A Moment Matching Calibration under the Bilateral Gamma Model and Its Application

This chapter provides a closed-form solution for a moment matching calibration under the Bilateral Gamma (BG) model. The BG model is an exponential Lévy model with as underlying distribution a bilateral gamma distribution. The famous Variance-Gamma (VG) model is a special case of this BG model. Compared with the variance-gamma distributions, bilateral gamma distributions allow more flexible tail behaviors. Model calibration is a vital and classical problem of option pricing. The standard calibration process applies a search algorithm to find an optimal parameter set by minimizing an objective function representing the error between the model and market quotes. However, numerical optimization often suffers from local minima, an ill-posed problem, and the optimal result is sensitive to the initial value of the search algorithm. Moment matching calibration follows an alternative approach. It estimates the model parameters by matching a series of moments of the model with the corresponding market-implied moments. Hence, instead of applying a search algorithm, it obtains the optimal parameters by solving the moments matching equations, and thus it avoids the mentioned issues. Previous studies have developed the moment matching calibration for the mentioned VG model. This chapter provides an analytical solution of the moments matching equations for the BG model. Next, we evaluate its performance in a numerical study and compare a few variations of the methodology. As theoretically expected, the moment matching calibration under the BG model outperforms the moment matching calibration under the VG model. In terms of minimizing root-mean-squared errors, also as expected, it is suboptimal by construction compared with the search algorithm under the BG model. We quantify the gap and note that the straightforward moment matching calibration is obtained almost instantaneously in contrast to the search algorithm, which is rather time-consuming. Accordingly, we combine the moment matching calibration and standard calibration by using the moment matching solution as a plausible initial value for the search algorithm and improve model performance as such.