# SFB 303 Discussion Paper No. B - 202

**Author**: Föllmer, Hans

**Title**: Probabilistic Aspects of Options

**Abstract**: In recent years, derivative securities such as options have generated a lot of interest, both on a practical and
on a theoretical level. This survey gives an introduction to those aspects of the theory of options which seem
particularly interesting from a probabilistic point of view. In section 1 we describe the mathematical model of a
complete financial market where contingent claims can be represented as stochastic integrals of the underlying
price fluctuation. A fundamental representation theorem of K. Itô implies that the standard diffusion model for
risky assets is indeed complete. In section 2 we consider situations which are incomplete: A typical claim now
carries an intrinsic risk, and hedging strategies can only reduce the actual risk to that intrinsic component.
Section 3 comments on some relations between option pricing, actuarial premium principles and economic
equilibrium analysis. In section 4 we take a closer look at the structure of the probability measure P which
models the price fluctuations of the underlying asset. We review some of the arguments in favour of geometric
Brownian motion and conclude with a few tentative remarks on possible modifications.

**Keywords**: Options, Financial markets, contingent claims, Brownian motion, Risk exchanges

**JEL-Classification-Number**:

**Creation-Date**: November 1991

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