Variance Reduction Applied to Machine Learning for Pricing Bermudan/American Options in High Dimension

In this paper, we propose an efficient method for computing the price of multi-asset American options, based on Machine Learning, Monte Carlo simulations and variance reduction technique. We consider specif- ically options written on a basket of assets, each of them following a Black-Scholes dynamics. In the wake of Ludkovski’s approach, we im- plement here a backward dynamic programming algorithm, based on a finite number of uniformly distributed exercise dates. On these dates, the option value is computed as the maximum between the exercise value and the continuation value, which is in turn obtained by means of Gaus- sian process regression technique and Monte Carlo simulations. Such a method performs well for low dimension baskets but remains inaccurate for very high dimension baskets. In order to improve the dimension range, we employ the European option price as a control variate, allowing us to treat very large baskets while reducing the variance of price estimators. Numerical tests demonstrate that the proposed algorithm is fast and reli- able, and able to handle American options on very large baskets of assets as well, thus overcoming the problem of the curse of dimensionality.