Roberto Renò

Discrete-Time Volatility Forecasting With Persistent Leverage Effect and the Link With Continuous-Time Volatility Modeling

We first propose a reduced-form model in discrete time for S&P 500 volatility showing that the forecasting performance can be significantly improved by introducing a persistent leverage effect with a long-range dependence similar to that of volatility itself. We also find a strongly significant positive impact of lagged jumps on volatility, which however is absorbed more quickly. We then estimate continuous-time stochastic volatility models that are able to reproduce the statistical features captured by the discrete-time model. We show that a single-factor model driven by a fractional Brownian motion is unable to reproduce the volatility dynamics observed in the data, while a multifactor Markovian model fully replicates the persistence of both volatility and leverage effect. The impact of jumps can be associated with a common jump component in price and volatility. This article has online supplementary materials.

On measuring volatility and the GARCH forecasting performance

Abstract We apply a new algorithm based on Fourier analysis to compute the volatility of a diffusion process. By using simulations of the continuous-time GARCH model, we show that our method performs well in computing integrated volatility. We show that linear interpolation of high frequency observations induces a downward bias in estimating integrated volatility. By measuring ex post volatility with our method, we find that the forecasting performance of the GARCH model is improved with respect to what is established when classical methods are employed. These results are confirmed by the analysis of exchange rate high frequency time series.

On measuring volatility of diffusion processes with high frequency data

Abstract We analyze a recently proposed method to estimate the volatility of a diffusion process with high frequency data. The method is based on Fourier analysis, all observations are included in the computation without any data manipulation. By Monte Carlo experiments, we evaluate its performance in measuring volatility under the assumption that the asset price evolves according to models belonging to the SR-SARV(1) class, which includes GARCH(1, 1) as a particular case. We compare the performance of the method to that associated with the cumulative squared intraday returns. The forecasting capability of the models is also evaluated.

Nonparametric Stochastic Volatility

Using recent advances in the nonparametric estimation of continuous-time processes under mild statistical assumptions as well as recent developments on nonparametric volatility estimation by virtue of market microstructure noise-contaminated high-frequency asset price data, we provide (i) a theory of spot variance estimation and (ii) functional methods for stochastic volatility modelling. Our methods allow for the joint evaluation of return and volatility dynamics with nonlinear drift and diffusion functions, nonlinear leverage effects, jumps in returns and volatility with possibly state-dependent jump intensities, as well as nonlinear risk-return trade-offs. Our identification approach and asymptotic results apply under weak recurrence assumptions and, hence, accommodate the persistence properties of variance in finite samples. Functional estimation of a generalized (i.e., nonlinear) version of the square-root stochastic variance model with jumps in both volatility and returns for the S&P500 index suggests the need for richer variance dynamics than in existing work. We find a linear specification for the variances diffusive variance to be misspecified (and inferior to a more flexible CEV specification) even when allowing for jumps in the variance dynamics.

Trading strategies in the Italian interbank market

Using a data set which includes all transactions among banks in the Italian money market, we study their trading strategies and the dependence among them. We use the Fourier method to compute the variance–covariance matrix of trading strategies. Our results indicate that well defined patterns arise. Two main communities of banks, which can be coarsely identified as small and large banks, emerge.

NONPARAMETRIC ESTIMATION OF THE DIFFUSION COEFFICIENT OF STOCHASTIC VOLATILITY MODELS

In this paper, new fully nonparametric estimators of the diffusion coefficient of continuous time models are introduced. The estimators are based on Fourier analysis of the state variable trajectory observed and on the estimation of quadratic variation between observations by means of realized volatility. The estimators proposed are shown to be consistent and asymptotically normally distributed. Moreover, the Fourier estimator can be iterated to get a fully nonparametric estimate of the diffusion coefficient in a bivariate model in which one state variable is the volatility of the other. The estimators are shown to be unbiased in small samples using Monte Carlo simulations and are used to estimate univariate and bivariate models for interest rates.

Threshold Estimation of Jump-Diffusion Models and Interest Rate Modeling

We reconstruct the level-dependent diffusion coefficient of a univariate semimartingale with jumps which is observed discretely. The consistency and asymptotic normality of our estimator are provided in presence of both finite and infinite activity (finite variation) jumps. Our results rely on kernel estimation, using the properties of the local time of the data generating process and the fact that it is possible to disentangle the discontinuous part of the state variable through those squared increments between observations exceeding a suitable threshold function. We also reconstruct the drift and the jump intensity coefficients when they are level-dependent and jumps have finite activity, through consistent and asymptotically normal estimators. Simulated experiments show that the newly proposed estimators are better performing in finite samples than alternative estimators, and this allows us to reexamine the estimation of a univariate model for the short term interest rate, for which we find less jumps and more variance due to the diffusion part than previous studies.

Spot Volatility Estimation Using Delta Sequences

We introduce a unifying class of nonparametric spot volatility estimators based on delta sequences and conceived to include many of the existing estimators in the field as special cases. The full limit theory is first derived when unevenly sampled observations under infill asymptotics and fixed time horizon are considered, and the state variable is assumed to follow a Brownian semimartingale. We then extend our class of estimators to include Poisson jumps or financial microstructure noise in the observed price process. This work makes different approaches (kernels, wavelets, Fourier) comparable. For example, we explicitly illustrate some drawbacks of the Fourier estimator. Specific delta sequences are applied to data from the S&P 500 stock index futures market.

Is volatility lognormal? Evidence from Italian futures

We study the unconditional volatility distribution of the Italian futures market, measuring it via Fourier analysis. Our data set consists of all tick-by-tick transactions in 2000 and 2001, a period characterized by unusually high volatility levels in its final part, because of the dramatic events following 11 September 2001. Our results show that the standard assumption of lognormal unconditional volatility has to be rejected for such a turbulent sample, since it is unable to capture the tail behavior of the distribution; a much better description is provided by a Pareto tail.

Unexpected volatility and intraday serial correlation

We study the impact of volatility on intraday serial correlation, at time scales of less than 20 minutes, exploiting a data set with all transactions on SPX500 futures from 1993 to 2001. We show that, while realized volatility and intraday serial correlation are linked, this relation is driven by unexpected volatility only, that is by the fraction of volatility that cannot be forecasted by a linear model. The impact of predictable volatility is instead found to be negative (LeBaron effect). Our results are robust to microstructure noise, and they confirm the leading economic theories on price formation.