Abstracts

Stochastic volatility models were introduced to reproduce well-established stylized facts like volatility smile and negative correlation of returns and volatility. Despite many successful application, stochastic volatility models in continuous and discrete time exhibit problems with fitting strike profile and term structure of implied volatility surface, especially for far in-the-money and out-of-the-money options. Many authors have suggested that the origin of these difficulties might be due to the heterogeneity of agents acting in the market. Investors with different time horizons have different impact on instantaneous volatility and as a consequence a single factor of volatility, running on a single time scale, is simply not sufficient for describing the dynamics of the volatility process. This argument has been empirically confirmed and has led to the development of models with multi-component volatility structure, where each component of volatility corresponds to different time-scale. Recently it has been advocated that also modelling leverage effect needs multi-component structure.
We focus our research on discrete-time models, which have the inherent advantage that volatility is readily observable from the history of asset prices by filtering (e.g. GARCH models) or by precise non-parametric measurement based on intra-day data (Realized Volatility approach). In the current literature, the analytical tractability of discrete time option pricing models is guarantee only for rather specific type of models and pricing kernels. We propose a very general and fully analytical option pricing framework encompassing a wide class of discrete time models featuring multiple components structure in both volatility and leverage and a flexible pricing kernel with multiple risk premia. The proposed framework is general enough to include either GARCH-type volatility, Realized Volatility or a combination of the two. In our presentation we will apply our framework to two particular models: multi-component GARCH and HARG-RV with extended heterogeneous structure of leverage. In both cases we will apply multi-dimensional pricing kernel, taking into account various risk premia and obtaining closed-form solutions for option prices.