Efficient Variance Reduction for American Call Options Using Symmetry Arguments

François-Michel Boire,Lars Stentoft,R Mark Reesor

doi:10.3390/jrfm14110504

François-Michel Boire, Lars Stentoft + Show 1 more

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https://doi.org/10.3390/jrfm14110504

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Abstract

Recently it was shown that the estimated American call prices obtained with regression and simulation based methods can be significantly improved on by using put-call symmetry. This paper extends these results and demonstrates that it is also possible to significantly reduce the variance of the estimated call price by applying variance reduction techniques to corresponding symmetric put options. First, by comparing performance for pairs of call and (symmetric) put options for which the solution coincides, our results show that efficiency gains from variance reduction methods are different for calls and symmetric puts. Second, control variates should always be used and is the most efficient method. Furthermore, since control variates is more effective for puts than calls, and since symmetric pricing already offers some variance reduction, we demonstrate that drastic reductions in the standard deviation of the estimated call price is obtained by combining all three variance reduction techniques in a symmetric pricing approach. This reduces the standard deviation by a factor of over 20 for long maturity call options on highly volatile assets. Finally, we show that our findings are not particular to using in-sample pricing but also hold when using an out-of-sample pricing approach.

Highlights

Key to valuing American options with a dynamic programming approach is estimating a continuation value that determines the optimal exercise strategy
Since control variates is always more effective for puts than calls, and since symmetric pricing already offers some variance reduction, we demonstrate that a drastic reduction of the standard deviation of the call option price is obtained by combining variance reduction techniques with a symmetric pricing approach
We note that when using control variates, either alone or in combination with other techniques, efficiencies are always larger for put options than for the corresponding call options and we demonstrate that the joint or total effect of using variance reduction techniques together with symmetric pricing for call options can lead to price estimates with substantially lower variance

Summary

Introduction

Key to valuing American options with a dynamic programming approach is estimating a continuation value that determines the optimal exercise strategy. For this task, simulation and regression-based methods are nowadays often preferred to other deterministic algorithms, like finite differences and multinomial trees, because they are easy to implement and because of their flexibility. Like other Monte Carlo pricing methods the LSM method is numerically costly and reducing its variance is important. This paper examines the relationship between the efficiency of variance reduction techniques and option features like moneyness, maturity, and asset volatility when pricing. Three classical variance reduction techniques are studied in the context of LSM pricing: antithetic sampling, control variates, and importance sampling. We restrict our attention to these three techniques because of their popularity, and because they do not require simulating additional paths

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