Mathematical Finance, Vol. 17, No. 1 (January 2007), 31–57 SELF-DECOMPOSABILITY AND OPTION PRICING PETER CARR Courant Institute, New York University H´ ELYETTE GEMAN Birkbeck, University of London and ESSEC Business School DILIP B. MADAN Robert H. Smith School of Business, University of Maryland MARC Y OR Laboratoire de Probabilit´ es et Mod` eles Al´ eatoires, Universit´ e Pierre et Marie Curie The risk-neutral process is modeled by a four parameter self-similar process of in- dependent increments with a self-decomposable law for its unit time distribution. Six different processes in this general class are theoretically formulated and empirically in- vestigated. We show that all six models are capable of adequately synthesizing European option prices across the spectrum of strikes and maturities at a point of time. Consid- erations of parameter stability over time suggest a preference for two of these models. Currently, there are several option pricing models with 6–10 free parameters that deliver a comparable level of performance in synthesizing option prices. The dimension reduc- tion attained here should prove useful in studying the variation over time of option prices. KEY WORDS: additive processes, scaling, Background Driving L´ evy Processes, Ornstein–Uhlenbeck Processes 1. INTRODUCTION The standard models for portfolio allocation (Merton 1973) and for option pricing (Black and Scholes 1973) both assume that continuously compounded returns are normally distributed. The central limit theorem is often invoked as a primary motivation for this assumption. By this theorem, the normal distribution arises as the limiting distribution for the sum of n independent random variables, when the sum is divided by √ n. Hence, if returns are realized as the sum of a large number of independent influences, then one can anticipate that returns will in fact be normally distributed. Dilip Madan would like to thank Ajay Khanna, Yong Ren, and Rick Shypit for discussions on the subject matter of this paper. Manuscript received August 2005; final revision received January 2006. Address correspondence to Dilip B. Madan, Robert H. Smith School of Business, University of Maryland; e-mail: dbm@rhsmith.umd.edu. C  2007 The Authors. Journal compilation C  2007 Blackwell Publishing Inc., 350 Main St., Malden, MA 02148, USA, and 9600 Garsington Road, Oxford OX4 2DQ, UK. 31