Introduction

Aim of the chapter

The aim of this chapter is to introduce the general theory of derivative pricing in a simple and intuitive, but rather unconventional, way. The usual presentation, which can be found in all the available books on the subject, relies on particular models where it is possible to construct riskless hedging strategies, which replicate exactly the corresponding derivative product. Since the risk is strictly zero, there is no ambiguity in the price of the derivative: it is equal to the cost of the hedging strategy. In the general case, however, these ‘perfect’ strategies do not exist. Not surprisingly for the layman, zero risk is the exception rather than the rule. Correspondingly, a suitable theory must include risk as an essential feature, which one would like to minimize. The following chapters thus aim at developing simple methods to obtain optimal strategies, residual risks, and prices of derivative products, which take into account in an adequate way the peculiar statistical nature of financial markets, that have been described in Chapters 6, 7, 8.

Strategies in uncertain conditions

A derivative product is an asset the value of which depends on the price history of another asset, the ‘underlying’. The best known examples, on which the following chapters will focus in detail, are futures and options.