Processing math: 100%

Saturday, March 9, 2013

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:
  • Δt=Tt 
  • 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(Tt)eB(Tt)r

where: A(Tt)=[2γe(a+γ)(Tt)/2(γ+a)(eγ(Tt)1]2ab/σ2

and
B(Tt)=2(eγ(Tt)1)(γ+a)(eγ(Tt)1)+2γ



Vacisek Model:
P(t,T)=A(Tt)eB(Tt)r


where: A(Tt)=exp[(B(Tt)T+t)(a2bσ2/2)a2σ2B(Tt)24a]

and
B(Tt)=1ea(Tt)a


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


Sources: 

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

No comments:

Post a Comment