# What is the reasoning to derive this financial model called the Vasicek Model?

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

$$\mathrm{d}r_t = a(b-r_t)\: \mathrm{d}t + \sigma \: \mathrm{d}W_t$$

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, $\sigma$, determines the Volatility (finance) of the interest rate and in a way characterizes the amplitude of the instantaneous randomness inflow. The typical parameters b, a and $\sigma$, together with the initial condition $r_0$, completely characterize the dynamics, and can be quickly characterized as follows, assuming a to be non-negative:

• $b$: "long term mean level". All future trajectories of $r$ will evolve around a mean level b in the long run;
• a: "speed of reversion". a characterizes the velocity at which such trajectories will regroup around b in time;
• $\sigma$: "instantaneous volatility", measures instant by instant the amplitude of randomness entering the system. Higher $\sigma$ implies more randomness

From the description of Wikipedia

What is the mathematical reasoning behind this formula for the finance professional to introduce this?

• Great stuff posting it here! Hopefully you will get a brilliant answer from the guys over here :). Good luck. – Chinny84 Aug 21 '14 at 19:50
• Well to model interest rates... Do you mean: why model them like that? – Bob Jansen Aug 21 '14 at 20:11
• @BobJansen - Yes, I think so – Victor Aug 21 '14 at 20:17
• The equation at the start of your question is not a model preposition. It's just a model, it has some useful properties but doesn't follow from the usual axioms. It's not the truth. Why we want to use it is in the Discussion section there and in the papers linked in at the bottom of the Wiki-page. – Bob Jansen Aug 21 '14 at 20:31
• As I understand it, it was a pretty basic rationale - the short interest rate is mean-reverting, and they just supposed Gaussian diffusion. The OU process was well-known, tractable, so why not? – experquisite Aug 21 '14 at 23:43

I think the rationale behind it is that if $r$ is the short rate, the the price of the bond is $P(t,T) = \mathbf{E}e^{- \int_t^T r_s ds }.$ As is well known by know is easy to calculate expectations of random variables of the form $e^Z$, where $Z$ is Gaussian.