: "long term mean level". All future trajectories of will evolve around a mean level b in the long run;
: "speed of reversion". characterizes the velocity at which such trajectories will regroup around in time;
: "instantaneous volatility", measures instant by instant the amplitude of randomness entering the system. Higher implies more randomness
The following derived quantity is also of interest,
: "long term variance". All future trajectories of will regroup around the long term mean with such variance after a long time.
and tend to oppose each other: increasing increases the amount of randomness entering the system, but at the same time increasing amounts to increasing the speed at which the system will stabilize statistically around the long term mean with a corridor of variance determined also by. This is clear when looking at the long term variance, which increases with but decreases with. This model is an Ornstein–Uhlenbeck stochastic process. Making the long term mean stochastic to another SDE is a simplified version of the cointelation SDE.
Discussion
Vasicek's model was the first one to capture mean reversion, an essential characteristic of the interest rate that sets it apart from other financial prices. Thus, as opposed to stock prices for instance, interest rates cannot rise indefinitely. This is because at very high levels they would hamper economic activity, prompting a decrease in interest rates. Similarly, interest rates do not usually decrease below 0. As a result, interest rates move in a limited range, showing a tendency to revert to a long run value. The drift factor represents the expected instantaneous change in the interest rate at time t. The parameter b represents the long-run equilibrium value towards which the interest rate reverts. Indeed, in the absence of shocks, the interest rate remains constant when rt = b. The parameter a, governing the speed of adjustment, needs to be positive to ensure stability around the long term value. For example, when rt is below b, the drift term becomes positive for positive a, generating a tendency for the interest rate to move upwards. The main disadvantage is that, under Vasicek's model, it is theoretically possible for the interest rate to become negative, an undesirable feature under pre-crisis assumptions. This shortcoming was fixed in the Cox–Ingersoll–Ross model, exponential Vasicek model, Black–Derman–Toy model and Black–Karasinski model, among many others. The Vasicek model was further extended in the Hull–White model. The Vasicek model is also a canonical example of the affine term structure model, along with the Cox–Ingersoll–Ross model.
