Deriving derivatives of derivative securities

Peter Carr

Abstract

ABSTRACT

Various techniques are used to simplify the derivations of "greeks" of path-independent claims in the Black-Merton-Scholes model. First, delta, gamma, speed, and other higher-order spatial derivatives of these claims are interpreted as the values of certain quantoed contingent claims. It is then shown 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, and this is illustrated for the case of European options.