Product Options

24 Pages Posted: 12 Jul 2021

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: July 9, 2021


Options paying the product of put and or call option payouts at different strikes on two underlying assets are observed to synthesize joint densities and replicate differentiable functions of two underlying asset prices. The pricing of such options is undertaken from three perspectives. The first uses a geometric two dimensional Brownian motion model. The second inverts two dimensional characteristic functions. The third uses a bootstrapped physical measure to propose a risk charge minimizing hedge using options on the two underlying assets. The options are priced at the cost of the hedge plus the risk charge.

Keywords: Multivariate Bilateral Gamma, Fast Fourier Transform, Distorted Expectations, Acceptable Risks.

JEL Classification: G10, G11, G12

Suggested Citation

Madan, Dilip B. and Wang, King, Product Options (July 9, 2021). 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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