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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.

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.

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.

Certain topological and geometrical properties of data driven local coordinates (DDLC) for state-space systems as introduced in (Wolodkin et al. , 1997) and (McKelvey and Helmersson, 1999) are derived. First the special case of SISO systems with McMillan degree n = 1 is discussed in order to provide some insights into the geometry of the DDLC construction. Then for the MIMO case with general n it is shown that the set of transfer functions corresponding to the parameter space contains a nonvoid open subset of the manifold of transfer functions of order n and that the estimation problem is locally well posed. Moreover, it is stated that the parameter space always contains points corresponding to non minimal systems and a result on the number of disconnected components of the equivalence classes in the space ℝ n2 +n ( m+s ) (obtained by an embedding of the system matrices ( A,B,C )) concludes this contribution.

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.

In this paper we consider cointegrated I (1) processes in the state-space framework. We introduce the state-space error correction model (SSECM) and discuss in detail how to estimate SSECMs by (pseudo) maximum likelihood methods, including reduced rank regression techniques which allow for a successive reduction of the number of parameters in the original constrained likelihood optimization problem. In doing so, we follow very closely the Johansen approach for the VAR case; see Johansen (1995). Finally, the remaining free parameters will be represented using a local parametrization technique and we show how efficient gradient-type algorithms can be employed for the numerical optimization in the resulting lower dimensional unconstrained problem. Simulation studies and applications are presented in a separate 'companion paper'; see Ribarits and Hanzon (2014).

Economic capital (ECap) modeling is a fundamental part of Pillar II of the Basel framework. Indeed, 'sophisticated' financial institutions need to have in place internal models for the assessment of the level of the overall capital buffer which is deemed sufficient to cover the risk of their business activities. On top, ECap models are also frequently used for pricing purposes on an ex-ante basis: financial institutions need to know the incremental economic capital (IECap), i.e. the size by which the overall capital buffer needs to be increased after addition of e.g. a single new loan to the existing portfolio. This is important in order to be able to price such additional loan accordingly. Finally, ECap contributions (ECapC) are also required ex-post in order to break down the overall capital buffer to the individual obligors, products etc. within the portfolio. Simulation of IECap and ECapC can be computationally expensive and unstable, but it appears that closed form approximations provide accurate, consistent and quick solutions in many cases. The formula introduced here is based on the multi-factor approximation from [Pykhtin, 2004] applicable to a default-mode Merton type model. As such, default correlations between obligors (stemming from a multi-factor-model) are taken into account, but the formula also captures specific amortization schedules and loss given default (LGD) values for each individual position -- a feature which is of practical relevance, but often neglected in standard default-mode models. For the time being, credit-risk mitigants such as the existence of guarantors on individual loans, are not captured by the formula, neither are credit migrations or correlations between default probabilities and LGDs. Our formula allows for approximation of all ECap contributions without extra computational cost. After calculation of the ECap of the original portfolio, IECap can be computed within few seconds and more accurately than in standard linear approximations based on ECap contributions.

We consider a new state-space parametrization for linear time series models: data driven coordinates (DDC), which provides an atlas for the manifold of (stable) p x m transfer functions of fixed McMillan degree n. Hence, DDC has similar desirable properties as more traditional overlapping parametrizations and better than classical canonical forms. Moreover, the choice of charts can be done in a data-driven manner in a very simple way. Althugh not yet as good numerically as the parametrization by data driven local coordinates (DDLC), this parametrization has the advantage of not being local. The application of DDC to maximum likelihood identification is exemplified.

We describe a novel approach, called data driven local coordinates (DDLC), for parametrizing linear systems in state space form, and we analyze some of its properties which are relevant for e.g. maximum likelihood estimation. In addition we describe how this idea can be used for a concentrated likelihood function, obtained by a least squares type concentration step, which gives the so called sls (separable least squares) DDLC approach. Both approaches give favourable results in numerically optimizing the likelihood function in simulation studies.

A new approach to the question of parametrizing state-space systems is considered. The approach consists of using input-normal state-space representations. These representations are unique up to an isometric state isomorphism. By decomposing the tangent space of the set of normalized controllable matrix pairs into the tangent space of the equivalence class and the orthogonal complement one obtains an interesting set of local coordinates with a number of remarkable properties. Here the approach is presented in the context of a separable least squares approach to maximum likelihood estimation of a linear system.