Vasicek Model - Details

Details

The model specifies that the instantaneous interest rate follows the stochastic differential equation:

where W_t is a Wiener process under the risk neutral framework modelling the random market risk factor, in that it models the continuous inflow of randomness into the system. The standard deviation parameter, determines the volatility of the interest rate and in a way characterizes the amplitude of the instantaneous randomness inflow. The typical parameters and, together with the initial condition, completely characterize the dynamics, and can be quickly characterized as follows, assuming to be non-negative:

• : "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.