Pricing American-Style Options by Monte Carlo Simulation ...

Table VII presents the case of a max-call option on ten assets. 7 Almost all methodsperform poorly. MMPP does relatively better but still does not consistently achieveaccurate estimates. 8D. Overall Evaluation of the Five TestcasesTables VIII and IX summarize the results for all the estimation methods and all thetestcases. Each estimation method is given a grade of “+”, “–” , or “o”, dependingon whether it performed well, poorly, or average. Below each method the four columnscorrespond to 100, 1,000, 5,000, and 10,000 simulation paths. Table VIII summarizesthe results for all the sets of basis functions, while Table IX focuses on the case of thesets of basis functions with the largest number of functions.Overall we find that for a small number of paths OLS regression performs poorly.The MMPP method consistently outperforms OLS regression in the case of 100 and1,000 paths. As the number of paths increases, OLS regression tends to catch up to theother methods. This is not a surprise since the price estimated using OLS regressionconverges to the asymptotic price in the limit of a large number of simulation paths.The regression tree method performs poorly for most of the specifications. Checkingthe approximate continuation value function generated by this method shows that thesize of the tree turns out to be too small to capture the structure of the continuationvalue function.For Testcases 1, 2, 4, and 5 with two assets, the quantile regression method convergesto a price significantly below the asymptotic price. This suggests a systematic bias thatmay be caused by the convexity of the continuation value that leads to positive skewnessof the distribution of option prices.7 Results for the cases of three and five assets are similar and are available from the authors.8 We know of no easy way to compute European option prices for this case, and for this reason wehave not included European option prices to the basis.23