Four distinct (but related) research areas have been selected to showcase topics at various levels of modeling development and mathematical complexity. Each course will consist of a theoretical part and a computational lab part. The courses will be taught in a unified computational setting (Python and its free packages). The lectures are designed to build off one another, such that material taught in later in the school builds on material taught earlier in the school. For example, Monte Carlo simulation methods are at the heart of many of the proposed courses: they are used extensively in quantitative risk management for scenario analysis and calculation of risk measures; in games to recover the numerical solution of Hamilton–Jacobi PDEs yielding the mean field dynamics; and in commodity markets to value energy derivatives and hedge energy risk. Each day will end with a student-led summary of the main takeaways from the day.
Quantitative Risk Management in Finance
Delivered by Prof. Ludovic Tangpi, an expert on stochastic analysis and probability theory, with applications to risk management and model uncertainty. Quantitative risk management is an active area of research and is of significant relevance to industry, especially due to the post-crisis regulations. There is, over a decade later, still the need to build better techniques that recognize, quantify, and mitigate risk, taking into consideration the complex interaction of financial institutions and risk factors. Most of the methodologies taught transcend the specific application to finance, and are broadly applicable to environmental and energy risk.
Day 1 – Monday, 1st week
The focus of the morning session will be on the axiomatization of risk measures, and their dual representation in terms of solutions to optimization problems. In the afternoon, the focus will be on laboratory sessions conducted by students, who will be split into groups and compute risk measures on portfolios of stocks, options and bonds using simulations.
Morning session: (i) Introduction to risk measures (monetary and set-valued) and model risk, (ii) computational schemes for value-at-risk (VaR) and expected shortfall (ES) of credit default and interest rate swaps portfolios.
Afternoon session: (i) development of high-performance computational procedures for risk valuation of high-dimensional portfolios through extensive Monte-Carlo simulations.
Both topics will be demonstrated through a lab session: Students will be provided with historical quotes of the term structure of credit default swap contracts. The instructor will provide students with code that can be used to calibrate risk models to these term structures of data. Students will be asked to compute conditional and unconditional loss distributions of derivative portfolios, and to perform principal component analysis (PCA) to estimate the driving factors of more complex derivatives portfolios.
|9:00 – 10:15||Introduction to risk measures|
|11:00 – 12:00||ES and portfolio management|
|13:30 – 15:00||Monte-Carlo simulation of risk measures|
Day 2 – Tuesday, 1st week
The focus of the morning session will be on the theory of systemic risk measures and its connections to real markets. The focus of the afternoon session will be on laboratory sessions conducted by students, who will be split into groups.
Morning session: (i) Introduction to systemic risk (origins of the topic, and relation to the global 2007-2008 financial crisis), (ii) Mathematics of systemic risk (network models of financial contagion, computational methods for learning network structures and recovering the Nash equilibria of games between banks and regulators), and (iii) Numerical valuation of network policies, including structural (clearinghouses) and resolution (bailouts) policies, and analysis of network risks (numerical linear algebra techniques for spectral analysis of interconnectedness matrices capturing financial network vulnerabilities).
Afternoon session: (i) calibration of network models to interbank liability data downloadable from public sources (e.g., Federal Reserve Bank repositories), and (ii) performance evaluation of calibrated network models under stress test scenarios.
The topics of the afternoon will be demonstrated through a lab session.
Students will be provided by the instructor with a numerical approximation of the network of interbanking loans from the German market. Students will be given Python code that implements payments exchanged between banks when they clear their liabilities, given their initial balance sheet structure. Students will then be asked to simulate shocks to the balance sheet structure. They will be asked to report summary statistics on the systemic effects, including the size of default cascades, the form of contagion triggered by reduced payments of defaulting banks, and to measure the network resiliency in the presence of bailout policies using well-known systemic and early warning indicators.
|9:00 – 10:15||Introduction to a general theory of risk measures|
|11:00 – 12:00||Toward simulation of general risk measures|
|13:30 – 15:00||Introduction to systemic risk|
|15:45 – 16:45||Mathematics of systemic risk and networks|
Energy and Commodity Markets
Delivered by Prof. Mike Ludkowski, a former chair of the SIAG/FME, and an internationally recognized leader in the area of stochastic models for energy and commodities. He co-edited the volume “Commodities, Energy and Environmental Finance” – a key reference for teaching and practical implementation.
Commodity Markets (including oil, gas, electricity, metals, CO2 emissions) are vitally important for the world economy. Commodities involve physical delivery, storage and transportation, and hence differ in fundamental ways from other asset classes. The markets experience the tension between suppliers and customers, who make/take delivery of the physical commodity, and financial traders, who treat commodities as a new asset class for investment. As a result, structural aspects of supply-demand equilibrium, long-term strategic behavior by the participants, and market design by regulators are intertwined with the financial issues of optimal investment and no-arbitrage.
Day 1 – Wednesday, 1st week
The focus of the morning session will be on introducing stochastic models for commodity spot and futures markets, and stochastic control techniques related to real-option valuation and intra-day trading of commodities. In the afternoon, the focus will be on laboratory sessions conducted by students including computation, simulation and data analysis. The students will then work in teams on three case studies that will include computation, simulation and data analysis. The first lab session will teach the basics of Regression Monte Carlo, including the foundational Longstaff–Schwartz procedure and its variations. The second lab session will involve calibrating reduced-form models to historical electricity prices.
Morning session: (i) Building mathematical models via reduced-form and structural approaches for commodity spot and futures prices, including case-studies for electricity, natural gas and carbon emissions. (ii) Tools, including stochastic control and optimal stopping methods, for valuation and risk-management of common commodity contracts, such as continuous delivery periods, swing options, and gas storage.
Afternoon session: (i) in-depth discussion about recent numerical developments related to the problems from the morning lecture, such as Monte Carlo Regression. (ii) numerical approaches for calibrating 1 – and 2 – factor models to historical electricity prices using real data.
|9:00 – 10:15||Modeling commodity risk|
|11:00 – 12:00||Pricing commodity contracts|
|13:30 – 15:00||Intro to Regression Monte Carlo|
|15:45 – 16:45||Calibrating to electricity price data|
Day 2 – Thursday, 1st week
In the morning the focus will be on real options in commodity finance, covering both the classical single-agent formulation, as well as competitive investment in an oligopoly. The afternoon lab sessions conducted by students will focus on implementing the least squares Monte Carlo simulation approach to value and hedge a swing (multiple exercise) option on electricity delivery, as well as a real option to invest in additional generation capacity.
Morning session: (i) real options for investing in long-term projects and capacity expansion (power plants, mines, etc); and (ii) competitive investment, with examples from renewable energy generation, investment in R&D and technology, and competition between electricity market sectors.
Afternoon session: Lab sessions will explore the numerical implementation of the discussed frameworks. In the first part students will use regression Monte Carlo (RMC) simulation approach to value and hedge a swing (multiple exercise) option on electricity delivery based on the calibrated model from Day 1. Treating the swing option price as the value function of a multiple optimal stopping problem, students will explore various machine learning tools for fitting the corresponding functional approximators. The second part will then consider applying RMC to solve a capacity expansion problem for an energy asset, where the investor dynamically picks the timing and size of the expansion.
|9:00 – 10:15||Real Options w/single agent|
|11:00 – 12:00||Competition in commodity Real Options|
|13:30 – 15:00||Hands-on valuation of a swing option|
|15:45 – 16:45||Valuing a capacity expansion option|
Machine Learning and Financial Technology
Delivered by Prof. Matthew Dixon develops code for high performance architectures, is a co-author of the book “Machine Learning in Finance: From Theory to Practice” published by Springer, and is highly regarded as an expert on computational finance appearing in major media outlets including the Financial Times and Bloomberg Markets. The purpose of this course is to introduce students to the theory and practice of supervised and reinforcement learning problems in algorithmic and quantitative finance.
Machine learning(ML) is an active field in the finance industry. This course emphasizes the various mathematical and statistical frameworks for applying ML in quantitative finance, such as quantitative risk modeling with kernel learning and optimal investment with reinforcement learning. Laboratory examples shall demonstrate the implementation of various applications in finance using real market data. The lab sessions will be designed not only to reinforce the core concepts of the material, but also to provide exposure to real world datasets. Each lab session shall provide a mixture of toy examples and a real-world application. The toy examples allow the participants to gain familiarity with the implementation aspects of machine learning, using data visualization to develop intuition on the relationships in the data. Each real-world application is designed to take the toy example a step further by applying a larger-scale version of the models on real-word data, some preprocessing steps and more complex model validation procedures. Data will be provided from the instructor’s own data archive used for research. This will include high frequency limit order book data for futures, historical equity returns and fundamental factors, and single-name equity derivative chains. The lab sessions shall use Python notebooks combined with either TensorFlow, PyTorch or OpenAI Gym.
Day 1 – Monday, 2nd week
In the morning the focus is on key concepts in deep learning. In
the afternoon session the focus is on the implementation of deep learning for quantitative finance.
Morning session: (i) (a) Artificial and Deep Neural Networks; (b) back-propagation; and (c) stochastic gradient descent. (ii) an overview of the concepts arising in architecture design and fitting including VC dimension, bias-variance tradeoff, cross-validation and regularization; (iii) an overview of how deep learning is applied to applications in quantitative finance, including no- arbitrage derivative pricing and calibration, and factor modeling with interpretability.
Afternoon session: The topics in the afternoon will (i) provide an overview of how deep learning architectures are designed and fitted; and (ii) show how to fit deep networks to financial data and evaluate their performance in applications such as option pricing, calibration and factor modeling. Python notebooks shall be provided with data sets including the Russell 3000 fundamental factor exposure and stock price histories, and histories of option price surfaces.
|9:00 – 10:15||Foundations of Deep Learning|
|11:00 – 12:00||Deep Learning & Quantitative Finance|
|13:30 – 15:00||Implementing Deep Learning in TensorFlow|
|15:45 – 16:45||Applications in Quantitative Finance|
Day 2 – Tuesday, 2nd week
In the morning an overview of Markov Decision Processes and dynamic programming, with examples in financial modeling, shall be given. In the afternoon session the focus is on the implementation of Reinforcement Learning for financial modeling.
Morning session: (i) (a) Markov Decision Processes (b) Dynamic Programming; and (c) Reinforcement Learning (RL) and Inverse Reinforcement Learning (IRL). (ii) Examples formulating RL for market making, portfolio optimization, wealth management, and related optimal consumption problems.
Afternoon session: The topics in the afternoon will be demonstrated through one technical tutorial followed by a lab session using PyTorch and OpenAI Gym to fit Q-learning and related RL/IRLalgorithms to stock return histories . The lab shall begin with fitting Q-learning to simple toy planning problems in finance such as the financial cliff walking problem, optimal execution with market impact and market making. The session shall proceed to more advanced problems in portfolio optimization and wealth management.
|9:00 – 10:15||Introduction to Markov Decision Processes|
|11:00 – 12:00||Reinforcement learning / Inverse Reinforcement Learning & Finance|
|13:30 – 15:00||Implementing Reinforcement Learning in PyTorch / OpenAI Gym|
|15:45 – 16:45||Applications in Investment Management|
Students’ day – Wednesday, 2nd week
This day will consist of group sessions. Students will be split into four groups, each group supervised by a summer school lecturer. In the morning session, each team will be given an assignment. Students will brainstorm, interact with the team supervisor, and then propose their own solution to the problem. At the end of the session, each group should put together a short presentation, to show case their methodology and results. The objective of the afternoon session is to work on students’ presentation skills. The OC strongly believes that the ability of clearly, concisely, and enthusiastically convey in front of audience is of fundamental importance to students’ success. In particular, when applying for graduate/post-graduate student, tenure track faculty positions, or entering the private sector. Therefore, students will be asked to give a 10 minute presentation on a topic of their choice, possibly based on the topics learned during the summer school, their own research, or other related interests. The students will benefit from the earlier professional development session on presentations (see below) and will also be given guidance on how to prepare slides for the specific goal (e.g., each presentation should have at least one introductory slide, one slide introducing the question, one slide explaining the results, and one slide summarizing their findings; and slides should be over cluttered with material, nor have difficult to read graphs). Their presentations will be attended by the school lecturer supervising the team, and other fellow students. Each student will be encouraged to speak up and provide feedback to their colleagues. Students will gain confidence, and understand the importance of developing good presentation skills.
Mean field Games
Delivered by Prof. Roxana Dumitrescu, a leading expert in stochastic optimal control and mean field games, and Lecturer in Financial Mathematics at King’s College London, with an excellent publication record in the leading journals of FM.
Mean field games (MFGs) represent a recent breakthrough in game theory. In games with large populations of uncooperative agents, individual agents’ actions affect how the system behaves and the reward/costs of all agents. As a result of the complexities and size of these interactions, it is often difficult to construct, or even establish the existence of, Nash equilibria. The MFG approach approximates equilibria by studying the infinite player limit and establishes ε-Nash equilibria for the finite game. Many important applications of MFGs, which drive new mathematics in finance include: (i) systemic risk and financial networks, and (ii) algorithmic trading in electronic markets. Stochastic games also arise in many other problem areas in finance, economics, applied mathematics, and engineering.
Day 1 – Thursday, 2nd week
The focus of the morning session is on foundations. The focus of the afternoon session is on numerical implementation and simulation.
Morning session: (i) examples of stochastic games and sketch of the mathematical developments of MFG theory; (ii) examples of how finite games can be approximated by MFGs; and (iii) the dynamic programming principle for stochastic games.
Afternoon session: applications to financial networks with a large number of competing banks. Students will implement large scale simulations, and investigate numerically how agents lose value when they deviate from the Nash equilibria, as well as, how agents interact with one another and affect one another actions’ and their rewards.
|9:00 – 10:15||Intro to stochastic games|
|11:00 – 12:00||Intro to dynamic programming|
|13:30 – 15:00||Numerics of interbank borrowing/lending I|
|15:45 – 16:45||Numerics of interbank borrowing/lending II|
Day 2 – Friday, 2nd week
The focus of the morning session is on foundations. In the afternoon, the focus will be on numerical implementation and simulation.
Morning session: (i) understanding the mean field limit of stochastic games, and how the problem factors into a stochastic control problem together with a fixed-point problem for the distribution over states, and (ii) understanding the mean field game formulation for finite state/action space systems, e.g., those arising in ad auctions, and how to cast those problems in terms of discrete time Markov decision processes coupled with a fixed point problem.
Afternoon session: applications of MFGs to systemic risk and ad auctions. Students will be able to leverage the material learned from Day 2 of their Quantitative Risk Management course, which focuses on systemic risk and networks. As well, they will be able to leverage on their knowledge from Day 3 on machine learning techniques, as this session will focus on implementing the mean field Q-learning algorithm (a type of reinforcement learning paradigm) for numerically solving finite state/action MFGs.
In the afternoon sessions of both days, the goal will be to simulate the finite player game, and use numerical methods to investigate how well the MFG approximation to the finite player game performs. In addition, students will learn financial insights gained from the MFG approximations, which cannot be obtained directly from the simulation of finite player systems.
|9:00 – 10:15||Intro to Mean Field Games|
|11:00 – 12:00||Numerics of PDEs for MFGs|
|13:30 – 15:00||Modeling and Implementation of MFG systemic risk I|
|15:45 – 16:45||Modeling and Implementation of MFG systemic risk II|