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

Question

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?

Answer 1

The original Vasicek paper is "An equilibrium model of the term structure". If you google for it, you'll find it and you can read in his own words his motivation for developing it. In particular, what now is called the Vasicek model basically comes from applying his results to an Ornstein-Uhlenbeck model for the spot process, which he claims was proposed by Merton in 1971, in "Optimum Consumption and Portfolio Rules in a Continuous-time Mode", which is another reference you can track down on google. The clearest reference in there, says that:

"The first term in (120) implies a long-run, regressive adjustment of the expected rate of return toward a "normal" rate of return.... I will call the assumption of a price mechanism described by (119) and (120) the "De Leeuw" hypothesis for Frank De Leeuw who first introduced this type mechanism to explain interest rate behavior."

I couldn't track a specific reference down further back than that. I think a large part of the answer why people like the model is a combination of analytical tractability and obvious qualitative properties (i.e., mean reversion, meaning that interest rates levels "can't keep running away") that are consistent with behavior expected from interest rates.

Answer 2

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.

This model is the simplest example of a case in which the integral of the short rate as Gaussian distribution. Of course it has the gross disadvantage of allowing negative rates at any given time with positive probability, and eventually negative rates with probability 1.