Interest Rate Models

The class groups discussed two of term structure models: The Vacisek Model and the CIR term model.  These models are used to model interest rates.

What about the models:

• equilibrium type of model
• uses the Mean Reversion 
• offer benefits since they are numerically quite simple and easy to solve with computers
• very helpful in finding the important factors. 
• interesting and widely tested empirically since they offer closed form solutions of their conditional and steady state density functions.
• possible to get negative interest rates

Given the following:

• $\Delta t = T-t$ 
• $a$ as the speed of mean reversion
• $\mu$ as the long run average rate
• $r$ as the current state
• $\sigma$ as the volatility

We can get the models:

CIR Model:
\[ P(t,T) = A(T-t)e^{-B(T-t)r}\]
where: \[ A(T-t) = \left [   \frac{2 \gamma e^{(a+ \gamma)(T-t)/2}} {(\gamma +a)(e^{\gamma (T-t)}-1} \right ]^{2ab/ \sigma^2} \]
and
\[ B(T-t) = \frac {2(e^{\gamma (T-t)}-1)}{(\gamma+ a)(e^{\gamma (T-t)}-1)+2 \gamma}\]


Vacisek Model:
\[ P(t,T) = A(T-t)e^{-B(T-t)r}\]

where: \[ A(T-t) = \text{exp} \left [ \frac{ ( B(T-t) - T+t ) (a^2b-\sigma^2/2)}{a^2} -\frac{\sigma^2 B(T-t)^2}{4a}\right ] \]
and
\[ B(T-t) = \frac {1-e^{-a(T-t)}}{a}\]

Comparing these two models, they have the same process only that the parameters $ A(T-t)$ and  $B(T-t)$ are different.


Sources: 

• Hull book
• Groups ppt. They simplified the process.