Econometrics of jump-diffusion processes : approximation, estimation and forecasting

Abstract

In this thesis we consider the relationship between jump-diffusion processes and ARCH models with jump components. In the theoretical financial economics literature, jump-diffusion processes in continuous time have been used to model financial markets. Most empirical works use either directly discretised jump-diffusion processes or ARCH models with jump components to estimate the underlying processes. There is, however, no guarantee that those models used in empirical works are discrete counterparts of the continuous time jump-diffusion processes.

In Chapter 2, Survey on Jump-Diffusion Processes in Financial Econometrics, we survey the existing literature on jump-diffusion processes. During the 1980's and 90's, it started to draw more attention as an alternative tool to ARCH type models. The most significant theoretical developments and empirical findings are reviewed.

In Chapter 3, Approximation of Jump-Diffusion Processes, we show that a discrete time stochastic difference equation (e.g. ARCH with jumps) converges weakly to the continuous time stochastic differential equation (e.g. jump-diffusion limit) as the length of sampling interval goes to zero. It is shown that, as examples, GARCH(1,1)-M with jumps and EGARCH with jumps are discrete counterparts of their jump-diffusion limits.

In Chapter 4, Filtering with Jump-Diffusion Processes, we study the properties of the conditional covariance estimates generated by a misspecified model with jumps. We show that a misspecified model can correctly identify the true conditional covariance of underlying process. I.e., the difference between a conditional covariance estimate and the true conditional covariance converges to zero in probability as the length of sampling interval goes to zero.

In Chapter 5, Forecasting with Jump-Diffusion Processes, we investigate the forecasting ability of jump-diffusion processes. It is shown that forecasts generated by a sequence of misspecified models with jumps converge weakly to forecasts generated by the true data generating process as the sampling interval approaches to zero.

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