Minicourses
Antoine Jacquier:
Quantum Computing for Quantitative Finance
Quantum Computing, relegated for decades as a spooky distant myth, is now becoming a reality. To wit, quantum computers (albeit small in scale) are already available, developed by the likes of IBM, Rigetti, DWave, Google, Microsoft, ..... However, a quantum computer is not simply a bigger and more powerful computer, and requires a whole new set of algorithms to be written to perform useful tasks. These, and the underlying technology, draw from the laws of quantum mechanics, fundamentally different from our usual numerical toolbox. The goal of this course is to provide a mathematical introduction to Quantum Computing and to highlight applications in Quantitative Finance, in particular for Monte Carlo simulations, machine learning and optimisation. Numerical examples (through python) will also be introduced to provide a tangible reality.
Frank Riedel:
Knightian Uncertainty
Knightian Uncertainty has emerged as a major research topic in recent years. Frank Knight's pioneering dissertation on "Risk, Uncertainty, and Profit" distinguishes risk—a situation that allows for an objective probabilistic description—from uncertainty—a situation that cannot be modeled by a single probability distribution. By now, it is widely acknowledged that such Knightian uncertainty is crucial in many fields, including financial markets, climate economics, and pandemics. The lectures introduce the main decisiontheoretic models that have been developed. Applications to finance (risk measures and management, absence of arbitrage, robust portfolio choices) and economics (viability and equilibrium, game theory, mechanism design, climate change) are discussed in detail. Special invited lectures
Zorana Grbac:
Modeling of overnight and term interest rates after the Libor transition
In the recent reform of interbank interest rates the classical term rates such as Libor were discontinued and the overnight riskfree rates (RFRs) such as the secured overnight financing rate (SOFR) are now taking the lead role. Pertinent modelling of overnight rates and constructing of term rates have become major challenges in fixed income theory and practice. In this talk I would like to give some insights into these issues and present a modeling framework taking them into account. Firstly, I will give a brief overview of the reform and introduce the main modeling quantities. One of the key features of overnight rates is the presence of jumps and spikes occurring at predetermined dates which are caused by monetary policy interventions and liquidity constraints. These jumps correspond to the socalled stochastic discontinuities in the stochastic processes driving the dynamics of the rates. Next, I will discuss the way the forward term rates are produced in the market and the discrepancy between the theoretical term rates generated by overnight rates and the market term rates. We provide a mathematical model of this situation and propose a general framework in which the dynamics of the forward term rates are described via BSDEs with a given terminal condition. In a tractable setup based on affine semimartingales with stochastic discontinuities we study these BSDEs which in general may admit multiple solutions and provide sufficient conditions ensuring uniqueness. This reflects precisely the current market environment and allows for its better understanding and analysis. The resulting modeling framework is a HeathJarrowMorton (HJM) generalized framework incorporating possible stochastic discontinuities in the dynamics of overnight and term rates. In particular, when the term rate is generated by the overnight rate itself, we show that it solves a BSDE, whose driver is determined by the HJM drift restrictions. In the concluding part of the talk, I will present some simple specifications focusing on the overnight rate and illustrate pricing and hedging of interest rate derivatives in the presence of stochastic discontinuities. Based on joint works with C. Fontana, S. Gümbel and T. Schmidt.
Birgit Rudloff:
Epic Math Battles: Nash vs Pareto
Nash equilibria and Pareto optimization are two distinct concepts in multicriteria decision making. It is well known that the two concepts do not coincide. However, in this work we show that it is possible to characterize the set of all Nash equilibria for any noncooperative game as the set of all Pareto optimal solutions of a certain vector optimization problem. The characterization holds for all noncooperative games (nonconvex, convex, linear). This characterization opens a new way of computing Nash equilibria. It allows to use algorithms from vector optimization to compute resp. to approximate the set of all Nash equilibria, which is in contrast to the classical fixed point iterations that find just a single Nash equilibrium. This computation is straight forward in the linear case. In this talk we will discuss recent results in the convex case. An algorithm is proposed that computes for a given error distance epsilon, a subset of the set of epsilonNash equilibria such that it contains the set of all (true) Nash equilibria for convex games with either independent convex constraint sets for each player, or polyhedral joint constraints.
Sara SvalutoFerro:
Tractable infinite dimensional models: theory and applications
During this course we introduce, study, and apply several stochastic models in infinite dimensions. We focus in particular to the class of infinite dimensional polynomial (jump) diffusions, for which a set of readytouse mathematical tools have been developed. The first part of the course is dedicated to the theoretical aspects. After a presentation of the martingale problem we move to the socalled moment formula, providing a representation of some expected quantities in terms of an infinite dimensional linear ODE. In some cases of interest such ODE reduces to a PDE, has a representation à la FeymannKac, or can even be solved explicitly, for some or all initial conditions. In the second part we study two main applications: measure valued diffusions for energy markets and signature models in finance. We introduce a framework that allows to employ (nonnegative) measurevalued processes for energy market modeling, in particular for electricity and gas futures. Interpreting the process’ spatial structure as time to maturity, we show how the HeathJarrowMorton (HJM) approach can be translated to this framework, thus guaranteeing arbitrage free modeling in infinite dimensions. We derive an analog to the HJMdrift condition and then treat in a Markovian setting existence of nonnegative measurevalued diffusions that satisfy this condition. To analyze mathematically convenient classes we consider measurevalued polynomial and affine diffusions, where we can precisely specify the diffusion part in terms of continuous functions satisfying certain admissibility conditions. For calibration purposes, we illustrate how these functions can then be parameterized by neural networks yielding measurevalued analogs of neural SPDEs. By combining Fourier approaches or the moment formula with stochastic gradient descent methods, this then allows for tractable calibration procedures which we also test by way of example on market data. Finally, we introduce the concept of signature of a semimartingale and present its main properties. In particular, we show that the signature process of a (classical) polynomial process is a polynomial process and illustrate how the moment formula applies in this case. We then move to signature SDEs whose characteristics are linear functions of a primary underlying process, which can range from a (marketinferred) Brownian motion to a general multidimensional tractable stochastic process. Short lectures
Kristoffer Andersson:
Robust deep FBSDE methods
The deep BSDE method and its extensions have brought significant progress in approximating PDEs and FBSDEs, particularly in highdimensional settings. However, the method fails to converge for a large class of important problems. This presentation aims to provide the audience with an intuitive understanding of when and why the deep BSDE method encounters convergence issues. A focus will be placed on the class of FBSDEs arising from stochastic optimal control problems, discussing how mathematical structures from control theory can be utilized to adjust the deep BSDE method, ensuring convergence. Additionally, potential adjustments for general FBSDEs, where connections to stochastic control are absent, will be explored. The goal is to share insights into the theoretical aspects and practical implementations of these approaches, extending the applicability of the deep BSDE method to a broader range of problems.
Zhipeng Huang:
Generalized convergence for the Deep BSDE method
We are concerned with highdimensional coupled FBSDE systems approximated by the deep BSDE method of Han et al. (2018). It was shown by Han and Long (2020) that the errors induced by the deep BSDE method admit a posteriori estimate depending on the loss function, whenever the backward equation only couples into the forward diffusion through the Y process. We generalize this result to fullycoupled drift coefficients, and give sufficient conditions for convergence under standard assumptions. The resulting conditions are directly verifiable for any equation. Consequently, our convergence analysis enables the treatment of FBSDEs stemming from stochastic optimal control problems. In particular, we provide a theoretical justification for the nonconvergence of the deep BSDE method observed in recent literature, and present direct guidelines for when convergence can be guaranteed in practice. Our theoretical findings are supported by several numerical experiments in highdimensional settings. This is a joint work with Balint Negyesi and Cornelis W. Oosterlee.
Balint Negyesi :
A Deep BSDE approach for the simultaneous pricing and delta
gamma hedging of large portfolios consisting of highdimensional multi
asset Bermudan options
Direct financial applications of the methods introduced in Negyesi et al. (2024), Negyesi (2024) are presented. Therein, a new discretization of (discretely reflected) Markovian backward stochastic differential equations is given which involves a Gamma process, corresponding to secondorder sensitivities of the associated option’s price. The main contributions of this work is to apply these techniques in the context of portfolio risk management. Large portfolios of a mixture of European and Bermudan derivatives are cast into the framework of discretely reflected BSDEs. The resulting system is solved by a neural network regression Monte Carlo method proposed in the aforementioned papers. Numerical experiments are presented on highdimensional portfolios, consisting of several European, Bermudan and American options, which demonstrate the robustness and accuracy of the method. The corresponding ProfitandLoss distributions indicate an order of magnitude gain in the number of rebalancing dates needed in order to achieve a certain level of risk tolerance, when the secondorder conditions are also satisfied. The hedging strategies significantly outperform benchmark methods both in the case of standard delta and deltagamma hedging.
Jasper Rou:
A timestepping deep gradient flow method for option pricing
In this research, we consider a novel deep learning approach for pricing options by solving partial differential equations (PDE). More specifically, we consider a Timestepping Deep Gradient Flow method, where the PDE is solved by discretizing it in time and writing it as the solution of minimizing a variational problem. A neural network approximation is then trained to solve this minimization using stochastic gradient descent. This method reduces the training time compared to for instance the Deep Galerkin Method. The proposed scheme respects the asymptotic behavior of option prices for large levels of moneyness and adheres to known bounds for option prices. The accuracy and efficiency of the proposed method is assessed in a series of numerical examples, with particular focus in the lifted Heston model. Furthermore, we prove two things. First, that there exists a neural network converging to the solution of the PDE. This proof consists of three parts: 1) convergence of the time stepping; 2) equivalence of the solution of the discretized PDE and the minimizer of the variational formulation and 3) the approximation of the minimizer by a neural network by using a version of the universal approximation theorem. Second, we prove that when training the network we converge to the correct solution. This proof consists of two parts: 1) as the number of neurons goes to infinity we converge to some gradient flow and 2) as the training time goes to infinity this gradient flow converges to the solution. Poster presentations
