Search

The article aims to introduce concepts in option pricing and risk management. Pricing and risk management is one of the fundamental problems in financial mathematics. Then readers may explore further to understand how to use mathematical models in pricing and risk management. More specifically, our research introduces a new method called Monte Carlo-Tree (MC-Tree), for option pricing and risk management with high accuracy.

Firstly it will be shown that, using the concept of monomial orderings, the classical 1D theory for linear systems generalizes in a very natural way to (autonomous) Attasi nD-systems, giving the Attasi-Hankel matrix, the Attasi transfer function and Attasi state-space realizations. Secondly we will explain how one can associate an Attasi system to any set of polynomial equations having a finite number of solutions and how Attasi realization theory can be used to (1) describe a commutative matrix solution to the set of polynomial equations (this generalizes the Cayley-Hamilton theorem) and (2) find the (scalar) solutions to the set of polynomial equations by computing the joint eigenvalues and eigenvectors of the commuting matrices. Some remarks will be made about how the (discrete time) Attasi system equations can be solved recursively and how that can be used in large-scale eigensolvers. Using the associated Attasi system to a set of polynomial equations also helps to understand various different approaches to polynomial equation solving and multivariate polynomial and rational function minimization.

The advent of single molecule microscopy has rev- olutionized biological investigations by providing a powerful tool for the study of intercellular and intracellular trafficking processes of protein molecules which was not available before through conventional microscopy. In practice, pixelated detectors are used to acquire the images of fluorescently labeled objects moving in cellular environments. Then, the acquired fluorescence microscopy images contain the numbers of the photons detected in each pixel, during an exposure time interval. Moreover, instead of having the exact locations of detection of the photons, we only know the pixel areas in which the photons impact the detector. These challenges make the analysis of single molecule trajectories, from pixelated images, a complex problem. Here, we investigate the effect of pixelation on the parameter estimation of single molecule trajectories. In particular, we develop a stochastic framework to calculate the maximum likelihood estimates of the parameters of a stochastic differential equation that describes the motion of the molecule in living cells. We also calculate the Fisher information matrix for this parameter estimation problem. The analytical results are complicated through the fact that the observation process in a microscope prohibits the use of standard Kalman filter type approaches. The analytical framework presented here is illustrated with examples of low photon count scenarios for which we rely on Monte Carlo methods to compute the associated probability distributions

In this paper a direct generalisation of the recombining binomial tree model by Cox et al. (J Financ Econ 7:229–263, 1979) based on the Pascal’s simplex is constructed. This discrete model can be used to approximate the prices of derivatives on multiple assets in a Black–Scholes market environment. The generalisation keeps most aspects of the binomial model intact, of which the following are the most important: The direct link to the Pascal’s simplex (which specialises to Pascal’s triangle in the binomial case); the matching of moments of the (log-transformed) process; convergence to the correct option prices both for European and American options, when the time step length goes to zero and the completeness of the model, at least for sufficiently small time step. The goal of this paper is to present basic theoretical aspects of this approach. However, we also illustrate the approach by a number of example calculations. Further possible developments of this approach are discussed in a final section.

Tracking of objects in cellular environments has become a vital tool in molecular cell biology. A particularly important example is single molecule tracking, which enables the study of the motion of a molecule in cellular environments by locating the molecule over time and provides quantitative information on the behavior of individual molecules in cellular environments, which were not available before through bulk studies. Here, we consider a dynamical system where the motion of an object is modeled by stochastic differential equations (SDEs), and measurements are the detected photons, emitted by the moving fluorescently labeled object, that occur at discrete time points, corresponding to the arrival times of a Poisson process, in contrast to equidistant time points, which have been commonly used in the modeling of dynamical systems. The measurements are distributed according to the optical diffraction theory, and therefore, they would be modeled by different distributions, e.g., an Airy profile for an in-focus and a Born and Wolf profile for an out-of-focus molecule with respect to the detector. For some special circumstances, Gaussian image models have been proposed. In this paper, we introduce a stochastic framework in which we calculate the maximum likelihood estimates of the biophysical parameters of the molecular interactions, e.g., diffusion and drift coefficients. More importantly, we develop a general framework to calculate the Cramér-Rao lower bound (CRLB), given by the inverse of the Fisher information matrix, for the estimation of unknown parameters and use it as a benchmark in the evaluation of the standard deviation of the estimates. There exists no established method, even for Gaussian measurements, to systematically calculate the CRLB for the general motion model that we consider in this paper. We apply the developed methodology to simulated data of a molecule with linear trajectories and show that the standard deviation of the estimates matches well with the square root of the CRLB. We also show that equally sampled and Poisson distributed time points lead to significantly different Fisher information matrices.

International audience; Balanced realizations of lossless systems can be generated from the tangential Schur algorithm using linear fractional transformations. In discrete-time, canonical forms with pivot structures are obtained by interpolation at in finity. In continuous-time case interpolation at infi nity is on the stability boundary, leading to an angular derivative interpolation problem. In the scalar case, the balanced canonical form of Ober can be recovered in this way. Here we generalize to the multivariable case. It is shown that boundary interpolation can be regarded as a limit of classical interpolation with interpolation points tending to the imaginary axis. Some pivot structures can be generated, but no complete atlas is obtained. However, for input normal pairs an atlas of admissible pivot structures can be generated in a closely related way.

A Markov sequence is a non-zero sequence of complex numbers that satisfies a homogeneous linear difference equation with constant coefficients. The terms of such a sequence M admit of a representation of the form M(k) = cAkb, fc = 0,l,2,..., where ^4,6, c are matrices of orders n x n, n x 1, 1 x n, respectively, and n is least in the sense that b,cl are cyclic vectors for A and its transpose A1, respectively. In this article, we study the relationship between the long-term behaviour of M and the spectral properties of A. In particular, we determine the asymptotic behaviour of M(k) as k ?y co, and prescribe various conditions on M from which it can be decided that the spectral radius of A is one of its eigenvalues. For instance, we prove that this occurs when the terms of M are eventually nonnegative.

We consider a two-factor Heath–Jarrow–Morton (HJM) model under the risk neutral measure and show that it may be decoupled into a particular dynamic Nelson–Siegel (NS) model plus a somewhat counter-intuitive adjustment (lying outside the NS family) which keeps it arbitrage-free. We assess the importance of the adjustment for arbitrage-free pricing by comparing the HJM model with a novel NS model which is selected using projection techniques. We analyze forward curves and derivative prices generated by the HJM and projected NS model and consider two real-world case studies. Our analysis shows that the influence of the adjustment term on arbitrage-free evolution is small.

Anew finite atlas of overlapping balanced canonical forms for multivariate discrete-time lossless systems is presented. The canonical forms have the property that the controllability matrix is positive upper triangular up to a suitable permutation of its columns. This is a generalization of a similar balanced canonical form for continuous-time lossless systems. It is shown that this atlas is in fact a finite sub-atlas of the infinite atlas of overlapping balanced canonical forms for lossless systems that is associated with the tangential Schur algorithm; such canonical forms satisfy certain interpolation conditions on a corresponding sequence of lossless transfer matrices. The connection between these balanced canonical forms for lossless systems and the tangential Schur algorithm for lossless systems is a generalization of the same connection in the SISO case that was noted before. The results are directly applicable to obtain a finite sub-atlas of multivariate input-normal canonical forms for stable linear systems of given fixed order, which is minimal in the sense that no chan can be left out of the atlas without losing the property that the atlas covers the manifold. (c) 2007 Elsevier Inc. All rights reserved.

The problem of finding the global minimum of a so-called Minkowski-norm dominated polynomial can be approached by the matrix method of Stetter and Moller, which reformulates it as a large eigenvalue problem. A drawback of this approach is that the matrix involved is usually very large. However, all that is needed for modern iterative eigenproblem solvers is a routine which computes the action of the matrix on a given vector. This paper focuses on improving the efficiency of computing the action of the matrix on a vector. To avoid building the large matrix one can associate the system of first-order conditions with an nD system of difference equations. One way to compute the action of the matrix efficiently is by setting up a corresponding shortest path problem and solving it. It turns out that for large n the shortest path problem has a high computational complexity, and therefore some heuristic procedures are developed for arriving cheaply at suboptimal paths with acceptable performance.

In this paper we introduce two variants of a new parametrization for state–space systems which we will both call separable least squares data driven local co-ordinates (slsDDLC). SlsDDLC is obtained by modifying the parametrization by data driven local co-ordinates (DDLC). These modifications lead to analogous parametrizations, and we show how they can be used for a suitably concentrated likelihood criterion function. The concentration step can be done by an ordinary or generalized least squares step. An obvious consequence is the reduced number of parameters in the iterative search algorithm. The application of the parametrizations to maximum likelihood identification is exemplified. Simulations indicate that the usage of slsDDLC for concentrated likelihood functions has numerical advantages as compared to the usage of the more commonly used echelon canonical form or conventional DDLC for the likelihood function. Copyright © 2004 John Wiley & Sons, Ltd.

This is a tutorial introduction to the invited session entitled Links between financial econometrics and system identification

In this paper, the parametrization of state-space systems by data driven local coordinates as introduced by (McKelvey et al., 2003) is modified. This modification leads to an alternative analogous parametrization which can be used for a suitable concentrated likelihood-type criterion function, where the concentration step can be done by a generalized least squares step. An obvious consequence is the reduced number of parameters resulting in less computational burden, but, of course, the criterion function itself is changed by the concentration step. The resulting new parametrization is called slsDDLC, and its topological and geometrical properties are investigated in detail.

In the area of financial time series the Black-Scholes model is often used. However, it is well-known that although the volatility is assumed to be constant in the Black-Scholes model, in practice it is varying in time. This has led to the investigation of more general models in which the volatility is allowed to vary. In one class of such models the dynamic behavior of volatility is described by some stochastic process. A problem with such models is that it is generally very difficult to solve the volatility estimation problem for such models: the calculation of the conditional density of the volatility at some point in time, given the observations up till that point in time, is usually a difficult task for which there are no closed form expressions. In the present paper a class of models of the same type is presented, which however has the advantage that for these models the volatility estimation problem can be solved exactly. In our models all disturbances have a rational probability density function on the real line.

In this paper, we show that a recently defined cepstral norm for ARMA models equals, up to a constant factor, the square root of the area enclosed by the polar plot of the logarithm of the transfer function.

The problem of minimizing a polynomial function in several variables over Rn is considered and an algorithm is given. When the polynomial has a minimum the algorithm returns the global minimal value and finds at least one point in every connected component of the set of minimizers. A characterization of such points is given. When the polynomial does not have a minimum the algorithm computes its infimum. No assumption is made on the polynomial.

State space methods have proved to be powerful theoretical and computational tools in a number of areas of applications, in particular filtering and control theory. In this paper we advocate the use of state space methods for the study of discrete probability densities on the set {0,1,2,…}. The fundamental approach is to consider the class D of all discrete probability densities that can be represented as the impulse response/convolution kernel of a finite dimensional discrete time state space system. We show that all standard operations such as the calculation of moments, convolution, scaling, translation, product, etc. can be carried out using system representations.

The tangential Schur algorithm provides a means of constructing the class of multivariable discrete-time stable all-pass transfer functions of a fixed finite McMillan degree. In each iteration step a linear fractional transformation is employed which is associated with a J-inner rational matrix of McMillan degree 1. In this set-up, the emphasis is exclusively on transfer functions. In the present contribution we present a unified framework in which linear fractional transformations on transfer functions are represented by corresponding linear fractional transformations on state-space realization matrices. When applied to the case of the tangential Schur algorithm, minimal balanced realizations of stable all-pass systems in terms of the parameters used are obtained. The balanced state-space approach of (Hanzon and Peeters, 2000) is incorporated as a special case.

We propose what we believe to be a novel approach to performing calculations for rational density functions using state-space representations of the densities. By standard results from realization theory, a rational probability density function is considered to be the transfer function of a linear system with generally complex entries. The stable part of this system is positive-real, which we call the density summand. The existence of moments is investigated using the Markov parameters of the density summand. Moreover, explicit formulae are given for the existing moments in terms of these Markov parameters. Some of the main contributions of the paper are explicit state-space descriptions for products and convolutions of rational densities. As an application which is of interest in its own right, the filtering problem is investigated for a linear time-varying system whose noise inputs have rational probability density functions. In particular, state-space formulations are derived for the calculation of the prediction and update equations. The case of Cauchy noise is treated as an illustrative example.

This paper introduces in detail a new systematic method to construct approximate finite-dimensional solutions for the nonlinear filtering problem. Once a finite-dimensional family is selected, the nonlinear filtering equation is projected in Fisher metric on the corresponding manifold of densities, yielding the projection filter for the chosen family. The general definition of the projection filter is given, and its structure is explored in detail for exponential families. Particular exponential families which optimize the correction step in the case of discrete-time observations are given, and an a posteriori estimate of the local error resulting from the projection is defined. Simulation results comparing the projection filter and the optimal filter for the cubic sensor problem are presented. The classical concept of assumed density filter (ADF) is compared with the projection filter. It is shown that the concept of ADF is inconsistent in the sense that the resulting filters depend on the choice of a stochastic calculus, i.e. the Ito or the Stratonovich calculus. It is shown that in the context of exponential families, the projection filter coincides with the Stratonovich-based ADF. An example is provided, which shows that this does not hold in general, for non-exponential families of densities.

Through the use of balanced realizations it has been possible to derive parametrizations and canonical forms for various classes of minimal linear systems of given dimension. A possible problem of these parametrizations is that they are not overlapping. This could be a drawback for the application of balanced parametrizations in such areas as system identification, model reduction and optimization. It is the topic of this paper to derive overlapping parametrizations which are closely related to the existing balanced parametrizations. We first introduce input-normal canonical forms which are defined through a novel way of choosing nice selections of columns of the reachability matrix. These canonical forms provide overlapping parametrizations in the sense that they form a real analytic atlas of the manifold of systems which are considered. Then we introduce so-called block-balanced input normal forms which use the previously constructed input normal forms as building blocks. The classes of systems for which such parametrizations are given are the stable minimal systems, positive-real minimal systems, bounded-real minimal systems and the class of all minimal systems of given McMillan degree. The results include both the single-input single-output and the multivariable case. In the single-input single-output case, however, the issue of choosing nice selections of columns does not occur. Therefore in this case the derivation and presentation of the results is considerably simplified.

In this paper we present a simulation study as well as two applications of a newly proposed approach towards I(1) cointegration using so-called state-space error correction models (SSECMs). SSECMs have been introduced in Ribarits and Hanzon (2014) where questions of model selection, parametrization and maximum likelihood estimation are treated in detail. The present paper shows the potential of this new approach to I (1) cointegration analysis.