**5. Conclusions**

This paper proposed an innovative algorithm that significantly improves on the approximation of the optimal early exercise boundary obtained with simulation based methods for American option pricing like, e.g., the Least-Squares Monte Carlo (LSM) method of Longstaff and Schwartz (2001). The method successfully exploited and leveraged the information in multiple cross-sectional regressions to the fullest by averaging the individually obtained estimates at each early exercise step, starting from just before maturity, in the backwards induction algorithm. We referred to our method as a bootstrapping approach because of the similarities it had with how the term structure of interest rates was bootstrapped and the way bootstrapping was used for inference in regression models. With the bootstrapping approach, less errors were accumulated in the backward induction algorithm, and as a result of this, the price estimate was essentially unbiased even for long maturity options.

We compared the results from our bootstrapping approach to the regular LSM method, and the numerical results demonstrated large and significant improvements from our method. These findings were robust to the choice of simulation setup, the characteristics of the option, and the dimensionality of the problem. Finally, because our method naturally disassociated the estimation of the optimal early exercise boundary from the pricing of the option, significant efficiency gains could be obtained by using less simulated paths and repetitions to estimate the optimal early exercise boundary than with the regular method. To illustrate this, we priced a diverse sample of options with different moneyness, maturity, and levels of volatility of the underlying asset. The results showed that when pricing this sample of empirically relevant options, our bootstrapping method essentially yielded unbiased price estimates that were as precise as if the true optimal early exercise boundary had been used.

Our bootstrapping method should have wide applications empirically. First, the majority of the options traded are in fact American style, and the recent global financial crisis clearly demonstrated that considering more risk factors is essential to model the complex behavior of financial markets properly. Our proposed method can be used with such advanced models and will allow more efficient pricing than what is currently possible. Second, our results have important implications for other simulation based applications. One particularly important and challenging area in which the LSM method has been used is the field of real option valuation. In particular, the dynamics of the underlying assets for this type of options are often very complicated, and simulation is the only viable method. We conjecture that our proposed bootstrapping method will allow more efficient determination of the optimal controls and more precise valuation of these assets, benefiting the economy at large.

Besides the empirical applications outlined above, there are other important lines for future research. First, we are currently working on establishing the finite sample properties of our proposed bootstrapping method to demonstrate theoretically its usefulness and to assess if there are any limitations to the applicability of the methodology. Second, simulation based methods have recently been used to obtain estimates of option risk sensitivities or hedge ratios, the Greeks, for American options. We conjecture that our bootstrapping method would allow such quantities to be estimated with much more precision as well and are currently working on establishing this empirically. Finally, our general idea could be applied to other methods that use simulation and regression techniques for pricing early exercise style option like, e.g., the value function iteration method of Carriere (1996) and Tsitsiklis and Van Roy (2001), and to other dynamic programming methods used to solve optimal control type problems.

**Author Contributions:** The authors contributed equally to all aspects of this paper.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors would like to thank two anonymous referees for valuable comments and suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.
