**4. Results**

The motivating example in Section 2 clearly demonstrated that there might be significant value to pricing call options as put options using symmetry properties when using the LSMC algorithm. To test this further, we now price a large sample of call options with five different strike prices, *K* = [90, 95, 100, 105, <sup>110</sup>], maturities, *T* = [0.5, 1, 2, 3, 5] years, interest rates, *r* = [0.0%, 2.5%, 5.0%, 7.5%, 10%], dividend yields, *d* = [0.0%, 2.5%, 5.0%, 7.5%, 10%], and volatilities, *σ* = [10%, 20%, 30%, 40%, 50%], for a total of 5 × 5 × 5 × 5 × 5 = 3, 125 options. This sample arguably

<sup>5</sup> For now, we maintain the assumption that dynamics are governed by simple geometric Brownian motion. However, our results generalize to other models for which PCS holds, as we demonstrate in Section 5.

<sup>6</sup> We deal with the difference in, for example, the payoff when exercising the option by using negative values for the strike price and the stock prices for put options since *ZCall*(*<sup>S</sup>*, *K*) = *max*(*<sup>S</sup>* − *K*, 0) = *max*((−*<sup>K</sup>*) − (−*<sup>S</sup>*), 0) = *ZPut*(−*S*, <sup>−</sup>*<sup>K</sup>*).

<sup>7</sup> Unreported results, available upon request, show that this generalizes to the much larger sample of options we consider in Section 4.

<sup>8</sup> This follows from the Weierstrass approximation theorem, which states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function.

spans most of the important cases one would come across in real-life applications of option pricing. We first consider performance for various numbers of exercise possibilities *J* and across option characteristics, i.e., *K* and *T*. Next, we consider performance across model parameters, i.e., *r*, *d* and *σ*. Benchmark values are from the Cox et al. (1979) binomial model with 25,000 steps and *J* early exercise possibilities.

In this section, we use a slightly different setup for the LSMC algorithm in that we use monomials as regressors and we use simple "plain vanilla" Monte Carlo simulation. We choose monomials instead of Laguerre polynomials because they are simpler and faster to use. We choose a plain Monte Carlo simulation without any variance reduction techniques such that our results are not potentially dependent on a particular variance reduction method. We again report results with the LSMC method using *I* = 100 independent simulations with *N* = 100,000 paths and *L* = 3 regressors. We assess model performance using the *Bias* = *I*−<sup>1</sup> ∑*Ii*=<sup>1</sup> *P*ˆ*i* − *P* and *RMSE* = *<sup>I</sup>*−<sup>1</sup> ∑*Ii*=<sup>1</sup> *P*ˆ*i* − *P*2 error metrics where *P* ˆ *i* is the *i*th simulated estimate of the price *P*. Since we cannot report all the individual errors, we report average errors instead. We also consider the fraction of options for which the regular and symmetric method have the highest bias and RMSE, respectively.<sup>9</sup>

### *4.1. Performance across Option Characteristics*

Table 2 reports results for our benchmark implementation of the LSMC method for the large sample of call options considering different numbers of total early exercises, *J*, constant across the maturity, from *J* = 10 to *J* = 200 (a close approximation to the continuously-exercisable American option) and across different strike prices, *K*, and different maturities, *T*, for options with *J* = 50 early exercises. Figure 1 plots the relative performance of the symmetric method compared to the regular method for the four aggregate error metrics across these three dimensions.

Panel A of Table 2 first of all shows that using the regular method for this sample of options leads to significantly low biased price estimates. For example, when considering the case with *J* = 50 exercise times, the average bias is almost six cents with this method, whereas it is less than a cent if the symmetric method is used leading to an average improvement of 1 − |−0.0574/ − 0.0022| = 96.22%. The improvement in performance with the symmetric approach is large also for the RMSE. Moreover, the improvement in performance is not only large on average, but also across most of the options, as the counting metrics show. In particular, improvements occur for 83.90% and 88.19% of the options in terms of bias and the RMSE, respectively. Figure 1a shows that the improved performance is not limited to a particular choice of the number of early exercise possibilities, *J*, and improvements are found for all the reported values of *J*. Across the number of early exercises, the figure does indicate that the relative performance improves rapidly with *J* when there are only a small number of exercise possibilities. Once *J* reaches 50 or 100, the effect in terms or RMSE tapers off, and the relative improvement in performance does not change much when increasing the number of early exercise points further.<sup>10</sup>

Panel B of Table 2 shows the results across moneyness and demonstrates that the absolute errors of both methods decrease when the strike price increases. In terms of the counting metrics the panel shows that the symmetric method has the smallest errors for 98.4% of the out of the money options. For in the money options, where determining the early exercise strategy is of less importance, the improvement is relatively smaller, though the symmetric method continues to yield more precise price estimates on average and for the majority of the options. In relative terms, Figure 1b shows that the symmetric method performs better than the regular method across all strike prices. The relative

<sup>9</sup> Using the fraction of times a given method has the highest error metric ensures, as is the case with the bias and RMSE error metrics, that lower numbers are better.

<sup>10</sup> Note, though, that, e.g., the bias of both the regular and symmetric method increases in absolute terms when increasing the number of exercise points. This is likely related to the fact that dependence is introduced between the paths in the LSMC method because of the cross-sectional regression, and this dependence "accumulates" as we go backwards in time in the algorithm and becomes more and more important as the number of early exercise possibilities increases.

performance is best for options with low strike prices, i.e., call options that are out of the money, and for these options, the relative performance in terms of the counting metrics is quite extraordinary.

Finally, Panel C of Table 2 shows the results across maturity and documents clear and significant improvements in the relative performance of the symmetric method for all metrics when maturity increases. It is noteworthy that the symmetric method actually leads to more precise, in terms of RMSE, estimates for all subcategories. In terms of the counting metrics, the symmetric method also largely outperforms the regular method and leads to prices estimated with smaller errors in at least 72.16% of the cases, the worst relative performance being for the shortest maturity options. In relative terms, Figure 1c shows that the symmetric method performs better than the regular method across all maturities and error metrics. In fact, for the majority of the categories, that is for options with maturity of *T* = 2 years or more, the symmetric method leads to lower RMSE for at least nine out of 10 of the options. Thus, the results from Section 2 hold true in general for a much larger sample of options.


**Table 2.** Pricing errors across option characteristics.

This table shows pricing errors for the regular and symmetric method for various numbers of exercise possibilities, *J*, strike prices, *K*, and maturities, *T*. Results are based on *I* = 100 independent simulations with *N* = 100,000 paths and *L* = 3 regressors. In each panel, we report results for the bias and RMSE in terms of the average metrics and counting metrics, i.e., the fraction of times a given method has the highest error metric. Panels B and C use results for *J* = 50 early exercise points only.

*J. Risk Financial Manag.* **2019**, *12*, 59

 Relative performance across *J*

 Relative performance across *K*

(**c**) Relative performance across *T*

**Figure 1.** Relative pricing performance across option characteristics. This figure plots the relative performance of the symmetric method compared to the regular method across the number of early exercises, *J*, strike price, *K*, and maturity, *T*. Panels B and C use results for *J* = 50 early exercise points only.

### *4.2. Performance across Model Parameters*

We now consider the relative performance of the two methods for some interesting subgroups of model parameters like the interest rate *r*, the dividend yield *d* and the volatility of the underlying asset *σ*. Table 3 shows the results across interest rate, *r*, dividend yield, *d*, and volatility, *σ*, for our large sample of options. Figure 2 plots the relative performance of the symmetric method compared to the regular method across these three dimensions.

**Table 3.** Pricing errors across model characteristics.


30%

40%

50% −0.0382

−0.0694

−0.1598

 0.0411

 0.0716

 0.1652


**Table 3.** *Cont*.

This table shows pricing errors for the regular and symmetric method for various numbers of interest rates *r*, dividend yields, *d*, and volatility level, *σ*. Results are based on *I* = 100 independent simulations with *N* = 100,000 paths and *L* = 3 regressors. In each panel, we report results for the bias and RMSE in terms of the average metrics and counting metrics, i.e., the fraction of times a given method has the highest error metric.

 0.0114 0.8848

 0.0143 0.9328

 0.0161 0.9728  0.9152 0.1152

 0.9552 0.0672

 0.9776 0.0272  0.0848

 0.0448

 0.0224

−0.0025

−0.0037

−0.0047

Figure 2a,b along with Panels A and B of Table 3 show that the relative improvement from using the symmetric method is large across all the possible values of interest rates and dividend yields. This holds both in terms of the average metrics and in terms of the number of options for which the symmetric method has the smallest error. In terms of absolute errors, the relative performance of the symmetric method decreases slightly when the interest rate increases, though the method produces estimates with errors that are very small and never above one third of the errors obtained with the regular method. When the dividend yield increases, the relative performance of the symmetric method increases somewhat. Note that the case with *d* = 0 is special since in this situation, the American call option should never be exercised early.

Figure 2c and Panel C of Table 3 document clear and significant improvements in the relative performance of the symmetric method for all metrics in absolute, as well as relative terms when volatility increases. It is noteworthy that the symmetric method actually leads to more precise, in terms of RMSE, estimates for all subcategories. In terms of the counting metrics, the symmetric method also largely outperforms the regular method and leads to price estimates with smaller errors in at least 64.64% of the cases, the worst relative performance being for the lowest volatility options. For the majority of the categories, that is for options with volatility of *σ* = 0.30 or more, the symmetric method leads to lower RMSE for at least nine out of 10 of the options. Thus, the results from Section 2 hold true in general for a much larger sample of options.

*J. Risk Financial Manag.* **2019**, *12*, 59

 across

 across

(**c**) Relative performance across *σ*

**Figure 2.** Relative pricing performance across model characteristics. This figure plots the relative performance of the symmetric method compared to the regular method across interest rates, *r*, dividend yields, *d*, and volatility levels, *σ*.
