Nonparametric Finance

Chapter 15Pricing in Incomplete Models

We give an overview of various approaches to price derivatives in incomplete markets. In an incomplete market, there exists derivatives which cannot be exactly replicated. The second fundamental theorem of asset pricing (Theorem 13.3) states that if the market is arbitrage-free and complete, then there is only one equivalent martingale measure, and thus there is only one arbitrage-free price. When the arbitrage-free market is not complete, then there are many equivalent martingale measures, and thus there are many arbitrage-free prices.

In this chapter, we describe some approaches for choosing the equivalent martingale measure from a set of available equivalent martingale measures, in the case of an incomplete market. Chapter 16 is devoted to the study of quadratic hedging and pricing. In this chapter, we give only a short description of quadratic pricing, and concentrate to describe other methods.

Utility maximization provides a general method for the construction of an equivalent martingale measure. We show that the Esscher measure, which was used to prove the first fundamental theorem of asset pricing (Theorem 13.1), is related to the maximization of the expected utility, when the utility function is the exponential utility function. The concept of marginal rate of substitution provides a heuristic way to connect the utility maximization to the pricing of options. Minimizing the relative entropy between a martingale measure and the physical ...