**5. Robustness**

The previous section provides strong evidence in favour of using symmetric pricing for call options. In this section, we examine the robustness of these results along two dimensions. We first examine the importance of the choice of the number of paths, *N*, and the number of regressors, *L*, used in the Monte Carlo simulation and whether or not our reported results are robust to using so-called out-of-sample pricing. Next, we examine the robustness of our results to using alternative option pricing models. Here, we consider the case with multiple underlying assets and the case in which the underlying asset follows the stochastic volatility model of Heston (1993).

### *5.1. Alternative Choices for the Number of Paths and Regressors*

When implementing the LSMC method, one needs to choose the number of paths to simulate, *N*, and the number of regressors, *L*, to use in the cross-sectional regressions. While it is well known that the estimated prices converge to the true price when both *N* and *L* tend to infinity (see, e.g., Stentoft (2004b)), any real application involves choosing a finite number of paths and regressors. Table 4 shows the results across the number of simulated paths, *N*, and the number of regressors, *L*, for our large sample of options. Figure 3 plots the relative performance of the symmetric method compared to the regular method across these two dimensions.


**Table 4.** Pricing errors across algorithm characteristics.

This table shows pricing errors for the regular and symmetric method for various numbers of simulated paths *N* and number of regressors *L*. Results are based on *I* = 100 independent simulations. 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.

(**b**) Relative performance across *L*

**Figure 3.** Relative pricing performance across algorithm characteristics. This figure plots the relative performance of the symmetric method compared to the regular method across the number of simulated paths, *N*, and number of regressors, *L*.

Panel A of Table 4 shows the results when increasing the number of simulated paths from *N* = 20,000 to *N* = 500,000, while keeping the number of regressors fixed at *L* = 3. In this case, we know that the methods converge to a low estimate of the true value, one that is based on using a rather rough approximation of the conditional expectation function used to determine the optimal early exercise. The table confirms this numerically in that the bias for both methods, regular as well as symmetric, becomes more negative with increasing *N*. Note also that the regular method always yields price estimates with a low bias on average even when using as low as *N* = 20,000 paths, whereas the symmetric method yields high biased estimates for low *N*.

When comparing the two methods, it is noteworthy, though, that in all cases, the absolute bias and the RMSE is lowest with the symmetric method, and this method consistently outperforms the regular method across all choices of *N*, as can be seen from Figure 3a. When it comes to the number of times the symmetric method has lower errors, a pattern very similar to what was seen when increasing *T* or *σ* is found.

Panel B of Table 4 shows the results when increasing the number of regressors from *L* = 2 to *L* = 15, while keeping the number of simulated paths fixed at *N* = 100,000. In this case, we know that, everything else equal, the estimated prices should increase as the approximation gets better and better, although this may eventually result in a high bias because of over fitting the function on a finite number of simulated paths. The table confirms this numerically in that the bias for both methods, regular as well as symmetric, becomes more positive with increasing *L*. The change is most dramatic for the regular method, which goes from having an average negative bias of close to eight cents to having an average positive bias of more than six cents.

When comparing the two methods, the table shows that the symmetric method almost always provides estimates with smaller errors than does the regular method across the choice of *L*. The exception to this is when using *L* = 5 and looking at the average bias, where the absolute value from using the regular method is half that of using the symmetric method. When it comes to the number of times the symmetric method has lower errors, a pattern very similar to what was observed previously is found. The main difference is that, for the first configuration of all the ones considered up to this point, a case occurs where the regular method on average provides estimates that are better in terms of the RMSE. Unsurprisingly, this happens when *L* = 5 for which 60.99% of the estimated regular prices have smaller errors.

When looking at Panel B of the table, it is noteworthy that the performance of the symmetric method is better for a small, i.e., *L* ≤ 3, or a large, i.e., *L* ≥ 9, choice of regressors. A similar, though less pronounced, non-linear relationship is found in Panel A of the table when the number of simulated paths is increased. Given this concave relationship, as a function of *L*, and convex relationship, as a function of *N*, in the relative performance, it is indeed possible that one could find a combination of *L* and *N* for which the regular method would outperform the symmetric method for our large sample of options. This though would be largely due to luck (or would require one to consider a large number of possible combinations) and as such is not of much help or relevance.

One reason that the choice of polynomial is important is that the LSMC method mixes two types of biases: a low bias due to having to approximate the optimal stopping time with a finite degree polynomial and a high bias stemming from using the same paths to determine the optimal early exercise strategy and to price the option, potentially leading to over fitting to the simulated paths. As a result of this, the bias just happens to be somewhat smaller without symmetry, a value of −0.006, then when using symmetry, a value of 0.010 when using *L* = 5 regressors with *N* = 100,000 paths.<sup>11</sup> One easy way to control the sign of the bias is to conduct so-called out-of-sample pricing in which a new set of simulated paths is used to price the option instead of using the same set of paths that were used for determining the optimal early exercise strategy. Table 5 shows the results for the different configurations of *N* and *L* and, as expected and in line with the theory, shows that the bias of the estimates from the regular, as well as the symmetric method is negative, i.e., the estimates are low biased.

Compared to Table 4, Panel A of Table 5 shows that when using out-of-sample pricing, the estimated prices with the regular method improve significantly when the number of simulated paths, *N*, increases. The estimates from the symmetric method, however, are much less affected by the number of paths used. The reason for this is related to the over fitting and large variance in the

<sup>11</sup> Again, for all other values of the number of simulated paths with this number of regressors and when using other numbers of regressors with this number of simulated paths, the symmetric estimates are less biased.

estimates in the cross-sectional regressions with the regular method, which for a given choice of *L*, becomes less of an issue with increasing *N*. When using symmetric pricing, this is much less of an issue since the regressors are bounded. Compared to Figure 3, Figure 4 shows that when using out-of-sample pricing, the relative performance of the symmetric method is much less dependent on the choice of *N* and *L*. In particular, the symmetric method now improves significantly on the regular method irrespective of the choice of *L*. Panel B of Table 5 shows that once *L* = 3 or more regressors are used, the RMSE of the symmetric method is around 20% of the RMSE of the regular method. In terms of the number of times the symmetric method leads to the smallest RMSE, this is around 90% or more for all values of *L*.


**Table 5.** Pricing errors across algorithm characteristics using out-of-sample pricing.

This table shows pricing errors for the regular and symmetric method using out-of-sample pricing for various numbers of simulated paths *N* and number of regressors *L*. Results are based on *I* = 100 independent simulations. 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.

(**a**) Relative performance across *N* using out-of-sample pricing (**b**) Relative performance across *L* using out-of-sample pricing

**Figure 4.** Relative pricing performance across algorithm characteristics using out-of-sample pricing. This figure plots the relative performance of the symmetric method compared to the regular method using out-of-sample pricing across the number of simulated paths, *N*, and number of regressors, *L*.

### *5.2. Extensions to Other Option Pricing Models*

Until now, we have presented results for the simple Black–Scholes–Merton setup. The reason for this was obvious: we wanted to have fast and precise benchmark results available. Without these, it makes no sense to talk about one method being more efficient than another since we measure efficiency by loss functions such as the RMSE for which a benchmark is required. However, we have argued that since our results rely on nothing but simulation and regression, our conclusions should be valid for other settings in terms of asset dynamics and option payoffs for which PCS holds. We now consider two obvious alternatives and demonstrate that our previous conclusions indeed continue to hold. We first provide results when the option payoffs depend on the average of several assets in a multivariate model, and second, we consider the case where the asset dynamics are instead given by the stochastic volatility (SV) model of Heston (1993).

The case with options written on multiple assets is the most obvious generalization of the standard constant volatility case. Options can be written on the maximum, minimum or average of multiple assets. These types of payoff functions have been used widely in the literature, and the work in Boyle and Tse (1990) gave examples on where these types of options are traded. The work in Stentoft (2004a) demonstrated that as the dimension of the problem increases, simulation-based methods like the LSMC are the most efficient methods to use for pricing. While put-call symmetry properties have been established in several cases (see for example Detemple (2001)), the most clean-cut case occurs with options written on the geometric average, i.e., options for which the payoff is given by:

$$\mathcal{G} = \max\left(0, \left(\prod\_{l=m}^{M} S^m\right)^{\frac{1}{M}} - K\right),\tag{8}$$

where *Sm*, *m* = 1, .., *M* are the prices of the underlying assets, *M* being the dimension, and *K* is the strike price as before. The reason that this case is a "clean-cut" example is that since the product of lognormals is lognormal, the pricing problem essentially reduces to that of pricing single asset options on an asset that follows a (particular and slightly non-standard) GBM.<sup>12</sup>

In the LSMC method, we again consider the complete set of polynomials of order *L* = 3 or less and therefore use a total of 10, 20 and 56 regressors when the dimension of the problem, *M*, is 2, 3 and 5, respectively. In all cases, we use *I* = 100 independent simulations with *N* = 100,000 simulated paths. We consider options with different features and different asset dynamics. In particular, we consider three values of the strike price, *K* = [95, 100, <sup>105</sup>], the time to maturity, *T* = [0.5, 1, 2] years, the volatility, *σ* = [10%, 20%, 40%], the correlation between the assets, *ρ* = [0.25, 0.50, 0.75], and the number of assets, *M* = [2, 3, 5], for a total of 35 = 243 options. Pricing options in multiple dimensions quickly become computationally complex, and for this reason, we consider a smaller sample of options than in the benchmark case. Overall, our results show that the average RMSE obtained with the symmetric method is 36% smaller than that obtained with the regular method, and the RMSE is smallest for 87% of the individual options when using the symmetric method. These results clearly show that symmetry is valuable also for more advanced multivariate models.

Figure 5a and Panel A of Table 6 show the results across the number of assets, *M*, and demonstrate clearly that our suggestion of pricing call options as put options becomes more important as the dimension, and hence the computational complexity, of the option increases. In particular, the relative error of the symmetric method in terms of RMSE in Figure 5a goes from 0.890–0.545 as the dimension increases from 2–5. Moreover, when the dimension of the problem is high, the symmetric method almost always, in 96.3% of the cases, has the lowest RMSE, as shown in Panel A of Table 6.

<sup>12</sup> The working version of this paper contains the full details on how to derive these dynamics.

**Figure 5.** Relative pricing performance in a multivariate model. This figure plots the relative performance of the symmetric method compared to the regular method across the mean reversion rate *κ*, long-term variance, *θ*, initial volatility, *V*0, and correlation, *ρ*.


**Table 6.** Pricing errors across model characteristics in a multivariate model.

This table shows pricing errors for the regular and symmetric method in a multivariate model for various values of the number of stocks *M* and correlation *ρ*. Results are based on *I* = 100 independent simulations with *N* = 100,000 paths. 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.

Figure 5b and Panel B of Table 6 show the results across the correlation between the assets, *ρ*. From the figure, it is seen that symmetry becomes more important when correlations between assets increase. In particular, in terms of the absolute metrics, the regular method performs on par with the symmetric method when the correlation is low and *ρ* = 0.25, but when the correlation is high and *ρ* = 0.75, the error in pricing of the symmetric method is only around half that of the regular method, as shown in Figure 5b. Moreover, the symmetric method is always the method that yields the smallest errors for the largest fraction of the options. For example, when correlations are high among the underlying assets, the symmetric method has larger RMSE for only 4.94% of the options as shown in Panel B of Table 6.

The SV model of Heston (1993) is one of the most famous extensions to the constant volatility model. While it may not have been the first SV model, e.g., earlier examples include Hull and White (1987), Scott (1987) and Wiggins (1987), this particular model has emerged as the most important one and now serves as a benchmark against which many other SV models are compared. In the model of Heston (1993), the variance followed a Cox et al. (1985) process specified as:

$$d\upsilon\_t = \kappa(\theta - \upsilon\_t)dt + \sigma\sqrt{\upsilon\_t}dW\_{2,t}^Q. \tag{9}$$

Here, *κ* represents the mean reversion rate of the variance, *θ* is the long-term variance, *σ* is the volatility of volatility (vol of vol) and the stock dynamics and variance are allowed to be correlated with correlation coefficient *ρ*. Put-call symmetry also holds in this model as demonstrated by, e.g., Battauz et al. (2014), which also listed the appropriate changes of parameters. In particular, the paper shows that in the model of Heston (1993), the following parity holds between a call option and a put option:

$$\mathbb{C}(\mathbb{S}, r, q, \mathbb{K}, V\_0, \theta, \mathbb{x}, \sigma, \rho) = \mathbb{P}(\mathbb{K}, q, r, \mathbb{S}, V\_0, \mathbb{x}\theta / (\mathbb{x} - \sigma \rho), \mathbb{x} - \sigma \rho, \sigma, -\rho). \tag{10}$$

In other words, a call option can be priced as a put option in which the stock price and the strike price and the interest rate and the dividend yield are interchanged, as was the case in the constant volatility setting, and in which the mean reversion, *<sup>κ</sup>* , the long-term variance, *θ* , and the correlation, *ρ* , for the symmetric put option are changed to:

$$\mathbf{x}' = \mathbf{x} - \sigma \boldsymbol{\rho}, \; \boldsymbol{\theta}' = \boldsymbol{\theta} \mathbf{x}/\mathbf{x}', \; \text{and} \; \boldsymbol{\rho}' = -\boldsymbol{\rho}, \tag{11}$$

where *κ*, *θ* and *ρ* are the actual mean reversion, long-term variance and correlation.

We again price each of the individual options *I* = 100-times with independently-simulated state variables used in the LSMC method, which we implemented using *N* = 100,000 paths. In the cross-sectional regressions, we use the complete set of polynomials in two dimensions of order *L* = 3 or less as regressors for a total of 10 regressors including the constant term. We consider three different values of the strike price, *K* = [90, 100, <sup>110</sup>], the time to maturity, *T* = [0.5, 1, 2] years, the mean reversion rate, *κ* = [1, 3, 5], the long-term variance, *θ* = [0.01, 0.04, 0.16], the initial level of the variance, *V*0 = [0.01, 0.04, 0.16], the volatility of volatility, *σ* = [0.05, 0.1, 0.2], and the correlation between the two Brownian motions, *ρ* = [−0.5, 0, 0.5], for a total of 37 = 2187 options. Overall, our results show that the average RMSE obtained with the symmetric method is 14% smaller than that obtained with the regular method, and the RMSE is smallest for 61% of the individual options when using the symmetric method. These results clearly show that symmetry is valuable also for more advanced models, and though the results are somewhat closer to each other than with our benchmark model, this is to be expected since we are considering options here that have on average shorter maturities.

Figure 6 and Table 7 show the results across the various values of the mean reversion rate, *κ*, the long-term variance, *θ*, the initial level of the variance, *V*0, and the correlation, *ρ*. The first thing to notice from the table is that across all these interesting parameters and parameter values, the symmetric method most of the time outperforms the regular approach. The two exceptions to this are options with very low long-term variance and *θ* = 0.01 and when the initial level of the variance is very high and *V*0 = 0.16, where the RMSE is slightly lower with the regular method than with the symmetric method. In both of these cases, the symmetric method also has larger RMSE for more options than does the regular method, whereas in all the other cases, the symmetric method most often has the smallest RMSE.

In terms of performance across parameters, Figure 6 and Table 7 show that the symmetric method performs relatively better the faster the volatility mean reverts, i.e., the larger the value of *κ* in Figure 6a, and the larger the value of the long-term variance, i.e., the larger the value of *θ* in Figure 6b. The last of these findings is completely in line with the results from our benchmark model where symmetric pricing was most important for high values of the asset volatility. Figure 6d shows that the symmetric method performs relatively better when correlations are negative, which is indeed the empirically most relevant case in the stochastic volatility model of Heston (1993). When the correlation is highly negative and *ρ* = −0.5, the symmetric method has errors in terms of bias and RMSE that are around 25% lower than the regular method and has the largest errors for only around one third of the individual options. The effect on the relative performance of the initial variance, *V*0, is less clear cut though Figure 6c does indicate that the symmetric method performs relatively better when *V*0 is not too extreme, and in particular not too high.


**Table 7.** Pricing errors across model characteristics in a stochastic volatility model.

 0.5281 0.5322

 0.4719

 0.0100 0.4678

0.5 0.0036  0.0105 0.0081

This table shows pricing errors for the regular and symmetric method in a stochastic volatility model for various values of the mean reversion rate, *κ*, long-term variance, *θ*, initial volatility, *V*0, and correlation, *ρ*. Results are based on *I* = 100 independent simulations with *N* = 100,000 paths. 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.

**Figure 6.** Relative pricing performance in a stochastic volatility model. This figure plots the relative performance of the symmetric method compared to the regular method across the mean reversion rate *κ*, long-term variance, *θ*, initial volatility, *V*0, and correlation, *ρ*.
