Pricing high-dimensional Bermudan options with hierarchical tensor formats

An efficient compression technique based on hierarchical tensors for popular option pricing methods is presented. It is shown that the "curse of dimensionality" can be alleviated for the computation of Bermudan option prices with the Monte Carlo least-squares approach as well as the dual martingale method, both using high-dimensional tensorized polynomial expansions. This discretization allows for a simple and computationally cheap evaluation of conditional expectations. Complexity estimates are provided as well as a description of the optimization procedures in the tensor train format. Numerical experiments illustrate the favourable accuracy of the proposed methods. The dynamical programming method yields results comparable to recent Neural Network based methods.Comment: 26 pages, 3 figures, 5 tables, added affiliations and update acknowledgement

FuNVol: A Multi-Asset Implied Volatility Market Simulator using Functional Principal Components and Neural SDEs

Here, we introduce a new approach for generating sequences of implied volatility (IV) surfaces across multiple assets that is faithful to historical prices. We do so using a combination of functional data analysis and neural stochastic differential equations (SDEs) combined with a probability integral transform penalty to reduce model misspecification. We demonstrate that learning the joint dynamics of IV surfaces and prices produces market scenarios that are consistent with historical features and lie within the sub-manifold of surfaces that are essentially free of static arbitrage. Finally, we demonstrate that delta hedging using the simulated surfaces generates profit and loss (P&L) distributions that are consistent with realised P&Ls.Comment: 30 pages, 12 figures, 5 table

Deep Learning in a Generalized HJM-type Framework Through Arbitrage-Free Regularization

We introduce a regularization approach to arbitrage-free factor-model selection. The considered model selection problem seeks to learn the closest arbitrage-free HJM-type model to any prespecified factor-model. An asymptotic solution to this, a priori computationally intractable, problem is represented as the limit of a 1-parameter family of optimizers to computationally tractable model selection tasks. Each of these simplified model-selection tasks seeks to learn the most similar model, to the prescribed factor-model, subject to a penalty detecting when the reference measure is a local martingale-measure for the entire underlying financial market. A simple expression for the penalty terms is obtained in the bond market withing the affine-term structure setting, and it is used to formulate a deep-learning approach to arbitrage-free affine term-structure modelling. Numerical implementations are also performed to evaluate the performance in the bond market.Comment: 23 Pages + Reference

Machine Learning in Insurance

Machine learning is a relatively new field, without a unanimous definition. In many ways, actuaries have been machine learners. In both pricing and reserving, but also more recently in capital modelling, actuaries have combined statistical methodology with a deep understanding of the problem at hand and how any solution may affect the company and its customers. One aspect that has, perhaps, not been so well developed among actuaries is validation. Discussions among actuaries’ “preferred methods” were often without solid scientific arguments, including validation of the case at hand. Through this collection, we aim to promote a good practice of machine learning in insurance, considering the following three key issues: a) who is the client, or sponsor, or otherwise interested real-life target of the study? b) The reason for working with a particular data set and a clarification of the available extra knowledge, that we also call prior knowledge, besides the data set alone. c) A mathematical statistical argument for the validation procedure