Introduction

In Chapter 6 we showed that the payoff pricing functional – and also its extension, the valuation functional – can be represented either by state prices or by risk-neutral probabilities. In this chapter we derive another representation of the payoff pricing functional, the pricing kernel. The existence of the pricing kernel is a consequence of the Riesz representation theorem, which says that any linear functional on a vector space can be represented by a vector in that space.

We begin by introducing the concepts of inner product, orthogonality, and orthogonal projection. These concepts are associated with an important class of vector spaces, the Hilbert spaces, to which the Riesz representation theorem applies. In the finance context, the Riesz representation theorem implies that any linear functional on the asset span can be represented by a payoff. Two linear functionals are of particular interest: the payoff pricing functional and the expectations functional, which maps every payoff into its expectation. Their representations are the pricing kernel and the expectations kernel, respectively.

Hilbert space methods are important for the study of the Capital Asset Pricing Model and factor pricing in the following chapters. Our treatment of these methods here is mathematically superficial, for our interest is in arriving quickly at results that are applicable in finance. In particular, the finite-dimensional contingent claims space RS is for us the primary example of a Hilbert space. The most important applications of Hilbert space methods come when the payoff space is infinite-dimensional.