Friday August 29
| Title | On the Relation between Option Prices and Higher Order Moments |
| Speaker | Yuji Yamada |
| Graduate School of Business Sciences | |
| University of Tsukuba, Japan | |
| http://www.cds.caltech.edu/~yuji/ |
Abstract:
It is widely recognized today that there is a non-negligible discrepancy between the standard Black-Scholes model and real market behavior, which appears as the “smile effect” or “implied volatility smile” in derivative security markets, and option valuation/hedging techniques have been extended to more realistic assumptions in a number of ways. One of the most common approaches is to generalize the underlying stock process and/or the underlying stock distribution. Along this line, we first present a general model of stochastic dynamics that incorporates any given moments (or cumulants) of the underlying stock, and then apply mean square optimal hedging. A single parameterization of a multinomial lattice model is provided, where the first m moments are matched over each time step. It is shown that the proposed parameterization covers standard results for binomial and trinomial lattices and that any given moments can be matched using a multinomial lattice with an adequate number of branches. Next, we discuss the relations between option prices and higher order moments such as skewness and kurtosis. The term structure of skewness vs. smirk and the term structure of kurtosis vs. smile is investigated. Finally, we compare our approach with a jump-diffusion model, and mention about the implementation of the minimal entropy martingale measure pricing method for geometric Levy processes using a multinomial lattice.
Friday September 05
| Title | Ab Initio study of the mode coupling in N-methylacetamide: A model system for the peptide group in proteins |
| Speaker | Susan Gregurick |
| Chemistry and Biochemistry | |
| UMBC | |
| http://research.umbc.edu/~smith/chem/faculty/gregurick/skg.html |
Abstract:
The second-order Moeller-Plesset ab initio electronic structure method is used to compute points for the anharmonic mode-coupled potential energy surface of N-methylacetamide (NMA) in the trans_ct configuration, including all degrees of freedom. The anharmonic vibrational states and the spectroscopy are directly computed from this potential surface using the Correlation Corrected Vibrational Self-Consistent Field (CC-VSCF) method. The results are compared with CC-VSCF calculations using both the standard and improved empirical Amber-like force fields and available low temperature experimental matrix data.
Analysis of our calculated spectroscopic results show that:
- The excellent agreement between the ab initio CC-VSCF calculated frequencies and the experimental data suggest that the computed anharmonic potentials for N-methylacetamide are of a very high quality.
- For most transitions, the vibrational frequencies obtained from the ab initio CC-VSCF method are superior to those obtained using the empirical CC-VSCF methods, when compared with experimental data. However, the improved empirical force field yields better agreement with the experimental frequencies as compared with a standard AMBER-type force field.
- The improved empirical force field in particular overestimates anharmonic couplings for the amide II mode, the methyl asymmetric bending modes, the out-of-plane methyl bending modes, and the methyl distortions.
- Disagreement between the ab initio and empirical anharmonic couplings is greater than the disagreement between the frequencies, and thus the anharmonic part of the empirical potential seems to be less accurate than the harmonic contribution.
- Both the empirical and ab initio CC-VSCF calculations predict a negligible anharmonic coupling between the amide I and other internal modes.
The implication of this is that the intramolecular energy flow between the amide I and the other internal modes may be smaller than anticipated. These results may have important implications for the anharmonic force fields of peptides, for which N-methylacetamide is a model.
Friday September 12
| Title | Interpretations and Formulations for “Ill-Posed Problems” |
| Speaker | Thomas I. Seidman |
| Department of Mathematics and Statistics | |
| UMBC | |
| http://www.math.umbc.edu/~seidman |
Abstract:
Hadamard called problems “ill-posed” if there was not a continuous solution map [e.g., solving Ax = b when A is compact] and felt that such problems were hopeless. Nevertheless, such problems do arise in realistic and significant applications and are often handled successfully — although, as might be expected, this requires more effort in computation and more care in formulation and interpretation. In this talk we will primarily emphasize the issues involved and discuss what makes it possible to obtain useful results in an extremely sensitive context.
Friday September 19
No talk today
Friday September 26
| Title | Suface Motion by Surface Diffusion |
| Speaker | Ricardo Nochetto |
| Department of Mathematics | |
| University of Maryland College Park | |
| http://www.math.umd.edu/~rhn/ |
Abstract:
Surface diffusion is a 4th order (highly nonlinear) geometric driven motion of a surface with normal velocity proportional to the surface Laplacian of mean curvature. We present a novel variational formulation for the parametric case based on a mixed approach, which converts the system into four lower order PDE. We develop a finite element method, and propose a Schur complement approach to solve the resulting linear systems. We also introduce a new graph formulation and show stability and an optimal a priori error estimate. We illustrate key features of both formulations with several significant simulations, some with pinch-off in finite time. This is joint work with E. Baensch and P. Morin.
Friday October 03
| Title | Electrostatic-Elastic Interactions, Theory and Experiment |
| Speaker | John Pelesko |
| Department of Mathematics | |
| University of Delaware | |
| http://www.math.udel.edu/~pelesko/ |
Abstract:
In 1968, in the context of investigating fundamental equations in electrohydrodynamics, G.I. Taylor studied the electrostatic deflection of elastic membranes. Utilizing soap film as the membrane material and applying a fixed high voltage potential difference between two supported circular membranes, Taylor showed experimentally that at a critical voltage the two membranes snap together and touch. That is, the equilibrium state where the membranes remained separate that existed at smaller voltages either became unstable or failed to exist. This instability is familiar to researchers in the MEMS (microelectromechanical systems) and NEMS (nanoelectromechanical systems) fields where it is known as the “pull-in” instability. In fact, in an interesting historical coincidence H.C. Nathanson and his coworkers studied this instability in the context of a primitive MEMS device at roughly the same time as Taylor was conducting his studies. Nathanson is responsible for the “pull-in” nomenclature and the analysis of a mass-spring model of this effect. Taylor, in conjunction with R.C. Ackerberg developed and numerically analyzed a more accurate membrane based model of electrostatic deflection. Recently, a rigorous analysis of this model was completed. Surprisingly, even this simple model of electrostatic deflection contains a rich solution set exhibiting a bifurcation diagram with infinitely many folds.
In this talk, we provide an overview of recent results on the interaction of elastic membranes with electrostatic fields. We discuss a re-creation of the Taylor experiment, some new experimental results and discuss the relevance of this research to MEMS and NEMS systems.
Friday October 10
| Title | Adaptive Numerical Methods for Sensitivity Analysis of Differential-Algebraic Equations and Partial Differential Equations |
| Speaker | Linda Petzold |
| Mechanical and Environmental Engineering | |
| U.C. Santa Barbara | |
| http://www.me.ucsb.edu/dept_site/people/new_faculty_pages/petzold_page.html |
Abstract:
Sensitivity analysis of differential-algebraic equation (DAE) systems generates essential information for design optimization, parameter estimation, optimal control, model reduction, process sensitivity and experimental design. Recent work on methods and software for sensitivity analysis of DAE systems has demonstrated that forward sensitivities can be computed reliably and efficiently. However, for problems which require the sensitivities with respect to a large number of parameters, the forward sensitivity approach is intractable and the adjoint (reverse) method is advantageous. In this talk we give the adjoint system for general DAEs and investigate some of its fundamental analytical and numerical properties. We describe our new adjoint DAE software and outline some issues which are critical to the implementation.
Defining the adjoint sensitivity system and writing the appropriate software to describe it can be a very challenging problem for large-scale engineering systems, particularly when it comes to finding appropriate boundary conditions for the adjoint partial differential equation (PDE) system. Therefore our goal for both DAE and PDE systems has been the development of methods and software in which generation and solution of the sensitivity system are transparent to the user. This has been largely achieved for DAE systems. We will propose a solution to this problem for PDE systems solved with adaptive mesh refinement.
Friday October 17
| Title | Text Mining, Clustering, and Beyond… |
| Speaker | Jacob Kogan |
| Department of Mathematics and Statistics | |
| UMBC | |
| http://www.math.umbc.edu/~kogan |
Abstract:
Clustering is a fundamental problem which has numerous applications in many disciplines. Large and often high dimensional data sets are now increasingly common and available. Clustering techniques are used to discover natural groups in data sets, and to identify abstract structures that might reside there, without having any background knowledge of the characteristics of the data.
In this talk we focus on Text Mining applications and provide a gentle introduction to this exciting research area. Performance of a number of clustering algorithms will be discussed and illustrated.
Friday October 24
| Title | Exponential boundary stabilization of a 3-D Navier-Stokes equation in the neighborhood of an unstable equilibrium |
| Speaker | Irena Lasiecka |
| Department of Mathematics | |
| University of Virginia | |
| http://www.math.virginia.edu/Faculty/lasiecka/ |
Abstract:
We consider the Navier-Stokes equation defined on a bounded domain Ω ⊂ R3 with smooth boundary. We are interested in uniform stability of solutions in the neighborhood of an unstable equilibrium.
The aim of this talk is to show that solutions near the equilibrium can be exponentially stabilised by means of a finite dimensional feedback supported on the boundary ∂Ω. The proof of this stabilization result is based on optimization techniques. The main tool used is the construction of an appropriate “badly behaving” Riccati Equation (RE) with strongly unbounded coefficients. A distinct feature of this RE is that its solution (Riccati operator or value function) is an unbounded operator on the underlying state space. This unboundedness allows us to control topological issues related to supercriticality of the nonlinearity appearing in 3-D Navier-Stokes dynamics.
This talk is based on joint work with Viorel Barbu.
Friday October 31
| Title | Duality in Nonconvex Quadratic Problems |
| Speaker | Marc Teboulle |
| School of Mathematical Sciences | |
| Tel-Aviv University | |
| http://www.math.tau.ac.il/~teboulle/ |
Abstract:
Quadratic problems are currently one of the most active areas of research in optimization. These problems arise in a broad range of fields from mathematics, engineering and control to hard combinatorial problems. In this talk we concentrate on the role and use of convex duality in the analysis of some classes of nonconvex quadratic problems and other closely related topics such as: the convexity of the range of quadratic maps, the S-procedure, and semidefinite relaxations of combinatorial optimization problems.
The main theme is to show that elementary convex duality arguments can be successfully used to: identify classes of problems where the dual bound is exact, derive global optimality conditions, detect “hidden” convexity in seemingly hard nonconvex problems, demonstrate that well known semidefinite relaxations of combinatorial problems (e.g., max-cut, stable set) are particular instances of the dual approach, show that dual approximations problems are somehow providing best computationally tractable bounds, and to formulate some open and challenging questions in the aforementioned topics.
The talk is intended to a wide audience. We will assume no prior knowledge in continuous optimization, duality, and on the other presented topics.
Friday November 07
| Title | Generating Set Search for Nonlinear Programming |
| Speaker | Virginia Torczon |
| Department of Computer Science | |
| College of William and Mary | |
| http://www.cs.wm.edu/~va/ |
Abstract:
Set Search (GSS) defines a class of direct search methods that rely on a set of generators for the cone of feasible descent directions. Using this observation leads to a unifying framework that lends itself to a variety of convergence results. Stationarity results for derivative-free, GSS methods for unconstrained optimization will be the focus of this talk. The principles underlying the analysis for the unconstrained case can be generalized to handle bound constraints and linear constraints, as well as extensions to problems with nonlinear constraints.
A particular measure of stationarity will be shown to be of the same order as the step length at an identifiable subset of the iterations. Thus, even in the absence of explicit knowledge of the derivatives of the objective function, there is information about stationarity. These results help clarify the fundamental geometrical ideas underlying several classes of direct search algorithms. In addition, these results validate a practical stopping criterion for such algorithms and lead to local convergence results.
This is joint work with Tamara G. Kolda, Sandia National Laboratories, and Robert Michael Lewis, College of William & Mary.
Friday November 14
| Title | Solving Semidefinite Programs via Nonlinear Programming |
| Speaker | Samuel Burer |
| College of Business | |
| The University of Iowa | |
| http://www.biz.uiowa.edu/sburer/profile.html |
Abstract:
The field of semidefinite programming (SDP) has received considerable attention in the last decade due to its numerous applications and nice theoretical properties. Although standard interior-point methods can theoretically solve SDPs in polynomial-time and have proven effective on small- to medium-scale problems, their practicality on large-scale problems is currently limited due to high computational requirements.
In this talk, we discuss recent efforts to solve large-scale SDPs using algorithms other than the standard interior-point methods. In particular, we show how the primal SDP can be reformulated to allow the use of fast first-order nonlinear programming algorithms such as the limited memory BFGS approach. Finally, we provide computational evidence demonstrating the considerable progress that these methods have made on large-scale SDPs.
Friday November 21
| Title | Integral Equation Methods, Fast Algorithms, and Applications |
| Speaker | Jingfang Huang |
| Department of Mathematics | |
| University of North Carolina at Chapel Hill | |
| http://www.amath.unc.edu/Faculty/huang/ |
Abstract:
Integral equation methods are particularly appropriate for the solution of many fundamental equations of mathematical physics for several reasons: they are unconditionally stable, they are insensitive to the complexity of the geometry, and they do not require the artificial truncation of the computational domain as do finite difference and finite element techniques. Methods of this type have not become widespread due to the high cost of evaluating/solving the dense linear systems. In the last twenty years, however, this is changing rapidly due to the introduction of fast algorithms.
In this talk, we first study a simple model problem, and compare integral equation methods with finite difference and finite element methods. We then give an overview of integral equation formulations for the Poisson, Helmholtz, and diffusion equations, and discuss the basic ideas for fast algorithms including the fast multipole method (FMM), the particle mesh Ewald (PME), and the precorrected fast Fourier transform (pFFT). We conclude the talk by discussing several current research topics and their applications.
This talk should be accessible to graduate students.
Friday November 28
Thanksgiving
Friday December 05
| Title | Long-Distance Access Network Design |
| Speaker | S. Raghavan |
| School of Business | |
| University of Maryland College Park | |
| http://www.glue.umd.edu/~raghavan |
Abstract:
Long-distance telephone companies in the United States (U.S.) pay access fees to local telephone companies to transport calls that originate and terminate on their networks. These charges form the largest portion of the cost of providing long-distance service. Recent changes in the structure of access rates, which were mandated by the Federal Communications Commission (FCC), have created opportunities for long-distance companies to better manage access costs. In this paper, we develop an optimization-based approach to the economic design of access networks. Our novel solution approach combines stochastic aspects of the problem with a challenging discrete facility location problem in a three-phase algorithm. Computational results indicate a potential cost savings of hundreds of millions of dollars annually for long-distance companies.
(Paper available at http://www.glue.umd.edu/~raghavan/access.pdf)
The above paper is to appear in Management Science, and was awarded second place in the INFORMS Junior Faculty Paper competition this year. This demonstrates a nice application on queuing, Integer programming, and networks to a real-world application.