The Leverage Effect in Stochastic Volatility: New Models, Bayesian Inference, and Model Selection

A striking empirical feature of many financial time series is that when the price drops, the future volatility increases. This negative correlation between the financial return and future volatility processes was initially addressed in Black 76 and explained based on financial leverage, or a firm's debt-to-equity ratio: when the price drops, financial leverage increases, the firm becomes riskier, and hence, the future expected volatility increases. The phenomenon is, therefore, traditionally been named the leverage effect..;In a discrete time Stochastic Volatility (SV) model framework, the leverage effect is often modelled by a negative correlation between the innovation processes of return and volatility equations. These models can be represented as state space models in which the returns and the volatilities are considered as the observed and the latent state variables respectively. Including the leverage effect in the SV model not only results in a better fit of the model to several types of financial time series, but also yields significant improvements in option pricing applications.;Statistical inference and model selection for such models are, in general, involved, mainly due to the fact that the model likelihood is not available in analytical closed-form and appears as a high-dimensional integral over the latent volatility vector. In this thesis, we apply Markov chain Monte Carlo (MCMC) methods (based on the influential work of Jacquier et al. (1994)) for parameter estimation. For model selection purposes, based on the Deviance Information Criterion (DIC) (introduced by Spiegelhalter et al. (2002)), we utilize a version of the DIC in which the latent volatility variables are integrated out. We then elaborate on new models for the leverage effect.;The important contributions of this thesis can be outlined as follows:;• In Section 2.1 Bayesian Model Selection we demonstrate, through extensive simulations, that when applying the DIC to the SV model selection problem, the latent volatility variables must be integrated out, otherwise the model selection conclusions will be potentially misleading.;• We use particle filtering algorithm (based on sequential Monte Carlo methods) to estimate the marginal likelihood of the SV model. We then use numerical maximization techniques to calculate the Akaike/Bayesian information criteria to test for the leverage effect in a selection of real financial time series. This method is applied as an alternative to the DIC model selection method and is shown to yield similar results (Section 2.2).;• In order to capture a potential nonlinear extreme value dependence structure between the returns and volatilities, in Chapter 3 we propose a new model that incorporates copulas into SV models. We develop the corresponding Bayesian inference methodology for the new model and show that for a wide selection of stock prices and indices, a Gaussian copula is favoured to the extreme value copulas.;• In Section 4.1 we propose a new model which assumes that the leverage effect is a time-varying stochastic process. Specifically, we consider an intuitively-supported high-persistence autoregressive process for the leverage effect. We then develop the corresponding Bayesian inference methodology and show that our new model outperforms the conventional fixed-leverage SV models in terms of the DIC.;• In Section 4.2 we propose a model that incorporates the realized volatility (which is an accurate estimator of the true volatility) into the SV model. Our model improves upon an existing similar model by correcting a subtle leverage timing issue and capturing the correlation between the realized volatility and the returns.;We test all our new models on simulated data as well as several real financial time series including stock prices, stock indices, and foreign exchange time series.