Deriving Better Second-Order Derivatives

Abstract

In his pioneer paper traced back to 1993, “Deriving Derivatives of Derivatives Securities,” Peter Carr used the operator calculus to show that that all partial derivatives of path independent claims can be represented in terms of the spatial derivatives. We generalized these results for multiasset situations. Reversing the relationships and expressing the higher-order Greeks (as gamma or cross gamma) in terms of the first-order Greeks leads to better numerical stability and convergence properties. We apply the results to evaluation and risk of an important energy asset as storage. In addition, we consider Greeks for the CEV model and the stochastic volatility case. At the time of our discussions, dating back in 2010–2011, I was mostly interested in applications for commodity derivatives. Peter suggested including the exponential Lévy model, his favorite subject; the CEV models; and the stochastic volatility case. In preparing this article, I kept the original draft, dated December 2011, of the write-up we worked out together. I reworked the write-up, and added storage models, numerical examples, and derivations.