Complexity reduction for calibration to American options

American put options are among the most frequently traded single stock options, and their calibration is computationally challenging since no closed-form expression is available. Due to their higher flexibility compared with European options, the mathematical model involves additional constraints, and a variational inequality is obtained. We use the Heston stochastic volatility model to describe the price of a single stock option. In order to speed up the calibration process, we apply two model-reduction strategies. First, we introduce a reduced basis method. We thereby reduce the computational complexity of solving the parametric partial differential equation drastically, compared with a classical finite-element method, which makes applications of standard minimization algorithms for the calibration significantly faster. Second, we apply the so-called de-Americanization strategy. Here, the main idea is to reformulate the calibration problem for American options as a problem for European options and to exploit closed-form solutions. These reduction techniques are systematically compared and tested for both synthetic and market data sets.