The mathematics of pricing contingent claims in incomplete markets using discrete stochastic models

This thesis focuses on pricing derivatives securities such as stock options in incomplete financial markets. The goal is to determine arbitrage free prices for these securities. For this we consider a finite state, discrete time stochastic model of a financial market known as the finite market model. We restrict our attention to derivatives securities known as European contingent claims, those that can only be exercised on the expiration date. In the early chapters, we define the model precisely and also summarize the pricing theory for complete markets. In this case, it turns out that there is a unique way to price arbitrage freely. This unique price can be computed as a certain conditional expected value under the associated equivalent martingale measure. The larger goal of this thesis is to give a thorough exposition of the pricing theory for incomplete markets. We will show that in these markets, arbitrage free prices exist, but unique pricing cannot always be obtained. When a particular price is not unique, there is an open interval over which the price can vary freely. The left ( resp. right) end points of this interval can be characterized as an infimum (resp. a supremum) of a certain conditional expected value over the set of associated equivalent martingale measures. Keywords: Financial Markets, Incomplete Markets, European Contingent Claims, Discrete Stochastic Models, and Arbitrage Free Pricing