The Financial Markets

Abstract

In this chapter we introduce the financial markets which will appear in our applications. In Section 3.1 a financial market in discrete time is presented. A prominent example is the binomial model. However, we do not restrict to finite probability spaces in general. We will define portfolio strategies and characterize the absence of arbitrage in this market. In later chapters we will often restrict to Markov asset price processes in order to be able to use the Markov Decision Process framework. In Section 3.2 a special financial market in continuous time is considered which is driven by jumps only. More precisely the asset dynamics follow so-called Piecewise Deterministic Markov Processes. Though portfolio strategies are defined in continuous time here we will see in Section 9.3 that portfolio optimization problems in this market can be solved with the help of Markov Decision Processes. In Section 3.3 we will briefly investigate the relation of the discrete-time financial market to the standard Black-Scholes-Merton model as a widely used benchmark model in mathematical finance. Indeed if the parameters in the discrete-time financial market are chosen appropriately, this market can be seen as an approximation of the Black-Scholes-Merton model or of even more general models. This observation serves as one justification for the importance of discrete-time models. Other justifications are that trading in continuous time is not possible or expensive in reality (because of transaction cost) and that continuous-time trading strategies are often pretty risky. In Section 3.4 utility functions and the concept of expected utility are introduced and discussed briefly. The last section contains some notes and references.