This chapter introduces optimization—a methodology for selecting an optimal strategy given an objective and a set of constraints. Optimization appears in a variety of financial applications, including portfolio allocation, trading strategies, identifying arbitrage opportunities, and pricing financial derivatives. In this chapter, we motivate the discussion with a simple example, and describe how optimization problems are formulated and solved.

Let us recall the retirement example from Chapter 5.2.3. We showed how to compute the realized return on the portfolio of stocks and bonds if we allocate 50% of the capital in each of the two investments. Can we obtain a “better” portfolio return with a different allocation? (As we discussed in Chapter 5, a “better” return is not well defined in the context of uncertainty, so for the sake of argument, let us assume that “better” means higher expected return.) We found that if the allocation in stocks and bonds is 30% and 70%, respectively, rather than 50% and 50%, we end up with a lower portfolio expected return, but also lower portfolio standard deviation. What about an allocation of (60%, 40%)? Of (80%, 20%)?

In this example, we are dealing with only two investments (the two asset classes), and we have no additional requirements on the portfolio structure. It is, however, still difficult to enumerate all the possibilities and find those that provide the optimal tradeoff of return and risk. In practice, ...

Start Free Trial

No credit card required