Additive Processes with Bilateral Gamma Marginals

23 Pages Posted: 25 Feb 2020 Last revised: 8 May 2020

See all articles by Dilip B. Madan

Dilip B. Madan

University of Maryland - Robert H. Smith School of Business

King Wang

Morgan Stanley

Date Written: January 30, 2020


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.

Keywords: Bilateral Gamma, Self Decomposable Laws, Sato Process

JEL Classification: G10, G11, G12

Suggested Citation

Madan, Dilip B. and Wang, King, Additive Processes with Bilateral Gamma Marginals (January 30, 2020). Available at SSRN: or

Dilip B. Madan (Contact Author)

University of Maryland - Robert H. Smith School of Business ( email )

College Park, MD 20742-1815
United States
301-405-2127 (Phone)
301-314-9157 (Fax)

King Wang

Morgan Stanley ( email )

1585 Broadway
New York, NY 10036
United States

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