Asset Pricing Under Jump Diffusion PDF Download

Continuous-time Markov processes are widely used to model a variety of variables in financial economics. When estimating the parameters of a continuous-time Markov model the method of choice, from a classical perspective, is maximum likelihood. However, in most cases the transition density of the process is not known in closed form and so the likelihood is uncomputable in closed form. In the first chapter of this dissertation I construct a closed form series expansion for the unknown likelihood for jump-diffusion models. In particular I can treat jump-diffusions with very little restriction on the state dependency of the jump distribution and this potentially allows for the construction of flexible models for state variables such as nominal interest rates or volatilities that have a natural finite boundary. It is well known that GARCH models, when viewed as filters and not as the data generating process, can consistently filter the unobservable volatility state of a diffusion process with stochastic volatility. However although the use of GARCH models remains widespread, if one accepts that in most applications the underlying process is likely to exhibit jumps then it is not clear what, if anything, the GARCH model is estimating. The second chapter of this dissertation shows that GARCH models retain their consistency for the diffusive volatility when the data generating process has jumps, provided that the diffusive volatility follows a diffusion. In a situation where ultra high frequency data is unavailable a GARCH type model is likely to be appropriate for volatility estimation. The result of this paper implies that in the presence of jumps the GARCH type model is still applicable provided the jumps are included in the quasi-likelihood of the time series model. Finally in the third chapter I construct a measure of "jumpiness" that does not require intra-day data and is robust to a realistic amount of error in the filtering of the diffusive volatility. This allows me to design a test for the presence of jumps that is applicable in the absence of ultra-high frequency data. An application to USD swap rate data indicates that jumps are prevalent in the yield curve and that jumps account for roughly a quarter of the variation in 10 year USD swap rates.