**3. Results**

The previous section demonstrated how our bootstrapping method can be implemented and shows that it leads to much more precise estimates of the optimal early exercise boundary and results in estimated prices that are essentially unbiased for a benchmark option. In this section, we test the robustness of these findings along several dimensions. First, we show that our results are robust across choices in the simulation setup for the number of regressors and the number of paths used and across option characteristics like the moneyness and maturity of the option and the volatility of the underlying asset. To illustrate this, we consider first options on a single stock in a simple model with Black–Scholes–Merton dynamics because this allows us to characterize the true optimal early exercise boundary, which can be used to obtain Monte Carlo benchmark values. Finally, we show that our results generalize to the case with multiple underlying assets. We consider four different payoffs, and though the pricing performance varies across payoffs and depends on the order of the polynomial approximation, our bootstrapped method performs the best across different approximations.<sup>9</sup>

### *3.1. Robustness to the Simulation Setup*

In Section 2.3, we demonstrated how to implement the bootstrapping method for a given number of regressors, *M* = 9, and number of simulated paths, *N* = 100,000. However, both *M* and *N* are choice parameters in the simulation setup that need to be picked when implementing the method. Thus, the first thing we analyze is the robustness of our proposed method to the choice of the number of simulated paths, *N*, used for determining the estimated optimal early exercise boundary and subsequently for pricing in case the OS method is used. The option we consider here has a strike price of *K* = 40, a maturity of *T* = 1 year, and *J* = 50 early exercise points per year. The initial stock price is fixed at *S*(0) = 40; the volatility is *σ* = 20%; and the interest rate is *r* = 6%. For now, we continue to use a polynomial of order *M* = 9 in the cross-sectional regressions.

Figure 2 compares the results from using the three different methods to determine the estimated optimal early exercise boundary: the regular LSM method, the regular average of the individual LSM boundaries, and the recursively bootstrapped boundaries, as a function of the number of simulated paths, *N*. The red lines represent the average price over *I* = 100 independent repetitions of the LSM method, each of which uses *N* paths. The line above the benchmark reference uses the IS method, whereas the line below uses the OS method. To eliminate the sampling bias from using a finite number of paths in the Monte Carlo simulation, we compare the price estimates to what would be obtained using the true early exercise boundary backed out from a binomial model with 50,000 steps on the two sets of simulated paths. The green lines use the regular average over the *I* = 100 individual LSM

<sup>9</sup> All simulations were conducted using MATLAB.

repetitions using *N* paths, and the blue lines use the recursive average over *I* = 100 individual LSM repetitions using *N* paths.

**Figure 2.** Convergence of price estimates as a function of sample size, *N*. This figure shows the price estimates from *I* = 100 simulations with different numbers of paths *N*. Individually, early exercise boundaries are estimated with *M* = 9 regressors and a constant term in the cross-sectional regressions. The option has a strike price of *K* = 40, a maturity of *T* = 1 year, and *J* = 50 early exercise points per year. The initial stock price is fixed at *S*(0) = 40; the volatility is *σ* = 20%; and the interest rate is *r* = 6%. The red lines report the results for the standard LSM method. The green lines report the results when the early exercise boundary is estimated from the average of the *I* = 100 continuation value approximations. The blue lines report the results when the early exercise boundary is backed out from our bootstrapped continuation values. The horizontal black line shows the result when the true early exercise boundary estimated with a binomial model with 50,000 steps is used.

First, the figure clearly shows that the regular IS LSM price can be strongly biased if the number of paths is too low. This is due to using the same paths to determine the individual optimal early exercise boundary and for calculating payoffs, which as we discussed earlier, leads to a high bias in the estimated option price. The OS prices, i.e., from the method that uses a new set of paths for pricing, are guaranteed to be low biased as shown in the figure. As the number of paths used in the individual simulation increases, the IS and OS results converge to a value that is (slightly) lower than the benchmark value, but corresponds to the true approximate value obtained when the approximation *FM X tj* based on a polynomial of order *M* = 9 of the true conditional expectation *F X tj* is used.

Next, the figure shows that using the regular average method over the *I* = 100 repetitions improves on the results when a low number of paths is used. This is the situation where the early exercise boundary is estimated with the most noise, and averaging helps counter that somewhat. Note that the IS prices for this method are also low biased because of the averaging. As expected, both price estimates converge to the true approximated price as the number of paths used increases and in the limit, which in our setting corresponds to using *N* = 512,000 paths; averaging has very little effect on the estimated prices.

The final and most interesting thing the figure shows is that using the recursive average dramatically improves on the results across all values of *N*. In fact, Figure 2 essentially shows that the recursive method is virtually unbiased, even when using as little as *N* = 1000 paths to approximate the conditional expectations. Using only *N* = 1000 paths for OS pricing with the estimated early exercise boundary also leads to unbiased estimates, although both of these estimates will naturally have quite large variances.

In addition to *N* being a choice parameter for the LSM method, so is *M*. In Figure 3, we plot equivalent results to those in Figure 2, but using *M* = 5 and *M* = 15, respectively. The first thing to notice is that the two plots in Figure 3 show a pattern that is very similar to that obtained when using *M* = 9 in Figure 2. For example, the LSM method is high biased when using IS pricing and low biased when using OS pricing with *M* = 5, as well as with *M* = 15. Figure 3, however, does indicate that these biases are larger for a given value of *N* when *M* = 15 than when *M* = 5. This is expected since the degree of overfitting increases with *M*. This also affects the method that averages the optimal early exercise boundaries somewhat, although our bootstrapping method is much less effected. In particular, our proposed method works very well for both *M* = 5 and *M* = 15, even when a low number of simulated paths, *N*, is used in the simulation.

**Figure 3.** Convergence of price estimates for other values of regressors, *M*. This figure shows the price estimates from *I* = 100 simulations with different numbers of paths *N*. Individually, early exercise boundaries are estimated with *M* = 5 and *M* = 15, respectively, regressors" and a constant term in the cross-sectional regressions. The option has a strike price of *K* = 40, a maturity of *T* = 1 year, and *J* = 50 early exercise points per year. The initial stock price is fixed at *S*(0) = 40; the volatility is *σ* = 20%; and the interest rate is *r* = 6%. The red lines report the results for the standard LSM method. The green lines report the results when the early exercise boundary is estimated from the average of the *I* = 100 continuation value approximations. The blue lines report the results when the early exercise boundary is backed out from our bootstrapped continuation values. The horizontal black line shows the result when the true early exercise boundary estimated with a binomial model with 50,000 steps is used.

### *3.2. Robustness across Option Characteristics*

We now demonstrate that our previous findings are robust to considering options with different moneyness, maturity, and volatility of the underlying asset. To examine this, we consider additional options with strike prices of *K* = 36 and *K* = 44, maturities of *T* = 0.5 and *T* = 2 years, and volatilities of the underlying of *σ* = 10% and *σ* = 40%. Again, we assume that the options have *J* = 50 early exercise points per year, such that the option with *T* = 0.5 years maturity has 25 early exercise points and the option with *T* = 2 years maturity has 100 early exercise points, respectively. We keep the number of regressors fixed at *M* = 9, but plot the resulting price estimates as a function of the number of simulated paths, *N*.

Figure 4 shows the performance of the three methods for an Out of The Money (OTM) option with *K* = 36 in the left hand plot and for an In The Money (ITM) option with *K* = 44 in the right hand plot. The first thing to notice is that the two plots in Figure 4 show a pattern that is very similar to that obtained for the At The Money (ATM) option with *K* = 40 in Figure 2. Figure 4 does indicate, though, that the biases in the LSM method are larger for a given value of *N* when pricing an OTM option than when pricing an ITM option. This is expected since the degree of overfitting increases as we go out of the money where less paths are used in the cross-sectional regressions. However, our proposed method continues to work very well, and much better than the regular LSM method, for both OTM and ITM options even when a low number of simulated paths, *N*, is used in the simulation.

**Figure 4.** Convergence of price estimates for other values of the strike price, *K*. This figure shows the price estimates from *I* = 100 simulations with different numbers of paths *N*. Individually, early exercise boundaries are estimated with *M* = 9 regressors and a constant term in the cross-sectional regressions. The option has a strike price of *K* = 36 and *K* = 44, respectively, a maturity of *T* = 1 year, and *J* = 50 early exercise points per year. The initial stock price is fixed at *S*(0) = 40; the volatility is *σ* = 20%; and the interest rate is *r* = 6%. The red lines report the results for the standard LSM method. The green lines report the results when the early exercise boundary is estimated from the average of the *I* = 100 continuation value approximations. The blue lines report the results when the early exercise boundary is backed out from our bootstrapped continuation values. The horizontal black line shows the result when the true early exercise boundary estimated with a binomial model with 50,000 steps is used.

Figure 5 shows the performance of the three methods for a Short Maturity (ST) option with *T* = 0.5 years to maturity and 25 early exercise points in the left hand plot and for a Long Maturity (LT) option with *T* = 2 years to maturity and 100 early exercise points in the right hand plot. The first thing to notice is again that the two plots in Figure 5 show a pattern that is very similar to that obtained for the option with a maturity of *T* = 1 year in Figure 2. Figure 5 does indicate, though, that the biases in the LSM method are larger for a given value of *N* when the maturity of the option increases. This is to be expected since errors in the approximation of the conditional expectations accumulate in the backward algorithm, and we thus expect larger accumulated errors for longer maturities. However, our proposed method continues to work very well, and much better than the regular LSM method, for both ST and LT options even when a low number of simulated paths, *N*, is used in the simulation.

Figure 6 shows the performance of the three methods for an option on an asset with a low volatility of *σ* = 10% in the left hand plot and for an option on an asset with a high volatility of *σ* = 40% in the right hand plot. Again, the first thing to notice is that the two plots in Figure 6 show a pattern that is very similar to that obtained for the option on an underlying asset with a volatility of *σ* = 20% in Figure 2. Figure 6 does indicate, though, that the biases in the LSM method are larger for a given value of *N* when the volatility of the underlying asset increases. However, our proposed method continues to work very well, and much better than the regular LSM method, for both options on low and high volatility assets even when a low number of simulated paths, *N*, is used in the simulation.

### *3.3. Robustness to the Dimensionality of the Problem*

We now demonstrate that our previous findings for options on a single asset are robust when increasing the number of underlying assets. To examine this, we consider options on three underlying assets with payoffs on the arithmetic average, the geometric average, the maximum, and the minimum of the underlying assets. We consider options that are at the money with *K* = 40 and a maturity of *T* = 0.5 years, and have *J* = 25 early exercises in total. The underlying asset prices are *Si*(0) = 40; the volatilities are *σi* = 40%, for *i* = 1, 2, 3; the correlation between all assets is *ρ* = 0.5; and the interest rate is *r* = 6%. Benchmark prices are obtained with a binomial model with 2000 steps a year. Since it is

difficult to characterize explicitly the optimal early exercise boundary for these options, we compare to the true price, and we therefore cannot take the Monte Carlo error into consideration.

**Figure 5.** Convergence of price estimates for other maturities, *T*. This figure shows the price estimates from *I* = 100 simulations with different numbers of paths *N*. Individually, early exercise boundaries are estimated with *M* = 5 and *M* = 15 regressors and a constant term in the cross-sectional regressions, respectively. The option has a strike price of *K* = 40, a maturity of *T* = 0.5 and *T* = 2 years, respectively, and *J* = 50 early exercise points per year. The initial stock price is fixed at *S*(0) = 40; the volatility is *σ* = 20%; and the interest rate is *r* = 6%. The red lines report the results for the standard LSM method. The green lines report the results when the early exercise boundary is estimated from the average of the *I* = 100 continuation value approximations. The blue lines report the results when the early exercise boundary is backed out from our bootstrapped continuation values. The horizontal black line shows the result when the true early exercise boundary estimated with a binomial model with 50,000 steps is used.

**Figure 6.** Convergence of price estimates for other values of the volatility, *σ*. This figure shows the price estimates from *I* = 100 simulations with different numbers of paths *N*. Individually early exercise boundaries are estimated with *M* = 9 regressors and a constant term in the cross-sectional regressions. The option has a strike price of *K* = 40; a maturity of *T* = 1 year; and *J* = 50 early exercise points per year. The initial stock price is fixed at *S*(0) = 40, the volatility is *σ* = 10% and *σ* = 40%, respectively, and the interest rate is *r* = 6%. The red lines report the results for the standard LSM method. The green lines report the results when the early exercise boundary is estimated from the average of the *I* = 100 continuation value approximations. The blue lines report the results when the early exercise boundary is backed out from our bootstrapped continuation values. The horizontal black line shows the result when the true early exercise boundary estimated with a binomial model with 50,000 steps is used.

Figure 7 shows the price estimates for options with the four different payoff across the number of paths used in the simulation, *N*. In all cases, the complete set of polynomials of order *M* = 9 and a

constant term are used as regressors in the cross-sectional regressions. The first thing to note is that in all cases the algorithms converge, though for options on the maximum and minimum, this is to a somewhat low biased estimate. This makes sense since the conditional expectations are more difficult to approximate for these options; see also Stentoft (2004a). More importantly, though, for all the payoffs, our proposed bootstrapping method delivers the least biased price estimates. Thus, the results clearly show that our proposed method is robust to increases in the dimension of the pricing problem.

**Figure 7.** Convergence of price estimates for options on three assets. This figure shows the price estimates from *I* = 100 simulations with different numbers of paths *N*. Individually, early exercise boundaries are estimated with the complete set of polynomials of order *M* = 9 and a constant term as regressors in the cross-sectional regressions. The option has a strike price of *K* = 40, a maturity of *T* = 0.5 year, and *J* = 50 early exercise points per year. The initial stock price is fixed at *S*(0) = 40; the volatility is *σ* = 40%; and the interest rate is *r* = 6%. The red lines report the results for the standard LSM method. The green lines report the results when the early exercise boundary is estimated from the average of the *I* = 100 continuation value approximations. The blue lines report the results when the early exercise boundary is backed out from our bootstrapped continuation values. The horizontal black line shows the benchmark price obtained with a binomial model with 2000 steps per year.

To demonstrate further our method's robustness, Figure 8 shows the corresponding price estimates when the maximum order of the complete polynomial is *M* = 15, in which case a total of 816 regressors are used in the cross-sectional regressions. It is possible that by using other regressors, e.g., functions of the maximum asset value, than the complete set of polynomials, better results could be obtained with less regressors; however, for consistency and simplicity, we chose to stay with monomials as the basis. The figure shows that when *M* is increased, the asymptotic bias, apparent when using a very large number of simulated paths *N*, decreases, and this is particularly so for the maximum and minimum

options. Note though that for all payoffs, our proposed bootstrapping method again continues to perform the best and deliver price estimates with the lowest bias of all the methods reported.

**Figure 8.** Convergence of price estimates for options on three assets, *M* = 15. This figure shows the price estimates from *I* = 100 simulations with different numbers of paths *N*. Individually, early exercise boundaries are estimated with the complete set of polynomials of order *M* = 15 and a constant term as regressors in the cross-sectional regressions. The option has a strike price of *K* = 40, a maturity of *T* = 0.5 year, and *J* = 50 early exercise points per year. The initial stock price is fixed at *S*(0) = 40; the volatility is *σ* = 40%; and the interest rate is *r* = 6%. The red lines report the results for the standard LSM method. The green lines report the results when the early exercise boundary is estimated from the average of the *I* = 100 continuation value approximations. The blue lines report the results when the early exercise boundary is backed out from our bootstrapped continuation values. The horizontal black line shows the benchmark price obtained with a binomial model with 2000 steps per year.
