**6. Discussion**

The fact that pricing call options using the symmetry method works best for most and along some dimensions almost all of the options considered is grea<sup>t</sup> news. However, since it does not perform the best for all the options, it leaves the obvious question of when to choose one method over the other. As it is, the only solid recommendations that arise from Sections 4 and 5 are that using the symmetric method with standard choices of the number of paths and number of regressors used in the LSMC method is relatively better the longer the maturity and the larger the volatility and that the methods become more similar when simulating a very large number of paths, e.g., when *N* is as large as 500,000, and that they diverge when using a large number of regressors, e.g., when *L* is as large as 15. In this section, we first examine the performance of the individual methods in terms of a relative efficiency measure, which compares the performance of a method to what could have been obtained optimally. Then, using properties of the out-of-sample method for pricing, we propose a method for selecting which specification to use across the methods and the number of regressors, which is simple to implement and achieves a very high degree of efficiency.

### *6.1. Efficiency as an Alternative Metric*

Until know, we have compared performance metrics, i.e., the RMSE or the number of times a method works the best or worst, for the regular and symmetric pricing methods, respectively. An alternative and perhaps more interesting metric for "practitioners" is what one stands to lose in terms of increased pricing errors by picking and sticking to one particular method instead of using the optimal method for a given individual option in our sample. To examine this, we now consider a metric, which we will refer to as the "efficiency" given by the ratio of a specific method's RMSE to the optimal and infeasible RMSE that could be obtained if one knew which method to use for each of the individual options.

Table 8 shows the efficiency of the two methods using in-sample pricing for various values of *N* and *L* in Columns 5 and 6. For comparison, the fraction of the options for which a particular model performs the best in terms of having the lowest RMSE is also reported in Columns 7 and 8.<sup>13</sup> Panel A of Table 8 clearly shows that the symmetric method performs extremely well across the number of simulated paths, and one would never lose more than 7% from using this method. In fact, for most realistic specifications, i.e., when *N* ≥ 100,000, the loss is less than 2%. The regular method, on the other hand, often has an efficiency of just around 20%, meaning that if this method was used to price the sample of options, one would lose around 80% compared to what could optimally be obtained.


**Table 8.** Efficiency across the number of paths *N* and regressors *L*.

This table shows the efficiency across the number of paths, *N*, and regressors, *L*. Efficiency is measured as the optimal RMSE, conditional on knowing which of the two methods yields the lowest RMSE, as a fraction of the RMSE of the regular or symmetric method, respectively. Panel A reports results for different values of the number of simulated paths, *N*, and Panel B reports results for different values of the number of regressors, *L*. In addition to the efficiency, the table also reports the fraction of times for which the regular and symmetric method provide the lowest RMSE, respectively.

Panel B of the table is, given the results in the previous section on robustness, even more interesting. In particular, the previous results showed that for some specification, i.e., when picking *L* = 5, the symmetric method actually has larger RMSE than the regular method for most options. The row labelled *L* = 5 in Table 8, however, shows that even in this case where the symmetric RMSE is the lowest for only 39% of the options, the method's efficiency is above 84%. That is, even for settings when the regular method is the best, measured by minimizing the RMSE, for 61% of the options when

<sup>13</sup> These numbers are the "inverse" of the counting metrics used in previous tables.

using the symmetric method, you would not lose more than 16% compared to what could be optimally obtained had you known what would be the best method to use for the individual options. It is also striking that if you, on the other hand, would use the regular method for all options, the efficiency is only around 49% in spite of the fact that this is the method that has the lowest RMSE for most of the options.

Table 9 shows the efficiency of the two methods using out-of-sample pricing for various values of *N* and *L* in Columns 5 and 6. The first thing to notice form this table is that when using out-of-sample pricing, i.e., when a new set of paths is used for pricing, the efficiency of the symmetric method is extremely high, and often above 99%, across both the choice of the number of paths, *N*, and the number of regressors, *L*. Compared to the in-sample results in Table 8, the efficiency of the symmetric method is most of the time improved, the exception being when using *N* = 200,000 paths in the simulation. For the regular method, on the other hand, efficiency is generally much lower, as low as 15%, and does not improve in any systematic way when using out-of-sample pricing. In conclusion, although the symmetric method is not always the model that has the smallest RMSE, the efficiency of this method is generally very high, always significantly higher than that of the corresponding regular method, and therefore, the costs of using this method are always reasonably low. Our suggestion is therefore very naturally to use the symmetric method for call option pricing.


**Table 9.** Efficiency across number of paths *N* and regressors *L* using OSpricing.

This table shows the efficiency across the number of paths, *N*, and regressors, *L*, using out-of-sample pricing. Efficiency is measured as the optimal RMSE, conditional on knowing which of the two methods yields the lowest RMSE, as a fraction of the RMSE of the regular or symmetric method, respectively. Panel A reports results for different values of the number of simulated paths, *N*, and Panel B reports results for different values of the number of regressors, *L*. In addition to the efficiency, we also report the fraction of times for which the regular and symmetric method provide the lowest RMSE, respectively.

### *6.2. Picking the Best Configuration*

Although we recommend to always use the symmetric method for call option pricing, you may still wonder if it is possible to improve on this recommendation, i.e., if it is possible to pick the "right" model using some "observables". This is essentially a question of classification. A straightforward classification variable is the estimated price. In particular, we know that when using the out-of-sample pricing technique, estimates are in expectation low biased. Moreover, while we expect the estimates to increase when increasing the number of regressors, *L*, initially, as this improves the polynomial approximation, when *L* becomes very large and over fitting to the paths used to determine the optimal exercise strategy becomes a problem, the estimated out-of-sample price could decrease. When comparing results for several different values of *L* and different methods, i.e., regular versus symmetric,

one could therefore propose to choose the method that maximizes the out-of-sample price. In particular, this should result in picking the method that has the smallest bias, and this would potentially also be the method with a small RMSE. The results from implementing this classification strategy are shown in Table 10.


**Table 10.** Efficiency across the number of regressors *L* using OS pricing.

This table shows the relative efficiency of the two methods using out-of-sample pricing across the number of regressors, *L*. Efficiency is measured as the optimal RMSE, conditional on knowing which of the methods yields the lowest RMSE for a given individual option in our large sample of 3125 options, as a fraction of the RMSE of the regular and symmetric method, respectively. Panel A report results for different values of the number of regressors used in the cross-section regression, *L*, and Panel B reports results across all the values of *L* in Panel A. The result in the column headed "Optimal" corresponds to selecting the method with the minimum RMSE either for a given *L*, in Panel A, or across all values of *L*, in Panel B. The results in the column headed "Classified" correspond to what would be obtained if the method with the highest price is used, either for a given *L* or across all values of *L* and reports the resulting RMSE, the fraction of options for which the method with the lowest RMSE was actually picked and the efficiency of this method compared to the corresponding optimal method. In Panel A, "Local Efficiency" is measured relative to the optimal RMSE for a given value of *L*. In Panel B, "Local Efficiency" is measured relative to the minimum RMSE obtained for the regular and symmetric method, respectively, across all values of *L*. In both panels, results in the column headed "Global Efficiency" report the efficiency of the two methods compared to the best possible RMSE of 0.0073 reported in Panel B.

Panel A in Table 10 reports results for individual values of *L*, i.e., when the method, regular or symmetric, that has the highest price for a given value of *L* is picked. The first thing to notice from this panel is that the right method for a given option is picked at least 93% of the time, and the efficiency of this method is always above 99%. The symmetric method clearly performs the best on average for all values of *L*, and this method does have a very high "local" efficiency, that is compared to the optimal RMSE for a particular value of *L*. The regular method, on the other hand, has a much lower efficiency. Compared to the efficiency of the individual methods, the panel shows that classification according to maximum price does improve on the RMSE in all but one case. In terms of "global" efficiency though, the performance of the methods varies greatly across *L*.

Panel B in Table 10 reports results across all the values of *L* used in Panel A, i.e., in this panel, the optimal RMSE is picked across both methods and the values of *L*. The first thing to notice from this panel is that the classification method performs very well and picks the right method more than 90% of the time and has a very high efficiency of close to 99%. Picking the symmetric method that has the highest price across *L* also results in estimates that are very efficient, although the optimal RMSE is slightly lower. The efficiency of the regular method is below 20% when compared to the globally optimal method, although measured locally across *L* picking the method with the highest price results in estimates with an RMSE very close to the minimum RMSE for this method.

The results above show that when using out-of-sample pricing, it is possible to derive a simple classification algorithm that achieves very high efficiency. In reality, the algorithm ends up picking the symmetric method most of the time, and if you only pick within this method, the loss in efficiency is very small, i.e., it decreases from 98.98–98.36%. Thus, it is possible to save on the computational time by only considering this method. Moreover, for most of the options, the highest price is achieved with *L* = 5 when using the symmetric method, and if, instead of picking among all the possible values of *L* in the table, you only consider *L* ≤ 7, the global efficiency decreases only marginally to 98.24%.<sup>14</sup> This approach thus yields very good estimates across our large sample of options, and it is easy to implement. It is not possible to come up with a similar approach when using in-sample pricing though.
