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.

中文翻译：

---

连续时间亚洲期权希腊人的基于准蒙特卡罗的条件 Malliavin 方法

尽管提出了许多计算离散时间亚洲期权希腊人的方法，但很少有人知道计算连续时间亚洲期权希腊人的方法。在本文中，我们在多维 Malliavin 微积分中开发了一个分部积分公式，并将其应用于多资产情况下的简单和复杂连续时间亚洲期权的希腊公式。我们讨论了 Malliavin 衍生品对希腊人连续时间亚洲期权的模拟估计的渐近收敛。我们将传统的 Malliavin 方法与拟蒙特卡罗方法相结合来计算希腊人。我们建议使用条件准蒙特卡罗方法来平滑 Malliavin 希腊人，并表明条件期望的分析计算对于许多常见的亚洲期权是可行的。我们证明希腊人的新估计具有良好的平滑性。例如，对于二元亚洲期权、亚洲看涨期权和涨停亚洲看涨期权，我们的估计是无限次可微的。数值实验证明了所提出的方法的显着效率提升，特别是对于具有不连续收益函数的选项。