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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).)