In this note we consider the approximation of the Greeks Delta and Gamma of American-style options through the numerical solution of time-dependent partial differential complementarity problems (PDCPs). This approach is very attractive as it can yield accurate approximations to these Greeks at essentially no additional computational cost during the numerical solution of the PDCP for the pertinent option value function. For the temporal discretization, the Crank-Nicolson method is arguably the most popular method in computational finance. It is well-known, however, that this method can have an undesirable convergence behaviour in the approximation of the Greeks Delta and Gamma for American-style options, even when backward Euler damping (Rannacher smoothing) is employed. In this note we study for the temporal discretization an interesting family of diagonally implicit Runge-Kutta (DIRK) methods together with the two-stage Lobatto IIIC method. Through ample numerical experiments for one- and two-asset American-style options, it is shown that these methods can yield a regular second-order convergence behaviour for the option value as well as for the Greeks Delta and Gamma. A mutual comparison reveals that the DIRK method with suitably chosen parameter $\theta$ is preferable.

翻译：本文主要研究通过求解时间依赖偏微分互补问题（PDCPs）的数值解来逼近美式期权的Delta和Gamma值。该方法极具吸引力，因为在求解相关期权价值函数的PDCP过程中，几乎无需额外计算成本即可获得这些Greeks值的精确近似。在时间离散方面，Crank-Nicolson方法当属计算金融领域最流行的技术。然而众所周知，该方法在逼近美式期权的Delta和Gamma时可能出现不理想的收敛行为，即便采用向后欧拉阻尼（Rannacher平滑）处理也无法避免。本文研究了时间离散中一类有趣的隐式对角Runge-Kutta（DIRK）方法族，以及两级Lobatto IIIC方法。通过针对单资产和双资产美式期权的充分数值实验表明，这些方法能够使期权价值及其Delta和Gamma值呈现正则的二阶收敛行为。相互比较显示，参数θ选择得当的DIRK方法更具优势。