Conserving Capital by Adjusting Deltas for Gamma in the Presence of Skewness

23 Pages Posted: 20 Sep 2010 Last revised: 23 Nov 2010

Dilip B. Madan

University of Maryland - Robert H. Smith School of Business

Date Written: August 19, 2010

Abstract

An argument for adjusting Black Scholes implied call deltas downwards for a gamma exposure in a left skewed market is presented. It is shown that when the objective for the hedge is the conservation of capital ignoring the gamma for the delat position is expensive. The gamma adjustment factor in the static case is just a function of the risk neutral distribution. In the dynamic case one may precompute at the date of trade initiation a matrix of delta levels as a function of the underlying for the life of the trade and subsequently one just has to look up the matrix for the hedge. Also constructed are matrices for the capital reserve, the profit, leverage and rate of return remaining in the trade as a function of the spot at a future date in the life of the trade. The concepts of profit, capital, leverage and return are as described in Carr, Madan and Vicente Alvarez (2010). The dynamic computations constitute an application of the theory of nonlinear expectations as described in Cohen and Elliott (2010).

Keywords: Acceptable Risks, Distorted Expectations, Non Linear Expectations, Markov Chains

JEL Classification: G10, G12, G13

Suggested Citation

Madan, Dilip B., Conserving Capital by Adjusting Deltas for Gamma in the Presence of Skewness (August 19, 2010). Robert H. Smith School Research Paper No. RHS 06-132. Available at SSRN: https://ssrn.com/abstract=1679519 or http://dx.doi.org/10.2139/ssrn.1679519

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)

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