Notation and Definitions for Single - Period Models

These notes develop the modern theory of martingale pricing in a discrete-time, discrete-space framework. This theory is also important for the modern theory of portfolio optimization as the problems of pricing and portfolio optimization are now recognized as being intimately related. We choose to work in a discrete-time and discrete-space environment as this will allow us to quickly develop results using a minimal amount of mathematics: we will use only the basics of linear programming duality and martingale theory. Despite this restriction, the results we obtain hold more generally for continuous-time and continuous-space models once various technical conditions are satisfied. This is not too surprising as one can imagine approximating these latter models using our discrete-time, discrete-space models by simply keeping the time horizon fixed and letting the number of periods and states go to infinity in an appropriate manner.