Quasi-Monte Carlo-based conditional pathwise method for option Greeks

The calculation of option Greeks is important in financial risk management. However, the traditional pathwise method is not applicable to options with discontinuous payoffs. In this paper, using the idea of the conditional quasi-Monte Carlo method to smooth the payoff functions, we generalize the traditional pathwise method to calculate the first- and high-order Greeks. By taking a conditional expectation, the discontinuous integrand is smoothed. More importantly, the interchange of expectation and differentiation is proved to be possible. We show that the calculation of conditional expectations and then taking the derivatives with respect to the parameter of interest analytically is feasible for many common options. The new estimates for Greeks have good smoothness. For Asian and binary Asian options, for instance, our estimates are infinitely differentiable. So using the quasi-Monte Carlo method to estimate expectations improves the efficiency significantly. We also study the relationship of our method with several others in the literature, and show that our method is an extension of these methods. Numerical experiments are performed to demonstrate the high efficiency of the proposed method.