The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 632 Wednesday, February 4, 2004 1:33 PM

632 The Mathematics of Financial Modeling and Investment Management

If intermediate payoffs are zero the previous formula becomes

T–r ud

Q ∫t u

St = Et e ST

Given the final price ST, there is a pricing function in the sense that

T–r ud

Q ∫t u

St = Fi( t, t) = Et e ST

The pricing function satisfies a Feynman-Kac formula and is the solu-

tion of a PDE. It tells us that the price St is a function of time t and of

the interest rate at time t.

Multifactor Term Structure Model

The above discussion presented the derivation of the term structure

from the interest rate process. We say that, under this assumption, the

term structure model is a one-factor model because it depends on one

single process. Empirical analysis has shown that one factor is insuffi-

cient. Principal component analysis of the term structure of the U.S.

Treasury market, as well as other country government bond markets,

has shown that three factors are sufficient to explain 98% of the term

structure fluctuations. The three factors are the level, slope, and curva-

ture of the yield curve. Typically 90% of the term structure is explained

by changes in the level of interest rates. Around 8% is explained by

changes in the slope, or steepness, of the spot rate curve. Exhibit 20.4

provides a summary of these studies.^10

Multifactor models of the term-structure have been proposed. Note

that multifactor models described in the literature and currently used by

practitioners might use variables such as the long-term interest rate and

the short-term interest rate. This might give the impression that the

short-term interest rate is not sufficient to determine the term structure.

This is not true. The short-term rate is indeed sufficient to completely

determine the term structure. Conversely, given the term structure,

(^10) In addition to the references in Exhibit 20.4, there is the study from which the ex-
hibit is reproduced: Lionel Martellini, Philippe Priaulet, and Stéphane Priaulet, “An
Empirical Analysis of the Domestic and Euro Yield Curve Dynamics,” Chapter 24 in
Frank J. Fabozzi and Moorad Choudhry (eds.), The Handbook of European Fixed
Income Markets (Hoboken, NJ: John Wiley & Sons, 2004).