Deep optimal stopping

The main focus of this thesis is to explain a new pricing technique for complex derivative products, published by Sebastian Becker, Patrick Cheridito, and Arnulf Jentzen in the paper named ”Deep Optimal Stopping” in 2019, which involves the use of machine learning techniques. This method is called ”Deep Optimal Stopping.” In recent decades, researchers have tried to solve option pricing problems involving multi-dimensional assets with techniques like the Monte Carlo Least Square methods. Nonetheless, these pricing methods have proven to be computationally expensive when there are hundreds of assets and have shown to be affected by the so-called curse of dimensionality. These issues have been the primary reason that induced Sebastian Becker, Patrick Cheridito, and Arnulf Jentzen to develop the Deep Optimal Stopping method. We provide, in the first part of Chapter 1, a general background on stochastic processes and some key properties related to Markov processes. In the second part of the chapter, we describe the main settings and properties of an optimal stopping problem, some key results that characterize the solution to such problems, their connection to PDEs, and a practical application. In Chapter 2, we start by discussing Monte Carlo methods in general. We then move to the explanation and analysis of the Monte Carlo Least Square method, outlining some key steps of its proof, and conclude the chapter with two practical applications. In the first part of Chapter 3, we analyze the general structure of a neural network, its key operations, and one of the possible ways used to calibrate it. In the second part of the chapter, we describe the theory on which the Deep Optimal Stopping method is based, how its neural network is built and calibrated, how the lower and upper bounds of the pricing problem we try to solve are computed, how the confidence intervals for our estimated values are built, and finally, we show a couple of applications of the Deep Optimal Stopping method. Finally, in Chapter 4, we compare all the results obtained in the previous chapters. Furthermore, we added an Appendix that enables the reader to understand the background theory associated with the Monte Carlo Least Square method, and all the examples are built with Python code.