Claudio Fontana

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. Numerical results illustrate how accounting for attack clustering leads to more responsive and effective investment policies, offering significant improvements over static and Poisson-based benchmark strategies. Our findings underscore the value of incorporating realistic threat dynamics into cybersecurity risk management.

We study an extension of the Cox-Ingersoll-Ross (CIR) process that incorporates jumps at deterministic dates, referred to as stochastic discontinuities. Our main motivation stems from short-rate modelling in the context of overnight rates, which often exhibit jumps at predetermined dates corresponding to central bank meetings. We provide a formal definition of a CIR process with stochastic discontinuities, where the jump sizes depend on the pre-jump state, thereby allowing for both upwarrd and downward movements as well as potential autocorrelation among jumps. Under mild assumptions, we establish existence of such a process and identify sufficient and necessary conditions under which the process inherits the affine property of its continuous counterpart. We illustrate our results with practical examples that generate both upward and downward jumps while preserving the affine property and non-negativity. In particular, we show that a stochastically discontinuous CIR process can be constructed by applying a determinisitic cadlag time-change of a classical CIR process. Finally, we further enrich the affine framework by characterizing conditions that ensure infinite divisibility of the extended CIR process.

We study the problems of consistency and of the existence of finite-dimensional realizations for multi-curve interest rate models of Heath-Jarrow-Morton type, generalizing the geometric approach developed by T. Björk and co-authors in the classical single-curve setting. We characterize when a multi-curve interest rate model is consistent with a given parameterized family of forward curves and spreads and when a model can be realized by a finite-dimensional state process. We illustrate the general theory in a number of model classes and examples, providing explicit constructions of finite-dimensional realizations. Based on these theoretical results, we perform the calibration of a three-curve Hull-White model to market data and analyse the stability of the estimated parameters.

In this paper, we consider a generic interest rate market in the presence of roll-over risk, which generates spreads in spot/forward term rates. We do not require classical absence of arbitrage and rely instead on a minimal market viability assumption, which enables us to work in the context of the benchmark approach. In a Markovian setting, we extend the control theoretic approach of Gombani & Runggaldier (2013) and derive representations of spot/forward spreads as value functions of suitable stochastic optimal control problems, formulated under the real-world probability and with power-type objective functionals. We determine endogenously the funding-liquidity spread by relating it to the risk-sensitive optimization problem of a representative investor.

In this paper, we consider a generic interest rate market in the presence of roll-over risk, which generates spreads in spot/forward term rates. We do not require classical absence of arbitrage and rely instead on a minimal market viability assumption, which enables us to work in the context of the benchmark approach. In a Markovian setting, we extend the control-theoretic approach of Gombani and Runggaldier (Math. Finance23 (2013) 659–686) and derive representations of spot/forward spreads as value functions of suitable stochastic optimal control problems, formulated under the real-world probability and with power-type objective functionals. We determine endogenously the funding–liquidity spread by relating it to the risk-sensitive optimisation problem of a representative investor.

Overnight rates, such as the Secured Overnight Financing Rate (SOFR) in the United States, are central to the current reform of interest rate benchmarks. A striking feature of overnight rates is the presence of jumps and spikes occurring at predetermined dates due to monetary policy interventions and liquidity constraints. This corresponds to stochastic discontinuities (i.e., discontinuities occurring at ex ante known points in time) in their dynamics. In this work, we propose a term structure modeling framework based on overnight rates and characterize absence of arbitrage in a generalized Heath–Jarrow–Morton (HJM) setting. We extend the classical short‐rate approach to accommodate stochastic discontinuities, developing a tractable setup driven by affine semimartingales. In this context, we show that simple specifications allow to capture stylized facts of the jump behavior of overnight rates. In a Gaussian setting, we provide explicit valuation formulas for bonds and caplets. Furthermore, we investigate hedging in the sense of local risk‐minimization when the underlying term structures feature stochastic discontinuities.

Variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) entitle the policy holder to periodic withdrawals together with a terminal payoff linked to the performance of an equity fund. In this paper, we consider the valuation of a general class of GMWB annuities, allowing for step-up, bonus and surrender features, taking also into account mortality risk and death benefits. When dynamic withdrawals are allowed, the valuation of GMWB annuities leads to a stochastic optimal control problem, which we address here by dynamic programming techniques. Adopting a Hull-White interest rate model, correlated with the equity fund, we propose an efficient tree-based algorithm. We perform a thorough analysis of the determinants of the market value of GMWB annuities and of the optimal withdrawal strategies. In particular, we study the impact of a low/negative interest rate environment. Our findings indicate that low/negative rates profoundly affect the optimal withdrawal behaviour and, in combination with step-up and bonus features, increase significantly the fair values of GMWB annuities, which can only be compensated by large management fees.