Statistical Properties of Forward Libor Rates

historical forward rates are used to calibrate the lognormal forward rate model - as advocated by Hull and White (1999, 2000), Longstaff, Santa Clara and Schwartz (1999), Rebonato (1999a,b,c), Rebonato and Joshi (2001) and many others - a Libor yield curve needs to be fit to the available data on spot libor rates, forward rate agreements (FRAs) or futures, and swap rates. This paper compares the statistical properties of the time series of forward rates that are obtained using three different yield curve fitting techniques. Introduced by McCulloch (1975), Steely (1991) and Svensson (1994), each of the three techniques is well known for its application to the construction of bond yield curves. Our work focuses on the eigenstructure of estimated forward rate correlation matrices. These are shown to be dominated by the semi-parametric or parametric form that is used in the yield curve model. The spectral decomposition of forward rate correlation - and covariance - matrices is considered in some detail, and in particular we test the common principal component hypothesis of Flury (1988), which has been applied to the lognormal forward rate model by Alexander (2003). We conclude that, if historical data are used to calibrate the lognormal forward rate model, it is best to use Svensson forward rate correlation matrices. However, the empirical evidence is strongly in favour of the common principal component hypothesis, where the three principal eigenvectors in all correlation matrices of the same dimension are identical. Hence we further conclude that a parsimonious parameterisation of forward rate correlations is possible, and this allows for direct calibration of forward rate correlations to market data, so historical data are not necessary.Yield curve fitting, common principal component analysis, volatility, correlation, covariance, lognormal formal rate model

Mapping networks of influence: tracking Twitter conversations through time and space

The increasing use of social media around global news events, such as the London Olympics in 2012, raises questions for international broadcasters about how to engage with users via social media in order to best achieve their individual missions. Twitter is a highly diverse social network whose conversations are multi-directional involving individual users, political and cultural actors, athletes and a range of media professionals. In so doing, users form networks of influence via their interactions affecting the ways that information is shared about specific global events. This article attempts to understand how networks of influence are formed among Twitter users, and the relative influence of global news media organisations and information providers in the Twittersphere during such global news events. We build an analysis around a set of tweets collected during the 2012 London Olympics. To understand how different users influence the conversations across Twitter, we compare three types of accounts: those belonging to a number of well-known athletes, those belonging to some well-known commentators employed by the BBC, and a number of corporate accounts belonging to the BBC World Service and the official London Twitter account. We look at the data from two perspectives. First, to understand the structure of the social groupings formed among Twitter users, we use a network analysis to model social groupings in the Twittersphere across time and space. Second, to assess the influence of individual tweets, we investigate the ageing factor of tweets, which measures how long users continue to interact with a particular tweet after it is originally posted. We consider what the profile of particular tweets from corporate and athletes’ accounts can tell us about how networks of influence are forged and maintained. We use these analyses to answer the questions: How do different types of accounts help shape the social networks? and, What determines the level and type of influence of a particular account

Joint statistics of amplitudes and phases in Wave Turbulence

Random Phase Approximation (RPA) provides a very convenient tool to study the ensembles of weakly interacting waves, commonly called Wave Turbulence. In its traditional formulation, RPA assumes that phases of interacting waves are random quantities but it usually ignores randomness of their amplitudes. Recently, RPA was generalised in a way that takes into account the amplitude randomness and it was applied to study of the higher momenta and probability densities of wave amplitudes. However, to have a meaningful description of wave turbulence the RPA properties assumed for the initial fields must be proven to survive over the nonlinear evolution time, and such a proof is the main goal of the present paper. We derive an evolution equation for the full probability density function which contains the complete information about the joint statistics of all wave amplitudes and phases. We show that, for any initial statistics of the amplitudes, the phase factors remain statistically independent uniformly distributed variables. If in addition the initial amplitudes are also independent variables (but with arbitrary distributions) they will remain independent when considered in small sets which are much less than the total number of modes. However, if the size of a set is of order of the total number of modes then the joint probability density for this set is not factorisable into the product of one-mode probabilities. In the other words, the modes in such a set are involved in a ``collective'' (correlated) motion. We also study new type of correlators describing the phase statistics.Comment: 27 pages, uses feynmf packag

Excitation of interfacial waves via near---resonant surface---interfacial wave interactions

We consider interactions between surface and interfacial waves in the two layer system. Our approach is based on the Hamiltonian structure of the equations of motion, and includes the general procedure for diagonalization of the quadratic part of the Hamiltonian. Such diagonalization allows us to derive the interaction crossection between surface and interfacial waves and to derive the coupled kinetic equations describing spectral energy transfers in this system. Our kinetic equation allows resonant and near resonant interactions. We find that the energy transfers are dominated by the class III resonances of \cite{Alam}. We apply our formalism to calculate the rate of growth for interfacial waves for different values of the wind velocity. Using our kinetic equation, we also consider the energy transfer from the wind generated surface waves to interfacial waves for the case when the spectrum of the surface waves is given by the JONSWAP spectrum and interfacial waves are initially absent. We find that such energy transfer can occur along a timescale of hours; there is a range of wind speeds for the most effective energy transfer at approximately the wind speed corresponding to white capping of the sea. Furthermore, interfacial waves oblique to the direction of the wind are also generated