AbstractsMathematics

Includes abstract. We give a review of regression-based Monte Carlo methods for pricing high-dimensional American and Bermudan options for which backwards methods such as lattice and PDE methods do not work. The continuous-time pricing problem is approximated in discrete time and the problem is formulated as an optimal stopping problem. The optimal stopping time can be expressed through continuation values (the price of the option given that the option is exercised after time j conditioned on the state process at time j). Regression-based Monte Carlo methods apply regression estimates to data generated by artificial samples of the state process in order to approximate continuation values. The resulting estimate of the option price is a lower bound. We then look at a dual formation of the optimal stopping problem which is used to generate an upper bound for the option price. The upper bound can be constructed by using any approximation to the option price. By using an approximation that arises from a lower bound method we have a general method for generating valid confidence intervals for the price of the option. In this way, the upper bound allows for a better estimate of the price to be computed and it provides a way of investigating the tightness of the lower bound by indicating whether more effort is needed to improve it.