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. Peter Carr Banc of America Securities 9 West 57th Street, 40th floor New York, NY 10019 [email protected] Office Phone: (212) 583-8529 Fax Phone: (212) 583-8569 Current Version: September 23, 1999 I thank Warren Bailey, Akash Bandyopadhyay, George Constantinides, Peter Cyrus, Louis Gagnon, Jerry Hass, David Heath, Ali Hirsa, Neal Horrell, Eric Jacquier, Robert Jarrow, Herb Johnson, David Lando, Keith Lewis, Francis Longstaff, Dilip Madan, Ramesh Menon, Van Nguyen, Shivagi Rao, Eric Reiner, Steve Santini, Jeremy Staum, and especially George Pastrana, and Ravi Viswanathan for their comments. I also thank Joseph Cherian and especially Zhenyu Duanmu for excellent research assistance. The usual disclaimer applies. Deriving Derivatives of Derivative Securities