Systemic risk governance in a dynamical model of a banking system with stochastic assets and liabilities

We consider the problem of governing systemic risk in an assets–liabilities dynamical model of a banking system. In the model considered, each bank is represented by its assets and liabilities. The net worth of a bank is the difference between its assets and liabilities and bank is solvent when its net worth is greater than or equal to zero; otherwise, the bank has failed. The banking system dynamics is defined by an initial value problem for a system of stochastic differential equations whose independent variable is time and whose dependent variables are the assets and liabilities of the banks. The banking system model presented generalizes those discussed in Fouque and Sun (in: Fouque, Langsam (eds) Handbook of systemic risk, Cambridge University Press, Cambridge, pp 444–452, 2013) and Fatone and Mariani (J Glob Optim 75(3):851–883, 2019) and describes a homogeneous population of banks. The main features of the model are a cooperation mechanism among banks and the possibility of the (direct) intervention of the monetary authority in the banking system dynamics. By “systemic risk” or “systemic event” in a bounded time interval, we mean that in that time interval at least a given fraction of banks have failed. The probability of systemic risk in a bounded time interval is evaluated via statistical simulation. Systemic risk governance aims to maintain the probability of systemic risk in a bounded time interval between two given thresholds. The monetary authority is responsible for systemic risk governance. The governance consists in the choice of assets and liabilities of a kind of “ideal bank” as functions of time and in the choice of the rules for the cooperation mechanism among banks. These rules are obtained by solving an optimal control problem for the pseudo mean field approximation of the banking system model. Governance induces banks in the system to behave like the “ideal bank”. Shocks acting on the banks’ assets or liabilities are simulated. Numerical examples of systemic risk governance in the presence and absence of shocks acting on the banking system are studied.

     

An explicitly solvable multi‐scale stochastic volatility model: Option pricing and calibration problems

We introduce an explicitly solvable multiscale stochastic volatility model that generalizes the Heston model. The model describes the dynamics of an asset price and of its two stochastic variances using a system of three Ito stochastic differential equations. The two stochastic variances vary on two distinct time scales and can be regarded as auxiliary variables introduced to model the dynamics of the asset price. Under some assumptions, the transition probability density function of the stochastic process solution of the model is represented as a one‐dimensional integral of an explicitly known integrand. In this sense the model is explicitly solvable. We consider the risk‐neutral measure associated with the proposed multiscale stochastic volatility model and derive formulae to price European vanilla options (call and put) in the multiscale stochastic volatility model considered. We use the thus‐obtained option price formulae to study the calibration problem, that is to study the values of the model parameters, the correlation coefficients of the Wiener processes defining the model, and the initial stochastic variances implied by the “observed” option prices using both synthetic and real data. In the analysis of real data, we use the S&P 500 index and to the prices of the corresponding options in the year 2005. The web site http://www.econ.univpm.it/recchioni/finance/w7 contains some auxiliary material including some animations that helps the understanding of this article. A more general reference to the work of the authors and their coauthors in mathematical finance is the web site http://www.econ.univpm.it/recchioni/finance . © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:862–893, 2009