The continuous-time Markov chain (CTMC) approximation method is a powerful tool that has recently been utilized in the valuation of derivative securities, and it has the advantage of yielding closed-form matrix expressions suitable for efficient implementation. For two types of popular path-dependent derivatives, the arithmetic Asian option and the occupation-time derivative, this paper obtains explicit closed-form matrix expressions for the Laplace transforms of their prices and the Greeks of Asian options, through the novel use of pathwise method and Malliavin calculus techniques. We for the first time establish the exact second-order convergence rates of the CTMC methods when applied to the prices and Greeks of Asian options. We propose a new set of error analysis methods for the CTMC methods applied to these path-dependent derivatives, whose payoffs depend on the average of asset prices. A detailed error and convergence analysis of the algorithms and numerical experiments substantiate the theoretical findings.

连续时间马尔可夫链（CTMC）逼近方法是一种功能强大的工具，最近已用于衍生证券的估值中，并且具有产生适合有效实现的封闭形式矩阵表达式的优势。对于两种流行的路径依赖型导数，算术亚洲期权和占领时间导数，本文通过新颖地使用pathwise，获得了价格的拉普拉斯变换和亚洲期权希腊人的显式闭式矩阵表达式。方法和Malliavin演算技术。当我们将其应用于亚洲期权的价格和希腊语时，我们首次确定了CTMC方法的确切二阶收敛速度。我们为应用于这些路径相关导数的CTMC方法提出了一套新的误差分析方法，其收益取决于资产价格的平均值。算法和数值实验的详细误差和收敛性分析证实了理论发现。