Inference for stochastic volatility models based on Lévy processes

The standard Black-Scholes model is a continuous time model to predict asset movement. For the standard model, the volatility is constant but frequently this model is generalised to allow for stochastic volatility (SV). As the Black-Scholes model is a continuous time model, it is attractive to have a continuous time stochastic volatility model and recently there has been a lot of research into such models. One of the most popular models was proposed by Barndorff-Nielsen and Shephard (2001b) (BNS), where the volatility follows an Ornstein-Uhlenbeck (OU) equation and is driven by a background driving Lévy process (BDLP). The correlation in the volatility decays exponentially and so the model is able to explain the volatility clustering present in many financial time series. This model is studied in detail, with assets following the Black-Scholes equation with the BNS SV model. Inference for the BNS SV models is not trivial, particularly when Markov chain Monte Carlo (MCMC) is used. This has been implemented in Roberts et al. (2004) and Griffin and Steel (2003) where a Gamma marginal distribution for the volatility is used. Their focus is on the difficult MCMC implementation and the performance of different proposals, mainly using training data generated from the model itself. In this thesis, the four main new contributions to the Black-Scholes equation with volatility following the BNS SV model are as follows:- (1) We perform the MCMC inference for generalised Inverse Gaussian and Tempered Stable marginal distributions, as well as the special cases, the Gamma, Positive Hyperbolic, Inverse Gamma and Inverse Gaussian distributions. (2) Griffin and Steel (2003) consider the superposition of several BDLPs to give quasi long-memory in the volatility process. This is computationally problematic and so we allow the volatility process to be non-stationary by allowing one of the parameters, which controls the correlation in the volatility process, to vary over time. This allows the correlation of the volatility to be non-stationary and further volatility clustering. (3) The standard Black-Scholes equation is driven by Brownian motion and a generalisation of this allowing for long-memory in the share equation itself (as opposed to the volatility equation), which is based on an approximation to fractional Brownian motion, is considered and implemented. (4) We introduce simulation methods and inference for a new class of continuous time SV models, with a more flexible correlation structure than the BNS SV model. For each of (1), (2) and (3), our focus is on the empirical performance of different models and whether such generalisations improve prediction of future asset movement. The models are tested using daily Foreign Exchange rate and share data for various different countries and companies.EThOS - Electronic Theses Online ServiceGBUnited Kingdo