Motivated by the evidence that real-world networks evolve in time and may exhibit non-stationary features, we propose an extension of the Exponential Random Graph Models (ERGMs) accommodating the time variation of network parameters. Within the ERGM framework, a network realization is sampled from a static probability distribution defined parametrically in terms of network statistics. Inspired by the fast growing literature on Dynamic Conditional Score-driven models, in our approach, each parameter evolves according to an updating rule driven by the score of the conditional distribution. We demonstrate the flexibility of the score-driven ERGMs, both as data generating processes and as filters, and we prove the advantages of the dynamic version with respect to the static one. Our method captures dynamical network dependencies, that emerge from the data, and allows for a test discriminating between static or time-varying parameters. Finally, we corroborate our findings with the application to networks from real financial and political systems exhibiting non stationary dynamics.

# Category Archives: Preprints

# D. Di Gangi, G. Bormetti, F. Lillo (2019) ,

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L.M. Calcagnile, F. Corsi, S. Marmi, * Entropy and efficiency of the ETF market**
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We investigate the relative information efficiency of financial markets by measuring the entropy of the time series of high frequency data. Our tool to measure efficiency is the Shannon entropy, applied to 2-symbol and 3-symbol discretisations of the data. Analysing 1-minute and 5-minute price time series of 55 Exchange Traded Funds traded at the New York Stock Exchange, we develop a methodology to isolate true inefficiencies from other sources of regularities, such as the intraday pattern, the volatility clustering and the microstructure effects. The first two are modelled as multiplicative factors, while the microstructure is modelled as an ARMA noise process. Following an analytical and empirical combined approach, we find a strong relationship between low entropy and high relative tick size and that volatility is responsible for the largest amount of regularity, averaging 62% of the total regularity against 18% of the intraday pattern regularity and 20% of the microstructure.

arXiv preprint arXiv:1609.04199

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P.Mazzarisi, P.Barucca, F.Lillo, D.Tantari,

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P.Barucca, P.Mazzarisi, F.Lillo, D.Tantari (2017),

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Letizia E., Lillo F. (2017).

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P.Mazzarisi, P.Barucca, F.Lillo, D.Tantari, *A dynamic network model with persistent links and node-specific latent variables, with an application to the interbank market*

We propose a dynamic network model where two mechanisms control the probability of a link between two nodes: (i) the existence or absence of this link in the past, and (ii) node-specific latent variables (dynamic fitnesses) describing the propensity of each node to create links. Assuming a Markov dynamics for both mechanisms, we propose an Expectation-Maximization algorithm for model estimation and inference of the latent variables. The estimated parameters and fitnesses can be used to forecast the presence of a link in the future. We apply our methodology to the e-MID interbank network for which the two linkage mechanisms are associated with two different trading behaviors in the process of network formation, namely preferential trading and trading driven by node-specific characteristics. The empirical results allow to recognise preferential lending in the interbank market and indicate how a method that does not account for time-varying network topologies tends to overestimate preferential linkage.

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P.Barucca, P.Mazzarisi, F.Lillo, D.Tantari (2017), * Disentangling group and link persistence in dynamic stochastic block models*

We study the inference of a model of dynamic networks in which both communities and

links keep memory of previous network states. By considering maximum likelihood inference from

single snapshot observations of the network, we show that link persistence makes the inference of

communities harder, decreasing the detectability threshold, while community persistence tends to make

it easier. We analytically show that communities inferred from single network snapshot can share a

maximum overlap with the underlying communities of a specific previous instant in time. This leads

to time-lagged inference: the identification of past communities rather than present ones. Finally

we compute the time lag and propose a corrected algorithm, the Lagged Snapshot Dynamic (LSD)

algorithm, for community detection in dynamic networks. We analytically and numerically characterize

the detectability transitions of such algorithm as a function of the memory parameters of the model.

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Letizia E., Lillo F. (2017). *Corporate payments networks and credit risk rating*

This paper provides empirical evidences that corporate firms risk assessment could benefit from taking quantitatively into account the network of interactions among firms. Indeed, the structure of interactions between firms is critical to identify risk concentration and the possible pathways of propagation of financial distress. In this work, we consider the interactions by investigating a large proprietary dataset of payments between Italian firms. We first characterise the topological properties of the payment networks, and then we focus our attention on the relation between the networks and the risk of firms. Our main finding is to document the existence of an homophily of risk, i.e. the tendency of firms with similar risk profile to be statistically more connected among themselves. This effect is observed when considering both pairs of firms and communities or hierarchies identified in the network. We leverage this knowledge to demonstrate that network properties of a node can be used to predict the missing rating of a firm.

# 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.

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2899105

# Schneider, M. and Lillo, F. (2016) Cross-Impact and No-Dynamic-Arbitrage

**Abstract**

We extend the “No-dynamic-arbitrage and market impact”-framework of Jim Gatheral [Quantitative Finance, 10(7): 749-759 (2010)] to the multidimensional case where trading in one asset has a cross-impact on the price of other assets. From the condition of absence of dynamical arbitrage we derive theoretical limits for the size and form of cross-impact that can be directly verified on data. For bounded decay kernels we find that cross-impact must be an odd and linear function of trading intensity and cross-impact from asset i to asset j must be equal to the one from j to i. To test these constraints we estimate cross-impact among sovereign bonds traded on the electronic platform MOT. While we find significant violations of the above symmetry condition of cross-impact, we show that these are not arbitrageable with simple strategies because of the presence of the bid-ask spread.

# Schneider, M., Lillo, F., and Pelizzon, L. (2016) How Has Sovereign Bond Market Liquidity Changed? – An Illiquidity Spillover Analysis. SAFE Working Paper No. 151

**Abstract**

Amid increasing regulation, structural changes of the market and Quantitative Easing as well as extremely low yields, concerns about the market liquidity of the Eurozone sovereign debt markets have been raised. We aim to quantify illiquidity risks, especially such related to liquidity dry-ups, and illiquidity spillover across maturities by examining the reaction to illiquidity shocks at high frequencies in two ways:

a) the regular response to shocks using a variance decomposition and,

b) the response to shocks in the extremes by detecting illiquidity shocks and modeling those as ultivariate Hawkes processes.

We find that:

a) market liquidity is more fragile and less predictable when an asset is very illiquid and,

b) the response to shocks in the extremes is structurally different from the regular response.

In 2015 long-term bonds are less liquid and the medium-term bonds are liquid, although we observe that in the extremes the medium-term bonds are increasingly driven by illiquidity spillover from the long-term titles.

# Taranto, D. E., Bormetti, G., Bouchaud, J.-P., Toth, B., and Lillo, F. (2016). Linear models for the impact of order flow on prices II. The Mixture Transition Distribution model

**Abstract**

Modeling the impact of the order flow on asset prices is of primary importance to understand the behavior of financial markets. Part I of this paper reported the remarkable improvements in the description of the price dynamics which can be obtained when one incorporates the impact of past returns on the future order flow. However, impact models presented in Part I consider the order flow as an exogenous process, only characterized by its two-point correlations. This assumption seriously limits the forecasting ability of the model. Here we attempt to model directly the stream of discrete events with a so-called Mixture Transition Distribution (MTD) framework, introduced originally by Raftery (1985). We distinguish between price-changing and non price-changing events and combine them with the order sign in order to reduce the order flow dynamics to the dynamics of a four-state discrete random variable. The MTD represents a parsimonious approximation of a full high-order Markov chain. The new approach captures with adequate realism the conditional correlation functions between signed events for both small and large tick stocks and signature plots. From a methodological viewpoint, we discuss a novel and flexible way to calibrate a large class of MTD models with a very large number of parameters. In spite of this large number of parameters, an out-of-sample analysis confirms that the model does not overfit the data.

# Taranto, D. E., Bormetti, G., Bouchaud, J.-P., Toth, B., and Lillo, F. (2016). Linear models for the impact of order flow on prices I. Propagators: Transient vs. History Dependent Impact

**Abstract**

Market impact is a key concept in the study of financial markets and several models have been proposed in the literature so far. The Transient Impact Model (TIM) posits that the price at high frequency time scales is a linear combination of the signs of the past executed market orders, weighted by a so-called propagator function. An alternative description — the History Dependent Impact Model (HDIM) — assumes that the deviation between the realised order sign and its expected level impacts the price linearly and permanently. The two models, however, should be extended since prices are a priori influenced not only by the past order flow, but also by the past realisation of returns themselves. In this paper, we propose a two-event framework, where price-changing and non price-changing events are considered separately. Two-event propagator models provide a remarkable improvement of the description of the market impact, especially for large tick stocks, where the events of price changes are very rare and very informative. Specifically the extended approach captures the excess anti-correlation between past returns and subsequent order flow which is missing in one-event models. Our results document the superior performances of the HDIMs even though only in minor relative terms compared to TIMs. This is somewhat surprising, because HDIMs are well grounded theoretically, while TIMs are, strictly speaking, inconsistent.

# Corsi, F., Lillo, F. and Pirino, D. (2015). Measuring Flight-to-Quality with Granger-Causality Tail Risk Networks.

**Abstract**

We introduce an econometric method to detect and analyze events of flight-to-quality by financial institutions. Specifically, using the recently proposed test for the detection of Granger causality in risk (Hong et al. 2009), we construct a bipartite network of systemically important banks and sovereign bonds, where the presence of a link between two nodes indicates the existence of a tail causal relation. This means that tail events in the equity variation of a bank helps in forecasting a tail event in the price variation of a bond. Inspired by a simple theoretical model of flight-to-quality, we interpret links of the bipartite networks as distressed trading of banks directed toward the sovereign debt market and we use them for defining indicators of flight-to-quality episodes. Based on the quality of the involved bonds, we distinguish different patterns of flight-to-quality in the 2006-2014 period. In particular, we document that, during the recent Eurozone crisis, banks with a considerable systemic importance have significantly impacted the sovereign debt market chasing the top-quality government bonds. Finally, an out of sample analysis shows that connectedness and centrality network metrics have a significant cross-sectional forecasting power of bond quality measures.

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2576078

# Sabelli, C., Pioppi, M., Sitzia, L. and Bormetti, G. (2014). Multi-curve HJM modelling for risk management.

**Abstract**

We present a HJM approach to the projection of multiple yield curves developed to capture the volatility content of historical term structures for risk management purposes. Since we observe the empirical data at daily frequency and only for a finite number of time to maturity buckets, we propose a modelling framework which is inherently discrete. In particular, we show how to approximate the HJM continuous time description of the multi-curve dynamics by a Vector Autoregressive process of order one. The resulting dynamics lends itself to a feasible estimation of the model volatility-correlation structure. Then, resorting to the Principal Component Analysis we further simplify the dynamics reducing the number of covariance components. Applying the constant volatility version of our model on a sample of curves from the Euro area, we demonstrate its forecasting ability through an out-of-sample test.

Available at: http://arxiv.org/abs/1411.3977

# Marmi, S., Nassigh, A. and Regoli, D. (2014). Sovereign Ratings Implied by Coupled CDS-Bond Market Data

**Abstract
**We propose an approach to sovereign market implied ratings based on information coming both from Credit Default Swap spreads and bond spreads in a unified way. Operationally speaking, we implement a Support Vector Machine type of selection in the plane CDS-bond. Our numerical results seem to confirm that introducing the bond dimension accounts for implied ratings more accurate and with greater predictive power with respect to the 1-dimensional CDS implied ratings.

Available at http://ssrn.com/abstract=2512238

# Bandi, F., Pirino, D. and Renò, R. (2013). EXcess Idle Time.

**Abstract
**We introduce a novel stochastic quantity, named excess idle time (EXIT), measuring the extent of sluggishness in observed high-frequency financial prices. Using a limit theory robust to market microstructure noise, we provide econometric support for the fact that high-frequency transaction prices are, coherently with liquidity and asymmetric information theories of price determination, generally stickier than implied by the ubiquitous semimartingale assumptions (and its microstructure noise-contaminated counterpart). EXIT provides, for every asset and each trading day, a proxy for the extent of frictions (liquidity and asymmetric information) which is conceptually different from traditional price-impact measures. We relate it to existing measures and show its favorable performance under realistic data generating processes. We conclude by showing that EXIT uncovers an economically-meaningful short-term and long-term liquidity premium in market returns.

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2199468

# Bottazzi G. and Pirino D. (2010). Measuring Industry Relatedness and Corporate Coherence.

**Abstract
**Since the seminal contribution of Teece et al. (1994), the strength, scope and quality of corporate diversification is often detected comparing the observed value of some statistics derived from the diversification patterns of a sample of firms, with its expected value. The latter is obtained under a null hypothesis which assumes some random assignment procedure of sectors to firms. The approaches generally adopted in the literature present two problems. First, being based on the observed value of a statistic, these methods could lead, depending on the nature of the sample, to noisy and non-homogeneous estimates. Second, the benchmark value used to identify the presence and strength of deterministic patterns are obtained under specific and privileged null hypothesis. Both effects could lead to the erroneous classification of spurious random effects as deterministic. This paper shows that the adoption of p-scores as measure of relatedness strongly alleviate the first problem, leading to cleaner and more homogeneous estimates. We design and implement a null hypothesis which rules out random artifacts and effectively identify new features in firm diversification pattern. Using the NBER database on patents, we apply our results to the study of the relationship between the coherence and the scope of corporate patent portfolios.

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1831479