Chapter 18 - Modeling Term Structure and Related Concepts

Abstract

The first arbitrage relation that we need to study is the one between investment in very short-term savings accounts and bonds. Suppose both of these are default-free. How would the long-term bond prices relate to depositing money in a short-term savings account and then rolling this over continuously? It is clear that when one buys a longer term bond, the commitment is for more than one night, or one month. During this “long” period, several risky events may occur, and these may affect the price of the bond adversely. Yet, the overnight investment will be mostly immune to the risky events because the investor's money is returned the next “day,” and hence can be reinvested at a higher overnight rate. Thus, it appears that longterm bonds should pay a premium relative to overnight money, in order to be held by risk averse investors. In the Black–Scholes world the switch to the risk-neutral measure eliminated these risk premia and gave us a pricing equation. Can the same be done with interest-sensitive securities and random spot rates? We will see that the answer is yes. In fact, the classical approach to pricing interest-sensitive securities exploits this particular arbitrage relation extensively. The second arbitrage relation is specific to fixed income. Fixed income markets provide many liquid instruments that are almost identical except for their maturity. For example, we have a spectrum of discount bonds that are differentiated only by their maturity. Similarly, we have forward rates of different maturities. It turns out that this multidimensional aspect of interest-sensitive instruments permits writing down complex arbitrage relations between a set of zero-coupon bonds and a set of forward rates. In fact, if we have a k-dimensional vector of bond prices, we can relate this to a vector of forward rates using arbitrage arguments. These arbitrage relations form the basis of the Heath–Jarrow–Morton approach to pricing interest-sensitive securities. Thus, one way or another, the material in the present chapter should be regarded as a necessary background to discussing pricing of interest sensitive securities.