Mathematical (Sustainable) Finance and Risk Management

Content

1. General Course Description
This course provides a mathematically rigorous and application-oriented introduction to financial risk management, with a strong emphasis on sustainability, systemic risk, and the modeling of emerging asset classes such as energy products. Designed for students with a basic background in probability and financial mathematics, the course integrates theoretical modeling with computational techniques and data-driven analysis.

Students will learn to model, quantify, and manage financial risks (e.g., market, credit, operational, systemic and climate risks) using data-informed methods grounded in stochastic modeling and risk theory. Sustainability is treated not as a side topic, but as an integral part of modern risk management through modules on climate stress testing, ESG factor models, and the dynamics of renewable energy markets. The course emphasizes both conceptual understanding and practical skills through a combination of lectures and hands-on computational labs, preparing students to engage with current challenges in modern risk management and sustainable finance.

Key topics include:

• Foundations of Financial Risk Management: Roles within the banking ecosystem, introduction to risk measures, and the distinction between real-world (P-measure) and risk-neutral (Q-measure) modeling.
• Risk Measures and Tail Risks: Value-at-Risk (VaR), Expected Shortfall, default probabilities, hazard rates, heavy-tailed distributions, and Monte Carlo methods.
• Interest Rates and Fixed-Income Instruments: Short-rate models (e.g., Vasicek, Hull–White), yield curve construction, pricing of bonds and mortgages, prepayment modeling, and valuation of swaps and futures under a no-arbitrage framework.
• Core Risk Categories and Model Validation: In-depth study of market risk, credit risk (Merton and reduced-form models), operational/reputational risk, and best practices in model calibration, back-testing, and governance.
• Energy Markets and Renewable Risk: Spot and forward markets in electricity, gas, and oil; mean-reverting and jump-diffusion models; renewable energy uncertainty; seasonal calibration; and hedging strategies using futures and forwards.
• Sustainability and Systemic Risk: Climate-related risks (physical and transition), ESG-aware portfolio modeling, green bonds, carbon pricing, stress testing, and copulas.

2. Learning Goals
By the end of this course, students will be able to:

1. Understand the ecosystem and foundations of financial risk management

• Explain the structure of the banking system and the roles of front office, back office, and risk management functions.
• Distinguish between real-world (P-measure) and risk-neutral (Q-measure) models, and understand their use in pricing and risk evaluation.
• Recognize the relevance of regulatory frameworks (e.g., Basel) and the importance of scenario analysis and stress testing.

        2. Estimate and interpret quantitative risk measures

• Compute and interpret Value-at-Risk (VaR), Expected Shortfall, and semi-variance.
• Understand and model default probabilities and hazard rates, and apply them to credit risk modeling.
• Recognize and estimate heavy-tailed risks using appropriate statistical techniques.

        3. Model and assess different core categories (market, credit, and operational) of financial risk

• Analyze market risk across asset classes, including equities, and interest rates assets.
• Apply structural and reduced-form models to credit risk, including modeling correlations and portfolio default dependencies.
• Understand and model operational and reputational risks using extreme-value and real-world incident data.

        4. Validate and calibrate financial models

• Perform calibration and back-testing of risk models, especially for VaR and Expected Shortfall.
• Assess model risk through governance practices and understand challenges related to data limitations.

        5. Model interest rate and fixed-income products

• Describe and apply short-rate models such as Vasicek and Hull–White.
• Construct and interpret yield curves using bond data.
• Model mortgage cash flows and prepayment risk and understand their implications for mortgage-backed securities.
• Value interest rate swaps and futures using a no-arbitrage framework.

        6. Understand and model energy markets

• Model energy spot prices using stochastic mean-reverting and jump-diffusion processes.
• Analyze electricity markets and the impact of renewable energy on pricing and risk.
• Understand the structure and pricing of forward and futures contracts in energy markets.
• Evaluate hedging strategies used by producers and consumers, incorporating effects of storage and transportation.

        7. Integrate sustainability and ESG into financial modeling

• Identify and quantify climate-related financial risks (physical and transition risks).
• Conduct scenario-based climate stress testing within risk models.
• Extend standard factor models to include ESG considerations.
• Understand the structure and valuation of green bonds and carbon pricing mechanisms.

        8. Assess and model systemic risk

• Understand the role of macroprudential tools and systemic stress testing.
• Apply copulas and other dependence structures to model systemic risk.

        9. Apply computational and data-driven methods for financial modeling and analysis

• Implement simulations, estimators, and calibrations for stochastic models.
• Analyze and work with real-world financial and energy market data.
• Use computational tools to perform scenario analysis and visualize portfolio risk.

3. Literature/References
The course will be based on slides and lecture notes authored, among others, by the lecturers. The following references have been useful for preparing these notes and are recommended for further studies:

• The book "Quantitative Risk Management: Concepts, Techniques and Tools" by Alexander J. Mcneil, Rüdiger Frey, Paul Embrechts, Thierry Roncalli, Link
• The book "Handbook of Sustainable Finance" by Thierry Roncalli, Link
• The book "Stochastic modelling of electricity and related markets" by Benth, F.E., Benth, J.S. and Koekebakker, S., 2008. Link

4. Prerequisites

• A prerequisite for this course is the course "Introduction to Financial Mathematics (WISB373)" or similar.
• Introductory courses on numerical analysis and probability.
• Experience with computer programming in Python (or a similar language, for example, Matlab, ...): this will be needed for the assignments/homeworks.

5 Course Organisation and Course Team

• Lecture times: Wednesday 10:00–12:45

• Lecturer: Kees Oosterlee, Lech Grzelak, Chiheb Ben Hammouda

6 Evaluation: Homeworks, Final Project, Final Exam, and Grading Rules
Student assessment in this course consists of three main components: biweekly homework assignments, a final exam, and a final project. Each component plays a crucial role in reinforcing different aspects of the learning process—conceptual and theoretical understanding, computational and numerical proficiency, and both oral and written communication skills.

• Homework Assignments (20% of the final grade): Homework will be assigned approximately every two weeks and may be completed individually or in groups of two. Each group is required to submit a written report for each assignment. For each assignment, one group will be selected to present their solution in class (30–45 minute presentation). All group members must be fully familiar with the complete solution, as both presenters and non-presenting members may be questioned during the session. The presenting group may submit a draft version of their slides 2–3 days before the scheduled presentation to receive preliminary feedback from the teaching team.

• Final Exam (40% of the final grade): The final exam will be a closed-book, in-class examination testing students’ understanding of theoretical concepts, modeling approaches, and key computational techniques.
• Final Project (40% of the final grade): The final project will involve a practical case study—potentially based on real-world data or an industry example—in which students will apply the theoretical concepts, mathematical modeling techniques, numerical methods, and programming skills acquired throughout the course. Projects are to be completed in groups of two and must culminate in a well-structured written report. Optional oral presentations of project outcomes may be scheduled during the final week of the course.
• Grading and Retake Policy: Students must achieve a minimum score of 5.0 on the final exam and an overall satisfactory average grade (i.e., ≥ 5.5) to pass the course. A retake of the final exam will be offered if necessary; it will be a closed-book, in-class exam. Grades from the homework assignments and final project will continue to count toward the final grade after the retake.