A smooth estimator for MC/QMC methods in finance

We investigate the effect of martingale control as a smoother for MC/QMC methods. Numerical results of estimating low-biased solutions for American put option prices under the Black–Scholes model demonstrate that using QMC methods can be problematic. But it can be fixed by adding a (local) martingale control variate into the least-squares estimator to gain accuracy and efficiency. In examples of estimating European option prices under multi-factor stochastic volatility models, randomized QMC methods improve the variance by merely a single digit. After adding a martingale control, the variance reduction ratio raise up to 700 times for randomized QMC and about 50 times for MC simulations. When the delta estimation problem is considered, the efficiency of the martingale control variate method decreases. We propose an importance sampling method which performs better particularly in the presence of rare events.