A note on the numerical approximation of Greeks for American-style options

TL;DR

The paper tackles accurate numerical approximation of Delta and Gamma for American options by solving time-dependent partial differential complementarity problems (PDCPs). It compares Crank–Nicolson, diagonally implicit Runge–Kutta (DIRK), and Lobatto IIIC temporal discretizations, all via a penalty approach on one- and two-asset Black–Scholes models with a nonuniform time grid to avoid order reduction. The results show that CN-based schemes suffer from degraded convergence for Greeks, unless very fine time steps are used, whereas DIRK methods with theta = 1 - 0.5*sqrt(2) and theta = 1/3, and Lobatto IIIC, achieve regular second-order convergence for both option values and Greeks, with the DIRK variants generally the best in accuracy and robustness. The findings provide practical guidance for stable, efficient Greeks computation in American-option PDCPs, enabling reliable risk assessment with moderate computational overhead in multi-asset settings.

Abstract

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 $θ$ is preferable.

Figures (11)

• Figure 1: Sample spatial grid for $m=50$, $K=100$, $S_{\rm max} = 500$.
• Figure 2: The value, Delta and Gamma functions of an American put option for $t=T$ and parameter set \ref{['1Dputpar']}.
• Figure 3: American put and parameter set \ref{['1Dputpar']}. Temporal discretization errors of the BE-P (dark blue bullets), CN-P (green diamonds), DIRKa-P (light blue squares), DIRKb-P (orange squares) and Lobatto-P (red bullets) methods for $m=200$ in the case of a uniform temporal grid (left) and the nonuniform temporal grid \ref{['variablestep']} (right).
• Figure 4: American put and parameter set \ref{['1Dputpar']}. Temporal discretization errors of the BE-P, CN-P, DIRKa-P, DIRKb-P and Lobatto-P methods for $m=100$ (left column) and $m=200$ (right column) for the option value (top row), the Delta (middle row) and the Gamma (bottom row). Nonuniform temporal grid \ref{['variablestep']}.
• Figure 5: American put and parameter set \ref{['1Dputpar']}. Temporal discretization errors of the BE-P, CN-P, DIRKa-P, DIRKb-P and Lobatto-P methods for $m=300$ (left column) and $m=400$ (right column) for the option value (top row), the Delta (middle row) and the Gamma (bottom row). Nonuniform temporal grid \ref{['variablestep']}.