Additive Processes with Bilateral Gamma Marginals

Abstract

The Sato process associated with self decomposable laws at unit time is further generalized to an additive process with arbitrary innovation term structures. A second generalization to additive processes consistent with bilateral gamma marginal distributions is also made. The Sato process is a parametric special case of the two generailzations. This feature is exploited in defining calibration starting values. Calibration results are presented for 1255 days of daily data on SPY options. The deterministic innovation variance model makes a median improvement of 15% in root mean square error over the Sato process. The comparable value for the general additive process is 40%. The Sato process relative to the general additive process overprices negative moves and underprices positive ones. The underpricing of negative moves decreases with maturity. On the positive side the overpricing decreases with maturity. For negative moves the overpricing is larger for smaller moves, while for positive moves the underpricing is larger for the larger moves.