Conserving Capital by Adjusting Deltas for Gamma in the Presence of Skewness
Dilip B. Madan
University of Maryland - Robert H. Smith School of Business
August 19, 2010
Robert H. Smith School Research Paper No. RHS 06-132
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).
Number of Pages in PDF File: 23
Keywords: Acceptable Risks, Distorted Expectations, Non Linear Expectations, Markov Chains
JEL Classification: G10, G12, G13
Date posted: September 20, 2010 ; Last revised: November 23, 2010
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