**7. Conclusions**

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 path-wise 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 measured by, for example, the Root Mean Squared Error (RMSE). Our results show that, for a large sample of options with characteristics of relevance in real-life applications, the symmetric method on average performs much better than the regular pricing method, 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 classification algorithm 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 optimal method that minimizes the RMSE across all estimates. Our results also show that the relative importance of using the symmetric method increases with option maturity and with asset volatility and using symmetry methods 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 therefore conjecture that pricing such options using the symmetric method could improve the estimates significantly by decreasing their bias and RMSE by orders of magnitude.

### **Funding:** This research received no external funding.

**Acknowledgments:** I thank Pascal Francois for bringing the put-call symmetry property to my attention and the three reviewers for excellent comments that led to a much improved paper. Dayi Li provided expert research assistance. For additional numerical results and further details, please consult the online working paper version, which is available at https://papers.ssrn.com/sol3/papers.cfm?abstract\_id=3362426. More than 5,000,000 artificial individual options were priced in this paper, which would have been impossible without the computational resources made available by the Canadian Foundation for Innovation (CFI).

**Conflicts of Interest:** The author declares no conflict of interest.
