Author Archives: Luca Cattivelli

About Luca Cattivelli

Random walks on graphs, Econometrics

Cattivelli, Luca, and Davide Pirino. “A SHARP model of bid-ask spread forecasts.” (2017).

In this paper we propose an accurate and fast-to-estimate forecasting model for discrete valued time series with long memory and seasonality. The modelisation is achieved with an autoregressive Poisson process that features seasonality and heterogeneous autoregressive components (whence the acronym SHARP: Seasonal Heterogeneous AutoRegressive Poisson). Motivated by the prominent role of the bid-ask spread as a transaction cost for trading, we apply the SHARP model to forecast the bid-ask spread of a large sample of NYSE equity stocks. Indeed, the possibility of having a good forecasting model is of great importance for many applications, in particular for algorithms of optimal execution of orders.

We define two possible extensions of the model in order to investigate the possibility of increasing the forecasting accuracy of the original SHARP approach. The first extension features the presence of spillovers in the spread dynamics among equity stocks while the second is inspired by the Realized GARCH model of Hansen, Huang and Shek (2012), and features a measurement equation which relates the observed intra-minute (weighted) average spread with the unobserved instantaneous conditional Poisson intensity. We conclude with an application of our models by showing how bid-ask spread forecasts can be exploited to reduce the total cost incurred by a trader that is willing to buy (or sell) a given amount of an equity stock.

Agliari, Elena, et al. “Two-particle problem in comblike structures.” Physical Review E 93.5 (2016): 052111.

Encounters between walkers performing a random motion on an appropriate structure can describe a wide variety of natural phenomena ranging from pharmacokinetics to foraging. On homogeneous structures the asymptotic encounter probability between two walkers is (qualitatively) independent of whether both walkers are moving or one is kept fixed. On infinite comblike structures this is no longer the case and here we deepen the mechanisms underlying the emergence of a finite probability that two random walkers will never meet, while one single random walker is certain to visit any site. In particular, we introduce an analytical approach to address this problem and even more general problems such as the case of two walkers with different diffusivity, particles walking on a finite comb and on arbitrary bundled structures, possibly in the presence of loops. Our investigations are both analytical and numerical and highlight that, in general, the outcome of a reaction involving two reactants on a comblike architecture can strongly differ according to whether both reactants are moving (no matter their relative diffusivities) or only one is moving and according to the density of shortcuts among the branches.

Luca Cattivelli, Elena Agliari, Fabio Sartori, and Davide Cassi. 2015 Lévy flights with power-law absorption. Phys. Rev. E 92, 042156

We consider a particle performing a stochastic motion on a one-dimensional lattice with jump lengths distributed according to a power law with exponent μ+1. Assuming that the walker moves in the presence of a distribution a(x) of targets (traps) depending on the spatial coordinate x, we study the probability that the walker will eventually find any target (will eventually be trapped). We focus on the case of power-law distributions a(x)xα and we find that, as long as μ<α, there is a finite probability that the walker will never be trapped, no matter how long the process is. This result is shown via analytical arguments and numerical simulations which also evidence the emergence of slow searching (trapping) times in finite-size system. The extension of this finding to higher-dimensional structures is also discussed.

Agliari, E., Sartori, F., Cattivelli, L. and Cassi, D., 2015. Hitting and trapping times on branched structures. Physical Review E, 91(5), p.052132.

In this work we consider a simple random walk embedded in a generic branched structure and we find a close-form formula to calculate the hitting time H(i,f) between two arbitrary nodes i and j. We then use this formula to obtain the set of hitting times {H(i,f)} for combs and their expectation values, namely, the mean first-passage time, where the average is performed over the initial node while the final node f is given, and the global mean first-passage time, where the average is performed over both the initial and the final node. Finally, we discuss applications in the context of reaction-diffusion problems.