Fulvio Corsi

The volatility of realized volatility

In recent years, with the availability of high-frequency financial market data modeling realized volatility has become a new and innovative research direction. The construction of “observable” or realized volatility series from intra-day transaction data and the use of standard time-series techniques has lead to promising strategies for modeling and predicting (daily) volatility. In this article, we show that the residuals of commonly used time-series models for realized volatility and logarithmic realized variance exhibit non-Gaussianity and volatility clustering. We propose extensions to explicitly account for these properties and assess their relevance for modeling and forecasting realized volatility. In an empirical application for S&P 500 index futures we show that allowing for time-varying volatility of realized volatility and logarithmic realized variance substantially improves the fit as well as predictive performance. Furthermore, the distributional assumption for residuals plays a crucial role in density forecasting.

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.

Realized Correlation Tick-By-Tick

We propose the Heterogeneous Autoregressive (HAR) model for the estimation and prediction of realized correlations. We construct a realized correlation measure where both the volatilities and the covariances are computed from tick-by-tick data. As for the realized volatility, the presence of market microstructure can induce significant bias in standard realized covariance measure computed with artificially regularly spaced returns. Contrary to these standard approaches we analyse a simple and unbiased realized covariance estimator that does not resort to the construction of a regular grid, but directly and efficiently employs the raw tick-by-tick returns of the two series. Montecarlo simulations calibrated on realistic market microstructure conditions show that this simple tick-by-tick covariance possesses no bias and the smallest dispersion among the covariance estimators considered in the study. In an empirical analysis on S&P 500 and US bond data we find that realized correlations show significant regime changes in reaction to financial crises. Such regimes must be taken into account to get reliable estimates and forecasts.

Discrete sine transform for multi-scale realized volatility measures§

In this study we present a new realized volatility estimator based on a combination of the multi-scale regression and discrete sine transform (DST) approaches. Multi-scale estimators similar to that recently proposed by Zhang (2006) can, in fact, be constructed within a simple regression-based approach by exploiting the linear relation existing between the market microstructure bias and the realized volatilities computed at different frequencies. We show how such a powerful multi-scale regression approach can also be applied in the context of the Zhou [Nonlinear Modelling of High Frequency Financial Time Series, pp. 109–123, 1998] or DST orthogonalization of the observed tick-by-tick returns. Providing a natural orthonormal basis decomposition of observed returns, the DST permits the optimal disentanglement of the volatility signal of the underlying price process from the market microstructure noise. The robustness of the DST approach with respect to the more general dependent structure of the microstructure noise is also shown analytically. The combination of the multi-scale regression approach with DST gives a multi-scale DST realized volatility estimator similar in efficiency to the optimal Cramer–Rao bounds and robust against a wide class of noise contamination and model misspecification. Monte Carlo simulations based on realistic models for price dynamics and market microstructure effects show the superiority of DST estimators over alternative volatility proxies for a wide range of noise-to-signal ratios and different types of noise contamination. Empirical analysis based on six years of tick-by-tick data for the S&P 500 index future, FIB 30, and 30 year U.S. Treasury Bond future confirms the accuracy and robustness of DST estimators for different types of real data. §Earlier versions of this paper were circulated under the title ‘A discrete sine transform approach for realized volatility measurement’.

When Micro Prudence Increases Macro Risk: The Destabilizing Effects of Financial Innovation, Leverage, and Diversification

By exploiting basic common practice accounting and risk-management rules, we propose a simple analytical dynamical model to investigate the effects of microprudential changes on macroprudential outcomes. Specifically, we study the consequence of the introduction of a financial innovation that allows reducing the cost of portfolio diversification in a financial system populated by financial institutions having capital requirements in the form of Value at Risk (VaR) constraint and following standard mark-to-market and risk-management rules. We provide a full analytical quantification of the multivariate feedback effects between investment prices and bank behavior induced by portfolio rebalancing in presence of asset illiquidity and show how changes in the constraints of the bank portfolio optimization endogenously drive the dynamics of the balance sheet aggregate of financial institutions and, thereby, the availability of bank liquidity to the economic system and systemic risk. The model shows that when financial innovation reduces the cost of diversification below a given threshold, the strength (because of higher leverage) and coordination (because of similarity of bank portfolios) of feedback effects increase, triggering a transition from a stationary dynamics of price returns to a nonstationary one characterized by steep growths (bubbles) and plunges (bursts) of market prices.

Modelling Systemic Price Cojumps with Hawkes Factor Models

Instabilities in the price dynamics of a large number of financial assets are a clear sign of systemic events. By investigating portfolios of highly liquid stocks, we find that there are a large number of high-frequency cojumps. We show that the dynamics of these jumps is described neither by a multivariate Poisson nor by a multivariate Hawkes model. We introduce a Hawkes one-factor model which is able to capture simultaneously the time clustering of jumps and the high synchronization of jumps across assets.

Efficient Estimation of Volatility using High Frequency Data

The limitations of volatilities computed with daily data as well as simple statistical considerations strongly suggest to use intraday data in order to obtain accurate volatility estimates. Under a continuous time arbitrage-free setup, the quadratic variations of the prices would allow us, in principle, to construct an approximately error free estimate of volatility by using data at the highest frequency available. Yet, empirical data at very short time scales differ in many ways from the arbitrage-free continuous time price processes. For foreign exchange rates, the main difference originates in the incoherent structure of the price formation process. This market micro-structure effect introduces a noisy component in the price process leading to a strong overestimation of volatility when using naive estimators. Therefore, to be able to fully exploit the information contained in high frequency data, this incoherent effect needs to be discounted. In this contribution, we investigate several unbiased estimators that take into account the incoherent noise. One approach is to use a filter for pre-whitening the prices, and then using volatility estimators based on the filtered series. Another solution is to directly define a volatility estimator using tick-by-tick price differences, and including a correction term for the price formation effect. The properties of these estimators are investigated by Monte Carlo simulations. A number of important real-world effects are included in the simulated processes: realistic volatility and price dynamic, the incoherent effect, seasonalities, and random arrival time of ticks. Moreover, we investigate the robustness of the estimators with respect to data frequency changes and gaps. Finally, we illustrate the behavior of the best estimators on empirical data.

Intraday LeBaron effects

We study the relation at intraday level between serial correlation and volatility of the Standard and Poor (S&P) 500 stock index futures returns. At daily and weekly levels, serial correlation and volatility forecasts have been found to be negatively correlated (LeBaron effect). After finding a significant attenuation of the original effect over time, we show that a similar but more pronounced effect holds by using intraday measures, by such as realized volatility and variance ratio. We also test the impact of unexpected volatility, defined as the part of volatility which cannot be forecasted, on the presence of intraday serial correlation in the time series by employing a model for realized volatility based on the heterogeneous market hypothesis. We find that intraday serial correlation is negatively correlated to volatility forecasts, whereas it is positively correlated to unexpected volatility.

A Score-Driven Conditional Correlation Model for Noisy and Asynchronous Data: An Application to High-frequency Covariance Dynamics

We propose a new multivariate conditional correlation model able to deal with data featuring both observational noise and asynchronicity. When modelling high-frequency multivariate financial time-series, the presence of both problems and the requirement for positive-definite estimates makes the estimation and forecast of the intraday dynamics of conditional covariance matrices particularly difficult. Our approach tackles all these challenging tasks within a new Gaussian state-space model with score-driven time-varying parameters that can be estimated using standard maximum likelihood methods. Similarly to DCC models, large dimensionality is handled by separating the estimation of correlations from individual volatilities. As an interesting outcome of this approach, intra-day patterns are recovered without the need of any cross-sectional averaging, allowing, for instance, to estimate the real-time response of the market covariances to macro-news announcements.

Term Structure of Variance Risk Premium and Returns' Predictability

We derive an analytic relation between equity risk premium and the term structure of variance risk premium (VRP). Motivated by this result, we estimate the VRP term structure using a general and fully analytical discrete-time option pricing framework featuring multiple volatility components and multiple risk premia. We confirm the importance of VRP in improving option pricing performances and show the ability of multi-component GARCH models to produce realistic hump-shaped VRP term structure. We finally uncover the strong predictive power of the shape of the VRP term structure, summarized by its slope, on future stock-index returns.