Neural network regression for Bermudan option pricing

The pricing of Bermudan options amounts to solving a dynamic programming principle, in which the main difficulty, especially in high dimension, comes from the conditional expectation involved in the computation of the continuation value. These conditional expectations are classically computed by regression techniques on a finite-dimensional vector space. In this work, we study neural networks approximations of conditional expectations. We prove the convergence of the well-known Longstaff and Schwartz algorithm when the standard least-square regression is replaced by a neural network approximation, assuming an efficient algorithm to compute this approximation. We illustrate the numerical efficiency of neural networks as an alternative to standard regression methods for approximating conditional expectations on several numerical examples.

Nearest Neighbor Based Estimation Technique for Pricing Bermudan Options

Bermudan option is an option which allows the holder to exercise at pre-specified time instants where the aim is to maximize expected payoff upon exercise. In most practical cases, the underlying dimensionality of Bermudan options is high and the numerical methods for solving partial differential equations as satisfied by the price process become inapplicable. In the absence of analytical formula a popular approach is to solve the Bermudan option pricing problem approximately using dynamic programming via estimation of the so-called continuation value function. In this paper we develop a nearest neighbor estimator based technique which gives biased estimators for the true option price. We provide algorithms for calculating lower and upper biased estimators which can be used to construct valid confidence intervals. The computation of lower biased estimator is straightforward and relies on suboptimal exercise policy generated using the nearest neighbor estimate of the continuation value function. The upper biased estimator is similarly obtained using likelihood ratio weighted nearest neighbors. We analyze the convergence properties of mean square error of the lower biased estimator. We develop order of magnitude relationship between the simulation parameters and computational budget in an asymptotic regime as the computational budget increases to infinity.