The following article is reprinted from the Quarter 2, 2007 issue
of On the Edge, the Interactive Data Fixed Income Analytics Quarterly newsletter.
Focus on Term Structure Models
Teri Geske
Senior Vice President, Product Development
This brief article summarizes a key concept in fixed income analytics, the notion of how to describe, within an analytical framework, the evolution of interest rates over time. This concept of a "term structure model" is an attempt to model mathematically the process that determines changes in interest rates.
Term Structure Models in Fixed Income Analytics
Interest rate risk is the major market risk that fixed income investors face. Over the last several decades, many term structure models (also known as interest rate models) have been proposed to characterize the evolution of interest rates, such as: Vasicek (V, 1977), Ho-Lee (HL, 1986), Hull-White (HW, 1990); Cox-Ingersoll-Ross (CIR, 1985); Dothan (D, 1980), Black-Karasinski (BK, 1991), Mercurio-Moraleda (MM, 2000); Longstaff-Schwartz (LS, 2001), Chen (C, 1996), etc.
Normal or Log-normal?
All of the above approaches assume that interest rates follow some underlying probability distribution. Although there is debate about which distribution best describes changes in interest rates, most models assume either a normal (e.g., V, HL, HW) or log-normal distribution (e.g., D, BK, MM) for the analytical tractability or the ease of implementation. The pros and cons of models with either underlying distribution assumptions can be better understood by looking at the definitions and characteristics of, and the relationship between these two distributions.
1) Normal and log-normal distributions are closely related. If X has a normal distribution, then Y=exp(X) will have a log-normal distribution. In other words, if Y is log-normally distributed, X=log(Y) will be normally distributed.
2) A normal random variable can assume all real numbers whereas a log-normal random variable can only be positive. Although log-normal models guarantee interest rates to be positive, they are often less tractable and they can also lead to model explosion (implausibly high rates) if not handled properly. Normal models usually offer good analytical tractability even though they can allow the possibility of negative rates. Parameters can be chosen to minimize the probability of negative rates occurring.
3) A normal random variable has a symmetric density function about its mean but this symmetry does not hold in a log-normal distribution (Figure 1).
Figure 1. Probability Density Distributions
4) The sum of independent normally distributed random variables is still normally distributed whereas the product of log-normal random variables has a log-normal distribution. Therefore, the normal distribution is often used to model a random variable that is additive with respect to many small random factors whereas the log-normal distribution is used to model a random variable that is the multiplicative product of many small independent random factors. For example, the log-normal distribution can be used to model long-term rates, since they can be viewed as the product of many short forward rates. However, empirical analysis of interest rate movements in the market indicates an actual distribution more normal than log-normal, adding credibility to the use of normal models.
References:
- Levin, A.: Interest Rate Model Selection, A conscientious Choice for Mortgage Investors, The Journal of Portfolio Management, Winter 2004, 74-86
- Brigo, D. and Mercurio, F.: Interest rate Models: Theory and Practice, Springer (2000)
- Lee, H.H.: Interest Rate Risk Models: Similarities and differences, Online document
- Limpert, E, Stahel, W.A and Abbt, M.: Log-normal Distributions Across the Sciences: Keys and Clues, BioSciences, 51(5), 341-352
- Baxter, M. and Rennie, A.: Financial Calculus : An Introduction to Derivative Pricing, Cambridge University Press (1996)
- Kazarian, D., Joneja, D. and Ren, W.H.: Yield Curve Model: Implications for MBS, Lehman Brothers Research (1998)
