THE PURPOSE OF THIS CHAPTER

In this chapter we discuss how the modelling framework presented in the previous chapters can (must?) bemodified when nominal rates are very close to zero. The insight is based on Black's last paper (Black, 1995). The title of the paper (“Interest Rates as Options”) clearly suggests that option-like non-linearities enter (and complicate) the treatment of rate-dependent products, such as bonds. (The ‘option’ in the title, by the way, is the option to put one's money under the mattress if rates became substantially negative, rather than investing it at a certain negative rate of nominal return.)

The chapter starts with a brief descriptions of the macroeconomic implications of close-to-zero rates, of the options open to central banks faced with these macroeconomic conditions, and of the relevance of the attending policy responses (such as ‘quantitative easing’) for the shape of the yield curve.

From the technical side, first we highlight the technical problems brought about by a zero floor. Then we show that the zero floor affects, albeit to different extents, the whole yield curve, not just the very short end. Indeed, we show that the greatest effect need not be at the shortest-maturity end of the yield curve. Finally, we show how these complications can be easily and effectively overcome using an approximate procedure recently introduced by Wu and Xia (2014). This, of course, is not the only solution that has been proposed to handle the problem, but, as usual, we prefer to look at one model in detail, rather than providing a bird's eye view of the existing approaches. For completeness and guidance, we also provide in the closing section a space-shuttle view of the existing literature on the topic.

Before presenting the computational results, we explain in the opening sections why estimating the ‘shadowrate’ (ie, the rate thatwould apply if zerowere no lower bound) is important for macroeconomic applications, for predicting excess returns, and in the calibration of affine models.