THE PURPOSE OF THIS CHAPTER

This chapter deals with expectations. Expectations about what? At a very general level, about the future paths of all the state variables that describe our model. But, ultimately, the expectations that ‘matter most’ are expectations about the future paths of the short rate. Why so? Because, as we will see, yields are convexity-adjusted path averages of the short rate.

Speaking of expectations, however, does not make much sense unless the measure (roughly speaking, the probability distribution) under which the expectation is taken is clearly specified. Therefore in the text that follows we look below at expectations of the short rate both under the real-world and under the risk-neutral measures.

Building on this, we provide a first heuristic argument to show that a bond price is the expectation in the risk-neutralmeasure of the exponential of (minus) the path of the short rate from evaluation time to maturity. At this stage we do not introduce with any degree of precision of conditions of the no-arbitrage – obtaining these will be the topic of Part III, which is by far the most heavygoing of the book. However, exactly because Part III is rather abstract and in parts more demanding, we believe that building as early as possible a good intuition of the bond-as-expectation result will put the reader in good stead when the going gets, if not necessarily tougher, certainly more abstract.

We then focus on affine models, and we show how expectations can be specified with thesemodels.We conclude the chapter with a word of caution against overconfident fitting of the expectation part of affine models to bond prices.

LINKING EXPECTATIONS WITH NO-ARBITRAGE

We have argued that if we just assign expectations, term premia and convexity in what we happen to think is a ‘plausible’ and ‘reasonable’ way, there is no guarantee that the resulting bond prices will be free of arbitrage.