Financial Markets

Abstract

This chapter introduces the basic building blocks and assumptions used to set up a consistent framework for describing financial markets. We do this by starting with a general model of the basic financial instruments, called the primary traded assets, in Section 3.1. Their prices are described by stochastic processes or, to be more precise, by a corresponding stochastic differential equation. Trading with these financial instruments requires some basic trading principles defining the possibilities of how we can put the assets together (over time) and build a so-called portfolio (process). Given a specific financial product, we may be interested in replicating the cash flow paid by this product over time using our primary traded assets. In a heavily traded market we would hope that there will be no riskless profit which could be earned by selling the financial product and replicating it with the primary traded assets over time. If so, the market is called arbitrage-free. In Section 3.2 we show under which conditions there are no arbitrage opportunities in the financial market. We will then change the price scale of the primary traded assets and rather observe the prices relative to a specific unit price or numéraire. If the resulting normalized or discounted market prices can be described by martingales with respect to a so-called martingale measure we will show that the financial market is free of arbitrage opportunities. Furthermore, we will show that today’s price of a (discounted) primary traded asset is equal to the expected value of any corresponding and adequately discounted future market price where the expectation is taken with respect to the martingale measure. Another important characteristic of a financial market is its completeness, i.e. the possibility of replicating every financial product traded in the market. As we will learn in Section 3.3, this important feature is closely related to the uniqueness of the martingale measure. In Section 3.4 we will prove that financial derivatives can be uniquely priced if the financial market is complete. As an application, one of the most famous market models, the Black-Scholes model, is discussed in Section 3.5. It is used to derive the prices for (European) options on contingent claims by taking expectations with respect to the corresponding martingale measure. However, depending on the specific financial product, it may be convenient to change the numéraire or unit price and corresponding measure to significantly simplify the calculation of the expected value or price. This important technique is described and applied in Sections 3.6 and 3.7.