Motivated by empirical studies showing the patterns of volatility in financial time series, in the last decades many stochastic volatility models have been proposed.
The class of the stochastic volatility (SV) models was established with the focus on accounting for the well-known stylized facts of asset returns, such as time varying volatility, volatility clustering and leverage effect.
The multi-factor stochastic volatility (MFSV) models allow to take into account ”new” stylized facts observed in asset returns, such as stochastic skewness (Carr and Wu, 2007), stochastic leverage (Veraart and Veraart, 2012), stochastic volatility of volatility (Barndorff-Nielsen and Veraart, 2009), and the existence of a time-varying risk premium (Bollerslev et al., 2009) and are very successful in the context of option pricing (Christoffersen et al., 2009).
Recent theoretical and empirical studies on volatility modeling have pointed out the existence of volatility feedback effects. Volatility feedback and leverage effects are related to the same phenomena: the leverage effect explains why a negative return causes an increase in the volatility and was first discussed in (Black, 1976) and (Christie, 1982); in contrast, the notion of volatility feedback effect is based on the argument that volatility is priced and an increase in the volatility raises the required return on the asset, which can only be produced by an immediate decline in the asset price as observed by (French et al., 1987). The volatility feedback rate is a second order quantity which is supposed to describe the facility for the market to absorb small perturbations and can be used to explain the irregular behavior and instability of financial markets.
Despite the ability of the MFSV models to describe the dynamics of the asset returns, such models are difficult to calibrate to market information and then oversimplified models are commonly used in quantitative analysis. The recent financial crises, however, highlight that we can no longer use simplified models and then we have to improve our statistical methodologies.
The seminar will focus on the estimation of stochastic volatility of volatility, stochastic leverage and volatility feedback effects using high frequency data. We propose a Fourier estimator that needs only to pre-estimate the Fourier coefficients of the volatility process from the observations of a (noisy) price process and does not require a preliminary estimation of the instantaneous volatility. In fact, in the Fourier methodology the computation of the Fourier coefficients of the unobservable instantaneous variance process from observed prices allows us to reconstruct the volatility process in the frequency domain and thus to handle this process as an observable variable, so that second order effects can be estimated.
