The Foresight Bias in Monte-Carlo Pricing of Options with Early Exercise: Classification, Calculation & Removal

In this paper we investigate the so called foresight bias that may appear in the Monte-Carlo pricing of Bermudan and compound options if the exercise criteria is calculated by the same Monte-Carlo simulation as the exercise values. The standard approach to remove the foresight bias is to use two independent Monte-Carlo simulations: One simulation is used to estimate the exercise criteria (as a function of some state variable), the other is used to calculate the exercise price based on this exercise criteria. We shall call this the numerical removal of the foresight bias. In this paper we give an exact definition of the foresight bias in closed form and show how to apply an analytical correction for the foresight bias. Our numerical results show that the analytical removal of the foresight bias gives similar results as the standard numerical removal of the foresight bias. The analytical correction allows for a simpler coding and faster pricing, compared to a numerical removal of the foresight bias. Our analysis may also be used as an indication of when to neglect the foresight bias removal altogether. While this is sometimes possible, neglecting foresight bias will break the possibility of parallelization of Monte-Carlo simulation and may be inadequate for Bermudan options with many exercise dates (for which the foresight bias may become a Bermudan option on the Monte-Carlo error) or for portfolios of Bermudan options (for which the foresight bias grows faster than the Monte-Carlo error). In addition to an analytical removal of the foresight bias we derive an analytical correction for the suboptimal exercise due to the uncertainty induced by the Monte-Carlo error. The combined correction for foresight bias (biased high) and suboptimal exercise (biased low) removed the systematic bias even for Monte-Carlo simulations with very small number of paths.