The Fundamental Theorem of Arbitrage Pricing

The Black-Scholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages (the term will be defined shortly) do not exist in efficient markets. Although this is never completely true in practice, it is a useful basis for pricing theory, and we shall limit our attention (at least for now) to efficient (that is, arbitrage-free) markets. We shall see that absence of arbitrage sometimes leads to unique determination of prices of various derivative securities, and gives clues about how these derivative securities may be hedged. In particular, we shall see that, in the absence of arbitrage, the market imposes a probability distribution, called a risk-neutral or equilibrium measure, on the set of possible market scenarios, and that this probability measure determines market prices via discounted expectation. This is the Fundamental Theorem of arbitrage pricing. Before we state the Fundamental Theorem formally, or consider its ramifications, we shall consider several simple examples of derivative pricing in which the Efficient Market Hypothesis allows one to directly determine the market price.