almost Block Diagonal (BABD) linear systems that arise during the numerical solution of Boundary Value Ordinary Differential Equations (BVODEs). Two of the algorithms, SLF-QR and SLF-LU, were discovered independently by us and by S.J. Wright who presented the algorithms and analyzed their stability in two papers in the 1990s. In these papers Wright proves SLF-QR is stable and shows SLF-LU is stable under certain assumptions. The third algorithm, RSCALE, is based on a different numerical technique which reduces fill-in and local operation counts. For many problems, RSCALE is approximately 1.2 and 2.2 times faster than SLF-LU and SLF-QR, respectively. Moreover, we show that SLF-LU is potentially unstable on a surprising number of randomly-generated, yet well-posed, problems. Since these problems pose no difficulty for either of the other two algorithms, we believe that SLF-QR, not SLF-LU, may be RSCALE's nearest competitor in terms of both speed and reliability.
4:05	Iris R. Wang, University of Waterloo Robust Numerical Valuation of European and American Options under the CGMY Process
	Recently, option pricing under exponential Levy process has been extensively studied. In this talk we present an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Levy process. For processes of finite variation, quadratic convergence is obtained as the mesh and time step are refined. For infinite variation processes, better than first order accuracy is achieved. The jump component in the neighborhood of log jump size zero is specially treated using Taylor expansion approximation and the advection part is dealt with using the semi-Lagrangian scheme. The resulting partial integro-differential equation (PIDE) is then computed using preconditioned BiCGSTAB method coupled with a fast Fourier transform. The convergence properties of the BiCGSTAB scheme are discussed and compared with a fixed point iteration. Numerical results on convergence and performance of pricing European and American options under processes of finite and infinite variation are presented.
4:30	Maja Omanovic, University of Waterloo Efficient Least-Squares Matching of Dental X-rays for Forensic Identification Purposes
	Biometric identifiers such as dental records have been widely utilized as tools in forensic identification. With the vast volume of cases that need to be investigated by forensic odontologists, a computer aided dental identification system is inevitable. The semi-automatic approach to dental identification is described. The least squares template matching is used in order to efficiently match dental x-rays in the frequency domain. Given a dental x-ray with a marked region if interest (ROI), we search the database of x-rays in order to retrieve a closest match with respect to some salient feature. Different features such as dental work and root contours have been utilized. To express the degree of similarity/overlap between the two dental radiographs we propose the use of the slightly extended sum of squared differences (SSD) cost function. Efficient evaluation of this cost function using the Fast Fourier Transform (FFT) makes it feasible to simultaneously find the optimal shift (alignment) and the optimal intensity remapping between the two radiographs. Initial experimental results on a small database of dental x-rays indicate that matching dental records using the extended SSD cost function is a viable method, sensitive and precise enough for human dental identification.
4:55	Robert Enenkel, IBM Toronto Lab Can software floating-point division be faster than hardware?
	The hardware implementation of floating-point division on many modern pipelined processors uses a Newton iteration or Taylor series-based algorithm. Although the hardware divide instruction generally provides the shortest-latency full-accuracy result available, there are situations where software implementations are preferable, and are automatically generated by compilers. These include cases where full accuracy, full operand range, or strict exception handling are not required. Perhaps less obvious, however, are cases where opportunities for concurrency exist and maximum performance depends more on maximizing throughput than on minimizing latency. This talk gives an overview of hardware and software floating-point division algorithms used by IBM(TM) PowerPC(TM) processors and compilers, gives performance statistics, and compares the advantages of various approaches.
5:30	Reception (at Faculty Club)