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.
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