Don H. Kim

2008-31

**Abstract: **
This paper points out that several known ways of modeling non-negative
nominal interest rates lead to different implications for the risk-neutral
distribution of the short rate that can be checked with options data.
In particular, Black's boundary models ("interest rates as options")
imply a probability density function (pdf) that contains a Dirac delta
function and a cumulative distribution function (cdf) that is nonzero
at the zero boundary, while models like the CIR and positive-definite
quadratic-Gaussian (QG) models have a zero cdf at the boundary. Eurodollar
futures options data are found to favor Black's boundary models: the CIR/QG
models, even multifactor versions, have difficulty capturing option prices
accurately not only in low interest rate environments but also in higher
interest rate environments, and data in early 2008 provide an almost tangible
signature of the Dirac delta function in Black's boundary pdf models. Options
data also contradict the prediction of well-known models whose cdf is zero
at the zero boundary, namely that the risk-neutral pdf is always positively
skewed.

**Keywords:** Zero bound, term structure of interest rates, options, probability density functions