Deriving derivatives of derivative securities

We use various techniques to simplify the derivations of "greeks" of path-independent claims in the Black-Merton-Scholes model. We first interpret delta, gamma, speed, and other higher order spatial derivatives of these claims as the values of certain quantoed contingent claims. We then show that all partial derivatives of such claims can be represented in terms of these spatial derivatives. These observations permit the rapid deployment of high order Taylor series expansions, which we illustrate for European options.