Fabio Gobbi

Identifying The Brownian Covariation From The Co-Jumps Given Discrete Observations

In this paper we consider two semimartingales driven by Wiener processes and (possibly infinite activity) jumps. Given discrete observations we separately estimate the integrated covariation IC from the sum of the co-jumps. The Realized Covariation (RC) approaches the sum of IC with the co-jumps as the number of observations increases to infinity. Our threshold (or truncated) estimator \hat{IC}_n excludes from RC all the terms containing jumps in the finite activity case and the terms containing jumps over the threshold in the infinite activity case, and is consistent. To further measure the dependence between the two processes also the betas, \beta^{(1,2)} and \beta^{(2,1)}, and the correlation coefficient \rho^{(1,2)} among the Brownian semimartingale parts are consistently estimated. In presence of only finite activity jumps: 1) we reach CLTs for \hat{IC}_n, \hat\beta^{(i,j)} and \hat \rho^{(1,2)}; 2) combining thresholding with the observations selection proposed in Hayashi and Yoshida (2005) we reach an estimate of IC which is robust to asynchronous data. We report the results of an illustrative application, made in a web appendix (on www.dmd.unifi.it/upload/sub/persone/mancini/WebAppendix3.pdf), to two very different simulated realistic asset price models and we see that the finite sample performances of \hat{IC}_n and of the sum of the co-jumps estimator are good for values of the observation step large enough to avoid the typical problems arising in presence of microstructure noises in the data. However we find that the co-jumps estimators are more sensible than \hat{IC}_n to the choice of the threshold. Finding criteria for optimal threshold selection is object of further research.

Diffusion covariation and co-jumps in bidimensional asset price processes with stochastic volatility and infinite activity Lévy jumps

In this paper we consider two processes driven by diffusions and jumps. The jump components are Levy processes and they can both have finite activity and infinite activity. Given discrete observations we estimate the covariation between the two diffusion parts and the co-jumps. The detection of the co-jumps allows to gain insight in the dependence structure of the jump components and has important applications in finance. Our estimators are based on a threshold principle allowing to isolate the jumps. This work follows Gobbi and Mancini (2006) where the asymptotic normality for the estimator of the covariation, with convergence speed &sqrt;h, was obtained when the jump components have finite activity. Here we show that the speed is &sqrt;h only when the activity of the jump components is moderate.

A Convolution-Based Autoregressive Process

We propose a convolution-based approach to the estimation of nonlinear autoregressive processes. The model allows for state-dependent autocorrelation, that is different persistence of the shocks in different phases of the market and dependent innovations, that is drawn from different distributions in different phases of the market.

A Copula-Based Model for Spatial and Temporal Dependence of Equity Markets

In this contribution we provide a consistent pricing setting for multivariate equity derivatives. Consistently with the prescriptions of the Efficient Market Hypothesis and of the martingale pricing approach, we provide a model in which prices are martingales both with respect to their own filtration and to the enlarged multivariate filtration. We show that if the log-prices follow processes with independent increments and each one of them is not Granger caused by the others, the pricing procedure can be performed by simply: i) generating time series of each asset; ii) linking assets at each time with a prescribed copula function. We provide applications to multivariate digital options and spread options.

Estimation of Copula Models

In this chapter, we introduce copula functions and their main properties. For a more detailed study, we refer the interested reader to Joe (Multivariate models and dependence concepts, 1997), Nelsen (Introduction to copulas, 2006), and Durante and Sempi (Principles of copula theory, 2015).

Tail behavior of a sum of two dependent and heavy-tailed distributions

Abstract We consider the problem of a sum of two dependent and heavy tailed distributions through the C-convolution. The C-convolution provides the distribution of the sum of two random variables whose dependence structure is described by a copula function. Moreover, to investigate the role of heavy tails we use three different marginal distributions characterized by this property: Cauchy, Levy and Pareto. We show that the tail behavior of the C-convolution measured by level-q quantiles for q = 0.01, 0.05 (left tail) and q = 0.95, 0.99 (right tail) is strongly affected by the copula function which links the marginals and by the tail heaviness of marginals themselves.

Application to Interest Rates

There is a large literature investigating the nonlinear dynamics of the short-term rate. It mainly dates back to the last decade of the last century. Most of this literature was about persistence or mean reversion, linearity or nonlinearity, Gaussian or non-Gaussian innovations. Moreover, it is all about extensions and distortions of the linear AR(1) model, that is the subject addressed in this book. It is then the appropriate application to show how our approach works in practice, and maybe to stimulate new research on the subject.

Convolution-Based Processes

In what follows, we consider a random vector (X, Y) and we study the distribution of \(X+Y\) and the copula associated to the random vector \((X,X+Y)\). Since this represents the basic concept of the book, we include proofs, even if they are also presented in Cherubini et al., (Dynamic copula methods in finance, 2012) (see also Cherubini et al., Journal of Multivariate Analysis, 2011).

Copulas and Estimation of Markov Processes

In this section, we briefly introduce a central result due to Darsow, Nguyen, and Olsen (see Darsow et al., Illinois Journal of Mathematics, 36, 600–642, 1992 for the original and complete result) that allows to characterize a Markov process through the dependence structure of the finite dimensional levels independently of their marginal distributions.

The Dynamics of Economic Variables

In 1957 Pablo Picasso painted a series of interpretations of an old and famous painting by Velazquez of 1656 called Las Meninas, portraying the court of the Infanta Margarita Teresa. He reinterpreted, partitioned, and distorted the image of the painting in many new images.