THE PURPOSE OF THIS CHAPTER

In this chapter we are going to look at the conditions of no-arbitrage from a more general, but more abstract, perspective – the so-called ‘modern’ approach. In doing so, we will re-derive the results obtained in Chapter 12 (in particular, the expression for the market price of risk) from a different perspective. In order to build her intuition, the reader may want to look at the results we derive in Chapter 15, but, in case the reader were to find the treatment in that chapter a bit heavy-going, we make the present derivation self-contained.

Why complicate things, when the derivation presented in Chapter 11 was so simple and so intuitively obvious? Mainly for three reasons.

First, the ‘modern’ approach applies to any asset, and allows us to see bonds as a special case of security that has a particular payoff in some states of the world. When we look at term premia, the concept of securities that perform well or badly in ‘states of the world’ becomes crucial; for instance, we will be naturally led to ask questions such as: ‘Does a bond portfolio perform well or poorly when equities fall or rise? What about inflation?’ Therefore, looking at bonds as one state-dependent component of the universe of assets available to the investor affords a fruitful perspective.

Second, it is almost impossible to read the contemporary literature on termstructure modelling without a basic understanding of, say, the stochastic discount factor, or of the main properties of the state-price deflator. Studying this chapter won'tmake the reader an expert in the topic, but will allow her to understand, for instance, the links between the Sharpe Ratio and the volatility of the state-price deflator, why the latter grows at (minus) the short rate and why its volatility is equal to (minus) the market price of risk. Given this bewildering variety of ways of looking at, and talking about, what is essentially the same phenomenon, I would therefore like to think that I can serve as a ‘liaison officer, making reciprocally intelligible voices speaking provincial languages.’