Daily Papers

Researchers have been using Neural Networks and other related machine-learning techniques to price options since the early 1990s. After three decades of improvements in machine learning techniques, computational processing power, cloud computing, and data availability, this paper is able to provide a comparison of using Google Cloud's AutoML Regressor, TensorFlow Neural Networks, and XGBoost Gradient Boosting Decision Trees for pricing European Options. All three types of models were able to outperform the Black Scholes Model in terms of mean absolute error. These results showcase the potential of using historical data from an option's underlying asset for pricing European options, especially when using machine learning algorithms that learn complex patterns that traditional parametric models do not take into account.

We apply a physics-informed deep-learning approach the PINN approach to the Black-Scholes equation for pricing American and European options. We test our approach on both simulated as well as real market data, compare it to analytical/numerical benchmarks. Our model is able to accurately capture the price behaviour on simulation data, while also exhibiting reasonable performance for market data. We also experiment with the architecture and learning process of our PINN model to provide more understanding of convergence and stability issues that impact performance.

We consider two data driven approaches, Reinforcement Learning (RL) and Deep Trajectory-based Stochastic Optimal Control (DTSOC) for hedging a European call option without and with transaction cost according to a quadratic hedging P&L objective at maturity ("variance-optimal hedging" or "final quadratic hedging"). We study the performance of the two approaches under various market environments (modeled via the Black-Scholes and/or the log-normal SABR model) to understand their advantages and limitations. Without transaction costs and in the Black-Scholes model, both approaches match the performance of the variance-optimal Delta hedge. In the log-normal SABR model without transaction costs, they match the performance of the variance-optimal Barlett's Delta hedge. Agents trained on Black-Scholes trajectories with matching initial volatility but used on SABR trajectories match the performance of Bartlett's Delta hedge in average cost, but show substantially wider variance. To apply RL approaches to these problems, P&L at maturity is written as sum of step-wise contributions and variants of RL algorithms are implemented and used that minimize expectation of second moments of such sums.

We present Legal Argument Reasoning (LAR), a novel task designed to evaluate the legal reasoning capabilities of Large Language Models (LLMs). The task requires selecting the correct next statement (from multiple choice options) in a chain of legal arguments from court proceedings, given the facts of the case. We constructed a dataset (LAR-ECHR) for this task using cases from the European Court of Human Rights (ECHR). We evaluated seven general-purpose LLMs on LAR-ECHR and found that (a) the ranking of the models is aligned with that of LegalBench, an established US-based legal reasoning benchmark, even though LAR-ECHR is based on EU law, (b) LAR-ECHR distinguishes top models more clearly, compared to LegalBench, (c) even the best model (GPT-4o) obtains 75.8% accuracy on LAR-ECHR, indicating significant potential for further model improvement. The process followed to construct LAR-ECHR can be replicated with cases from other legal systems.

We present a methodology to price options and portfolios of options on a gate-based quantum computer using amplitude estimation, an algorithm which provides a quadratic speedup compared to classical Monte Carlo methods. The options that we cover include vanilla options, multi-asset options and path-dependent options such as barrier options. We put an emphasis on the implementation of the quantum circuits required to build the input states and operators needed by amplitude estimation to price the different option types. Additionally, we show simulation results to highlight how the circuits that we implement price the different option contracts. Finally, we examine the performance of option pricing circuits on quantum hardware using the IBM Q Tokyo quantum device. We employ a simple, yet effective, error mitigation scheme that allows us to significantly reduce the errors arising from noisy two-qubit gates.