DoCRA project 
Doctoral Colloquium on Risk Analytics

Note: Students admitted to this session must have a knowledge of probability theory that includes at least what is treated in chapters 1-4 in Olofsson and Andersson, "Probability, Statistics, and Stochastic Processes", 2nd edition, 2012.

Week 1 (Monday 24/2 to Friday 28/2):
Measuring and comparing risks

• Lecturer: Alfred Muller, U. Siegen
• Contents: We cover modern mathematical methods for measuring and comparing risks, motivated by properties that are desirable in applications. We start with the comparison of risks, leading to the mathematical notion of order relations for probability measures (also known as stochastic orders). We discuss these concepts in detail, give an overview of recent research on their generalizations, and examine their robustness. We move on to the comparison of multivariate risks, where the dependence structure also plays a role. We discuss in particular the supermodular stochastic order to compare dependence structures, which includes the study of copulas and comonotonicity. We also consider the general relation between stochastic orders and the modern topic of optimal transport, which is a relatively new field of research.
• The second main topic are measures of risk that play an important role in actuarial or financial risks, in the form of premium principles or of regulatory capital requirements. We follow an axiomatic approach and discuss mathematical properties of such risk measures like coherence, convexity and elicitability. We discuss the properties of well-known examples including value-at-risk, conditional value at risk, expectiles, certainty equivalents, as well as principles based on mean and variance.

Guest Lecture (Friday 28/2):
Anatomy and physiology of a cat-risk model

• Speaker: Mario Martina, IUSS Pavia

Week 2 (Monday 3/3 to Friday 7/3):
Emerging risks for actuaries: NatCat insurance and climate change

• Lecturer: Hansjörg Albrecher, U. Lausanne
• Contents: Insurance of natural catastrophies: what is specific about it, what makes it different from other insurance sectors (premiums, capital) — Analysis of very heavy tails (some extreme value analysis, insurability when only finitely many moments exist) — Dependence models in this context — Quantify the pooling potential (diversification benefit in space and in time) — Assess the degree of climate change (a little flavor of time series analysis, non-stationarities, change points versus trend analysis) — Quantitative effects on the sensitivity of premiums and solvency capital for NatCat risks under climate change — Theory, concrete calculations, some case studies, exercises.

Note: Students admitted to this session must have basic knowledge in theoretical and empirical asset pricing (e.g., see topics reported in the J Chocrane book, "Asset Pricing"); financial econometrics (multivariate linear regressions; time-series econometrics; forecasting; GMM; GARCH; estimation of APT models).

Week 1 (Monday28/7 to Friday 1/8): MacroFinTech

• Lecturer: Max Croce, U. Bocconi

Contents: Modern macrofinance models often take into account the role of news shocks. For news shocks to be priced, preferences should go beyond the expected utility framework. This course has two applied goals. First, familiarize with asset pricing models with recursive preferences in both endowment and production economies. Second, solve macrofinance DSGE models through standard perturbation methods. Becoming knowledgeable about these techniques is extremely beneficial as it increases both the quantity of research papers that can be produced in a short amount of time.

This course is mainly applied, in the sense that we will devote most of our time learning how to solve a model and get results. We will see only a minimum set of theoretical concepts that are essential to check the correctness of the solution and debug our codes. By the end of the course, solving a dynamic stochastic general equilibrium (DSGE) model should be a routine task.

We will work with dynare++.exe, a free stand-alone package that solves stochastic systems of smooth equations. We will also learn how to integrate dynare++.exe with Matlab in order to generate nice tables and figures in an efficient way.
We will see recent applications related to:

• MacroFinTech for emissions regulation;
• Supply Chain Uncertainty & Growth;
• Currencies.

Depending on enrollment, some presentations will be assigned to students. We will have 1 or 2 homeworks that will require students to replicate existing papers using dynare++ and Matlab.

Some references:

• Tallarini (2000, JME);
• Bansal and Yaron (2004, JF);
• Kung and Schmit (2013, JF);
• Croce et al (2024), WP, ‘Supply Chain Uncertainty’;
• Croce et al (2025), WP, ‘Green Coins’.

Friday 1/8: Guest Lecture: on Horizon risk, what it is and how to tackle it in a dymanic way

• Speaker: Giulia di Nunno, U. Oslo

Contents: Horizon risk is the assessing the financial exposure by a risk measure that is not adequate to the actual time horizon of the position. We clarify that dynamic risk measures are subject to horizon risk, so we propose to work with fully-dynamic risk measures. We shall combine horizon risk with other uncertainties of the future market scenarios, such as interest rates uncertainty, thus we propose and justify the use of a cash non-additive version. We then construct such risk measures via backward stochastic differential equations or via shortfall-type representation. As illustration, we introduce the class of hq-entropic risk measures.

Week 2 (Monday 4/8 to Friday 8/8): Model free pricing and estimation of financial risks

• Lecturer: Fabio Trojani, U. Torino and SFI

Contents: This course covers selected recent research topics in theoretical and empirical asset pricing. The common theme of the lectures is the construction of suitable model-free Stochastic Discount Factors (SDFs) that minimise convenient notions of SDF variability subject to the relevant asset pricing constraints. We cover univariate and multivariate markets, settings with frictions or ambiguity, recoveries based on option price information summarised by suitable portfolios replicating higher moment risk, and further reaching implications for the understanding of asset markets, investors beliefs and asset pricing factors. In parallel, we introduce a systematic convex analysis framework for understanding penalised estimation and inference about minimum dispersion SDFs in presence of pricing errors. Exploiting this estimation and inference approach, we present a new methodology for testing asset pricing models in presence of useless or weak factors. Instead of widely covering several individual approaches and particular directions in the literature, we focus on essential methodologies for applying these techniques with success in a structured and general way.

Reading Material:

• Own slides
• Selected research articles

Topics covered:

1. Model-free SDFs and asset pricing bounds
   • Minimum dispersion SDFs in univariate and multivariate markets
   • Asset pricing bounds
   • Implications for international asset pricing

2. Model-free trading of higher-moment risk
   • Realised measures of higher-moment risk
   • Replication of higher moment risk
   • Implied moment surfaces
   • Implications for the pricing of jump risk

3. Almost model-free SDF recovery
   • SDF recovery problems
   • Almost model-free recovery
   • Implications for conditional SDF modelling

4. Beyond frictionless markets: Smart SDFs
   • Arbitrage-free pricing with non zero pricing errors
   • Minimum dispersion Smart SDFs
   • Relation to APT pricing
   • Implications for international asset pricing

5. A convex analysis framework for penalised estimation and inference
   • Basic elements of convex analysis
   • Penalised estimation and proximal estimation
   • From regular to singular designs
   • Inference and Oracle estimation with irregular designs
   • Insights for asset pricing

6. Tradeable factor risk premia and tests of asset pricing models
   • Intrinsic factor risk premia
   • Sample intrinsic factor risk premia
   • Oracle intrinsic factor risk premium estimation
   • Intrinsic factor selection and inference with the factor zoo

Application opened

Week 1 (Monday 15/9 to Tuesday 23/9):
Predictive uncertainty in Machine Learning with conformal inference

• Lecturer: Stefano Favaro, U. Torino and Collegio Carlo Alberto

Guest Lecture: Introduction to CyberRisk

• Speaker: Alessandro Armando, U. Genoa and IMT Lucca

Week 2 (Monday 21/9 to Friday 25/9):
Concepts of Deep Learning and applications to Finance and Risk Management

• Lecturer: Christa Cuchiero, U. Vienna

Application will open on 1 June 2025

Week 1 (Monday 19/1 to Friday 23/1, 2026): Title TBD

• Lecturer: Remco Hofstad, Eindhoven U. of Technology

Guest Lecture: Title TBD

• Speaker: Monica Billio, U. Ca’ Foscari Venezia

Week 2: (Monday 26/1 to Friday 30/1, 2026): Title TBD

• Lecturer: Alireza Tahbaz-Salehi, Northwestern University

Application will open on 15 October 2025