Determination of the Probability Distribution Measures from Market Option Prices Using the Method of Maximum Entropy in the Mean

Abstract

We consider the problem of recovering the risk-neutral probability distribution of the price of an asset, when the information available consists of the market price of derivatives of European type having the asset as underlying. The information available may or may not include the spot value of the asset as data. When we only know the true empirical law of the underlying, our method will provide a measure that is absolutely continuous with respect to the empirical law, thus making our procedure model independent. If we assume that the prices of the derivatives include risk premia and/or transaction prices, using this method it is possible to estimate those values, as well as the no-arbitrage prices. This is of interest not only when the market is not complete, but also if for some reason we do not have information about the model for the price of the underlying.     

Social Network Discovery by Mining Spatio-Temporal Events

Knowing patterns of relationship in a social network is very useful for law enforcement agencies to investigate collaborations among criminals, for businesses to exploit relationships to sell products, or for individuals who wish to network with others. After all, it is not just what you know, but also whom you know, that matters. However, finding out who is related to whom on a large scale is a complex problem. Asking every single individual would be impractical, given the huge number of individuals and the changing dynamics of relationships. Recent advancement in technology has allowed more data about activities of individuals to be collected. Such data may be mined to reveal associations between these individuals. Specifically, we focus on data having space and time elements, such as logs of people's movement over various locations or of their Internet activities at various cyber locations. Reasoning that individuals who are frequently found together are likely to be associated with each other, we mine from the data instances where several actors co-occur in space and time, presumably due to an underlying interaction. We call these spatio-temporal co-occurrences events, which we use to establish relationships between pairs of individuals. In this paper, we propose a model for constructing a social network from events, and provide an algorithm that mines these events from the data. Experiments on a real-life data tracking people's accesses to cyber locations have also yielded encouraging results. Hady W. Lauw is a graduate student at the School of Computer Engineering, Nanyang Technological University, Singapore. His research interests include spatio-temporal data mining, social network discovery, and link analyisis. He has a BEng in computer engineering from Nanyang Technological University. Ee-Peng Lim is an Associate Professor with the School of Computer Engineering, Nanyang Technological University, Singapore. He received his PhD from the University of Minnesota, Minneapolis in 1994 and B.Sc. in Computer Science from National University of Singapore. Ee-Peng's research interests include information integration, data/text/web mining, digital libraries, and wireless intelligence. He is currently an Associate Editor of the ACM Transactions on Information Systems (TOIS), International Journal of Digital Libraries (IJDL) and International Journal of Data Warehousing and Mining (IJDWM). He was the Program Co-Chair of the ACM/IEEE Joint Conference on Digital Libraries (JCDL 2004), and Conference/Program Co-Chairs of International Conference on Asian Digital Libraries (ICADL 2004). He has also served in the program committee of numerous international conferences. Dr Lim is a Senior Member of IEEE and a Member of ACM. HweeHwa Pang received the B.Sc.—with first class honors—and M.S. degrees from the National University of Singapore in 1989 and 1991, respectively, and the PhD degree from the University of Wisconsin at Madison in 1994, all in Computer Science. He is currently an Associate Professor at the Singapore Management University. His research interests include database management systems, data security and quality, operating systems, and multimedia servers. He has many years of hands-on experience in system implementation and project management. He has also participated in transferring some of his research results to industry. Teck-Tim Tan is an IT Manager (Operations) at the Centre for IT Services, Nanyang Technological University (NTU), Singapore. He administers and oversees NTU's campus-wide wireless LAN infrastructure which facilitates access to the University's vast IT resources and services practically anywhere on campus.     

Variations on a theme: introducing new representations of fraction into two classrooms

Abstract

The interplay between generalisations and particular instances—examples—is an essential feature of mathematics teaching and learning. In this paper, we bring together our experiences of personal and classroom mathematics activity, and demonstrate that examples do not always fulfil their intended purpose (to point to generalisations). A distinction is drawn between ‘empirical’ and ‘structural’ generalisation, and the role of generic examples is discussed as a means of supporting the second of these qualities of generalisation.     

Some Undergraduate Students'' Perceptions of Using Technology for Mathematics: Tales of Resistance

We describe and report some results from a project designed to evaluate the use of information technology (IT) for teaching and learning on a range of undergraduate mathematics courses at a U.K. university. The project involved qualitative methods of enquiry (diaries and seminars). We emphasise that many of the students were positive about much of their use of IT; however in this article we focus particularly on their tales of resistance. These tales were told both by students whose overall perspective about the use of IT in the learning of mathematics was very positive as well as by those who were reluctant users. We have organised this article around two themes: the desire for understandingand technology and power — who is in control? We suggest that these tales spring from non-trivial concerns and that we are likely to be better able to support the development of authoritative learners of mathematics if we heed them.This revised version was published online in September 2005 with corrections to the Cover Date.     

The New Stony Brook

Abstract

Except for modest sections of a few professional journals, the disciplines of most college faculty offer the world little evidence that most of us spend most of our time teaching. Political scientists not only are representative of college faculty in this regard, but are perhaps a “worst case.” Because higher education has the role of preparing citizens to think critically and act democratically, one would expect that political scientists would provide the world extensive evidence that they teach and that they teach for and about democratic society. The contents of political science, social science, and other disciplinary journals, however, give little hint of either role. They suggest instead that those of us in academe make our living grappling with each other—in print and, increasingly, electronically—over age-old dilemmas of human prospects, current conundrums, and new math models to explain the actions of nations and people.

Truth be told, this is how we gain professional reputations even though most of us gain our bread and butter by teaching. The dichotomy of our ordinary work and our system of professional recognition is only one irony in the professional life of faculty. We also face preparing citizens to think critically and act democratically while we work in bureaucratic, hierarchical, and/or elitist institutions with a contrary contextual curriculum.     

Stochastic Ordering of Multivariate Normal Distributions

We show an interesting identity for Ef(Y) – Ef(X), where X, Yare normally distributed random vectors and f is a function fulfilling some weak regularity condition. This identity will be used for a unified derivation of sufficient conditions for stochastic ordering results of multivariate normal distributions, some well known ones as well as some new ones. Moreover, we will show that many of these conditions are also necessary. As examples we will consider the usual stochastic order, convex order, upper orthant order, supermodular order and directionally convex order.     

Analyzing Telephone Network Data

Abstract

As our reliance on computer networks grows, it becomes increasingly important that we understand the behavior of the traffic that they carry. Because of the speeds and sizes involved, working with traffic collected from a computer network usually means working with large datasets. This article describes our experience with traffic collected from the common channel signaling (CCS) network. I will briefly describe the data and how they are collected and discuss similarities with and differences from other large datasets. Next, I will describe our analysis tools and outline the reasons that they have been found useful. Finally, I will discuss the challenges facing us as we strive for a better understanding of more data from faster networks. Although my emphasis in this article is on the CCS network, it has been my experience that both the problems and the solutions generalize to data from other networks.     

Exterior algebra of a Banach space

As far as we know, the exterior product with any norm has not been studied for Banach spaces. Especially, no studies have been done on Grassmann manifolds in Banach spaces. We think it is important to study these because simple m-vectors can be thought of as m-dimensional subspaces scaled in some way according to our work. We hope Banach space norms of simple m-vectors will yield metric information about their associated subspaces. In fact, this is the case with m-uniform convexity and m-uniform rotundity which are associated with area (in Banach spaces).     

Proximal Point Algorithm On Riemannian Manifolds

Abstract

In this paper we consider the minimization problem with constraints. We will show that if the set of constraints is a Riemannian manifold of nonpositive sectional curvature, and the objective function is convex in this manifold, then the proximal point method in Euclidean space is naturally extended to solve that class of problems. We will prove that the sequence generated by our method is well defined and converge to a minimizer point. In particular we show how tools of Riemannian geometry, more specifically the convex analysis in Riemannian manifolds, can be used to solve nonconvex constrained problem in Euclidean, space.     

Dynamic structural percolation model of loss distribution for cyber risk of small and medium-sized enterprises for tree-based LAN topology

Cyber risk due to breach can be seen as a risk of a financial loss due to breach of an institution’s IT infrastructure by unauthorized parties and exploiting, taking possession of, or disclosing data assets, thus creating financial and/or reputation damage. In this paper, as a primary contribution to the existing body of actuarial literature, we propose a structural model of aggregate loss distribution for cyber risk of small and medium-sized enterprises under the assumption of a tree-based LAN topology. Up to our knowledge, there exist no theoretical models of an aggregate loss distribution for cyber risk in this setting. To achieve our goal, we contextualize the problem in the probabilistic graph-theoretical framework using percolation models. We assume that the IT network topology is represented by a random graph allowing for heterogeneous loss topology and providing instructive numerical examples.     

Cesaro Summability with Respect to Two-Parameter Walsh Systems

 The one- and two-parameter Walsh system will be considered in the Paley as well as in the Kaczmarz rearrangement. We show that in the two-dimensional case the restricted maximal operator of the Walsh–Kaczmarz (C, 1)-means is bounded from the diagonal Hardy space H ^p to L ^p for every . To this end we consider the maximal operator T of a sequence of summations and show that the p-quasi-locality of T implies the same statement for its two-dimensional version T _α. Moreover, we prove that the assumption is essential. Applying known results on interpolation we get the boundedness of T _α as mapping from some Hardy–Lorentz spaces to Lorentz spaces. Furthermore, by standard arguments it will be shown that the usual two-parameter maximal operators of the (C, 1)-means are bounded from L ^p spaces to L ^p if . As a consequence, the a.e. convergence of the (C, 1)-means will be obtained for functions such that their hybrid maximal function is integrable. Of course, our theorems from the two-dimensional case can be extended to higher dimension in a simple way. (Received 20 April 2000; in revised form 25 September 2000)     

On the convergence of Newton's method when estimating higher dimensional parameters

In this paper, we consider the estimation of a parameter of interest where the estimator is one of the possibly several solutions of a set of nonlinear empirical equations. Since Newton's method is often used in such a setting to obtain a solution, it is important to know whether the so obtained iteration converges to the locally unique consistent root to the aforementioned parameter of interest. Under some conditions, we show that this is eventually the case when starting the iteration from within a ball about the true parameter whose size does not depend on n. Any preliminary almost surely consistent estimate will eventually lie in such a ball and therefore provides a suitable starting point for large enough n. As examples, we will apply our results in the context of M-estimates, kernel density estimates, as well as minimum distance estimates.     

A third grader's way of thinking about linear function tables

This paper is inscribed within the research effort to produce evidence regarding primary school students’ learning of algebra. Given the results obtained so far in the research community, we are convinced that young elementary school students can successfully learn algebra. Moreover, children this young can make use of different representational systems, including function tables, algebraic notation, and graphs in the Cartesian coordinate grid. In our research, we introduce algebra from a functional perspective. A functional perspective moves away from the mere symbolic manipulation of equations and focuses on relationships between variables. In investigating the processes of teaching and learning algebra at this age, we are interested in identifying meaningful teaching situations. Within each type of teaching situation, we focus on what kind of knowledge students produce, what are the main obstacles they find in their learning, as well as the intermediate states of knowledge between what they know and the target knowledge for the teaching situation. In this paper, we present a case study focusing on the approach adopted by a third grade student, Marisa, when she was producing the formula for a linear function while she was working with the information of a problem displayed in a function table containing pairs of inputs-outputs. We will frame the analysis and discussion on Marisa's approach in terms of the concept of theorem-in action (Vergnaud, 1982) and we will contrast it with the scalar and functional approaches introduced by Vergnaud (1988) in his Theory of Multiplicative Fields. The approach adopted by Marisa turns out to have both scalar and functional aspects to it, providing us with new ways of thinking of children's potential responses to functions.     

Gap functions and error bounds for nonsmooth convex vector optimization problem

Abstract

In this article, our main aim is to develop gap functions and error bounds for a (non-smooth) convex vector optimization problem. We show that by focusing on convexity we are able to quite efficiently compute the gap functions and try to gain insight about the structure of set of weak Pareto minimizers by viewing its graph. We will discuss several properties of gap functions and develop error bounds when the data are strongly convex. We also compare our results with some recent results on weak vector variational inequalities with set-valued maps, and also argue as to why we focus on the convex case.     

Local density of Caputo-stationary functions of any order

ABSTRACT

We show that any given function can be approximated with arbitrary precision by solutions of linear, time-fractional equations of any prescribed order. This extends a recent result by Claudia Bucur, which was obtained for time-fractional derivatives of order less than one, to the case of any fractional order of differentiation. In addition, our result applies also to the ψ-Caputo-stationary case, and it will provide one of the building blocks of a forthcoming paper in which we will establish general approximation results by operators of any order involving anisotropic superpositions of classical, space-fractional and time-fractional diffusions.     

Semi-empirical likelihood ratio confidence intervals for the difference of two sample means

We all know that we can use the likelihood ratio statistic to test hypotheses and construct confidence intervals in full parametric models. Recently, Owen (1988,Biometrika,75, 237–249; 1990,Ann. Statist.,18, 90–120) has introduced the empirical likelihood method in nonparametric models. In this paper, we combine these two likelihoods together and use the likelihood ratio to construct confidence intervals in a semiparametric problem, in which one model is parametric, and the other is nonparametric. A version of Wilks's theorem is developed.     

Monte Carlo construction of hedging strategies against multi-asset European claims

For evaluating a hedging strategy we have to know at every moment the solution of the Cauchy problem for a corresponding parabolic equation (the value of the hedging portfolio) and its derivatives (the deltas). We suggest to find these quantities by Monte Carlo simulation of the corresponding system of stochastic differential equations using weak solution schemes. It turns out that with one and the same control function a variance reduction can be achieved simultaneously for the claim value as well as for the deltas. As illustrations we consider a Markovian multi-asset model with an instantaneously riskless saving bond and also some applications to the LIBOR rate model of Brace, Gatarck, Musiela and Jamshidian.     

Zorn's Matrices and Finite Index Subloops#

Zorn's Algebra ?(R) has a multiplicative function called determinant with properties similar to the usual one. The set of elements in ?(R) with determinant 1 is a Moufang loop that we will denote by IΓ. In our main result we prove that if R is a Dedekind algebraic number domain that contains a unit of infinite order, each finite index subloop ?, such that IΓ has the weak Lagrange property relative to ?, is a congruence subloop.     

Intégration Symplectique des Variétés de Poisson Régulières

Asymplectic integration of a Poisson manifold (M, Λ) is a symplectic groupoid (Γ,η) whichrealizes the given Poisson manifold, i.e. such that the space of units Γ₀ with the induced Poisson structure Λ₀ is isomorphic to (M, Λ). This notion was introduced by A. Weinstein in [28] in order to quantize Poisson manifolds by quantizing their symplectic integration. Any Poisson manifold can be integrated by alocal symplectic groupoid ([4], [13]) but already for regular Poisson manifolds there are obstructions to global integrability ([2], [6], [11], [17], [28]). The aim of this paper is to summarize all the known obstructions and present a sufficient topological condition for integrability of regular Poisson manifolds; we will indeed describe a concrete procedure for this integration. Further our criterion will provide necessary and sufficient if we require Γ to be Hausdorff, which is a suitable condition to proceed to Weinstein’s program of quantization. These integrability results may be interpreted as a generalization of the Cartan-Smith proof of Lie’s third theorem in the infinite dimensional case.

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Recherche supportée par D.G.I.C.Y.T. Espagne (Proyecto PB90-0765) et Xunta de Galicia (Proxecto XUGA20704B90)     

Mathematics: a means of collaboration between industry and university

This paper is divided into two distinct but connected parts. In the first one we look for the best approach to the Mathematics teaching of the future engineer. This approach should have an eminently pragmatic character and should be directed towards teaching engineers to know and manage Mathematics at a tool. In the second part we illustrate how a collaboration with industry can be obtained from this kind of teaching.

Finally the conditions and consequences which should produce this cooperation are analysed, and some particular examples are mentioned.