*2.1. Regular Call Option Prices*

Table 1 shows the pricing results for the sample of call options. The first thing to notice is that the majority of the estimated prices, shown in Column 5, are close to the benchmark values provided by the binomial model, shown in Column 4, and the bias, shown in Column 6, is in most cases less than one cent. However, for the longer term options with high volatility, shown in the last five rows, biases are large and significant at all reasonable levels. The size of the bias increases with moneyness, and the ITM option has a bias of 15 cents. Moreover, for these options, the estimated price also has a very large standard deviation, StDev, shown in Column 7, and as a result, the RMSE, shown in Column 8, is very large. For example, the RMSE of the option with *K* = 36, *T* = 2 and *σ* = 0.40 is almost 40-times larger than the RMSE of any of the other deep ITM options.

The results in Table 1 may hint at why often only put options are studied: it is potentially difficult to price long maturity call options in high volatility settings using the LSMC method. However, in many situations where the LSMC method is used, e.g., for real option pricing, the options considered are exactly long maturity call options. Therefore, what could (and does) go wrong? The fact that the standard deviation of these estimates is larger by (almost) an order of magnitude than that of any of the shorter term options indicates that this is likely caused by numerical issues. This conjecture is further supported by the fact that the skewness and kurtosis of the independent simulations are very far away from what we would expect, i.e., zero skewness and no excess kurtosis, when using independent simulations.<sup>3</sup>

Therefore, why then would you have numerical issues? The LSMC method estimates the early exercise strategy by performing a series of cross-sectional regressions of future path-wise payoffs on transformations of the current values of the stock price for the paths that are in the money, and the most obvious explanation for the numerical issues arising is that these regressions "break down" in one way or another. In particular, the properties of the input to the regression are very different when pricing calls, where regressors are unbounded, compared to when pricing puts, where regressors are bounded above by the strike price. Thus, one may end up performing regressions with regressors that have very large numerical values, and the probability of this happening increases with maturity and volatility. Note that this issue does not vanish when increasing the number of simulated paths, *N*.

### *2.2. Call Options Priced by Symmetry*

When pricing call options using the "symmetric" method, the regressions carried out to price the, now, put option may be expected to be better behaved. In particular, the independent variable and the regressors are now bounded above by the strike price when using only the paths that are in the money. Columns 10–13 of Table 1 show the resulting price estimates, which may be compared directly to the estimates from the "regular" method in Columns 5–8. The first thing to notice is that with this approach, the estimated prices for the long-term high volatility options are now much closer to the benchmark values, and in fact, none of them are statistically different from the benchmark values provided by the binomial model. Note that some of the biases, five to be precise, are slightly larger for the symmetric method than for the regular method, ye<sup>t</sup> in all cases, they are very small.

<sup>3</sup> Although the path-wise payoffs obtained with the LSMC method for a given Monte Carlo simulation are dependent and could be very far from normally distributed, the price estimates we report in the table are averages of *I* = 100 independent simulations and should therefore be normally distributed by a central limit theorem. The actual values for the skewness and excess kurtosis are not shown in the table, but are available upon request.


The Money (ITM) paths. Benchmark values are from the Cox et al. (1979) binomial model with 25,000 steps

and *J* = 50 early exercise possibilities per year.

**Table 1.** Call option prices.

However, not only is the bias of these estimates comparable across options, the standard deviation of the estimates is also similar across options. More importantly, the standard errors of the estimates are always lower than what is obtained with the regular method, and this is so even for the short-term options with low volatility in the first five rows. Across the 20 options, the regular method yields estimates with a standard error that is on average three-times larger, with the best case being roughly 26% worse (the option with *K* = 36, *T* = 1 and *σ* = 0.20), and the worst case having a standard deviation almost nine-times larger.

Because of the low bias and the much lower standard deviation, the RMSE of the call price estimates obtained using symmetry is much lower than that obtained when pricing the option with the regular method across the benchmark sample. For half of the options, the RMSE is less than half that obtained with the regular method when using the symmetric method. In the best case across the 20 options, the regular method is only 9% worse than the symmetry method; however, one would never do worse when pricing this set of call options using symmetry than with the regular method. This is a very strong argumen<sup>t</sup> for using symmetry to price call options.
