Non-parametric estimation of stochastic volatility models: spot volatility, leverage and vol-of-vol. Four essays on asymptotic error distributions, finite-sample properties and empirical applications.

This thesis contains four essays on non-parametric estimators of the spot volatility, the leverage and the volatility-of-volatility. In particular, the focus of this thesis is on the study of the asymptotic properties of the estimators, the optimization of their finite-sample performance and the use of the resulting estimates in empirical applications. Specifically, in Chapter 2 we prove a central limit theorem for the estimator of the integrated leverage based on the Fourier method of Malliavin and Mancino (2009), showing that it reaches the optimal rate of convergence and a smaller variance with respect to different estimators based on a pre-estimation of the instantaneous volatility. Then, we exploit the availability of efficient Fourier-based estimates of the integrated leverage to show, using S&P500 prices over the period 2006-2018, that adding an extra term which accounts for the leverage effect to the Heterogeneous Auto-Regressive (HAR) volatility model by Corsi (2009) increases the explanatory power of the latter. In Chapter 3 we study the sensitivity of the leverage process to changes of the price and the volatility. In particular, under the Constant Elasticity of Variance (CEV) model by Beckers (1980), which is explicitly designed to capture leverage effects, we find that the derivatives of the leverage with respect to the log-price and the volatility can be expressed as the ratio of quantities that can be consistently estimated from sample prices, that is, as the ratio of the price-leverage covariation and, respectively, the volatility and the leverage. From the financial standpoint, this suggests that the price-leverage covariation may be interpreted as a gauge of the responsiveness of the leverage to the arrival of new information that causes changes in the price or the volatility. Additionally, we also find that the priceleverage covariation is equal to twice the vol-of-vol under the CEV model, thereby suggesting that the responsiveness of the leverage (i.e., the price-leverage covariation) is proportional to the amount of uncertainty about risk (i.e., the vol-of-vol). After reconstructing the trajectories of the volatility, the leverage, the vol-of-vol and the price-leverage covariation through the Fourier methodology by Malliavin and Mancino (2009), we provide empirical evidence supporting this financial interpretation of the price-leverage covariation in a model-free setting, using 1-second S&P500 prices over the period March, 2018-April, 2018. In Chapter 4, we perform an analytical study to identify the sources of the finite-sample bias that typically plagues the simplest and most natural vol-of-vol estimator, the Pre-estimated Spot-variance based Realized Variance (PSRV) by Barndorff-Nielsen and Veraart (2009). Based on the full knowledge of its analytical expression, we show that the finite-sample bias of the PSRV may be substantially reduced by allowing for the overlap of consecutive local windows to pre-estimate the spot variance. In particular, we provide a feasible analytical rule for the biasoptimal selection of the length of local windows when the volatility is a process in the Chan, Karolyi, Longstaff and Sanders (CKLS) class (see Chan et al. (1992)) and show that selections based on this analytical rule match some selections prescribed in the literature, based on simulations. In Chapter 5, we exploit efficient Fourier estimates of the path of the volatility to empirically investigate the functional link between the latter and the variance swap rate. Specifically, using S&P500 data over the period 2006-2018, we find overwhelming empirical evidence supporting the affine link analytically found by Kallsen et al. (2011) in the context of exponentially affine stochastic volatility models. Additionally, based on tests performed on yearly subsamples, we find that exponentially mean-reverting variance models provide a good fit during periods of extreme volatility, while polynomial models, introduced in Cuchiero (2011), are suited for years characterized by more frequent price jumps. These empirical results are confirmed when replacing Fourier estimates of the spot volatility with realized local estimates. Chapter 6 concludes, summarizing the main findings of the thesis.