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:
- Δt=T−t
- a as the speed of mean reversion
- μ as the long run average rate
- r as the current state
- σ 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)=[2γe(a+γ)(T−t)/2(γ+a)(eγ(T−t)−1]2ab/σ2
and
B(T−t)=2(eγ(T−t)−1)(γ+a)(eγ(T−t)−1)+2γ
Vacisek Model:
P(t,T)=A(T−t)e−B(T−t)r
where: A(T−t)=exp[(B(T−t)−T+t)(a2b−σ2/2)a2−σ2B(T−t)24a]
and
B(T−t)=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.
No comments:
Post a Comment