Modelling Hedge Fund Indices Using Levy Processes

In this thesis, the use of Levy processes to model the dynamics of Hedge fund indices is proposed. Merton (1976) and Kou (2002) models which differ on the specifcation of the jump components are employed to model hedge funds in continuous time. Secondly, an alternative to the Maximum Likelihood Estimation (MLE) method, Empirical Characteristic Function (ECF) estimation method, is explored in our analysis and compared to MLE. The Cumulant Matching Method (CMM) is used in getting the starting parameters; and the method that overcomes the major problem associated with this estimation method is outlined. Calibration shows that these two models t the data well, however, the empirical comparison shows that double exponential jumps are more consistent with the empirical data. Each fund's exposure to risk is calculated using Monte Carlo Value-at-Risk (VaR) estimation method.

Modelling Hedge Fund Indices Using Levy Processes

In this thesis, the use of Levy processes to model the dynamics of Hedge fund indices is proposed. Merton (1976) and Kou (2002) models which differ on the specifcation of the jump components are employed to model hedge funds in continuous time. Secondly, an alternative to the Maximum Likelihood Estimation (MLE) method, Empirical Characteristic Function (ECF) estimation method, is explored in our analysis and compared to MLE. The Cumulant Matching Method (CMM) is used in getting the starting parameters; and the method that overcomes the major problem associated with this estimation method is outlined. Calibration shows that these two models t the data well, however, the empirical comparison shows that double exponential jumps are more consistent with the empirical data. Each fund's exposure to risk is calculated using Monte Carlo Value-at-Risk (VaR) estimation method.

ON THE UNIFORM CONVERGENCE OF DECONVOLUTION ESTIMATORS FROM REPEATED MEASUREMENTS

This paper studies the uniform convergence rates of Li and Vuong’s (1998, Journal of Multivariate Analysis 65, 139–165; hereafter LV) nonparametric deconvolution estimator and its regularized version by Comte and Kappus (2015, Journal of Multivariate Analysis 140, 31–46) for the classical measurement error model, where repeated noisy measurements on the error-free variable of interest are available. In contrast to LV, our assumptions allow unbounded supports for the error-free variable and measurement errors. Compared to Bonhomme and Robin (2010, Review of Economic Studies 77, 491–533) specialized to the measurement error model, our assumptions do not require existence of the moment generating functions of the square and product of repeated measurements. Furthermore, by utilizing a maximal inequality for the multivariate normalized empirical characteristic function process, we derive uniform convergence rates that are faster than the ones derived in these papers under such weaker conditions.

Statistical Inference of Spot Correlation and Spot Market Beta under Infinite Variation Jumps*

Empirical evidence has revealed that the jumps in financial markets appear to be very frequent. This study considers the statistical inference of the spot correlation and the spot market beta between two different assets using high-frequency data, in a setting where both the cojumps and the individual jumps in the underlying driving processes could be of infinite variation. Starting from the estimation of the spot covariance, we propose consistent estimators of the spot correlation and the spot market beta when the jump processes involved are general semimartingales. The second-order approximation for the estimators, namely, the central limit theorems, is established under the assumption that the jumps around zero are of stable Lévy type. Our estimation procedure is based on the empirical characteristic function of the increments of the processes and the application of the polarization identity; the bias terms stemming from the jumps are removed iteratively. The finite sample performances of the proposed estimators and other existing estimators are assessed and compared by using datasets simulated from various models. Our estimators are also applied to some real high-frequency financial datasets.