**1. Introduction**

In a paper published some 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, some version of this 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.<sup>1</sup>

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)

<sup>1</sup> 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 et al. (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.

method proposed by Longstaff and Schwartz (2001). In particular, we show that using PCS with LSMC results in estimates that are much less biased and have significantly lower Root Mean Squared Error (RMSE) 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 a large sample of 3125 different options, the average RMSE of the estimates obtained with the symmetric method is only 17% of the RMSE of the estimates from the regular method, and the symmetric estimates have smaller RMSEs for 88% of the options in the sample.

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 lead to poorly-conditioned cross-sectional regressions and this in terms results in poor approximations of the optimal early exercise strategy and precisely determining this strategy is most important for out of the money options. Widely-dispersed 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 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 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 much smaller than for the regular method and 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, 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. For all other choices of the number of paths with this number of regressors and when using other numbers of regressors with this number of simulated paths, the symmetric estimates are less biased. An 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 classification algorithm 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 discount bonds and 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 improve significantly on the estimated call option prices obtained with a particular numerical scheme.<sup>2</sup>

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. 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 and discuss the implementation of the method proposed by Longstaff and Schwartz (2001) when combined with PCS. In Section 4, we perform a large-scale study on 3125 options, showing that our proposed method works extremely well. In Section 5, we conduct several robustness checks, and in Section 6, we propose a new metric for efficiency and sugges<sup>t</sup> a method for choosing the optimal specification to use for option pricing. Section 7 offers concluding remarks.
