Volatility estimate via Fourier analysis

From the preface: The aim of this Thesis is to study some selected topics on volatility estimation and modeling. Recently, these topics received great attention in the financial literature, since volatility modeling is crucial in practically all financial applications, including derivatives pricing, portfolio selection and risk management. Specifically, we focus on the concept of realized volatility, which became important in the last decade mainly thanks to the increased availability of high-frequency data on practically every financial asset traded in the main marketplaces. The concept of realized volatility traces back to an early idea of Merton (1980), and basically consists in the estimation of the daily variance via the sum of squared intraday returns, see Andersen et al. (2003). The work presented here is linked to this strand of literature but an alternative estimator is adopted. This is based on Fourier analysis of the time series, hence the term Fourier estimator, which has been recently proposed by Malliavin and Mancino (2002). Moreover, we start from this result to introduce a nonparametric estimator of the diffusion coefficient. The Thesis has two main objectives. After introducing the concept of quadratic variation and the Fourier estimator, we compare the properties of this estimator with realized volatility in a univariate and multivariate setting. This leads us to some applications in which we exploit the fact that we can regard volatility as an observable instead of a latent variable. We pursue this objective in Chapters 3 and 4. The second objective is to prove two Theorems on the estimation of the diffusion coefficient of a stochastic diffusion in a univariate setting, and this is pursued in Chapter 5. [...]