Forward Interest Rates and Volatility of Zero Coupon Yield in Affine Models

Abstract

The analysis of the forward rate curve for enough wide class of one factor affine models of the term structure that includes not only Vasiček’s Gaussian model and the square root model CIR but also models with any levels of the lower boundary of the short term (riskfree) interest rates is resulted. The multi-factor Gaussian model is discussed in details too. The special attention is given to the problem connected with the tendency for the term structure of long term forward rates to slope downwards.

For one-factor models with stochastic volatility the following results are derived: the probability that the forward rate curve slopes downwards for long term yield rates is found and is shown that this probability is influenced essentially not only by interest rate volatility but also by level of the lower boundary of short term rates and parameters of the risk premium; the expectations, variances and covariances for the forward rates and the yield process volatility are calculated; the correlation between the forward rates and the yield process volatility is always positive and does not depend on term to maturity; its lower boundary is found; the average slope of the forward rate curves is negative for all terms to maturity.

For one factor models with deterministic volatility (Gaussian models) the probability that the forward rate curve slopes downwards for long term yield rates is found; this probability always increases as the term to maturity increases but has the upper boundary that is dependent on the interest rate volatility; in the mean the slope of the forward rate curve is too negative independent on term to maturity.

For multifactor Gaussian models the representation of state variable process in the explicit form is derived and its covariance matrix is found; the probability that the forward rate curve slopes downwards is found also.

Keywords: forward rate curve, volatility of zero coupon yield, affine model, Vasiček’s Gaussian model, square root model CIR, term structure of long-term forward rates

JEL Classification: G12, C49, C32