Efficient Numerical Pricing of American Call Options Using Symmetry Arguments

This paper demonstrates that it is possible to improve significantly on the estimated call prices obtained with the regression and simulation based Least-Squares Monte-Carlo method of Longstaff &amp; Schwartz (2001) by using put-call symmetry. Results show that the symmetric method performs much better on average than the regular pricing method for a large sample of options with characteristics of relevance in real life applications, is the best method for most of the options, never performs poorly and, as a result, is extremely efficient compared to the optimal but unfeasible method that picks the method with the smallest Root Mean Squared Error (RMSE). A simple classification method is proposed that, by optimally selecting among estimates from the symmetric method with a reasonably small order used in the polynomial approximation, achieves a relative efficiency of more than 98%. The relative importance of using the symmetric method increases with option maturity and with asset volatility. Using the symmetric method to price, for example, real options, many of which are call options with long maturities on volatile assets, for example energy, could therefore improve the estimates significantly by decreasing their Bias and RMSE by orders of magnitude.


Introduction
In a paper published 20 years ago, McDonald and Schroder (1998) demonstrated that when the price of the underlying asset is governed by a Geometric Brownian Motion (GBM) the price of a call option with underlying asset price S, strike price K, interest rate r, and dividend yield d is equal to the price of an otherwise identical put option with asset price K, strike price S, interest rate d, and dividend yield r. The result for the GBM case has since been generalized to more realistic dynamics in Schroder (1999), among others, and essentially a version of put-call symmetry (PCS), potentially with other fundamental parameters changed accordingly, will hold for virtually all the models that have been considered in the existing literature on option pricing, for options with several different payoffs and which are written on multiple assets. 1 In this paper we show that this simple result can be used to improve on one of today's state of the art numerical option pricing methods, the well known Least-Squares Monte-Carlo (LSMC) method proposed by Longstaff and Schwartz (2001). In particular, we show that using PCS with LSMC results in estimates with significantly less variance and estimates with much less bias when pricing American call options for the set of options used in Longstaff and Schwartz (2001) and for a very large sample of options with realistic characteristics.
Using standard choices for the LSMC method, which we implement with N = 100, 000 paths and a polynomial of order L = 3, we price options with different strike prices and maturities in a world with different values for the interest rate, dividend yield, and volatility. For this large sample of 3, 125 different options the average root mean squared error (RMSE) of the estimates obtained using the symmetric method is only 17% of the RMSE of the estimates from the regular method and the symmetric estimates have smaller RMSE for 88% of the options in the sample.
1 For example, PCS also holds in the stochastic volatility model of Heston (1993) when the parameters of the volatility process and the correlation are changed appropriately. See, e.g., Battauz, De Donno, and Sbuelz (2014) for the exact specification, Grabbe (1983) for an intuitive explanation of how to derive the relationship using options on foreign exchange, and Detemple (2001) for extensions to derivatives on multiple assets.
Our results show that the relative performance of the symmetric method, i.e. when call options are priced as put options using PCS, improves as the time to maturity and volatility increase. Moreover, using the symmetric method is most effective for options that are out of the money. The simple intuition for this results is that when option maturity is long and volatility is high asset values along simulated paths may become "very" large and be spread out over a large interval. Large and widely spread out asset values leads to poorly conditioned cross sectional regressions, this in terms results in bad approximations of the optimal early exercise strategy and precisely determining this strategy is most important for out of the money options. Widely spread out asset values also lead to estimates that have higher variance because of the spread out payoffs being discounted back to estimate the price.
The magnitude of the relative improvement obtained with the symmetric method also depends on the choice of parameters used in the LSMC algorithm, that is the number of simulated paths N and the number of regressors used in the cross sectional regression L in a non-trivial way. In particular, while it is now well known, see for example Stentoft (2004b), that the option price estimated with the LSMC converges to the true value when the number of paths and the number of regressors tend to infinity, this is of little use with finite choices of the number of paths N and the order of the polynomial used in the regression L. However, even with the "worst possible" configuration for the symmetric method, which occurs when N = 100, 000 and L = 5 where the symmetric method only performs the best for roughly 39% of the individual options, the average RMSE is only 18% larger than what could have been obtained with an infeasible method that picks from the regular and symmetric method the one with the smallest RMSE.
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 coming 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. For example, while the standard error of the estimates is, across the choice of polynomial order, almost always lower with the symmetric method than without, the bias just happens to be somewhat smaller without symmetry, a value of −0.006, than when using symmetry, a value of 0.010 when using L = 5 regressors with N = 100, 000 paths. 2 One easy way to control 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 used for determining the optimal early exercise strategy.
When using out of sample pricing, the relative importance of the symmetric method is even more striking. In particular, the symmetric method almost always, and in some cases for more than 99% of the individual options, has the lowest RMSE and the average RMSE for the large sample of options is around 20% or less of what is obtained with the regular method for most configurations. The efficiency of the symmetric method, when compared to the infeasible optimal method, is extraordinary and in most cases above 99% across various values of the number of paths N and number of regressors L whereas the regular method only achieves an efficiency of around 25%. Finally, while it is difficult to pick the best method in general, in the case of out of sampling pricing we propose a simple method that, by optimally selecting among estimates from the symmetric method with a reasonably small order used in the polynomial approximation, achieves a relative efficiency of more than 98% compared to the infeasible method that minimizes the RMSE across all estimates.
As noted by Detemple (2001), PCS is a useful property of many option pricing models since it reduces the computational burden when implementing these model. Indeed, a consequence of the property is that the same numerical algorithm can be used to price put and call options and to determine their associated optimal exercise policy. Another benefit is that it reduces the dimensionality of the pricing problem for some payoff functions. Examples include exchange options or quanto options. PCS also provides useful insights about the economic relationship between derivatives contracts. Puts and calls, forward prices and 2 For all other choices of number of paths with this number of regressors and when using this number of regressors with more paths the symmetric estimates are less biased. discount bonds, exchange options and standard options are simple examples of derivatives that are theoretically closely connected by symmetry relations. Compared to this literature our objective is somewhat different. In particular, though PCS can be used to demonstrate theoretically the convergence of a particular numerical scheme for call option pricing using results for put options, our interest here is primarily of a numerical nature and the objective is to show that PCS can be used to significantly improve on the estimated call option prices obtained with a particular numerical scheme. 3 Our findings and proposed method for selecting optimally the configuration to use for option pricing should have broad implications. In particular, we show that improvements are found for a very large sample of options with reasonable characteristics and since the symmetric method never performs very poorly and simple classification methods can be used to achieve very high relative efficiency there are strong arguments for always using the symmetric method to price call options. Moreover, our results show that the relative importance of using the symmetric method increases with option maturity and asset volatility and using symmetry to price long term options in high volatility situations thus improves massively on the price estimates. Finally, our results generalize to dynamics with e.g stochastic volatility and to options written on multiple assets (see the Appendix). The LSMC method is routinely used to price real options, most of which are call options with long maturities on volatile assets, for example energy. We conjecture that pricing such options using the symmetric method could improve significantly on the estimates by decreasing their Bias and RMSE by orders of magnitude.
The rest of this paper is organized as follows: In Section 2 we provide motivating results for the small sample of simple vanilla options from Longstaff and Schwartz (2001). In Section 3 we briefly introduce the use of simulation methods for American option pricing in general, discuss the method proposed by Longstaff and Schwartz (2001) in detail, and discuss the implementation of the method when combined with PCS. In Section 4 we perform a large scale application showing that our proposed method works extremely well. In Section 5 we conduct several robustness checks and in Section 6 we discuss which factors determine when our proposed method should be used and we suggest a method for choosing the optimal specification to use for individual option pricing. Section 7 offers concluding remarks. The Appendix considers extensions to a stochastic volatility model and to multiple asset options.

Motivation
In this section we present results for a set of options similar to those used in Longstaff and Schwartz (2001) but to illustrate the effect of PCS we consider pricing both call options and put options. In all cases we use a current value of the stock of 40 and an interest rate and dividend rate of 6%. A non-zero dividend is needed to make the American call option pricing non-trivial and to have positive early exercise premia. Options range between being 10% in the money (ITM) and out of the money (OTM), have maturities of T = 1 or T = 2 years and have J = 50 early exercise possibilities per year. We also consider two levels of the volatility and set σ = 20% or σ = 40%. The values in the tables are based on 100 independent simulations, each of which uses M = 100, 000 paths and the first L = 3 weighted Laguerre polynomials and a constant term as regressors in the cross sectional regressions using only the ITM paths. 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 5 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, shown in column 7, and as a result the Root Mean Squared Error (RMSE), shown in column 8, is very large.

Regular call option prices
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. So 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 100 independent simulations is very far away from what we would expect, i.e. zero skewness and no excess kurtosis, when using independent simulations. 4 So why then would you get numerical issues? The LSMC method estimates the early exercise strategy by performing a series of cross sectional regressions of future pathwise 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 . 4 Although the pathwise 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 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 available from the author upon request.

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 ten to thirteen of Table 1 show the resulting price estimates which may be compared directly to the estimates from the "regular" method in columns five to eight. 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 yet in all cases they are very small.
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 5 rows. Across the 20 options the regular method yields estimates with a standard error that is on average 3 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 9 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 argument for using symmetry to price call options.

Results for put options
For completeness, Table 2 shows the corresponding results for put options. We first of all note that for the regularly priced put options we obtain results that are completely in line with those in Longstaff and Schwartz (2001). In particular, the estimates are quite close to the benchmark values, and always less than a cent off, and the standard deviation of the estimates is very low and does not depend on the characteristics of the option except for σ which, when increased, naturally increases the uncertainty of the estimated price.
The method that uses symmetry to price the put as a call on the other hand by "symmetry" to the results in Table 1 generally performs worse than the regular method both in terms of Bias, where only 20% of the estimates are less biased, and in terms of the standard deviation of the estimates, which in all cases is larger by about 25% or more. In fact, for the longest maturity deep ITM option with high volatility both the bias and the standard error are larger by an order of magnitude when priced using symmetry than when priced as a regular put option.
Because of the poor performance of the method that uses symmetry in terms of bias and standard deviation all the estimates from this method has larger RMSEs than what is obtained with the regular pricing methodology. In fact, on average this method gives errors that are less than half and could be as much as almost 100 times smaller and are never less than 21% smaller. Thus, for this sample of options one should never use symmetry to price put options.

Implementation
The first step in implementing any type of numerical algorithm to price American options is to assume that time can be discretized. Thus, we assume that the derivative considered may be exercised at J points in time. We specify the potential exercise points as t 0 = 0 < t 1 ≤ t 2 ≤ ... ≤ t J = T , with t 0 and T corresponding to the current time and maturity of the option, respectively. An American option can be approximated by increasing the number of exercise points J and a European option can be valued by setting J = 1. We assume a complete probability space (Ω, F, P) equipped with a discrete filtration (F (t j )) J j=0 . The derivative's value depends on one or more underlying assets which are modeled using a Markovian process, with state variables (X (t j )) J j=0 adapted to the filtration and with X (0) = x known. We denote by (Z (t j )) J j=0 an adapted payoff process for the derivative satisfying Z (t j ) = π (X (t j ) , t j ) for a suitable function π (·, ·), which is assumed to be square integrable. Following, e.g., Karatzas (1988) and Duffie (1996), in the absence of arbitrage we can specify the American option price as where T (t j ) denotes the set of all stopping times with values in {t j , ..., t J } and where it is therefore implicitly assumed that the option cannot be exercised at time t 0 In the literature, the problem of calculating the American option price in (1), i.e. with J > 1, is referred to as a discrete time optimal stopping time problem. The preferred way to solve such problems is to use the dynamic programming principle. Intuitively this procedure can be motivated by considering the choice faced by the option holder at time t j : either to exercise the option immediately or to continue to hold the option until the next period.
Obviously, at any time the optimal choice will be to exercise immediately if the value of this is positive and larger than the expected payoff from holding the option until the next period and behaving optimally from there on forward. To fix notation, in the following we let V (X (t j )) denote the value of the option for state variables X at a time t j prior to expiration. We define F (X (t j )) ≡ E[Z (τ (t j+1 )) |X (t j )] as the expected conditional payoff, where τ (t j+1 ) is the optimal stopping time. It follows that and it is easily seen that it is possible to derive the optimal stopping time iteratively using the following algorithm: Based on this, the value of the option in (1) can be calculated as The backward induction theorem of Chow, Robbins, and Siegmund (1971) (Theorem 3.2) provides the theoretical foundation for the algorithm in (3) and establishes the optimality of the derived stopping time and the resulting price estimate in (4).

Simulation and regression methods
The idea behind using simulation for option pricing is quite simple and involves estimating expected values and therefore option prices by an average of a number of random draws.
This is easiest to illustrate in the case of a European option for which it is optimal to exercise at time T and therefore τ (t 1 ) = T by definition. Substituting this into (1) we obtain the following formula where we use lower case to denote that this is the European price and where Z (T ) = π (X (T ) , T ) is the payoff from exercising the option at time T . From (5) it is clear, that all that is needed to price the option are the values of the state variables, X (T ), on the day the option expires. Thus, an obvious estimate of the true price in (5) can be calculated using N simulated paths asp where X (T, n) is the value of the state variables at the time of expiration T along path number n. That is, the price estimate is simply an average of discounted simulated payoffs, and if these are generated from independently simulated paths, this estimate will have all the usual nice properties and will, for example, generally be unbiased.
When the option is American, one needs to simultaneously determine the optimal early exercise strategy, and this complicates matters. In particular, it is generally not possible to implement the exact algorithm in (3) because the conditional expectations are unknown and therefore the price estimate in (4) is infeasible. Instead an approximate algorithm is needed.
Because conditional expectations can be represented as a countable linear combination of basis functions we may write F (X (t j )) = ∞ l=0 φ l (X (t j )) c l (t j ), where {φ l (·)} ∞ l=0 form a basis. 5 In order to make this operational we assume that it is possible to approximate well the conditional expectation function by using the first L + 1 terms such that F (X (t j )) ≈ F L (X (t j )) = L l=0 φ l (X (t j )) c l (t j ) and that we can obtain an estimate of this function bŷ whereĉ N l (t j ) are approximated or estimated using N ≥ L simulated paths. Based on the estimate in (7) we can derive an estimate of the optimal stopping time from: From the algorithm in (8) a natural estimate of the option value in (4) is given bŷ where Z n,τ N L (1, n) is the payoff from exercising the option at the optimal stopping timê τ N L (1, n) determined for path n according to (8).

Implementation of the LSMC method
When implementing the method outlined above one has to choose two things: how to generate the data, the simulated state variables, and how to approximate the value function, that is how to estimate the parameters in the approximation. The key contribution of Longstaff and Schwartz (2001) is to suggest, very cleverly so, that the coefficients in the approximation of the continuation value,ĉ N l (t j ), can be estimated in a simple cross sectional ordinary linear (OLS) regression, where the independent variable is the discounted pathwise future payoff and the dependent variables are functions, for example the weighted Laguerre polynomial, of the current state variables using the in the money paths, ITM. The reason for choosing OLS is that this is simple, the reason for choosing weighted Laguerre polynomials is that these are orthogonal on the positive real line where state variables typically live, and the reason for choosing ITM paths only is that this is the area of interest for which we need approximations of the conditional expectations. Longstaff and Schwartz (2001) also use antithetic simulation as a variance reduction technique. However, each and everyone of these choices have been, correctly so, examined in detail in the literature that has followed.
While the choice of weighted Laguerre polynomials in Longstaff and Schwartz (2001) is motivated by the fact that this family is orthogonal on the positive real line this is of little use when using only ITM paths, particular for put pricing, since the state variables are then by construction bounded, and while proper scaling can adjust this to make the family orthogonal regressors need to be weighted and this with the computationally expensive exponential function. Thus weighted Laguerre polynomials are "expensive" to use. Moreover, as shown in, e.g., Stentoft (2004a), while other families of polynomials, like Hermite or Chebyshev polynomials, could be used simple polynomials often perform very well and are efficient to implement. 6 For additional comparisons of different polynomial families see also Areal, Rodrigues, and Armada (2008) and Moreno and Navas (2003).
The simple OLS regression can be implemented using various different algorithms and this may results in different estimated prices. For example, Areal, Rodrigues, and Armada (2008) consider two different methods and document differences in estimated prices as well as large differences in the computational time. More generally, although OLS have very nice properties, minimizing the squared errors of the predicted values might not be an optimal criteria when the objective is to use the predicted values to make decisions about when to exercise an option. In particular, when a polynomial of higher order is used, the fit (in sample) is always improved, but this may lead to over fitting and coefficients estimated with large variances which may adversely affect the ability to estimate well the optimal stopping time. One potential solution to this is to use regularization methods as proposed by, e.g., Tompaidis and Yang (2014). The drawback of these methods is the increase in computational costs. An alternative method for improving the cross sectional regression that is arguably more intuitive is to impose more structure on the problem and treating the regression as a constrained optimization problem. This is done in, e.g., Letourneau and Stentoft (2014) where theoretical properties like monotonicity and slope of the value function are imposed.
This improves significantly on the estimated prices, in particular when using few paths.
While it is true by definition that the in sample fit in the cross sectional regression is improved when using only the ITM paths instead of all the paths this is, since in sample fit is not the objective, not a reason to pick one method above another. In fact, if one was to price several options ranging from deep ITM to deep OTM with the same number of paths and number of regressors the potential over fitting from using only the ITM paths would be much more servere for the OTM option than for the ITM option as there would be much fewer paths in the ITM region for OTM options. Moreover, not all paths in the cross sectional regression contain information of similar "quality" about the value function and weighting paths may improve the fit. This is, for example, used in Fabozzi, Palett, and Tunaru (2017) where a weighted least squares method that takes into consideration heteroscedasticity in the cross sectional data is proposed. Several nonparametric methods have also been used to improve on the cross sectional regression fit but often these methods increase computational time significantly. More generally though, it is not the global fit to the value function that is important, it is the "local fit" for values of the dependent variables close to the early exercise frontier. Recent papers have been examining this both using nonparametric methods, see Ludkovski (2017) which also develops a method for simulating paths in the region of importance, and classical techniques, see Ibanez and Velasco (2017) which develops an iterative version of the method for localizing the optimal early exercise boundary.
Finally, Longstaff and Schwartz (2001) use antithetic simulation as a variance reduction technique and any of the well-known techniques for improving the precision of estimates from Monte Carlo simulation could in theory be used. However, when using regression methods for estimating the optimal early exercise strategy it is not clear that these methods are all "obvious" candidates. For example, whereas antithetic simulation results in unbiased estimates of European prices, American option price estimates from methods such as the LSMC are generally biased and this bias depends in part on the properties of the simulated paths. Antithetic simulation introduces correlation between paths and this increases the probability of over fitting in the cross section regression. When using a reasonably large number of regressors the LSMC estimates are biased high and in this case using antithetic simulation will increase the size of the bias. 7 Other methods, like the use of quasi Monte Carlo simulation, that introduce dependency between the simulated paths will have similar unknown effects on the bias of the estimates and though they may reduce the variance of the estimates the root mean squared error may increase in finite samples. 8 Importance sampling 7 When a very small number of regressors are used LSMC is biased low and in this case using antithetic simulation will decrease the size of the bias. In general, however, the effect is unknown.
8 Asymptotically, though, when the number of paths tends to infinity variance reduction techniques will decrease the RMSE for given choice of the number of regressors. and the use of control variates have also been examined, and whereas the use of importance sampling will affect the bias of the estimates in a similar unknown way using standard control variates will not do so in general. 9

Implementation of the symmetric LSMC method
In this paper, we propose to merge the LSMC method with PCS and use the symmetric method when pricing call options. Thus, instead of simulating paths from a dynamic model with a risk free rate of r and dividend yield of d we simulate from the same dynamic model but with a risk free rate of d and dividend yield of r, and instead of pricing the option as a call option with a strike price of K and a current value of the underlying asset of S we price the option as if it had been a put option with a strike price of S and a current value of the underlying asset of K. 10 These changes are simple to make and involve no extra computational complexity or changes to the numerical procedure. In fact, for consistency it is important to note that we use the exact same numerical procedures for simulating the paths and to implement the cross sectional regression. 11 There are two very intuitive reasons for why using the symmetric method to price call options may work better than when pricing the call option using the regular method. First of all, as explained above in simulation based methods the option price, an expectation under the risk neutral measure, is approximated by the average of a number of random realizations of future payoffs, obtained from simulated values of the appropriate state variables. This mean obviously behaves better and the estimator will have a smaller variance when the possible realizations are bounded, as they are in the case of the payoff of a put option, than when they unbounded, as they are in the case of the payoff of a call option. Note that this holds even when pricing European options and our numerical results in the next section shows that the standard deviation of the estimates are always lowered when using the symmetric method. 12 For the American options standard deviations are also lower for almost all and often well above 99% of the options.
Secondly, and this applies to the American style options in particular, it is easier to approximate the continuation value when this is a bounded function on a bounded interval than when this is an unbounded function on an unbounded interval. In particular, theoretically it is straightforward to design a robust approximation scheme for the continuation value of a put option using only the simulated paths that are in the money. 13 For call options no general theoretical results exist to justify that this is in fact feasible and, though numerical schemes are available and polynomial families that have nice properties can be used, approximating the continuation value is likely much more complicated. A further complication with the continuation value of a call option, is that this is bounded above by the exercise value for large values of the underlying asset and thus asymptotically linear in the stock value. It is obviously difficult to use a polynomial of reasonably high order to approximate a function with these characteristics.
In the large scale application which follows we use the LSMC method with simple monomials as regressors and paths simulated without any "bells or whistles" and our results show that both the variance and the absolute size of the bias of the estimated prices are generally reduced when using the symmetric method. Thus, one immediate interpretation is that PCS can be used as a simple and straightforward variance reduction technique for call options. 14 Moreover, the fact that the size of the bias is lower for the symmetric method across various values of simulated paths and regressors used is a strong indication that the approximations 12 As the estimates are unbiased the average RMSE is also much lower for the symmetric method and lower for more than 90% of the European options.
13 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.
14 In fact, our results show that the variance of the estimates is reduced for 99% or more of the options irrespective of the choice of the number of simulated paths N and the number of regressors L used when using symmetric pricing. used to determine the optimal stopping time are indeed better behaved when pricing the call option as a symmetric put. Our robustness checks show that the relative performance of the symmetric method is similar for other regression methods used, choices of polynomial families, and even when using variance reduction methods like antithetic simulation. This means that our method could, we conjecture, be used with any and all of the proposed improvements to the "plain vanilla" LSMC method discussed above, though we leave this for future research.

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 in a large scale application we now price a large sample of call options with five different strike prices, K = [90,95,100,105,110], 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 spans most of the important cases one would come across in real life applications of option pricing. We first consider performance for various number of exercise possibilities J. Next we consider performance across option characteristics, i.e. K and T , and we finish by considering performance across model parameters, i.e. r, d and σ. Benchmark values are from the Cox, Ross, and Rubinstein (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.
Here we report results with the LSMC method using 100 independent simulations with N = 100, 000 paths and L = 3 regressors but in Section 5 we consider alternatives to this with various number of paths ranging from 20, 000 to 1, 000, 000 and various number of regressors from 2 to 15, and we examine the robustness of our results to using other regressors, e.g. Laguerre polynomials, and variance reduction techniques as well as using all the paths instead of only the in the money paths and from conducting out of sample pricing which uses a new set of paths for pricing.

Results with the LSMC method
Since we cannot report all the individual prices we report overall errors instead. We focus on the Bias, the Absolute Error (AbsEr) and the Root Mean Squared Error (RMSE) in absolute terms and in relative, to the true price, terms. We also consider the count of options for which the regular and symmetric method, respectively, has the highest Bias, Standard Deviation (StDev) and RMSE. 15 Table 3 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 = 1 (the European option) to J = 200 (a close approximation to the continuously exercisable American option). Table 3 first of all shows that using the regular method for this sample of options leads to significantly low biased price estimates in Panel A. For example, when considering the case with J = 50 exercise times, the average bias is almost 6 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 AbsEr and RMSE. When considering the relative pricing errors in Panel B the improved performance of the symmetric method in terms of Bias is even more impressive. For AbsEr and RMSE the errors with this method remain much smaller at 24.13% and 27.34%, respectively. Finally, the improvement in performance is not only large on average but also across most of the options as is shown in Panel C. In particular, this panel shows that improvements occur for 83.90% and 99.55% of the options in terms of Bias and StDev, respectively, and as a result the symmetric method yields more precise estimates, in terms of RMSE, for almost 9 out of 10 of the options, 88.19% to be precise. Table 3 also shows that the improved performance does not result from a particular choice of the number of early exercise possibilities J. In fact, improvements are found for all the reported values of J and even when pricing European options, i.e. options for which the optimal stopping time strategy is known and involves exercising at maturity if the option is in the money, improvements are found for all absolute metrics. Symmetric pricing also leads to significantly lower relative RMSE though the relative Bias and AbsEr are of the same size as when the regular method is used, which is to be expected since theoretically the estimates are unbiased. Across the number of early exercises the table does indicate that using symmetry in pricing leads to larger relative improvements for options with early exercise and for a small number of exercise possibilities the relative performance improves rapidly. Once J reaches 50 the effect tapers off and the relative improvement in performance does not change much when increasing the number of early exercise points further. 16 In the following we consider J = 50 exercise possibilities.

Performance across option characteristics
The results above clearly demonstrates the improvements that can be obtained by pricing American call options using the symmetric method instead of the regular method. We now consider the relative performance of the two methods across option characteristics like the strike price K, implicitly the moneyness, and the maturity T . In particular, the results in Section 2 indicated that the relative improvements increase with the time to maturity.
Tables 4 and 5 shows the results across moneyness and maturity for our large sample of options.
16 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 increase.  Table 4 shows that the absolute errors of both methods decrease when the strike price increases whereas the relative errors are constant across strike price for the regular method and increase slightly for the symmetric method. In relative terms though, both panels show that the symmetric method performs better than the regular method across all strike prices.
The relative performance is best for options with low strike prices, i.e. call options that are out of the money and for these options Panel C shows that the symmetric method has 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. Table 5 documents clear and significant improvements in the relative performance of the symmetric method for all metrics in absolute as well as relative terms 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 in Panel C of Table 5. 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 9 out of 10 of the options. Thus, Table 5 confirms that the results from Section 2 hold true in general for a much larger sample of options.

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. The results in Section 2 indicated that the relative improvements increase with the volatility of the underlying asset. Tables 6, 7, and 8 show the results across interest rate, dividend yield, and volatility for our large sample of options.
Tables 6 and 7 show that the relative improvement from using the symmetric method is large across all 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 and relative errors Table 6 shows that 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, Table 7 shows that the relative performance of the symmetric method increases somewhat.
Note that the case with d = 0 is somewhat special since in this situation the American call option should never be exercised early. Table 8 documents clear and significant improvements in the relative performance of the symmetric method for all metrics in absolute as well as relative terms when volatility in- creases. 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 estimates with smaller errors in at least 64.64% of the cases, the worst relative performance being for the lowest volatility in Panel C of Table 8. 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 9 out of 10 of the options. Thus, Table 8 confirms that the results from Section 2 hold true in general for a much larger sample of options. Results are based on 100 independent simulations with N = 100, 000 paths and L = 3 regressors and with J = 50 exercise possibilities. Panel A reports the Bias, Absolute Error (AbsEr) and Root Mean Squared Error (RMSE) along with the performance of the symmetric method relative to the regular method, with values below one indicating that pricing by symmetry lowers the error. Panel B reports similar results for error metrics relative to the benchmark values. Panel C reports the fraction of times a given method has the highest error metric.

Robustness
The previous section provides strong evidence in favor of using symmetric pricing for call options. In this section we examine the robustness of these results along several 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. Next, we examine the effect of using different regression methods, alternative polynomials, implementing variance reduction techniques, and of using all paths for pricing instead of only the ITM paths. Lastly we show that the relative performance of symmetric pricing is even better when implementing the LSMC method with out of sample pricing, i.e. when a new set of paths is used for pricing instead of using the same paths that were used for determining the optimal early exercise strategy.

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. And, while it is well known that the estimated prices converge to the true price when both tend to infinity, see e.g. Stentoft (2004b), any real application involves choosing a finite number of paths and regressors. Table 9 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 value of the Bias, as well as the size of the two other absolute error metrics, is lowest with the symmetric method and this method consistently outperforms the regular method across all choices of N . In terms of relative errors the symmetric method also generally outperforms the regular method across N , with the exception being in terms of the Bias when using N = 20, 000 where the regular method results in a very small relative Bias of only 0.0001. With respect to these metrics though, the symmetric method improves on the regular method by at least 35% in terms of relative RMSE.
When it comes to the number of times the symmetric method has lower errors shown in Panel C a pattern very similar to what was seen when increasing T or σ is found. The one exception is in terms of the standard deviation of the estimates where the panel shows that as the number of paths increase, and when it gets as large as N = 500, 000, the fraction of options for which the symmetric method provides the most precise estimates, in terms of StDev, decreases from "almost always", that is around 99% of the time, to "very often", or just above 90% of the time. Table 10 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. 17 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 8 cents to having an average positive bias of more than 6 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.
The same holds when using the relative metrics and here the regular method even slightly outperforms the symmetric method when considering the AbsEr metric for this choice of the number of regressors as shown in the fourth row in Panel B.
When it comes to the number of times the symmetric method has lower errors shown in Panel C 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 which 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. The percentage is actually slightly higher in terms of Bias at 71.52% but for more than 99% of the options the StDev remains smaller with the symmetric method.
When looking at Panel B of Table 10 it is noteworthy that the relative 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 B of Table 9 when the number of simulated paths is increased. Given this concave relationship, 17 The case when the number of regressors equals the number of paths is a case in point as this would lead to a perfect fit and essentially resulting in using an algorithm with perfect foresight as described in Broadie and Glasserman (1997). 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.

Alternative specifications for the LSMC algorithm
We now examine wether the results in Section 4 are robust to the specification of the LSMC algorithm in terms of the method used for estimation, the choice of "basis functions" used for approximation, to the use of variance reduction method, and to using all paths for pricing instead of only the ITM paths. We present results for various different choices for how to implement the LSMC algorithm using a low number of regressors, L = 3, and a high number of regressors, L = 9, in Table 11 and 12, respectively.
The results above for the large scale application are all obtained using the MATLAB function POLYFIT, a function which is particularly well suited to perform polynomial approximations. 18 For ease of comparison we repeat these benchmark results in the first row of each of the panels in both tables (labelled Bnch). In the second row we report the results when a non-specialized and more generic regression algorithm is used in the cross sectional regressions (labelled Mon). 19 With respect to the choice of regression algorithm the tables clearly show that this may have an effect. In particular, though the results from using a generic regression algorithm are the same as for the benchmark algorithm when the number of regressors is low for both the regular and the symmetric method, when a large number of L = 9 regressors is used in the cross sectional regression, the generic regression algorithm breaks down unless symmetry is used. As a result of this, all the error metrics are signifi-18 The POLYFIT function uses QR decomposition of the Vandermonde matrix of regressors to perform the regression and is therefore very stable and efficient. See the MATLAB documentation for details on how this function is implemented. 19 We use here the MATLAB function REGRESS. If columns of the regressors are linearly dependent, REGRESS sets the maximum possible number of elements of the parameter vector to zero to obtain a basic solution. See the MATLAB documentation for details on how this function is implemented. This table shows pricing errors for the regular and symmetric method with different choices in the LSMC algorithm. We compare the previous benchmark results (Bnch) to results using simple monomial regressors (Mon), using weighted Lauguerre regressors (Lag), using antithetic simulation (Anti), and using all paths instead of only in the money paths (All). Results are based on 100 independent simulations with N = 100, 000 paths and L = 3 regressors and with J = 50 exercise possibilities. Panel A reports the Bias, Absolute Error (AbsEr) and Root Mean Squared Error (RMSE) along with the performance of the symmetric method relative to the regular method, with values below one indicating that pricing by symmetry lowers the error. Panel B reports similar results for error metrics relative to the benchmark values. Panel C reports the fraction of times a given method has the highest error metric.
cantly higher in the second rows of Panels A and B in Table 12 and the relative improvement from using symmetry is massive. Thus, our results appear to be robust to the method used for performing the cross sectional regression and may in fact be much better if more generic algorithms are used for this regression.
In the third row of the tables we report results when using weighted Laguerre polynomi- This table shows pricing errors for the regular and symmetric method with different choices in the LSMC algorithm. We compare the previous benchmark results (Bnch) to results using simple monomial regressors (Mon), using weighted Lauguerre regressors (Lag), using antithetic simulation (Anti), and using all paths instead of only in the money paths (All). Results are based on 100 independent simulations with N = 100, 000 paths and L = 9 regressors and with J = 50 exercise possibilities. Panel A reports the Bias, Absolute Error (AbsEr) and Root Mean Squared Error (RMSE) along with the performance of the symmetric method relative to the regular method, with values below one indicating that pricing by symmetry lowers the error. Panel B reports similar results for error metrics relative to the benchmark values. Panel C reports the fraction of times a given method has the highest error metric.

als instead of monomials as was suggested in Longstaff and Schwartz (2001) (labelled Lag).
With respect to the choice of "basis functions" the tables first of all show that the relative improvement from using the symmetry method may be much larger than what is obtained with the benchmark algorithm. In particular, when using a low order Laguerre polynomial in the approximation the errors from the regular method are very large and larger by orders of magnitude than what is obtained with either the benchmark method or with the symmetric method. Note that when using the symmetric method the difference between using regular polynomials and Laguerre polynomials is quite small. When a larger number of L = 9 Laguerre polynomials is used the regular method performs much better although the RMSE is still larger than with the benchmark algorithm. Note that compared to using regular polynomials, the generic regression algorithm with weighted Laguerre polynomials does not break down and continues to provide reasonable estimates of option prices. However, compared to using the symmetric method the estimates are on average much worse and the errors are larger for two thirds of the options when considering both the Bias and the RMSE. Thus, our results appear to be robust to using other "basis functions" as regressors and may in fact be much more important when using other polynomial families as regressors.
In the fourth row of the tables we report results when using antithetic simulation as a simple variance reduction method (labelled Anti). The first thing to note is that, as expected, when using a low number of L = 3 regressors implementing antithetic simulation decreases the RMSE of the estimates. The improvement in the average precision of the estimated prices is more important when using the regular method though. When using a large number of L = 9 regressors the precision of the estimates from the symmetric method are also improved with this type of variance reduction technique. However, the estimates with the regular method are in fact worse in terms of, e.g., the RMSE than with the benchmark method.
In particular, as Panel C of Table 12 shows when using antithetic simulation the regular method has larger Bias and Standard Deviation in 97% and 99% of the cases, respectively.
Thus, our results appear to be robust to using variance reduction techniques and may in fact be much more important when such methods are used.
Finally, in the fifth row of the tables we report results when using all the paths in the cross sectional regressions instead of using only the paths that are in the money (labelled All). The first thing to note is that, as expected, the estimated stopping time is much worse when using all the paths and as a results of this the errors in the estimated prices are much larger irrespective of the number of regressors used and irrespective of wether the regular or the symmetric method is considered. The errors are particularly large, relative to the benchmark algorithm, when using a low number of L = 3 regressors. When a larger number of L = 9 regressors is used the estimates from both the regular method and the symmetric method improve a lot and Table 12 shows that in this case the regular method actually performs slightly better on average than does the symmetric method and more often the regular method has the smallest errors. One obvious reason for this is that it is difficult to approximate well a bounded function, i.e. the payoffs from the symmetric call, with unbounded regressors whereas approximating the payoff of the regular call, which is unbounded, with unbounded regressors is relatively easier though still very difficult.

Out of sampling results
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 coming 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. For example, while the standard error of the estimates is, across the choice of polynomial order, almost always lower with symmetric pricing than without, 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. 20 One easy way to control 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. Tables 13 and 14 show the results for different configurations of N and L and, as expected and in line with theory, show 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 9, Table 13 shows that when using out of sample pricing the estimated prices with the regular method improves significantly when the number of simulated paths N increase. 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 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 Table 10, Table 14 shows that when using out of sample pricing the relative performance of the symmetric method is much less dependent on the choice of L. In particular, the symmetric method improves significantly on the regular method irrespective of the choice of L. 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.

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 great 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 is 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 they diverge when using a large number of regressors, e.g. when L is as large as 15. In this section we first analyse which factors significantly impact the relative performance of the method. Next we examine the performance of the individual methods in terms of a relative efficiency measure which compares performance of a method to what could have been obtained optimally. Finally, 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.

What affects the relative performance
The 3, 125 options considered varies in terms of their strike prices, maturities, interest rates, dividend yields, and volatilities, and for a given setup of the LSMC method these are therefore the factors that can impact the relative performance of the two methods. In particular, Section 4 showed that the relative performance of the symmetric method in terms of average performance and number of times the method had the smallest errors varied across most This table shows the results of logistic regressions of an indicator variable taking the value 1 when the RMSE of the symmetric method is smallest and 0 otherwise on the characteristics of the option, strike price K, time to maturity T , interest rate r, dividend yield d, asset volatility σ, and cross products. Each column reports the t-statistic for significance of a given variable for a particular set of values of N and L. Boldface denotes variables that have the same sign across the various specifications of N and L.
characteristics and in particular across maturity and volatility. To examine their effect further we now run logistic regressions using these factors in percentage deviations from a standard option, an ATM option, K = 100, with 2 years maturity, an interest rate and a dividend yield of 5%, and a volatility of 30%. We do so for various choices of the number of paths and the number of regressors used in the LSMC algorithm. The dependent variable takes the value 1 for options where the symmetric method has the smallest RMSE and 0 otherwise. Since we are after sign and significance and not the actual parameter estimates we only report the t-statistics in Table 15.
The results for our standard implementation of the LSMC algorithm with N = 100, 000 paths and L = 3 regressors are shown in column 2 (and 5) of Table 15. The first thing to notice is that several of the variables are significant in the regression and therefore affect the probability that the symmetric method yields the smallest RMSE. For example, and as expected, when the time to maturity and the volatility increase it is more likely that the symmetric method is better. When the strike price and the dividend yield increase the opposite is true whereas the interest rate is insignificant in affecting the relative performance of the two methods. In addition to these "direct" effects several of the cross products and squared variables are also significant. For example, while the interest is insignificant its square and cross product with the dividend yield and volatility are in fact significant.
When comparing across columns in the two panels of Table 15 it is seen that significance and even sign in the logistic regressions change for most of the parameters when the number of paths or the number of regressors is changed. Thus, the relationship above is not stable and this makes it difficult to draw general conclusions about when to use one method over the other based only on option characteristics or properties of the dynamic model being used. For example, time to maturity remains significantly positive when the number of regressors increase but when a large number of N = 500, 000 paths is used it becomes negative and significant indicating that in fact in this case it appears to become less likely that the symmetric method performs the best when maturity increase.

Efficiency as an alternative metric
Until know we have compared performance metrics, i.e. RMSE or 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 loose 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 This table shows the efficiency across the number of paths N (k) in thousands and regressors L. Efficiency is measured as the optimal RMSE, conditional on knowing which of the two methods yield 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 provides the lowest RMSE, respectively.
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 16 shows the efficiency of the two methods for various values of N and L using in sample pricing 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. 21 Panel A of Table 16 clearly shows that the symmetric method performs extremely well across the number of simulated paths and one would never loose 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 loose around 80% compared to what could optimally be obtained.
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 16, 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 using the symmetric method you would not loose 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 17 shows the efficiency of the two methods for various values of N and L using out of sample pricing 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 16 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 much lower, as low as 15%, and does not improve in any systematic way when using out of sample pricing.

Picking the best configuration
The previous sub-section showed that, 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 This table shows the efficiency across number of paths N (k) in thousands and regressors L using out of sample pricing. Efficiency is measured as the optimal RMSE, conditional on knowing which of the two methods yield 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 provides the lowest RMSE, respectively.
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. 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 and, as the logistic regressions above show, this is very difficult in this setting when using option characteristics and any recommendations would likely change with the choice of the number of simulated paths and the number of regressors used in the LSMC algorithm.
An alternative suggestion is to use characteristic of the option price estimates. In particular, we argued above that the reason the regular method "breaks down" should be related to the degree of over fitting in the cross sectional regression and the increased variance of the estimated coefficients in the approximating polynomial. As a result of this one would to use the estimated variance as a classification variable for example in the "worst case" considered above with N = 100, 000 paths and L = 5 regressors for which the efficiency of the symmetric method is only slightly above 84%. It turns out that using a "grid search" it is possible to improve the efficiency to 91% if you only use the symmetric model when the estimated variance is 1.57 times larger than the variance of the estimate from the regular method. However, if you use the same classification when using L = 3 regressors instead the efficiency, unfortunately, decreases to just above 85% compared to 98% when always using the symmetric method and thus such recommendations are also of little real use.
Another, and perhaps more 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 18. Table 18 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 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 yield the lowest RMSE for a given individual option in our large sample of 3, 125 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 the Panel A. The result in the column headed "Optimal" correspond 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" reports the efficiency of the two methods compared to the best possible RMSE of 0.0073 reported in Panel B.
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 a lot across L. Table 18 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 a RMSE very close to the minimum RMSE for this method.

Panel B in
The results above show that when using out of sample pricing it is possible to derive a It is not possible to come up with a similar approach when using in sample pricing though.

Conclusion
This paper shows that it is possible to improve significantly on the estimated call prices obtained with a state of the art algorithm, the Least Squares Monte Carlo (LSMC) method of Longstaff and Schwartz (2001), for option pricing using regression and Monte Carlo simulation by using put-call symmetry (PCS). PCS holds widely and in the classical Black-Scholes-Merton case, for example, implies that a call option has the same price as an otherwise similar put option where strike price and stock level and where interest rate and dividend yield are interchanged, respectively. The immediate implication is that pathwise payoffs in simulations are bounded above by the strike price instead of being unbounded, and for methods that use regression to determine the optimal early exercise strategy this leads to much improved estimates of the stopping time and more precise option price estimates as measure by, for example, root mean squared error (RMSE). Our results show that the symmetric method on average performs much better than the regular pricing method for a large sample of options with characteristics of relevance in real life applications, is the best method for most of the options, never performs very poorly and as a result is very efficient compared to an optimal but unfeasible method that picks the method with the smallest RMSE. When using out of sample pricing a simple method is proposed that, by optimally selecting among estimates from the symmetric method with a reasonably small order used in the polynomial approximation, achieves a relative efficiency of more than 98% compared to the infeasible but perfect method that minimizes the RMSE across all estimates. Results also show that the relative importance of using the symmetric method increases with option maturity and with asset volatility and using symmetry to price long term options in high volatility situations improves massively on the price estimates. The LSMC method is routinely used to price real options, many of which are call options with long maturity on volatile assets, for example energy. We conjecture that pricing such options using the symmetric method could improve the estimates significantly by decreasing their Bias and RMSE by orders of magnitude.

A Extensions
In the main section of the paper we presented results for the simple Black-Scholes-Merton setup. The reason for this is obvious: we want 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. In this appendix we consider two such alternatives and demonstrate that our previous conclusions hold. We first provide results when the dynamics are instead given by the stochastic volatility, or SV, model of Heston (1993). We then present results when the option payoffs depend on the average of several assets in a multivariate model.

A.1 The SV model of Heston
The SV model of Heston (1993) is one of the most famous extensions to the constant volatility model we have considered in the main body of this paper. 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 and now serves as a benchmark against which many other SV models are compared. In the Heston (1993) model the dynamics for the stock is given by where W Q 1,t a standard Brownian motion under the risk-neutral measure Q. Here r and q are the interest rate and the continuous dividend yield on the asset, respectively, which are again assumed to be constant. It is then further assumed that the variance follows a Cox, Ingersoll, and Ross (1985) process specified as where (κ, θ, σ) are assumed to be constant parameters with W Q 2,t also a standard Brownian motion under measure Q. Here κ represents the mean reversion rate of the variance, θ is the long-term variance and σ is the volatility of volatility (vol of vol). The two Brownian motions are allowed to be correlated with correlation coefficient ρ such that Notice that in order to ensure the variance is positive, the parameters need to satisfy the Feller condition and we assume that 2κθ > σ 2 .
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 it was the case in the constant volatility setting, and in which the mean reversion, κ , the long term variance, θ , and the correlation, ρ , for the put option are changed to where θ, κ, and ρ are the actual long term variance, mean reversion and correlation, respectively.

A.1.1 Issues related to implementation
An important challenge when considering the Heston (1993)   θ, the initial level of the variance, V 0 , and the correlation, ρ. The first thing to notice 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, 5.4% and 2.8%, respectively, 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 the tables show that the symmetric method performs relative better the faster the volatility mean reverts, i.e. the larger the value of κ in Table 19, and the larger the value of the long term variance, i.e. the larger the value of θ in Table 20. 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. Table 22 shows that the symmetric method performs relatively better when correlations are negative which is 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, Bias, AbsEr 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 This table shows pricing errors for the regular and symmetric method for options in a stochastic volatility model with different initial variances, V 0 , using the LSMC method. Results are based on 100 independent simulations with N = 100, 000 paths and correlations of V 0 = {0.01, 0.04, 0.16} and with J = 50 exercise possibilities. Panel A reports the Bias, Absolute Error (AbsEr) and Root Mean Squared Error (RMSE) along with the performance of the symmetric method relative to the regular method, with values below one indicating that pricing by symmetry lowers the error. Panel B reports similar results for error metrics relative to the benchmark values. Panel C reports the fraction of times a given method has the highest error metric.
the relative performance of the initial variance, V 0 , is less clear cut though Table 21 does indicate that the symmetric method performs relatively better when V 0 is not too extreme, and in particular not too high.

A.2 Multiple asset options
The case with options written on multiple assets is the most obvious generalization of the standard constant volatility case we considered in the main part of the paper. 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 Boyle and Tse (1990) give examples on where these types of options are traded. Stentoft (2004a) demonstrates 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 where S m , 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) Geometric Brownian Motion.
To be specific, consider M individual GBM processes and denote by S G the geometric average of these. Then it follows that the dynamics of S G are given by where σ i is the volatility of stock i, ρ ij is the correlation between stock i and j, and Z ∼ N (0, 1). If we assume that volatilities and correlations are identical (18) simplifies to Thus, exact formulas for the European price of an option on the geometric average can be derived and we can calculate benchmark values for any dimension using e.g. the BM in one dimension with, say 50, 000 steps.
In terms of put-call symmetry it is important to note that the above specification is not exactly that of a GBM which would instead have dynamics given by In other words the term involving (1 + (M − 1) ρ) /M is "missing" from the drift term.
Thus, when pricing the call option as a put option we simply "correct" for this by using This table shows pricing errors for the regular and symmetric method for options on different number of assets, M , using the LSMC method. Results are based on 100 independent simulations with N = 100, 000 paths and a dimension of M = {2, 3, 5} and with J = 50 exercise possibilities. Panel A reports the Bias, Absolute Error (AbsEr) and Root Mean Squared Error (RMSE) along with the performance of the symmetric method relative to the regular method, with values below one indicating that pricing by symmetry lowers the error. Panel B reports similar results for error metrics relative to the benchmark values. Panel C reports the fraction of times a given method has the highest error metric.

A.2.2 Results
We again consider options with several interesting features. In particular, we consider three   This table shows pricing errors for the regular and symmetric method for options on multiple underlying assets with different correlations, ρ, using the LSMC method. Results are based on 100 independent simulations with N = 100, 000 paths and correlations of ρ = {.025, 0.50, 0.75} and with J = 50 exercise possibilities. Panel A reports the Bias, Absolute Error (AbsEr) and Root Mean Squared Error (RMSE) along with the performance of the symmetric method relative to the regular method, with values below one indicating that pricing by symmetry lowers the error. Panel B reports similar results for error metrics relative to the benchmark values. Panel C reports the fraction of times a given method has the highest error metric.
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 performance of the symmetric method in terms of RMSE in Panel A goes from 0.0890 to 0.545 as the dimension increases from two to five. Moreover, when the dimension of the problem is high the symmetric method almost always has the lowest RMSE as Panel C shows. Table 23 shows the results across the correlation between the assets, ρ. From this table 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. Moreover, as Panel C shows the symmetric method is always the method the 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 around 5% of the options.