We present an analytical model to study the role of expectation feedbacks and overlapping portfolios on systemic stability of financial systems. Building on Corsi et al. (2016), we model a set of financial institutions having Value-at-Risk capital requirements and investing in a portfolio of risky assets, whose prices evolve stochastically in time and are endogenously driven by the trading decisions of financial institutions. Assuming that they use adaptive expectations of risk, we show that the evolution of the system is described by a slow-fast random dynamical system, which can be studied analytically in some regimes. The model shows how the risk expectations play a central role in determining the systemic stability of the financial system and how wrong risk expectations may create panic-induced reduction or over-optimistic expansion of balance sheets. Specifically, when investors are myopic in estimating the risk, the fixed point equilibrium of the system breaks into leverage cycles and financial variables display a bifurcation cascade eventually leading to chaos. We discuss the role of financial policy and the effects of some market frictions, as the cost of diversification and financial transaction taxes, in determining the stability of the system in the presence of adaptive expectations of risk.

# Author Archives: Stefano Marmi

# P. Mazzarisi, F.Lillo, S. Marmi (2019). When panic makes you blind: A chaotic route to systemic risk,

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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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L.M. Calcagnile, G. Bormetti, M. Treccani, S. Marmi, F. Lillo,

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N. Angelini, G. Bormetti, S. Marmi, F. Nardini,

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F. Corsi, S. Marmi, F. Lillo,

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G. Bormetti, L. M. Calcagnile, M. Treccani, F. Corsi, S. Marmi, F. Lillo,

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D.H. Kim, S. Marmi,

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S. Marmi, C. Pacati, R. Renò, W.A. Risso,

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G. Buccheri, S. Marmi, R.N. Mantegna,

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L.M. Calcagnile, G. Bormetti, M. Treccani, S. Marmi, F. Lillo, *Collective synchronization and high frequency systemic instabilities in financial markets*, Quantitative Finance 18 (2), 237-247

We present some empirical evidence on the dynamics of price instabilities in financial markets and propose a new Hawkes modelling approach. Specifically, analysing the recent high frequency dynamics of a set of US stocks, we find that since 2001 the level of synchronization of large price movements across assets has significantly increased. We find that only a minor fraction of these systemic events can be connected with the release of pre-announced macroeconomic news. Finally, the larger is the multiplicity of the event—i.e. how many assets have swung together—the larger is the probability of a new event occurring in the near future, as well as its multiplicity. To reproduce these facts, due to the self- and cross-exciting nature of the event dynamics, we propose an approach based on Hawkes processes. For each event, we directly model the multiplicity as a multivariate point process, neglecting the identity of the specific assets. This allows us to introduce a parsimonious parametrization of the kernel of the process and to achieve a reliable description of the dynamics of large price movements for a high-dimensional portfolio.

https://doi.org/10.1080/14697688.2017.1403141

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N. Angelini, G. Bormetti, S. Marmi, F. Nardini, *A Stylized Model for Long-Run Index Return Dynamics *, Essays in Economic Dynamics, 111-122

We introduce a discrete-time model of stock index return dynamics grounded on the ability of Shiller’s Cyclically Adjusted Price-to-Earning ratio to predict long-horizon market performances. Specifically, we discuss a model in which returns are driven by a fundamental term and an autoregressive component perturbed by external random disturances. The autoregressive component arises from the agents’ belief that expected returns are higher in bullish markets than in bearish markets. The fundamental term, driven by the value towards which fundamentalists expect the current price should revert, varies in time and depends on the initial averaged price-to-earnings ratio. The actual stock price may deviate from the perceived reference level as a combined effect of an idyosyncratic noise component and local trends due to trading strategies. We demonstrate both analytically and by means of numerical experiments that the long-run behavior of our stylized dynamics agrees with empirical evidences reported in literature.

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F. Corsi, S. Marmi, F. Lillo, *When micro prudence increases macro risk: The destabilizing effects of financial innovation, leverage, and diversification*, Operations Research 64 (5), 1073-1088

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.

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G. Bormetti, L. M. Calcagnile, M. Treccani, F. Corsi, S. Marmi, F. Lillo, *Modelling systemic price cojumps with Hawkes factor models *, Quantitative Finance 15 (7), 1137-1156

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.

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D.H. Kim, S. Marmi, *Distribution of asset price movement and market potential*, Journal of Statistical Mechanics: Theory and Experiment 2015 (7), P07001

In this article we discuss the distribution of asset price movements by introducing a market potential function. From the principle of free energy minimization we analyze two different kinds of market potentials. We obtain a U-shaped potential when market reversion (i.e. contrarian investors) is dominant. On the other hand, if there are more trend followers, flat and logarithmic potentials appear. By using the cyclically adjusted price-to-earning ratio, which is a common valuation tool, we empirically investigate the market data. By studying long term data we observe the historical change of the market potential of the US stock market. Recent US data show that the market potential looks more like a trend-following potential. Next, we compare the market potentials for 12 different countries. Though some countries have similar market potentials, there are specific examples like Japan which exhibits a very flat potential.

# 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

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S. Marmi, C. Pacati, R. Renò, W.A. Risso, *A quantitative approach to Faber’s tactical asset allocation*, International Journal of Computational Economics and Econometrics 3 (1-2), 91-101

Routinely, practitioners and academics alike propose the use of trading strategies with an

alleged improvement on the risk–return relation, typically entailing a considerably higher

return for the given level of risk. A very popular example is” A quantitative approach to

tactical asset allocation” by the fund manager M. Faber, a real hit in the SSRN online library.

Is this paper a counterexample to market efficiency? We reject this conclusion, showing that

a lot of caution should be used in this field, and we indicate a series of bootstrapping

experiments which can be easily implemented to evaluate the performance of trading

strategies.

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G. Buccheri, S. Marmi, R.N. Mantegna, *Evolution of correlation structure of industrial indices of US equity markets*, Physical Review E 88 (1), 012806

We investigate the dynamics of correlations present between pairs of industry indices of U.S. stocks traded in U.S. markets by studying correlation-based networks and spectral properties of the correlation matrix. The study is performed by using 49 industry index time series computed by K. French and E. Fama during the time period from July 1969 to December 2011, which spans more than 40 years. We show that the correlation between industry indices presents both a fast and a slow dynamics. The slow dynamics has a time scale longer than 5 years, showing that a different degree of diversification of the investment is possible in different periods of time. Moreover, we also detect a fast dynamics associated with exogenous or endogenous events. The fast time scale we use is a monthly time scale and the evaluation time period is a 3-month time period. By investigating the correlation dynamics monthly, we are able to detect two examples of fast variations in the first and second eigenvalue of the correlation matrix. The first occurs during the dot-com bubble (from March 1999 to April 2001) and the second occurs during the period of highest impact of the subprime crisis (from August 2008 to August 2009).