This talk will explain when it is (and when it is not) possible for a group of investors to gain mutual benefit from a collective pension design. We will see that investors can obtain mutual benefit by completing the market with additional insurance products and will estimate the potential benefit that collective designs can provide over traditional pension products

We review some empirical facts of financial markets that have motivated the rough volatility paradigm for modelling financial volatility, both from the point of view of financial time series and options pricing. Motivated by empirical evidence from the joint behavior of realized volatility time series, we propose to model the joint dynamics of log-volatilities using a multivariate fractional Ornstein-Uhlenbeck process. This model is a multivariate version of the Rough Fractional Stochastic Volatility model proposed in Gatheral, Jaisson, and Rosenbaum, Quant. Finance, 2018. It allows for different Hurst exponents in the different marginal components and non trivial interdependencies. We discuss the main features of the model, propose parameter estimators, derive their asymptotic theory and perform a simulation study that confirms the asymptotic theory in finite sample. We carry out an extensive empirical investigation on empirical realized volatility time series, showing that these time series are strongly correlated and can exhibit asymmetries in their empirical cross-covariance function, accurately captured by our model. These asymmetries lead to spillover effects, which we derive analytically within our model and compute based on empirical estimates of model parameters. Moreover, in accordance with the existing literature, we observe behaviors close to non-stationarity and rough trajectories.

We study the optimal control of mean-field systems with heterogeneous and asymmetric interactions. This leads to considering a family of controlled Brownian diffusion processes with dynamics depending on the whole collection of marginal robability laws. We prove the well-posedness of such systems and define the control problem together with its related value function. We next prove a law invariance property for the value function which allows us to work on the set of collections of probability laws. We show that the value function satisfies a dynamic programming principle (DPP) on the flow of collections of probability measures. We also derive a chain rule for a class of regular functions along the flows of collections of marginal laws of diffusion processes. Combining the DPP and the chain rule, we prove that the value function is a viscosity solution of a Bellman dynamic programming equation in a L²-set of Wasserstein space-valued functions. This talk is based on a joint work with A. De Crescenzo, M. Fuhrman and H. Pham.

Hawkes processes are a popular model for self-exciting phenomena, from earthquakes to finance. In this talk, I will first present them in a simple way, using a Poisson imbedding construction. I will then review what is known about their long-time behavior, through limit theorems for both linear and non-linear cases. The focus will be on three regimes that appear when the process has a long memory and the branching ratio gets close to or above one: the Nearly Unstable, the Weakly Critical, and the Supercritical Nearly Unstable Hawkes processes. These regimes have been studied qualitatively, but quantitative convergence results have been missing. I will explain how we obtain explicit convergence rates, relying on a coupling with a Brownian sheet, Fourier analysis, and a careful approximation of the absolute value function.
 

We develop a continuous-time stochastic model for optimal cybersecurity investment under the threat of cyberattacks. The arrival of attacks is modeled using a Hawkes process, capturing the empirically relevant feature of clustering in cyberattacks. Extending the Gordon-Loeb model, each attack may result in a breach, with breach probability depending on the system's vulnerability. We aim at determining the optimal cybersecurity investment to reduce vulnerability. The problem is cast as a two-dimensional Markovian stochastic optimal control problem and solved using dynamic programming methods. We perform a numerical study of the value function and the associated optimal investment strategy in cyber-security, highlighting the impact of randomly arriving clustered cyber-attacks. Based on a joint work with G. Callegaro, C. Fontana and C. Hillairet.
 

When returns are partially predictable and trading is costly, utility maximizing investors track a target portfolio at a constant trading speed. The target portfolio is optimal for a frictionless market, where asset returns are scaled back to account for trading costs and volatilities are adjusted to proxy the “execution risk” of holding assets that are costly to trade and exposed to volatile states. The trading speed solves an optimal execution problem, which describes how the legacy portfolio inherited from the past is traded towards the target portfolio in an optimal manner. Unlike for period-by-period mean-variance preferences as in Garleanu and Pedersen (2013), the target portfolio hedges changes in investment opportunities, and both it and the trading speed are linked and depend on execution risk. We set the problem out first in an “absolute” framework – price shocks independent of the price level and investors have CARA preferences – and then in a “relative” framework, with price shocks scaled by price levels and CRRA preferences.