Black Scholes Option Sensitivity Using High Order Greeks

Abstract

Option high order sensitivities have been presented by Carr P. as Greeks for geeks, though other authors have analyzed and insisted on the need to go beyond to the Delta-Gamma approximation usually considered in the practice of risk management. Actually in the stress-testing framework, as is required under Basel 3 bank regulation, adding high order Greeks may contribute to a good prediction of the option PL under extreme shocks. We revisit the Black-Scholes high order Greek parameters by providing their explicit formulas and proofs, which are expected to be more accessible for many readers. Limit of the use of these sensitivities are also analyzed here. Actually our main contribution in this work is on the introduction of an unified sensitivity approach with the ones used for other classes of assets as interest rates and commodities. This may be useful in the Credit Adjustment Valuation computation and hedging, where all aspects of risk (equity, interest rate, credit, commodities, …) need to be simultaneously considered.