This thesis contains four essays on nonparametric estimators of the spot volatility, the leverage and the volatilityofvolatility. In particular, the focus of this thesis is on the study of the asymptotic properties of the estimators, the optimization of their finitesample 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 preestimation of the instantaneous volatility. Then, we exploit the availability of efficient Fourierbased estimates of the integrated leverage to show, using S&P500 prices over the period 20062018, that adding an extra term which accounts for the leverage effect to the Heterogeneous AutoRegressive (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 logprice 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 priceleverage covariation and, respectively, the volatility and the leverage. From the financial standpoint, this suggests that the priceleverage 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 volofvol under the CEV model, thereby suggesting that the responsiveness of the leverage (i.e., the priceleverage covariation) is proportional to the amount of uncertainty about risk (i.e., the volofvol). After reconstructing the trajectories of the volatility, the leverage, the volofvol and the priceleverage covariation through the Fourier methodology by Malliavin and Mancino (2009), we provide empirical evidence supporting this financial interpretation of the priceleverage covariation in a modelfree setting, using 1second S&P500 prices over the period March, 2018April, 2018. In Chapter 4, we perform an analytical study to identify the sources of the finitesample bias that typically plagues the simplest and most natural volofvol estimator, the Preestimated Spotvariance based Realized Variance (PSRV) by BarndorffNielsen and Veraart (2009). Based on the full knowledge of its analytical expression, we show that the finitesample bias of the PSRV may be substantially reduced by allowing for the overlap of consecutive local windows to preestimate 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 20062018, 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 meanreverting 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.
Nonparametric estimation of stochastic volatility models: spot volatility, leverage and volofvol. Four essays on asymptotic error distributions, finitesample properties and empirical applications.
TOSCANO, Giacomo
2021
Abstract
This thesis contains four essays on nonparametric estimators of the spot volatility, the leverage and the volatilityofvolatility. In particular, the focus of this thesis is on the study of the asymptotic properties of the estimators, the optimization of their finitesample 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 preestimation of the instantaneous volatility. Then, we exploit the availability of efficient Fourierbased estimates of the integrated leverage to show, using S&P500 prices over the period 20062018, that adding an extra term which accounts for the leverage effect to the Heterogeneous AutoRegressive (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 logprice 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 priceleverage covariation and, respectively, the volatility and the leverage. From the financial standpoint, this suggests that the priceleverage 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 volofvol under the CEV model, thereby suggesting that the responsiveness of the leverage (i.e., the priceleverage covariation) is proportional to the amount of uncertainty about risk (i.e., the volofvol). After reconstructing the trajectories of the volatility, the leverage, the volofvol and the priceleverage covariation through the Fourier methodology by Malliavin and Mancino (2009), we provide empirical evidence supporting this financial interpretation of the priceleverage covariation in a modelfree setting, using 1second S&P500 prices over the period March, 2018April, 2018. In Chapter 4, we perform an analytical study to identify the sources of the finitesample bias that typically plagues the simplest and most natural volofvol estimator, the Preestimated Spotvariance based Realized Variance (PSRV) by BarndorffNielsen and Veraart (2009). Based on the full knowledge of its analytical expression, we show that the finitesample bias of the PSRV may be substantially reduced by allowing for the overlap of consecutive local windows to preestimate 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 20062018, 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 meanreverting 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.File  Dimensione  Formato  

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https://hdl.handle.net/20.500.14242/117514
URN:NBN:IT:SNS117514