**4. Discussion**

The results in Section 3 were obtained by averaging over *I* = 100 repeated samples with the same number of simulated paths, *N*, for both IS and OS pricing. In this section, we first demonstrate that it is in fact not necessary to average over such a large number of repeated samples when bootstrapping the optimal early exercise boundary. We also show that it is in fact not necessary to average over independently repeated simulations with a large number of simulated paths in the bootstrapping method either.<sup>10</sup> Finally, since the number of repeats, *IIS*, and the number of simulated paths, *NIS*, used for IS pricing can be disassociated from the number of repeats, *IOS*, and number of simulated paths, *NOS*, used for OS pricing, we propose to price options using our proposed bootstrapping method with reasonably low values for *IIS* and *NIS*, since this is enough to obtain unbiased results, and large values of, in particular, *NOS*, since this will deliver price estimates with a low variance.<sup>11</sup>

Figure 9 shows that the quality of the approximation of the estimated early exercise boundary very quickly improves, reflected in decreasing absolute bias of the estimated price, as *IIS* and *NIS* increase. In fact, if as little as *IIS* = 10 repeats and *NIS* = 10,000 simulated paths are used, the OS bias is very small, and when *IIS* = 10 repeats and *NIS* = 50,000 simulated paths are used, it is essentially eliminated. Note that estimating the optimal early exercise boundary with *IIS* = 10 repeats and *NIS* = 10,000 simulated paths can be done in roughly the same time as running the regular LSM method once with *N* = 100,000, a number typically used.<sup>12</sup>

**Figure 9.** OS bias in the bootstrapping method This figure shows the out-of-sample bias obtained using *IOS* = 100 and *NOS* = 100,000 paths when using early exercise boundaries determined with different numbers of in sample repeats, *IIS*, and paths, *NIS*, using our proposed bootstrapping method with a polynomial 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* = 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%.

To illustrate the efficiency of our method, we now report for all combinations of moneyness, maturity, and volatility, a total of 27 options, the estimated prices from implementing our bootstrapping method with *IIS IOS* and *NIS* < *NOS*. Specifically, we set *IIS* = 10 and *NIS* = 50,000 and price the options out of sample with *IOS* = 100 and *NOS* = 100,000, the standard choice in the literature. Compared to the regular LSM method, the IS results can be calculated in a fraction of the time (5% roughly) with our bootstrapping method. The results are reported in Table 2, which compares the bootstrapped results to the results that would be obtained had the true optimal early exercise boundaries been used.

<sup>10</sup> Given the results in, e.g., Figure 2, this should come as no surprise.

<sup>11</sup> Note that since the same approximations are used for all OS simulations, it does not matter if we simulate *I* = 100 times with 100,000 paths or once with *N* = 10,000,000 paths.

<sup>12</sup> Note that our proposed bootstrapping method is straightforward to implement on multiple cores, and more generally, it can be implemented on clusters to take advantage of available parallel computing resources.


**Table 2.** Option prices using bootstrapping.

This table shows estimated prices and standard deviations using our proposed bootstrapping method. The OS method is used with *IOS* = 100 and *NOS* = 100,000 with IS recursive averages calculated with *IIS* = 10 and *NIS* = 50,000 with *M* = 9 regressors, and a constant term in the cross-sectional regressions. The option characteristics are given in the first 3 columns. In all cases, the initial stock price is fixed at *S*(0) = 40; the options have *J* = 50 early exercise points per year; and the interest rate is *r* = 6%. The benchmark boundary denotes the results from a method in which the true early exercise boundary estimated from the binomial model with 50,000 steps is used in the Monte Carlo simulation. By comparing the results to the values from the benchmark boundary, the error coming from the Monte Carlo simulation is eliminated. The differences are shown in the last column.

Table 2 shows that all the estimated prices are low biased, which is expected since we are using OS pricing. However, the absolute size of the bias is indeed very small, and in all cases, the bias is statistically insignificant. Across the 27 options, the largest bias in absolute value is −0.0020, well below a cent, and the average bias across the sample of options is −0.0005. Moreover, the table shows that the standard deviations of our estimated prices are similar to what is obtained when applying the benchmark frontier. Thus, when pricing this sample of diverse and empirically relevant options, our bootstrapping method essentially yields unbiased price estimates that are as precise as if the true optimal early exercise boundary had been used. If the regular LSM method had been used instead, the corresponding biases, both the largest ones and the average across options, would have been much larger and more volatile.<sup>13</sup> Note that the option for which the bias is the largest is the long term in the money option on a high volatility underlying asset. The results in Section 3 demonstrate that this is the most challenging option to price with the regular LSM method, and in light of this, our bootstrapping method performs remarkably well.

The constant volatility Gaussian models considered above may not be adequate for empirical option pricing. Thus, to check the robustness of our results to more realistic alternatives, we now consider models with time varying volatility of the GARCH type. The work in Duan (1995) was

<sup>13</sup> These results are available from the authors upon request.

among the first to show how to price options in a (Gaussian) GARCH model, and this framework has since been widely used empirically. See Christoffersen et al. (2013) for a detailed survey of the use of GARCH option pricing models. In the GARCH option pricing model, returns under the pricing measure Q are given by:

$$R\_t = r - \frac{1}{2}h\_t + \varepsilon\_{t\_t} \tag{11}$$

where *εt* | F*<sup>t</sup>*−<sup>1</sup> ∼ *N* (0, *ht*), with F*t* denoting the information set at time *t* and where the conditional variance, *ht*, follows a NGARCH process given by:

$$
\mu h\_{t+1} = \omega + \beta h\_t + a \left( \varepsilon\_t - (\lambda + \gamma) \sqrt{h\_t} \right),
\tag{12}
$$

where *ω*, *β*, *α*, and *γ* are parameters governing the dynamics under the physical measure P and *λ* is the constant unit risk premium. The GARCH model is obtained when *γ* = 0, and the constant volatility model amounts to setting all parameters except *ω* equal to zero.

Table 3 shows the results for three different volatility specifications and thus examines the robustness of our results to using more general stochastic processes. The first thing to notice from the table is that our proposed bootstrapping method generates price estimates with insignificant biases irrespective of the dynamic model used. Thus, the table demonstrates that the excellent performance of our proposed method does not depend on the complexity of the dynamics. Using the ordinary LSM method, however, leads to significantly biased price estimates, and this is particularly so when the conditional volatility follows the more complicated NGARCH process. Our proposed method, on the other hand, continues to deliver statistically insignificant price estimates when more complicated and empirically relevant dynamics are used, and the estimates are much less biased than with the ordinary LSM method.

**Table 3.** Option prices with time varying volatility.


This table shows estimated prices and standard deviations using the individual LSM method and our proposed bootstrapping method for models with time varying volatility of the GARCH type. The OS method is used with *IOS* = 100 and *NOS* = 100,000 with the complete set of polynomials of order *MS* = 5 and *MV* = 3 in the stock price and the volatility, respectively, in the cross-sectional regressions. The individual LSM uses the same numbers in the IS method, whereas the recursive averages are obtained with *IIS* = 10 and *NIS* = 50, 000. The option strike prices are given in the first column, and all options have a maturity of *T* = 0.5 year and *J* = 252 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 following parameters were used: *β* = 0.92, *α* = 0.05, *γ* = 0.5, and *λ* = 0.1, and to ensure a risk neutral unconditional annual volatility of 20%, we set *ω* = 1 − *β* − *α* 1 + (*λ* + *γ*) 2 ∗ 0.20/ √252. Benchmark prices are calculated using *N* = 1,000,000 paths and the complete set of polynomials of order *MS* = 9 and *MV* = 5 in the stock price and the volatility, respectively, in the cross-sectional regressions.
