  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.
