Efficient upper bounds for American options: regression-based duals from backward primals

This article proposes a simple primal-dual Monte Carlo algorithm for approximating the prices of American and Bermudan options. The algorithm calculates the dual (upper bound) price of an option by applying least squares linear regression to extract the Doob martingale part of the Snell envelope from a set of primal (lower bound) paths generated by a stopping time approximation estimator. This allows for straightforward and fast approximations of upper bounds on the true option price, without the need for nested simulations. The key to the algorithm is the use of orthogonal projections to approximate both the value process and the conditional expectations along each path by projecting the primal paths at each timestep onto the subspace spanned by the same set of basis functions, evaluated one timestep apart. Numerical results for a variety of option pricing problems are presented, allowing for direct comparison with other algorithms found in the literature. The results suggest wide applicability of the algorithm to practical problems in option pricing.