Deep Learning for American Option Pricing

American Option pricing efficiency is still on interesting topic in Computational Mathematichs and Finance. These types of derivatives offer great flexibility in all financial and trading markets due to the possibility of an early exercise. However, there is not any closed form analytical valuation of American option because of the optimal exercise problem created by the early exercise. This is the reason why the problem is faced by a numerical approximation. Per contra, traditional numerical methods suffer from the curse of dimensionality. The classical approaches yield good results for up to 3 dimensional problem. To solve problems in higher dimensional space, Longstaff and Schwartz in [1] have proposed a regression based Monte Carlo method which approximates conditional expectation by projections on a finite set of basis function. The main problem of this method is that the number of paths required for convergence should grow exponentially with the number of basis functions. Consequently, the algorithm of Longstaff and Schwartz failed. During the last years, this problem has been treated through Artificial Intelligence tools. In [2], S. Becker, P. Cheridito and A. Jentzen have developed a Deep Learning method for optimal stopping problems that learns optimal stopping rule from Monte Carlo samples. By the virtue of this algorithm, they were able to achieve extraordinary results with a reasonable computational cost. Therefore, in the first chapter of this thesis we introduce the problem and its formulation from a theoretical point of view. Then we examine in depth the least square regression method of Longstaff and Schwartz. Finally we present the "Deep Optimal Stopping" method by S. Becker, P. Cheridito and A. Jentzen.