Dear colleagues,

you are all invited to participate in the following seminar organized by QFinLab – Department of Mathematics, Politecnico di Milano.

Wednesday, 2 April 2025, 12.15-13.15

Seminar room, third floor, building 14, Via Bonardi 9, Milano (Leonardo Campus)

Claudio Fontana (Università di Padova)

A stochastic Gordon-Loeb model for optimal security investment under clustered cyber-attacks.

Abstract: We propose a continuous-time extension of the Gordon-Loeb model for optimal investment in information security under the threat of cyber-attacks. The arrival of attacks is modeled using Hawkes processes, capturing the realistic feature of clustering in cyber-attacks. Each attack may lead to a system breach, with the probability of breach depending on the system’s vulnerability. We aim at determining the optimal investment in cyber-security to reduce the system’s vulnerability. The problem is formulated as a two-dimensional Markovian stochastic control problem and solved via dynamic programming techniques. 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. Hillairet, B. Ongarato.

Next seminar: Alessandro Sbuelz (Università Cattolica del Sacro Cuore), 7 April 12.15.

All news can be found on the QFinLab webpage.

The organizers: Michele Azzone and Alessandro Calvia

QFinLab promotes a series of thematic seminars on climate risk. Climate transformations have a deep impact on economic activity with implications ranging from the definition of green transition policies to the evaluation of financial assets, from the construction of innovative financial and insurance products to risk management, from the design of mechanisms incentive to asset management and much more. Themes that pose intriguing questions to the academic world, involving, those who deal with models to interpret a phenomenon that by its nature is very complex. The activities are organized through a series of double seminars with discussants. The third meeting is scheduled for

March 20, 2025

15.00-17.00

Department of Mathematics- Seminar room- III floor

Andrea Macrina (UCL-London)

Climate-Contingent Convertible Bonds (CloCo) – Seizing the Opportunity to Innovate and Adapt

Adjustments to governmental climate transition policies and uncertainties linked to the ability for industry sectors and consumers to manufacture and purchase low carbon technologies, respectively, have led to an increasing need to embed a richer framework for uncertainty in climate transition outcomes. Such uncertainties create financial risks for firms, their investors, the banking sector, and potentially sovereign risks. The need thus arises for the development of a bespoke pricing setup and financial instruments, which offer a mechanism to share climate transition risk. As an example, we propose the so-called climate-contingent convertible (CLoCo) bond. This instrument enables firms to reduce the risk of default due to adverse climate transition policies over the product’s lifetime and to respond to new investment opportunities by freeing up capital supporting more financial agility. The proposed financial innovation implies reduced risk of default for firms, thereby increasing the expected

firm value, but it also reduces the dependency of firm failures on the banking sector and potential bail-out costs incurred by sovereign nations.

Joint work with C. Cormack

Discussant: Emilio Barucci (Politecnico di Milano)

Luca Regis (Università di Torino)

Coordinating Dividend Taxes and Capital Regulation

We study the impact of state-contingent dividend taxes (and bans) and capital regulation on a firm’s optimal strategy and value. In the model, the firm generates stochastic income under time-varying macroeconomic conditions. Its manager distributes dividends and issues costly equity to maximize shareholder value. We solve the manager’s stochastic control problem and derive the firm’s reserve distribution in closed form. Imposing dividend taxes (or bans) during crises generates a trade-off, as it encourages reserve accumulation in bad states but promotes payouts in good ones. Also, the policy undermines financial stability by reducing the firm’s value and its recapitalization incentives across states. Coordinating dividend taxes with counter-cyclical capital regulation can mitigate value losses and ameliorate the trade-off, but it also creates additional recapitalization disincentives.

Joint work with Salvatore Federico and Andrea Modena

Discussant: Alessandro Calvia (Politecnico di Milano)

This event has been (partially) supported by MUR, Department of Excellence 2023-27

Dear colleagues,

you are all invited to participate in the following seminar organized by QFinLab – Department of Mathematics, Politecnico di Milano.

Wednesday, 5 March 2025, 12.00-13.00

Seminar room, third floor, building 14, Via Bonardi 9, Milano (Leonardo Campus)

Katia Colaneri (Università di Roma Tor Vergata)

Title: Expect the worst! Optimal emission abatement under tax policy uncertainty and stochastic differential games. 

Abstract: We study the problem of a profit maximizing electricity producer who has to pay carbon taxes and who decides on investments into technologies for the abatement of CO emissions in an environment where carbon tax policy is random and where the investment in the abatement technology is divisible, irreversible and subject to transaction costs.
We consider two approaches for modelling the randomness in taxes. First we assume a precise probabilistic model for the tax process, namely a pure jump Markov process (so-called tax risk); this leads to a stochastic control problem for the investment strategy.  
Second, we analyze the case of an uncertainty-averse producer who uses a differential game to decide on optimal production and investment. We carry out a rigorous mathematical analysis of the producer’s optimization problem and of the associated nonlinear PDEs in both cases. Numerical methods are used to study quantitative properties of the optimal investment strategy.   
We find that in the tax risk case the investment in abatement technologies is typically lower than in a benchmark scenario with deterministic taxes. However, there are a couple of interesting new twists related to production technology, divisibility of the investment, tax rebates and investor expectations. In the stochastic differential game on the other hand an increase in uncertainty might stipulate more investment.

Next seminar: Claudio Fontana (Università di Padova), 2 April 12.00.

All news can be found on the QFinLab webpage.

The organizers: Michele Azzone and Alessandro Calvia

Dear colleagues,

you are all invited to participate in the following seminar organized by QFinLab – Department of Mathematics, Politecnico di Milano.

Wednesday, 19 February 2025, 12.15-13.15

Seminar room, third floor, building 14, Via Bonardi 9, Milano (Leonardo Campus)

Marzia De Donno (Università Cattolica del Sacro Cuore)

Title: Short rate models with stochastic discontinuities: a PDE approach.

Abstract: With the recent reform of interest rate benchmarks, interbank offered rates (IBORs) like LIBOR have been replaced by risk-free rates (RFRs), such as the Secured Overnight Financing Rate (SOFR) in the U.S. and the Euro Short-Term Rate (€STR) in Europe. These rates exhibit characteristics like jumps and spikes that correspond to specific market events, driven by regulatory and liquidity constraints. To capture these characteristics, this paper considers a general short-rate model that incorporates discontinuities at fixed times with random sizes. Within this framework, we introduce a PDE-based approach for pricing interest rate derivatives. For affine models, we derive (quasi) closed-form solutions, while for the general case, we develop numerical methods to solve the resulting PDEs.

(Joint work with A. Calvia, C. Guardasoni, S. Sanfelici).

Next seminar: Katia Colaneri (Università di Roma Tor Vergata), 5 March 12.00.

All news can be found on the QFinLab webpage.

The organizers: Michele Azzone and Alessandro Calvia