With the availability of high-frequency financial data it becomes apparent that most financial asset prices contain jumps and therefore jumps should be incorporated in realistic asset pricing models. In this dissertation I study different issues related with the application of jump processes in finance and econometrics.

Chapter 1, which is coauthored with Professor George Tauchen, analyzes methods of simulation of Lévy-driven stochastic volatility models. The simulation schemes considered in the chapter are based on the shot-noise decomposition of the Lévy measure as given in Rosiński (2001). We show that the shot-noise representation of the Lévy measure gives a very easy way of direct simulation of moving averages of pure-jump Lévy processes and is convenient for generating dependence between jumps in multivariate jump processes.

In Chapter 2 I introduce and study the econometric properties of a new class of stochastic volatility models in which the stochastic volatility is purely jump-driven - it is modeled as a moving average of positive jumps. Some of the attractive features of these models are the natural way of generating sudden changes in volatility, the parsimonious way of generating persistence in the volatility and the generality of the dependence between price and variance jumps. The last feature of the models, the general jump dependence, is demonstrated with the estimation of GARCH-type stochastic volatility models in which the variance jumps are proportional to the squared price jumps.

Chapter 3 proposes an estimation method for continuous-time stochastic volatility models which can contain jumps. The estimation is based on matching moments of realized multipower variation statistics constructed from high-frequency data. These statistics provide a semiparametric estimate of latent quantities of the model such as integrated volatility and squared price jumps. In the estimation I treat the realized multipower variation statistics as their unobservable asymptotic limits (as the number of intraday observations increases). I derive easy-to-check conditions under which the error-in-variables problem that arises does not affect the asymptotic distribution of the resulting estimator.

Chapter 4 of the dissertation analyzes the pricing of jump risk and its implication for the variance risk premium. I use data on the VIX index to recover the expectation of future variance under the risk-neutral measure and high-frequency data on the stock market index to measure the quadratic variation and its continuous and discontinuous components. Working with a general semiparametric SV model and the estimation technique of Chapter 3, I uncover variance risk premium that shows significant variation. A central finding of the paper is the differential effect that jumps have on the risk premium and the price dynamics. The effect on the future price dynamics is short lived, while the effect on the risk premium is very persistent.