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LOS 34.k: Describe modern term structure READING 34: THE TERM STRUCTURE AND
models and how they are used. INTEREST RATE DYNAMICS
dr = a(b − r)dt + σdz MODULE 34.6: INTEREST RATE MODELS

Assume that the current short-term rate is r = 3% and the long-run value for the short-term rate is b = 8%. Also assume that the
speed of the adjustment factor is a = 0.40 and the annual volatility is σ = 2%. The Vasicek model provides the following formula
for the change in short-term interest rates, dr:

dr = 0.40(8% – r)dt + (2%)dz

The stochastic term, dz, is typically drawn from a standard normal distribution with a mean of zero and a standard
deviation of 1. Assume that a random number generator produced standard normal random error terms of 0.45, 0.18,
−0.30, and 0.25. The Vasicek model would produce the evolution of interest rates shown below:

Note that the simulation of interest rates Note that because
starts with an interest rate of 3%, which both the Vasicek
is well below the long-run value of 8%. model and the CIR
model require the
Interest rates generated by the model short-term rate to
move quickly toward this long-run follow a certain
value despite declining in the third process, the
time period, which reflects the mean estimated yield curve
reversion built into the model via the may not match the
drift term a(b – r)dt. observed yield curve.
But if the parameters
This example is stylized and intended of the models are
for illustrative purposes only. The believed to be
parameters used in practice typically correct, then
varysignificantly from those used here. investors can use
these models to
determine
mispricings.