A Discrete Time Approach to Option Pricing

Abstract

The goal of the PhD thesis is to propose a very general and fully analytical option pricing framework encompassing a wide class of discrete time models. The framework features multiple components structure in both volatility and leverage, and a flexible pricing kernel with multiple risk premia. The proposed approach is general enough to include either GARCH-type volatility, Realized Volatility or a combination of the two.Exploiting the general framework we propose three new option pricing models with original dynamics under physical measure. We start with an extension of HARG-RV model with heterogenous and analytically tractable leverage structure called LHARG-RV. Then we propose a model being mixture of LHARG-RV and GARCH approach acronymed GARCH-LHARG-RV and finally we consider an extension of LHARG-RV with jumps called JLHARG-RV. We also reconsider the CGARCH model of Christoffersen et al. (2008) by applying two-dimensional pricing kernel. Since our framework guarantees existence of semi-closed form formulas for option prices, the option pricing methodology is fast and efficient in implementation. In addition, by applying family of our fully analytically models with multi-component structure in volatility on a large sample of S&P 500 options, we show the superior ability of our models in pricing out-of-the-money (OTM) options compared to the existing benchmarks. Moreover, we derive an analytic relation between equity risk premium and the term structure of variance risk premium (VRP). Motivated by this result, we estimate the VRP term structure implied by CGARCH model and we show its ability to reproduce a realistic family of term structures of variance risk premium including the empirically observed hump-shaped curves. We finally uncover the strong predictive power of the shape of the VRP term structure, summarized by its slope, on future stock-index returns. Our findings reveal that stock investors should not only follow the popular Wall Street’s wisdom which suggests to look at the level of VIX Index but they also should look at the level of the term spread of variance risk premium, defined as the difference between monthly and annual VRP.A novel contribution of the thesis is an introduction of the new model, GARCH-LHARG-RV, (Section 3.3 and empirical results in Chapter 4). The new model, being combination of realized and latent volatility approaches, highlights the strength of the proposed general option pricing framework by delivering a superior option pricing performance. Moreover, thesis provides generalisation of result in Buhlmann et al. (1998) to multi-dimensional case (Theorem 10 in the thesis is showing that multi-dimensional Esscher transform satisfies Pareto equilibrium).