The LIBOR Market Model

Student Number : 0003819T - MSc dissertation - School of Computational and Applied Mathematics - Faculty of ScienceThe over-the-counter (OTC) interest rate derivative market is large and rapidly developing. In March 2005, the Bank for International Settlements published its “Triennial Central Bank Survey” which examined the derivative market activity in 2004 (http://www.bis.org/publ/rpfx05.htm). The reported total gross market value of OTC derivatives stood at $6.4 trillion at the end of June 2004. The gross market value of interest rate derivatives comprised a massive 71.7% of the total, followed by foreign exchange derivatives (17.5%) and equity derivatives (5%). Further, the daily turnover in interest rate option trading increased from 5.9% (of the total daily turnover in the interest rate derivative market) in April 2001 to 16.7% in April 2004. This growth and success of the interest rate derivative market has resulted in the introduction of exotic interest rate products and the ongoing search for accurate and efficient pricing and hedging techniques for them. Interest rate caps and (European) swaptions form the largest and the most liquid part of the interest rate option market. These vanilla instruments depend only on the level of the yield curve. The market standard for pricing them is the Black (1976) model. Caps and swaptions are typically used by traders of interest rate derivatives to gamma and vega hedge complex products. Thus an important feature of an interest rate model is not only its ability to recover an arbitrary input yield curve, but also an ability to calibrate to the implied at-the-money cap and swaption volatilities. The LIBOR market model developed out of the market’s need to price and hedge exotic interest rate derivatives consistently with the Black (1976) caplet formula. The focus of this dissertation is this popular class of interest rate models. The fundamental traded assets in an interest rate model are zero-coupon bonds. The evolution of their values, assuming that the underlying movements are continuous, is driven by a finite number of Brownian motions. The traditional approach to modelling the term structure of interest rates is to postulate the evolution of the instantaneous short or forward rates. Contrastingly, in the LIBOR market model, the discrete forward rates are modelled directly. The additional assumption imposed is that the volatility function of the discrete forward rates is a deterministic function of time. In Chapter 2 we provide a brief overview of the history of interest rate modelling which led to the LIBOR market model. The general theory of derivative pricing is presented, followed by a exposition and derivation of the stochastic differential equations governing the forward LIBOR rates. The LIBOR market model framework only truly becomes a model once the volatility functions of the discrete forward rates are specified. The information provided by the yield curve, the cap and the swaption markets does not imply a unique form for these functions. In Chapter 3, we examine various specifications of the LIBOR market model. Once the model is specified, it is calibrated to the above mentioned market data. An advantage of the LIBOR market model is the ability to calibrate to a large set of liquid market instruments while generating a realistic evolution of the forward rate volatility structure (Piterbarg 2004). We examine some of the practical problems that arise when calibrating the market model and present an example calibration in the UK market. The necessity, in general, of pricing derivatives in the LIBOR market model using Monte Carlo simulation is explained in Chapter 4. Both the Monte Carlo and quasi-Monte Carlo simulation approaches are presented, together with an examination of the various discretizations of the forward rate stochastic differential equations. The chapter concludes with some numerical results comparing the performance of Monte Carlo estimates with quasi-Monte Carlo estimates and the performance of the discretization approaches. In the final chapter we discuss numerical techniques based on Monte Carlo simulation for pricing American derivatives. We present the primal and dual American option pricing problem formulations, followed by an overview of the two main numerical techniques for pricing American options using Monte Carlo simulation. Callable LIBOR exotics is a name given to a class of interest rate derivatives that have early exercise provisions (Bermudan style) to exercise into various underlying interest rate products. A popular approach for valuing these instruments in the LIBOR market model is to estimate the continuation value of the option using parametric regression and, subsequently, to estimate the option value using backward induction. This approach relies on the choice of relevant, i.e. problem specific predictor variables and also on the functional form of the regression function. It is certainly not a “black-box” type of approach. Instead of choosing the relevant predictor variables, we present the sliced inverse regression technique. Sliced inverse regression is a statistical technique that aims to capture the main features of the data with a few low-dimensional projections. In particular, we use the sliced inverse regression technique to identify the low-dimensional projections of the forward LIBOR rates and then we estimate the continuation value of the option using nonparametric regression techniques. The results for a Bermudan swaption in a two-factor LIBOR market model are compared to those in Andersen (2000)

Progress in Experimental Research of Turbine Aeroacoustics

Modifications on the intermediate turbine duct in order to reduce noise emissions by changing interaction frequencies and/or modes capable to propagate are presented. Also different turbine exit casings are described that are optimised to reduce interaction noise that is propagating through the engine and is one of the major noise sources during landing (operating point approach). The most promising modifications to reduce sound power levels are described. Depending on different modifications at specific operating points, the reduction of sound power level is between 5 dB and 10 dB, which is a significant reduction. However, some of these measures show an increase in aerodynamic losses. Therefore, a compromise has to be found between higher losses during a short duration (e.g. landing) and significant noise reduction. The chapter focuses on experimental results obtained in the test facilities of the Institute for Thermal Turbomachinery and Machine Dynamics at Graz University of Technology

Self-Rated Health and its Relationship to Health/Life Problems and Coping Strategies in Members of the Professional Slovenian Armed Forces

The aim of this study was to test the association between self-rated health status (i.e. psychological and interpersonal welfare, physical health, coping mechanisms) and absence from work due to illness in the Slovenian armed forces. 390 military personnel were included in the study. Two groups of soldiers, healthy (G1-H) and sick/less healthy (G2-S), were created according to the median value of their annual sick leave. A third group consisted of soldiers on a mission (G3-M). A background questionnaire (demographic data, lifestyle habits, a list of life problems and a list of health problems in the last three years), a Self-Rated Health Scale and the Folkman-Lazarus Ways of Coping Questionnaire were administered. Self-rated physical health was best in group G1-H and worst in G2-S, with differences between the groups being statistically significant. No gender differences were found either between the groups or in the whole sample. The most common coping strategies amongst all the soldiers were found to be problem solving, positive re-evaluation of the situation and self-control. The groups differed only in their use of the distancing strategy. The self-rated health of all the participants was found to be in strong negative correlation with the escape/avoidance coping strategy. In group G2-S, more soldiers assessed their health as poor; the differences between the groups were statistically significant. Strong positive correlations between self-rated health and satisfaction with interpersonal relationships were found. Self-rated health was found to be significantly associated with the quality of interpersonal relationships and the socio-economic and psycho-physical conditions of the soldiers

Self-Rated Health and its Relationship to Health/Life Problems and Coping Strategies in Members of the Professional Slovenian Armed Forces

The aim of this study was to test the association between self-rated health status (i.e. psychological and interpersonal welfare, physical health, coping mechanisms) and absence from work due to illness in the Slovenian armed forces. 390 military personnel were included in the study. Two groups of soldiers, healthy (G1-H) and sick/less healthy (G2-S), were created according to the median value of their annual sick leave. A third group consisted of soldiers on a mission (G3-M). A background questionnaire (demographic data, lifestyle habits, a list of life problems and a list of health problems in the last three years), a Self-Rated Health Scale and the Folkman-Lazarus Ways of Coping Questionnaire were administered. Self-rated physical health was best in group G1-H and worst in G2-S, with differences between the groups being statistically significant. No gender differences were found either between the groups or in the whole sample. The most common coping strategies amongst all the soldiers were found to be problem solving, positive re-evaluation of the situation and self-control. The groups differed only in their use of the distancing strategy. The self-rated health of all the participants was found to be in strong negative correlation with the escape/avoidance coping strategy. In group G2-S, more soldiers assessed their health as poor; the differences between the groups were statistically significant. Strong positive correlations between self-rated health and satisfaction with interpersonal relationships were found. Self-rated health was found to be significantly associated with the quality of interpersonal relationships and the socio-economic and psycho-physical conditions of the soldiers

On the Influence of a Five-Hole-Probe on the Vibration Characteristics of a Low Pressure Turbine Rotor while Performing Aerodynamic Measurements

For many reasons it is essential to know and assess the flow field and its characteristics up- and downstream of a turbine stage. For these purpose measurements are conducted in test rigs such as the STTF-AAAI (subsonic test turbine facility for aerodynamic, acoustic, and aeroelastic investigations) at the Institute for Thermal Turbomachinery and Machine Dynamics at Graz University of Technology. A low pressure turbine is operated in engine relevant operating conditions. The turbine is experienced high mechanical loads and is excited to vibrate (forced response). In the rotor design process forced response predictions and structural assessments are performed. However, it is not common to include instrumentation (e.g. total pressure and temperature rakes, five-hole-probes, fast response aerodynamic pressure probes) in these forced response predictions. But, these measurement devices are essential and therefore this paper investigates the influence of such an instrumentation onto the vibrational behaviour of a low pressure turbine rotor of the STTF-AAAI. Several vibration measurements at distinct circumferential and radial positions of the five-hole-probe in the flow channel are conducted. These measurement results are compared to measurements performed without a five-hole-probe in the flow channel. A clear influence of the five-hole-probe on the vibration level is shown