Teaching Note 97-03:

One of the earliest stochastic models of the term structure was developed by Vasicek (1977). His model is based on the evolution of an unspecified short-term interest rate. Although the model is developed in a fairly general framework, it is most frequently applied in a more specific form that is presented here. The Vasicek model has many advantages, though it also has some shortcomings. The model assumes that the short rate follows the stochastic process ( ) dz dt r b a dr σ + − = . Here dz is a standard Wiener process and r is the current level of the interest rate. The parameter b is the long run normal interest rate. The model exhibits mean reversion, which means that if the interest rate is above the long run mean (r > b), then the coefficient a (>0) makes the drift become negative so that the rate will be pulled down in the direction of r. Likewise, if the rate is less than the long run mean, r < b, then the coefficient a makes the drift become positive so that the rate will be pulled up in the direction of r. The coefficient a is, thus, the speed of adjustment of the interest rate towards its long run normal level. This feature is particularly attractive because without it, interest rates could drift permanently upward the way stock prices do and this is simply not observed in practice. This particular type of stochastic process is referred to as an Ornstein-Uhlenbeck process. In the Vasicek model the risk is captured by assuming that the market price of interest rate risk, (μ r)/σ = λ, is constant across the term structure. This assumption is essentially the same as the no-arbitrage/equivalent martingale assumption and permits pricing in a risk-neutral framework. This results in the following formula for the price of a $1 face value zero coupon bond, ( ) ( ) ( )