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J. Risk Financial Manag. 2010, 3(1), 1-25; doi:10.3390/jrfm3010001
Article

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

Robert H. Smith School of Business, University of Maryland, College Park, MD. 20742, USA
Published: 31 December 2010
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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 delta 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 pro¯t, 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 pro¯t, 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: Bid and ask prices; concave distortions; non linear expectations; variance gamma model; non-uniform grids Bid and ask prices; concave distortions; non linear expectations; variance gamma model; non-uniform grids
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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MDPI and ACS Style

Madan, D.B. Conserving Capital by Adjusting Deltas for Gamma in the Presence of Skewness. J. Risk Financial Manag. 2010, 3, 1-25.

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