Today we are going to see another beautiful example called Vasicek model. It’s a type of one factor short rate model (which you will see) driven by the market risk. The model can be used in the valuation of interest rate derivatives, and has also been adapted for credit markets. It was introduced in 1977 by Oldřich Vašíček.
Let \(\mathbb{W(t),\ t \geq 0}\) be a Brownian motion. The Vasicek model for interest rate process \(\mathbb{R(t)}\) is given by-
\[\mathbb{dR(t)}\ =\ \mathbb{(\alpha\ -\ \beta\ R(t))\ dt\ +\ \sigma\ dW(t) }\] where \(\mathbb{\alpha}\), \(\mathbb{\beta}\), \(\mathbb{\sigma}\) are positive constants.
It is a Stochastic differential equation. It has a closed form solution. I am not going to find the closed form solution from the differential form.
But I am giving you the closed form and see if the differential equation matches.
\[\mathbb{R(t)}\ =\ \mathbb{e^{-\beta t}\ R(0)}\ +\ \mathbb{\frac{\alpha}{\beta}\ (1-e^{-\beta t})}\ +\ \mathbb{\sigma\ e^{-\beta t}\ \int_{0}^{t} e^{\beta s}\ dW(s)}\]
To check this, we need to calculate differential of the R.H.S. of the above equation. What we need to use is the Ito- Doeblin formula.
\[\begin{aligned} & \mathbb{f(t,x)}\ =\ \mathbb{e^{-\beta\ t}\ R(0)}\ +\ \mathbb{\frac{\alpha}{\beta}\ (1\ -\ e^{-\beta\ t})}\ +\ \mathbb{\sigma\ e^{-\beta\ t}\ x } \qquad \text{and} \\ & \mathbb{X(t)}\ =\ \mathbb{\int_{0}^{t}e^{-\beta\ s}\ dW(s)} \end{aligned}\]
But why have we taken this peculiar type of function to start with?
This is because, it helps us to write- \(\mathbb{R(t)}\ = \mathbb{f(t,X(t))}\), which is the combination of an ordinary function, \(\mathbb{f}\) taking dummy variables- \(\mathbb{t}\) and \(\mathbb{x}\).
Another function is an Ito process, \(\mathbb{X(t)}\).
These are some simple tricks; you will pick these up as you practice. Let’s see what will be my advantage for going in this route.
First, let me write the Ito-Doeblin formula for Ito process with usual notations-
\[\mathbb{df(t,X(t))}\ =\ \mathbb{f_{t}(t,X(t))\ dt}\ +\ \mathbb{f_{x}(t,X(t))\ dX(t)}\ +\ \frac{1}{2}\ \mathbb{f_{xx}(t,X(t))\ dX(t)\ dX(t)}\]
So, we need 3 things- \(\mathbb{f_{t}(t,x)}\), \(\mathbb{f_{x}(t,x)}\) and \(\mathbb{f_{xx}(t,x)}\). For the chosen \(\mathbb{f(t,x)}\), we have-
\[\begin{aligned} & \mathbb{f_{t}(t,x)}\ =\ \mathbb{-\beta\ e^{-\beta\ t}\ R(0)}\ +\ \mathbb{\alpha\ e^{-\beta\ t}}\ -\ \mathbb{\sigma\ \beta\ e^{-\beta\ t}\ x}\ =\ \mathbb{\alpha\ -\ \beta\ f(t,x)} \\ & \mathbb{f_{x}(t,x)}\ =\ \mathbb{\sigma\ e^{-\beta\ t}} \\ & \mathbb{f_{xx}(t,x)}\ =\ 0 \\ \end{aligned}\]
We will also need the differential of \(\mathbb{X(t)}\), i.e. \(\mathbb{dX(t)} = \mathbb{e^{\beta\ t}\ dW(t)}\). So the Ito-Doeblin formula states that-
\[\begin{aligned} \mathbb{df(t,X(t))}\ &=\ \mathbb{(\alpha\ -\ \beta\ f(t,x))\ dt}\ +\ \mathbb{\sigma\ e^{-\beta\ t}\ dX(t)}\ +\ \frac{1}{2}\ \mathbb{0.\ dX(t)\ dX(t)} \\ &=\ \mathbb{(\alpha\ -\ \beta\ f(t,x))\ dt}\ +\ \mathbb{\sigma\ e^{-\beta\ t}\ e^{\beta\ t}\ dW(t)} \\ &=\ \mathbb{(\alpha\ -\ \beta\ f(t,x))\ dt}\ +\ \mathbb{\sigma\ dW(t)} \\ \end{aligned}\]
Doesn’t it look similar to the equation we started with? Not only that- due to the choice of the function we have- \(\mathbb{f(0, X(0)) = R(0)}\).
So we must have- \(\mathbb{f(t, X(t)) = R(t)}\) for all \(\mathbb{t \geq 0}\).
Now from Ito integral for deterministic integrands in the previous blog we can write that- \[\mathbb{X(t)}\ =\ \mathbb{\int_{0}^{t}e^{-\beta\ s}\ dW(s)}\ \sim \mathbb{N(0, \int_{0}^{t}e^{-2\beta\ s}\ ds)}\ \equiv \mathbb{N(0,\frac{1}{2\beta}\ (e^{2\beta t}-1))}\]
Therefore- \[\mathbb{R(t)}\ \sim \mathbb{N(\mathbb{e^{-\beta t}\ R(0)}\ +\ \mathbb{\frac{\alpha}{\beta}\ (1-e^{-\beta t})}\ , \frac{\sigma^2}{2\beta}(1- e^{-2\beta t}) )}\]
Since \(\mathbb{R(t)}\) follows a Normal distribution, \(\mathbb{\Pr[R(t) < 0]} > 0\). An interest rate taking negative value with positive probability, not good for a interest rate model.
But the Vasicek model has an interesting property; the interest rate is mean-reverting.
i.e. if \(\mathbb{R(t)} = \frac{\alpha}{\beta}\), the drift term (the term multiplied to \(\mathbb{dt}\)) is zero.
So if \(\mathbb{R(t)} > \frac{\alpha}{\beta}\), the drift term is negative. So it pushes \(\mathbb{R(t)}\) down to \(\frac{\alpha}{\beta}\).
And if \(\mathbb{R(t)} < \frac{\alpha}{\beta}\), the drift term is positive, it pushes \(\mathbb{R(t)}\) back to \(\frac{\alpha}{\beta}\).
So ultimately we have-
If \(\mathbb{R(0)} = \frac{\alpha}{\beta}\), then \(\mathbb{E[R(t)|\ F(0)]} = \frac{\alpha}{\beta}\) for all \(\mathbb{t \geq 0}\).
If \(\mathbb{R(0)} \neq \frac{\alpha}{\beta}\), then \(\mathbb{\underset{t \rightarrow \infty}{\lim}\ E[R(t)]} = \frac{\alpha}{\beta}\).
Now that we have seen Vasicek model and understood it’s merits more importantly it’s demit which is- \(\mathbb{\Pr[R(t) < 0]} > 0\), we can improve the model. This improved model is CIR interest rate model which I will tell you in the next blog.