# Bootstrapping the Early Exercise Boundary in the Least-Squares Monte Carlo Method

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## Abstract

**:**

## 1. Introduction

## 2. Framework

#### 2.1. Simulation and Regression Methods

**Lemma**

**1**

#### 2.2. Regression and Optimal Early Exercise

#### 2.3. Bootstrapping the Early Exercise Boundary

**Corollary**

**1.**

**Proof.**

## 3. Results

#### 3.1. Robustness to the Simulation Setup

#### 3.2. Robustness across Option Characteristics

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

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | See also Boyer and Stentoft (2013) for applications of this method to longevity risk products in insurance. |

2 | See Ibanez and Zapatero (2004) for an example of a simulation method that directly approximates the optimal early exercise boundary. |

3 | We do not stress any further this difference as the literature on pricing early exercise options using simulation generally refers to these as American style options; see, e.g., Longstaff and Schwartz (2001). |

4 | This is justified when approximating elements of the ${L}^{2}$ space of square integrable functions relative to some measure. Since ${L}^{2}$ is a Hilbert space, it has a countable orthonormal basis (see, e.g., Royden 1988). |

5 | One of the important assumptions in Stentoft (2004b) is that the support is bounded. In Glasserman and Yu (2002), convergence was studied in the unbounded case. This complicates the analysis, and therefore, they limited their attention to the normal and lognormal cases. See also Gerhold (2011) for generalizations of the results to other processes and Belomestny (2011) for a proof using nonparametric local polynomial regressions. |

6 | For exotic options like stair-step options, there can be multiple intersections and multiple exercise regions. |

7 | Exercising when you should and not exercising when you should not have, on the other hand, no effect. |

8 | The one-sided p-value for a test of zero bias when compared to the estimate obtained with the benchmark boundary is $4.9\%$ and $5.5\%$ for the IS and OS prices, respectively. |

9 | All simulations were conducted using MATLAB. |

10 | Given the results in, e.g., Figure 2, this should come as no surprise. |

11 | 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. |

12 | 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. |

13 | These results are available from the authors upon request. |

**Figure 1.**Individual, average, and bootstrapped early exercise boundaries. This figure shows $I=100$ individually estimated early exercise boundaries using $N=$ 100,000 paths of simulated data with $M=9$ regressors and a constant term in the cross-sectional regressions. The right hand plot uses the proposed Initial State Dispersion (ISD) method from Rasmussen (2005). 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 $S\left(0\right)=40$; the volatility is $\sigma =20\%$; and the interest rate is $r=6\%$. The dotted blue line shows the early exercise boundary estimated from the average of the $I=100$ continuation value approximations at each time and the red line the frontier backed out from our bootstrapped recursively averaged continuation values. The dashed black line shows the true early exercise boundary estimated with a binomial model with 50,000 steps.

**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 $S\left(0\right)=40$; the volatility is $\sigma =20\%$; 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 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 $S\left(0\right)=40$; the volatility is $\sigma =20\%$; 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 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 $S\left(0\right)=40$; the volatility is $\sigma =20\%$; 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.**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 $S\left(0\right)=40$; the volatility is $\sigma =20\%$; 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, $\sigma $. 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 $S\left(0\right)=40$, the volatility is $\sigma =10\%$ and $\sigma =40\%$, 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.**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 $S\left(0\right)=40$; the volatility is $\sigma =40\%$; 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.

**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 $S\left(0\right)=40$; the volatility is $\sigma =40\%$; 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.

**Figure 9.**OS bias in the bootstrapping method This figure shows the out-of-sample bias obtained using ${I}_{OS}=100$ and ${N}_{OS}=$ 100,000 paths when using early exercise boundaries determined with different numbers of in sample repeats, ${I}_{IS}$, and paths, ${N}_{IS}$, 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 $S\left(0\right)=40$; the volatility is $\sigma =20\%$; and the interest rate is $r=6\%$.

**Table 1.**Option prices using In-Sample (IS) and Out-of-Sample (OS) methods. LSM, Least-Squares Monte Carlo.

Using IS Method | Using OS Method | Difference | |||
---|---|---|---|---|---|

Benchmark Boundary | 2.3150 | (0.0084) | 2.3137 | (0.0080) | 0.0013 |

Individual LSM | 2.3170 | (0.0087) | 2.3119 | (0.0079) | 0.0051 |

Regular Average | 2.3148 | (0.0085) | 2.3136 | (0.0079) | 0.0013 |

Recursive Average | 2.3150 | (0.0084) | 2.3138 | (0.0080) | 0.0012 |

Benchmark Boundary | Recursive Average | ||||||
---|---|---|---|---|---|---|---|

$\mathbf{K}$ | $\mathbf{T}$ | $\mathbf{\sigma}$ | Price | St. Dev. | Price | St. Dev | Difference |

36 | 0.5 | 10% | 0.0304 | (0.0006) | 0.0303 | (0.0006) | −0.0000 |

36 | 1 | 10% | 0.0896 | (0.0012) | 0.0894 | (0.0012) | −0.0001 |

36 | 2 | 10% | 0.1715 | (0.0018) | 0.1714 | (0.0018) | −0.0001 |

36 | 0.5 | 20% | 0.4974 | (0.0035) | 0.4971 | (0.0035) | −0.0003 |

36 | 1 | 20% | 0.9169 | (0.0052) | 0.9166 | (0.0052) | −0.0003 |

36 | 2 | 20% | 1.4317 | (0.0069) | 1.4311 | (0.0071) | −0.0006 |

36 | 0.5 | 40% | 2.1978 | (0.0104) | 2.1972 | (0.0102) | −0.0006 |

36 | 1 | 40% | 3.4360 | (0.0135) | 3.4355 | (0.0134) | −0.0006 |

36 | 2 | 40% | 4.9619 | (0.0168) | 4.9606 | (0.0169) | −0.0013 |

40 | 0.5 | 10% | 0.7344 | (0.0029) | 0.7343 | (0.0029) | −0.0000 |

40 | 1 | 10% | 0.8892 | (0.0031) | 0.8891 | (0.0032) | −0.0001 |

40 | 2 | 10% | 1.0235 | (0.0038) | 1.0234 | (0.0038) | −0.0001 |

40 | 0.5 | 20% | 1.7907 | (0.0072) | 1.7907 | (0.0071) | −0.0001 |

40 | 1 | 20% | 2.3137 | (0.0080) | 2.3133 | (0.0079) | −0.0005 |

40 | 2 | 20% | 2.8833 | (0.0099) | 2.8824 | (0.0099) | −0.0009 |

40 | 0.5 | 40% | 3.9708 | (0.0144) | 3.9697 | (0.0144) | −0.0010 |

40 | 1 | 40% | 5.3115 | (0.0170) | 5.3103 | (0.0168) | −0.0011 |

40 | 2 | 40% | 6.9149 | (0.0210) | 6.9131 | (0.0210) | −0.0017 |

44 | 0.5 | 10% | 3.9474 | (0.0019) | 3.9474 | (0.0019) | −0.0000 |

44 | 1 | 10% | 3.9473 | (0.0018) | 3.9472 | (0.0018) | −0.0001 |

44 | 2 | 10% | 3.9480 | (0.0020) | 3.9480 | (0.0019) | −0.0000 |

44 | 0.5 | 20% | 4.3079 | (0.0094) | 4.3078 | (0.0093) | −0.0001 |

44 | 1 | 20% | 4.6528 | (0.0098) | 4.6523 | (0.0098) | −0.0005 |

44 | 2 | 20% | 5.0804 | (0.0112) | 5.0796 | (0.0116) | −0.0008 |

44 | 0.5 | 40% | 6.3232 | (0.0176) | 6.3228 | (0.0176) | −0.0005 |

44 | 1 | 40% | 7.6107 | (0.0200) | 7.6095 | (0.0197) | −0.0011 |

44 | 2 | 40% | 9.1784 | (0.0232) | 9.1764 | (0.0233) | −0.0020 |

Panel A: Constant Volatility Model | |||||||||
---|---|---|---|---|---|---|---|---|---|

Individual LSM | Recursive Average | ||||||||

K | Price | St. Dev. | Bias | T-stat | Price | St. Dev. | Bias | T-stat | |

36 | 0.4983 | (0.0036) | −0.0011 | −2.8281 | 0.4990 | (0.0036) | −0.0004 | −1.0196 | |

40 | 1.7936 | (0.0068) | −0.0021 | −2.9162 | 1.7945 | (0.0066) | −0.0012 | −1.6947 | |

44 | 4.3158 | (0.0075) | −0.0018 | −2.2127 | 4.3171 | (0.0073) | −0.0005 | −0.6237 | |

Panel B: GARCH Model | |||||||||

Individual LSM | Recursive Average | ||||||||

K | Price | St. Dev. | Bias | T-stat | Price | St. Dev. | Bias | T-stat | |

36 | 0.5060 | (0.0037) | −0.0021 | −5.4783 | 0.5076 | (0.0037) | −0.0005 | −1.3202 | |

40 | 1.7725 | (0.0068) | −0.0031 | −4.3681 | 1.7747 | (0.0069) | −0.0009 | −1.2259 | |

44 | 4.2844 | (0.0074) | −0.0030 | −3.7376 | 4.2873 | (0.0074) | −0.0001 | −0.1330 | |

Panel C: NGARCH model | |||||||||

Individual LSM | Recursive Average | ||||||||

K | Price | St. Dev. | Bias | T-stat | Price | St. Dev. | Bias | T-stat | |

36 | 0.5822 | (0.0045) | −0.0026 | −5.4823 | 0.5842 | (0.0044) | −0.0005 | −1.1326 | |

40 | 1.7431 | (0.0072) | −0.0035 | −4.6303 | 1.7454 | (0.0071) | −0.0012 | −1.5857 | |

44 | 4.1799 | (0.0072) | −0.0036 | −4.7237 | 4.1826 | (0.0072) | −0.0010 | −1.2543 |

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## Share and Cite

**MDPI and ACS Style**

Létourneau, P.; Stentoft, L.
Bootstrapping the Early Exercise Boundary in the Least-Squares Monte Carlo Method. *J. Risk Financial Manag.* **2019**, *12*, 190.
https://doi.org/10.3390/jrfm12040190

**AMA Style**

Létourneau P, Stentoft L.
Bootstrapping the Early Exercise Boundary in the Least-Squares Monte Carlo Method. *Journal of Risk and Financial Management*. 2019; 12(4):190.
https://doi.org/10.3390/jrfm12040190

**Chicago/Turabian Style**

Létourneau, Pascal, and Lars Stentoft.
2019. "Bootstrapping the Early Exercise Boundary in the Least-Squares Monte Carlo Method" *Journal of Risk and Financial Management* 12, no. 4: 190.
https://doi.org/10.3390/jrfm12040190