Christmas Workshop 2023 in Stochastics and Finance

We are very greatful for the financial support of the Graduate Center LMU.

Date and Time

• 23.11.2023, 18:00 - 22:00
• 24.11.2023, 10:00 - 17:00

Venue

• Mathematical Insitut LMU, Theresienstraße 39, Room B349

Timetable - Thursday, 23.11.2023

Timetable - Friday, 24.11.2023

Confirmed Speakers

Overview Talks

Athena Picarelli - A deep solver for BSDEs with jumps

The aim of this work is to propose an extension of the deep solver by Han, Jentzen, E (2018) to the case of forward backward stochastic differential equations (FBSDEs) with jumps. As in the aforementioned solver, starting from a discretized version of the FBSDE and parametrizing the (high dimensional) control processes by means of a family of artificial neural networks (ANNs), the FBSDE is viewed as model-based reinforcement learning problem and the ANN parameters are fitted so as to minimize a prescribed loss function. We take into account both finite and infinite jumpactivity by introducing, in the latter case, an approximation with finitely many jumps of the forward process. We successfully apply our algorithm to option pricing problems in low and high dimension and discuss the applicability in the context of counterparty credit risk.

Andreas Søjmark - Weak convergence of the financial gains for tick-by-tick models

Continuous time financial models are idealizations of discrete tick-by-tick dynamics for the asset prices. Jacod and Ait-Sahalia declare these two worlds to be compatible if there is weak convergence of the tick-by-tick dynamics in Skorokhod’s M1 topology. In this talk, we will explore what can be said about the weak M1 convergence of the corresponding financial gains across general classes of trading strategies. We will pay special attention to so-called CTRW models, which have recently served to motivate an interesting option pricing framework of Jacquier and Torricelli.

Tim De Ryck - Mathematical guarantees for physics-informed machine learning

Physics-informed neural networks (PINNs) and their variants have been very popular in recent years as algorithms for the numerical simulation of both forward and inverse problems for partial differential equations. We provide an overview of currently available mathematical guarantees for PINNs and related models that constitute the backbone of physics-informed machine learning. A detailed review of available results on approximation, generalization and training errors and their behaviour with respect to the type of the PDE and the dimension of the underlying domain is presented.

Yadh Hafsi - Uncovering Market Disorder and Liquidity Trends Detection

The primary objective of this paper is to conceive and develop a new methodology to detect notable changes in liquidity within an order-driven market. We study a market liquidity model which allows us to dynamically quantify the level of liquidity of a traded asset using its limit order book data. The proposed metric holds potential for enhancing the aggressiveness of optimal execution algorithms, minimizing market impact and transaction costs, and serving as a reliable indicator of market liquidity for market makers. As part of our approach, we employ Marked Hawkes processes to model trades-through which constitute our liquidity proxy. Subsequently, our focus lies in accurately identifying the moment when a significant increase or decrease in its intensity takes place. We consider the minimax quickest detection problem of unobservable changes in the intensity of a doubly-stochastic Poisson process. The goal is to develop a stopping rule that minimizes the robust Lorden criterion, measured in terms of the number of events until detection, for both worst-case delay and false alarm constraint. We prove our procedure’s optimality in the case of a Cox process with simultaneous jumps, while considering a finite time horizon. Finally, this novel approach is empirically validated by means of real market data analyses.

Owen Futter - Signature Trading: A Path-Dependent Extension of the Mean-Variance Framework with Exogenous Signals

We derive a novel path-dependent extension for mean-variance portfolio optimisation by recasting the problem in terms of signature trading strategies, allowing to incorporate the evolution of the past signal-asset time series into the optimisation problem, avoiding the need for direct return prediction. The method is intuitive, interpretable and a lightweight alternative to modern machine learning methods whilst still being able to capture non-linearities for a large class of trading strategies (including momentum and pairs trading).

Lorenz Riess - The geometry of financial institutions - Wasserstein clustering of
financial data

Financial regulation requires the submission of diverse and often highly granular data from financial institutions to regulators. In turn, regulators face the challenge of condensing this data into a comprehensive map that captures the mutual similarity or distance between different institutions and identifies clusters or outliers based on features like size, credit portfolio, or business model. Additionally, missing data due to varying regulatory requirements for different types of institutions, can further complicate this task. To address these challenges, we interpret the credit data of financial institutions as probability distributions whose respective distances can be assessed through optimal transport theory. Specifically, we propose a variant of Lloyd's algorithm that applies to probability distributions and uses generalized Wasserstein barycenters to construct a metric space. Our approach provides a solution for the mapping of the banking landscape, enabling regulators to identify clusters of financial institutions and assess their relative similarity or distance.