Introduction

By the fundamental theorem of finance, the payoff pricing functional can be extended to a strictly positive (positive) valuation functional iff security prices exclude arbitrage (strong arbitrage). We show in this chapter that each strictly positive (positive) valuation functional can be represented by a vector of strictly positive (positive) state prices. State prices can easily be calculated as a strictly positive (positive) solution to a system of linear equations relating security prices and their payoffs. An implication of the existence of strictly positive (positive) state prices is the absence of arbitrage (strong arbitrage). An implication of the uniqueness of state prices is that markets are complete.

The valuation functional can also be represented by strictly positive (positive) probabilities of the states. These probabilities, commonly known as risk-neutral probabilities, are simple transforms of the state prices and therefore just as useful as those prices. Under the risk-neutral probabilities representation, the price of each security equals its expected payoff discounted by the risk-free return.

State Prices

In Chapter 3 we derived the state prices associated with given security prices under the assumption of complete markets. If markets are complete, the payoff pricing functional q is defined on the entire contingent claim space RS, and the state price vector q = (q1, …, qS) provides a representation of the functional q as q(z) = qz for every payoff z ∈ RS.