Financial Risk Management: Models, History, and Institutions

CHAPTER 4

Nonlinear Risks and the Treatment of Bonds and Options

In the previous chapter, we discussed risk measurement for single “well-behaved” securities, with returns that can appropriately be mapped to returns of one risk factor. In this basic case, the asset also has returns that move linearly, that is, one-for-one or in a constant proportion to some underlying asset or risk factor return. But many securities are not at all well-behaved in this sense. Rather, they have nonlinear returns that have much larger or smaller responses to some other asset returns, depending on the asset price level.

Nonlinearity can vitiate risk measurement techniques such as VaR that are designed primarily for linear exposures. In this chapter, we discuss nonlinearity, and how to measure risk in its presence. We'll focus on two important examples of nonlinear securities, options and bonds. Another reality that we will have to address is that many assets are complex, and are exposed to multiple risk factors, and that most real-world portfolios contain many positions and are exposed to multiple risk factors. In the next chapter, studying risk measurement for portfolios, we focus on assets that are sensitive to multiple risk factors.

Options and option-like exposures depart in both ways from the previous chapter's model: nonlinearity and dependence on several risk factors. First, the P&L of an option is a nonlinear function of returns on the underlying asset. A relatively small return on the underlying ...