An Introduction to Financial Markets

Chapter Two Basic Problems in Quantitative Finance

In the following chapters, we will discuss at length how quantitative finance methods may be used to solve practically relevant problems. Before embarking on a detailed investigation, it is important to get a broad overview of the most relevant themes and their mutual relationships. Bearing this in mind, in this chapter we introduce simplified versions of the following problems:

Portfolio optimization. Given a set of risky financial assets, whose return is uncertain, we must decide the fraction of wealth allocated to each of them, in order to find a satisfactory risk–reward tradeoff, while complying with a set of constraints.

Risk measurement and management. Measuring risk is not only essential for financial firms when selecting a portfolio of assets. Nonfinancial firms are subject to financial risk as well, in terms of exposure to adverse movements in interest or currency exchange rates. The next logical step is managing risk, which typically involves hedging some risk factors away by suitable policies, and possibly taking a position on those risk factors on which we believe we can place a reasonable bet.

Asset pricing. Finding the fair price of a financial asset is useful when we want to determine whether it is under- or overpriced, in order to drive portfolio decisions and, possibly, to detect arbitrage opportunities (a concept that we will formalize in this chapter). Another typical application is dealing with over-the-counter ...