Derivatives Analytics with Python: Data Analysis, Models, Simulation, Calibration and Hedging

CHAPTER 7 Valuation of American Options by Simulation

7.1 Introduction

Monte Carlo simulation (MCS) is a flexible and powerful numerical method to value financial derivatives of any kind (cf. Glasserman (2004)). As a forward evolving technique, it is per se not suited to address the valuation of American or Bermudan options which are valued in general by backwards induction (cf. Kohler (2009)). However, Longstaff and Schwartz (2001) provide a numerically efficient method to solve this problem by what they call Least-Squares Monte Carlo (LSM).¹ Their approach approximates continuation values for American options in backwards steps by an ordinary least-squares regression. Equipped with such approximations, the option is exercised if the approximate continuation value is lower than the value of immediate exercise. Otherwise it is not exercised. The LSM leads to a lower bound for the option’s value since the exercise decision is in any case sub-optimal (cf. Longstaff and Schwartz (2001)).

Haugh and Kogan (2004), among others, propose a dual formulation of the valuation problem for an American option which finally leads to a MCS estimator that represents an upper bound to the option’s value. In some situations, it is very helpful to have an upper bound in addition to a lower bound since the LSM does not allow to assess “how much too low the value estimate is.” Then, in the absence of alternative benchmarks, the accuracy of the LSM estimator cannot be judged.

This chapter proceeds ...