A Review on Regression-based Monte Carlo Methods for Pricing American Options

In this article we give a review of regression-based Monte Carlo methods for pricing American options. The methods require in a first step that the generally in continuous time formulated pricing problem is approximated by a problem in discrete time, i.e., the number of exercising times of the considered option is assumed to be finite. Then the problem can be formulated as an optimal stopping problem in discrete time, where the optimal stopping time can be expressed by the aid of so-called continuation values. These continuation values represent the price of the option given that the option is exercised after time t conditioned on the value of the price process at time t . The continuation values can be expressed as regression functions, and regression-based Monte Carlo methods apply regression estimates to data generated by the aid of artificial generated paths of the price process in order to approximate these conditional expectations. In this article we describe various methods and corresponding results for estimation of these regression functions. 1 Pricing of American Options as Optimal Stopping Problem In many financial contracts it is allowed to exercise the contract early before expiry. E.g., many exchange traded options are of American type and allow the holder any exercise date before expiry, mortgages have often embedded prepayment options such that the mortgage can be amortized or repayed, or life insurance contracts allow often for early surrender. In this article we are interested in pricing of options with early exercise features. It is well-known that in complete and arbitrage free markets the price of a derivative security can be represented as an expected value with respect to the so called martingale measure, see for instance Karatzas and Shreve (1998). Furthermore, the Michael Kohler, Department of Mathematics, Technische Universität Darmstadt, Schloßgartenstraße 7, 64289 Darmstadt, Germany e-mail: [email protected] L. Devroye et al. (eds.), Recent Developments in Applied Probability and Statistics, DOI 10.1007/978-3-7908-2598-5_2, c © Springer-Verlag Berlin Heidelberg 2010