\/ mathrisk An Introduction to Option Pricing and the Mathematical Theory of  Risk
Marco Avellaneda
Proceedings of the American Math. Society Winter Meeting 1997

This review  paper discussed the topic of option  pricing with emphasis  modeling  financial risk. The Black-Scholes formula is derived using  the classical  dynamic hedging argument. Dynamic hedging justifies the valuation of contingent claims based on the use of  risk-neutral, as opposed to ``frequential'' probabilities.  This still leaves open --  even in the simplest case of stock option  contracts -- the issue of  specifying the volatility parameter or  other characteristics of the model describing the evolution of market prices. This ``specification problem'' leads us to the issue of economic  uncertainty, or  risk,  the raison d'être of derivatives markets and financial intermediation. Thus, the valuation of contingent claims under uncertainty goes far beyond the exercise of computing expected cash-flows. After a discussion of the classical principles of option risk-management using differential sensitivities (``Greeks''), I review some more recent proposals for modeling uncertainty. The idea is to consider, as a starting point, a spectrum of risk-neutral probability measures spanning a set of beliefs and to construct option spreads to reduce uncertainty across all measures at once. This last part of the paper draws on work with my collaborators (Avellaneda, Levy and Paras (1995), Avellaneda and Paras (1996) and Avellaneda, Friedman, Holmes and Samperi (1997).)