Machine learning methods for pricing and hedging financial derivatives

Machine learning and deep learning have realized incredible success in areas such as computer vision and natural language processing. This thesis focuses on developing machine learning methods and models for financial data, which is a new application area in machine learning. It is challenging to successfully apply machine learning due to the substantial noise in financial data. As part of our approach to address this challenge of highly noisy data (which can cause machine learning models to easily overfit), we develop a pricing and hedging framework which merges classical stochastic differential equations (SDE) with neural networks. Optimization methods are developed to train these neural network-SDE models. The models are trained on large amounts of financial data using GPU high-performance computing. The neural-network-based SDE models demonstrate superiority over traditional mathematical option pricing models such as the Black-Scholes model, the Dupire's local volatility model, and the Heston model. We develop a new optimization algorithm that can take unbiased, stochastic gradient descent steps for optimizing the neural-network-based SDE for mean-squared-error objective functions and European options. This algorithm allows for rapid optimization of the models. For training with general objective functions and more complex options (e.g., American options), a partial differential equation (PDE) is derived to evaluate the objective function and we then optimize over the PDE. The training of models is mathematically and computationally challenging because one must optimize over an SDE/PDE for the European/American option price. Models are trained on large datasets (many contracts) and either large simulations (many Monte Carlo samples for the stock price paths) or large numbers of PDEs (a PDE must be solved for each contract). All numerical results are based on real market data including S&P500 index options, S&P100 index options, and single-stock American options. The neural-network-based SDE models are more accurate at out-of-sample pricing than the Black-Scholes model, the Dupire's local volatility model, and the Heston model. For out-of-the-money hedging, the neural-network-based SDE model outperforms Dupire's local volatility model and the Heston model while slightly under-performing the Black-Scholes model. For at-the-money hedging, the neural-network-based SDE model underperforms the other three models. Finally, this thesis also explores modeling limit order book data with deep neural networks. In this area, the challenges become how to extract useful information from terabytes of limit order books data and carefully design deep neural networks to improve the prediction accuracy of stock prices. This thesis focuses on developing convolutional neural networks (CNN) and recurrent neural networks (RNN) to predict the mid-price direction for 0.1 and 1 second. The models are tested on over 100 stocks data and achieve superior performance than traditional machine learning models