Quasi-Monte Carlo-Based Conditional Malliavin Method for Continuous-Time Asian Option Greeks

Although many methods for computing the Greeks of discrete-time Asian options are proposed, few methods to calculate the Greeks of continuous-time Asian options are known. In this paper, we develop an integration by parts formula in the multi-dimensional Malliavin calculus, and apply it to obtain the Greeks formulae for both simple and complex continuous-time Asian options in the multi-asset situation. We discuss the asymptotic convergence of simulation estimates for Greeks of continuous-time Asian options by Malliavin derivatives. We combine the traditional Malliavin method with the quasi-Monte Carlo method to calculate the Greeks. We propose to use the conditional quasi-Monte Carlo method to smooth Malliavin Greeks, and show that the calculation of conditional expectations analytically is viable for many common Asian options. We prove that the new estimates for Greeks have good smoothness. For binary Asian options, Asian call options and up-and-out Asian call options, for instance, our estimates are infinitely times differentiable. Numerical experiments demonstrate the large efficiency improvement of the proposed method, especially for options with discontinuous payoff functions.