Risk management under a two-factor model of the term structure of interest rates

This paper presents several applications to interest rate risk managementbased on a two-factor continuous-time model of the term structure of interestrates previously presented in Moreno (1996). This model assumes that defaultfree discount bond prices are determined by the time to maturity and twofactors, the long-term interest rate and the spread (difference between thelong-term rate and the short-term (instantaneous) riskless rate). Several newmeasures of ``generalized duration" are presented and applied in differentsituations in order to manage market risk and yield curve risk. By means ofthese measures, we are able to compute the hedging ratios that allows us toimmunize a bond portfolio by means of options on bonds. Focusing on thehedging problem, it is shown that these new measures allow us to immunize abond portfolio against changes (parallel and/or in the slope) in the yieldcurve. Finally, a proposal of solution of the limitations of conventionalduration by means of these new measures is presented and illustratednumerically.

Long-tailed trapping times and Lévy flights in a self-organized critical granular system

We present a continuous time random walk model for the scale-invariant transport found in a self-organized critical rice pile [K. Christensen et al., Phys. Rev. Lett. 77, 107 (1996)]. From our analytical results it is shown that the dynamics of the experiment can be explained in terms of Lvy flights for the grains and a long-tailed distribution of trapping times. Scaling relations for the exponents of these distributions are obtained. The predicted microscopic behavior is confirmed by means of a cellular automaton model.

Occupancy of a single site by many random walkers

We consider an infinite number of noninteracting lattice random walkers with the goal of determining statistical properties of the time, out of a total time T, that a single site has been occupied by n random walkers. Initially the random walkers are assumed uniformly distributed on the lattice except for the target site at the origin, which is unoccupied. The random-walk model is taken to be a continuous-time random walk and the pausing-time density at the target site is allowed to differ from the pausing-time density at other sites. We calculate the dependence of the mean time of occupancy by n random walkers as a function of n and the observation time T. We also find the variance for the cumulative time during which the site is unoccupied. The large-T behavior of the variance differs according as the random walk is transient or recurrent. It is shown that the variance is proportional to T at large T in three or more dimensions, it is proportional to T3/2 in one dimension and to TlnT in two dimensions.

Extreme times in financial markets.

We apply the theory of continuous time random walks (CTRWs) to study some aspects involving extreme events in financial time series. We focus our attention on the mean exit time (MET). We derive a general equation for this average and compare it with empirical results coming from high-frequency data of the U.S. dollar and Deutsche mark futures market. The empirical MET follows a quadratic law in the return length interval which is consistent with the CTRW formalism.

Statistics of depth probed by cw measurement of photons in a turbid medium

Photon migration in a turbid medium has been modeled in many different ways. The motivation for such modeling is based on technology that can be used to probe potentially diagnostic optical properties of biological tissue. Surprisingly, one of the more effective models is also one of the simplest. It is based on statistical properties of a nearest-neighbor lattice random walk. Here we develop a theory allowing one to calculate the number of visits by a photon to a given depth, if it is eventually detected at an absorbing surface. This mimics cw measurements made on biological tissue and is directed towards characterizing the depth reached by photons injected at the surface. Our development of the theory uses formalism based on the theory of a continuous-time random walk (CTRW). Formally exact results are given in the Fourier-Laplace domain, which, in turn, are used to generate approximations for parameters of physical interest.

Front propagation in the A+B->2A reaction under subdiffusion

We consider an irreversible autocatalytic conversion reaction A+B->2A under subdiffusion described by continuous-time random walks. The reactants transformations take place independently of their motion and are described by constant rates. The analog of this reaction in the case of normal diffusion is described by the Fisher-Kolmogorov-Petrovskii-Piskunov equation leading to the existence of a nonzero minimal front propagation velocity, which is really attained by the front in its stable motion. We show that for subdiffusion, this minimal propagation velocity is zero, which suggests propagation failure.

Loss of Prestress, Camber, and Deflection of Non-Composite and Composite Structures Using Different Weight Concretes, HR-137, 1970

General equations are presented for predicting loss of prestress and camber of both composite and non- composite prestressed concrete structures. Continuous time functins of all parameters needed to solve the equations are given, and sample results included. Computed prestress loss and camber are compared with experimental data for normal weight and lightweight concrete. Methods are also presented for predicting the effect of non-prestressed tension steel in reducing time-dependent loss of prestress and camber, and for the determination of short-time deflections of uncracked and cracked prestressed members. Comparisons with experimental results are indicated for these partially prestressed methods.

Renewal equations for option pricing

In this paper we will develop a methodology for obtaining pricing expressions for financial instruments whose underlying asset can be described through a simple continuous-time random walk (CTRW) market model. Our approach is very natural to the issue because it is based in the use of renewal equations, and therefore it enhances the potential use of CTRW techniques in finance. We solve these equations for typical contract specifications, in a particular but exemplifying case. We also show how a formal general solution can be found for more exotic derivatives, and we compare prices for alternative models of the underlying. Finally, we recover the celebrated results for the Wiener process under certain limits.

Moottorimalliton skalaarisäätö

Työssä tutkitaan nopeusanturittoman vaihtovirtakäytön skalaarisia ohjaus- ja säätömenetelmiä. Työn alussa esitetään perusteoriat taajuusmuuttajista ja oikosulkumoottoreista. Tämän jälkeen esitellään yleisimmin kirjallisuudessa esiintyneet skalaariohjaukset ja –säädöt. Vektorisäätöä ja erityisesti moottoriparametrien vaikutusta säädön toimivuuteen esitellään lyhyesti. Työn tavoitteena on ACS800 taajuusmuuttajan skalaarisäädön tutkiminen. ACS800:n nykyinen skalaarisäätö on liian sidoksissa vektorisäätöön, joten simulointien ja kirjallisuustutkimuksen tarkoituksena on täysin vektorisäädöstä eriytetyn skalaarisäädön kehitysmahdollisuuksien tutkiminen. Kirjallisuudessa esiintyneiden säätöjen avulla muodostetaan diskreettiaikainen toteutus skalaarisäädölle vaihtovirtakäytössä, jossa on käytössä virran ja välipiirijännitteen takaisinkytkentä. Säädettävää moottoria mallinnetaan jatkuvaaikaisella L-sijaiskytkennällä. Välipiirin mallinnus toimii myös jatkuva-aikaisena lukuun ottamatta välipiirin tasavirtakomponenttia, joka muodostetaan virran takaisinkytkennän ja PWM-modulaattorin kytkinasentojen avulla. Simuloinnin tarkoituksena on mallintaa skalaarisäädön suurimpia ongelmia, kuten virta- ja välijännitesäätöä. Tuloksista voidaan päätellä, että perussäädöt toimivat moitteettomasti, mutta erityisesti virtasäätöä tulisi kehittää.

Consumption, investment and life insurance strategies with heterogeneous discounting

[cat] En aquest treball s'analitza l'efecte que comporta l'introducció de preferències inconsistents temporalment sobre les decisions òptimes de consum, inversió i compra d'assegurança de vida. En concret, es pretén recollir la creixent importància que un individu dóna a la herència que deixa i a la riquesa disponible per a la seva jubilació al llarg de la seva vida laboral. Amb aquesta finalitat, es parteix d'un model estocàstic en temps continu amb temps final aleatori, i s'introdueix el descompte heterogeni, considerant un agent amb una distribució de vida residual coneguda. Per tal d'obtenir solucions consistents temporalment es resol una equació de programació dinàmica no estàndard. Per al cas de funcions d'utilitat del tipus CRRA i CARA es troben solucions explícites. Finalment, els resultats obtinguts s'il·lustren numèricament.

Consumption, investment and life insurance strategies with heterogeneous discounting

[cat] En aquest treball s'analitza l'efecte que comporta l'introducció de preferències inconsistents temporalment sobre les decisions òptimes de consum, inversió i compra d'assegurança de vida. En concret, es pretén recollir la creixent importància que un individu dóna a la herència que deixa i a la riquesa disponible per a la seva jubilació al llarg de la seva vida laboral. Amb aquesta finalitat, es parteix d'un model estocàstic en temps continu amb temps final aleatori, i s'introdueix el descompte heterogeni, considerant un agent amb una distribució de vida residual coneguda. Per tal d'obtenir solucions consistents temporalment es resol una equació de programació dinàmica no estàndard. Per al cas de funcions d'utilitat del tipus CRRA i CARA es troben solucions explícites. Finalment, els resultats obtinguts s'il·lustren numèricament.

Model for interevent times with long tails and multifractality in human communications: An application to financial trading

Social, technological, and economic time series are divided by events which are usually assumed to be random, albeit with some hierarchical structure. It is well known that the interevent statistics observed in these contexts differs from the Poissonian profile by being long-tailed distributed with resting and active periods interwoven. Understanding mechanisms generating consistent statistics has therefore become a central issue. The approach we present is taken from the continuous-time random-walk formalism and represents an analytical alternative to models of nontrivial priority that have been recently proposed. Our analysis also goes one step further by looking at the multifractal structure of the interevent times of human decisions. We here analyze the intertransaction time intervals of several financial markets. We observe that empirical data describe a subtle multifractal behavior. Our model explains this structure by taking the pausing-time density in the form of a superstatistics where the integral kernel quantifies the heterogeneous nature of the executed tasks. A stretched exponential kernel provides a multifractal profile valid for a certain limited range. A suggested heuristic analytical profile is capable of covering a broader region.

An Eigenfunction Approach for Volatility Modeling.

In this paper, we introduce a new approach for volatility modeling in discrete and continuous time. We follow the stochastic volatility literature by assuming that the variance is a function of a state variable. However, instead of assuming that the loading function is ad hoc (e.g., exponential or affine), we assume that it is a linear combination of the eigenfunctions of the conditional expectation (resp. infinitesimal generator) operator associated to the state variable in discrete (resp. continuous) time. Special examples are the popular log-normal and square-root models where the eigenfunctions are the Hermite and Laguerre polynomials respectively. The eigenfunction approach has at least six advantages: i) it is general since any square integrable function may be written as a linear combination of the eigenfunctions; ii) the orthogonality of the eigenfunctions leads to the traditional interpretations of the linear principal components analysis; iii) the implied dynamics of the variance and squared return processes are ARMA and, hence, simple for forecasting and inference purposes; (iv) more importantly, this generates fat tails for the variance and returns processes; v) in contrast to popular models, the variance of the variance is a flexible function of the variance; vi) these models are closed under temporal aggregation.

Asymptotic Approximations in the Near-Integrated Model with a Non-Zero Initial Condition

This paper considers various asymptotic approximations in the near-integrated firstorder autoregressive model with a non-zero initial condition. We first extend the work of Knight and Satchell (1993), who considered the random walk case with a zero initial condition, to derive the expansion of the relevant joint moment generating function in this more general framework. We also consider, as alternative approximations, the stochastic expansion of Phillips (1987c) and the continuous time approximation of Perron (1991). We assess how these alternative methods provide or not an adequate approximation to the finite-sample distribution of the least-squares estimator in a first-order autoregressive model. The results show that, when the initial condition is non-zero, Perron's (1991) continuous time approximation performs very well while the others only offer improvements when the initial condition is zero.

Aggregations and Marginalization of GARCH and Stochastic Volatility Models

The GARCH and Stochastic Volatility paradigms are often brought into conflict as two competitive views of the appropriate conditional variance concept : conditional variance given past values of the same series or conditional variance given a larger past information (including possibly unobservable state variables). The main thesis of this paper is that, since in general the econometrician has no idea about something like a structural level of disaggregation, a well-written volatility model should be specified in such a way that one is always allowed to reduce the information set without invalidating the model. To this respect, the debate between observable past information (in the GARCH spirit) versus unobservable conditioning information (in the state-space spirit) is irrelevant. In this paper, we stress a square-root autoregressive stochastic volatility (SR-SARV) model which remains true to the GARCH paradigm of ARMA dynamics for squared innovations but weakens the GARCH structure in order to obtain required robustness properties with respect to various kinds of aggregation. It is shown that the lack of robustness of the usual GARCH setting is due to two very restrictive assumptions : perfect linear correlation between squared innovations and conditional variance on the one hand and linear relationship between the conditional variance of the future conditional variance and the squared conditional variance on the other hand. By relaxing these assumptions, thanks to a state-space setting, we obtain aggregation results without renouncing to the conditional variance concept (and related leverage effects), as it is the case for the recently suggested weak GARCH model which gets aggregation results by replacing conditional expectations by linear projections on symmetric past innovations. Moreover, unlike the weak GARCH literature, we are able to define multivariate models, including higher order dynamics and risk premiums (in the spirit of GARCH (p,p) and GARCH in mean) and to derive conditional moment restrictions well suited for statistical inference. Finally, we are able to characterize the exact relationships between our SR-SARV models (including higher order dynamics, leverage effect and in-mean effect), usual GARCH models and continuous time stochastic volatility models, so that previous results about aggregation of weak GARCH and continuous time GARCH modeling can be recovered in our framework.